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--- abstract: 'Markov Chain Monte Carlo methods are widely used in signal processing and communications for statistical inference and stochastic optimization. In this work, we introduce an efficient adaptive Metropolis-Hastings algorithm to draw samples from generic multi-modal and multi-dimensional target distributions. The proposal density is a mixture of Gaussian densities with all parameters (weights, mean vectors and covariance matrices) updated using all the previously generated samples applying simple recursive rules. Numerical results for the one and two-dimensional cases are provided.' author: - \| David Luengo$^\dagger$, Luca Martino$^\ddagger$\ $^\dagger$Department of Circuits and Systems Engineering, Universidad Politécnica de Madrid.\ Carretera de Valencia Km. 7, 28031 Madrid, Spain.\ $^\ddagger$Department of Signal Theory and Communications, Universidad Carlos III de Madrid.\ Avenida de la Universidad 30, 28911 Leganés, Madrid, Spain.\ E-mail: [[email protected], [email protected]]{} bibliography: - 'bibliografia.bib' - 'refsAGM.bib' title: 'Fully Adaptive Gaussian Mixture Metropolis-Hastings Algorithm' --- Markov Chain Monte Carlo (MCMC), Gaussian mixtures, adaptive Metropolis-Hastings algorithms Introduction {#sec:intro} ============ Markov Chain Monte Carlo (MCMC) methods [@Liu04b; @Liang10] are ubiquitously used for performing inference and solving optimization problems in many scientific fields: statistics, digital communications, machine learning, signal processing, etc. [@Robert04; @Wang2002; @Andrieu2003; @Fitzgerald01]. MCMC approaches are able to generate samples virtually from any target distribution (known up to a normalizing constant) by using a simpler proposal distribution. The basic underlying idea of standard MCMC techniques is producing a Markov chain that converges to the target. The most famous MCMC technique is the Metropolis-Hastings (MH) algorithm [@Metropolis53; @Hastings70; @Robert04]. However, the main drawback of the MH method (and in general of all MCMC methods) is that the correlation among the samples in the Markov chain can be very high when the acceptance rate is low [@Liu04b; @Liang10; @MartinoSigPro10]. Correlated samples provide less statistical information and the resulting chain can remain trapped almost indefinitely in a local mode, meaning that convergence can be extremely slow. Therefore, since the correlation depends on the discrepancy between the target and proposal distributions, we would like it to be as close to the target as possible. Several extensions have been proposed in the literature to speed up the convergence and reduce the so called “burn-in” period. Among them, adaptive MH methods (i.e., MH algorithms with adaptive proposal distributions) are particularly interesting [@Liang10; @MartinoA2RMS]. Indeed, MCMC techniques usually need the selection of several parameters by the user before they can be applied to any particular problem. The use of adaptive proposals overcomes this issue, providing black-box algorithms with self-tuning capabilities. An adaptive MH technique improves the proposal distribution by learning at least some of its parameters from all the previously generated samples. Unfortunately, an important problem with the adaptation of the proposal is that the Markov property is lost and the invariant distribution of the chain could be disturbed. Hence, adaptive MH algorithms must be carefully designed to avoid this issue. An adaptive Metropolis that uses an adaptive random walk Gaussian proposal, was introduced in [@Haario01] (we will denote it as AM method). The covariance matrix of the proposal is updated using recursive empirical estimators applied to the samples generated by the chain. The AM algorithm is an example of a partially adaptive MH approach, since it only updates the covariance of the proposal, whereas the mean of the Gaussian jumps to the current state of the chain at each iteration. An attempt of extending the AM algorithm by using a mixture of Gaussians as a proposal and updating all of its parameters (thus obtaining a fully adaptive MH algorithm) can be found in [@Giordani10]. However, the resulting algorithm is quite complicated, and the proposal is only updated at some iterations. In this work, we introduce an independent MH technique where the proposal PDF is an adaptive mixture of Gaussians. All the parameters (weights, means and covariance matrices) of the Gaussians in the mixture are updated using empirical estimators with simple recursive formulas (i.e., our method is fully adaptive). After a training period, the proposal is adapted at every iteration. The resulting AGM-MH algorithm can be used to draw samples from arbitrary multi-modal and multi-dimensional targets, always improving the performance w.r.t. a non-adaptive MH scheme using the initial proposal. The rest of the paper is organized as follows. Section \[sec:problem\] shows the problem formulation. Section \[sec:algorithm\] presents the proposed AGM-MH algorithm. Sections \[sec:recursive\] and \[sec:blackBox\] describe efficient parameter update rules and black-box usage. Finally, Sections \[sec:results\] and \[sec:discussion\] show the results and conclude the paper. Problem Formulation {#sec:problem} =================== Let us assume that we need to draw samples from a (possibly multi-modal) generic $d$-dimensional target probability density function (PDF), $p_o({\bf x})$, with support $\mathcal{D} \subseteq \mathbb{R}^d$. The AM algorithm [@Haario01] uses an adaptive random walk MH with a Gaussian proposal, mean equal to the previous state of the chain (${\bm \mu} = {\bf x}_{t-1}$), and covariance matrix, ${\bf C}$, estimated from all previous states, i.e., $q({\bf x}\|{\bf x}_{t-1}, {\bf C}) \propto \mathcal{N}({\bf x}\| {\bf x}_{t-1}, {\bf C})$, where $$\mathcal{N}({\bf x}\| {\bm \mu}, {\bf C}) \propto \exp\left(-\frac{1}{2}({\bf x}-{\bm \mu})^{\top} {\bf C}^{-1} ({\bf x}-{\bm \mu})\right) \label{eq:gaussian}$$ denotes a standard multi-variate Gaussian PDF. In order to improve the performance of the AM algorithm, here we consider a mixture of $N$ Gaussians as the proposal PDF, i.e., $$q({\bf x}\|{\bf w}, {\bm \mu}_{1:N}, {\bf C}_{1:N}) = \sum_{i=1}^{N}{w_i \mathcal{N}({\bf x}\| {\bm \mu}_i, {\bf C}_i)}, \label{eq:GaussianProposal}$$ where $\mathcal{N}({\bf x}\| {\bm \mu}_i, {\bf C}_i)$ is given by , ${\bm \mu}_i=[\mu_{i,1},...,\mu_{i,d}]^T$ is the $d \times 1$ mean vector, ${\bf C}_i$ is the $d \times d$ positive definite covariance matrix, and ${\bf w}=[w_1,...,w_N]^T$ are the normalized weights (i.e., $w_1+...+w_N=1$). Moreover, we define ${\bm \mu}_{1:N}=[{\bm \mu}_1,...,{\bm \mu}_N]$ and ${\bf C}_{1:N}=[{\bf C}_1,...,{\bf C}_N]$. We note that our approach is a fully adaptive MH algorithm, since (unlike the AM algorithm, which only updates the covariance) all the parameters in the mixture are learnt from all the previously generated samples. The resulting algorithm is very simple, since the adaptation is based on empirical estimators that can be implemented efficiently using recursive formulas. Since the adaptation could disturb the convergence of the generated chain to the target PDF, we consider the possibility of stopping it at an iteration $T_{stop}$. Hence, for $t>T_{stop}$ our algorithm is a standard MH with an improved proposal PDF w.r.t. the initial choice, thus providing a better performance and guaranteeing convergence. However, the numerical results described in Section \[sec:results\] show that the algorithm seems to maintain the correct ergodicity properties, so we always use $T_{stop}=T_{tot}$. A theoretical convergence proof is under development and will be included in future works. Finally, we note that degeneracy problems can appear during the first iterations in the update of the covariance matrices if we have a poor initialization. In order to avoid this issue, we allow the method to use a few iterations ($t=1,\ldots,T_{train}$) to collect information about the target, assigning the produced state of the chain to the closest Gaussian in the mixture, as in [@Haario01]. AGM-MH Algorithm {#sec:algorithm} ================ \[sec\_Alg\] The proposed AGM-MH algorithm is described below. First of all, notice that, during the first $T_{train}$ time steps the algorithm just assigns the current state ${\bf x}_t$ of the chain to a Gaussian among the $N$ in the mixture, according to the minimum Euclidean distance between ${\bf x}_t$ and the means ${\bm \mu}_i^{(t-1)}$, $i=1,...,N$. Afterwards, the algorithm updates all the parameters in the mixture until $t=T_{stop}$, when adaptation is stopped. In the description of the algorithm, the parameters are updated using a block procedure, but efficient recursive update formulas can be obtained, as shown in Section \[sec:recursive\]. 1. [Initialization:]{} 1. [Time instants:]{} Set $t=0$. Choose also the values ${\bf x}_0\in \mathcal{D}$, $T_{train}<T_{tot}$ and $T_{train}<T_{stop}<T_{tot}$. Let be $T_{tot}$ the number of total iteration of the chain. 2. [Proposal:]{} Choose the number of Gaussians $N$, as well the initial settings for ${\bm \mu}_{1:N}^{(0)}=[{\bm \mu}_1^{(0)},...,{\bm \mu}_N^{(0)}]$ and ${\bf C}_{1:N}^{(0)}=[{\bf C}_1^{(0)},...,{\bf C}_N^{(0)}]$. Set ${\bf w}^{(0)}=\frac{1}{N}{\bf 1}$. 3. [Auxiliary parameters:]{} Define ${\bf S}_i^{(0)}\triangleq [{\bf s}_i^{(1)}={\bm \mu}_{i}^{(0)}]$, and $m_i=1$ represents the number of columns of ${\bf S}_i^{(0)}$, with $i=1,...,N$. Let $\epsilon$ a small constant value and ${\bf I}_d$ a identity matrix. 2. [MH steps:]{} 1. Sample ${\bf x}'$ from a mixture of Gaussian PDFs, $${\bf x}'\sim q_t({\bf x}' \| {\bf w}^{(t)},{\bm \mu}_{1:N}^{(t)}, {\bf C}_{1:N}^{(t)} ).$$ 2. Accept ${\bf x}_{t+1}={\bf x}'$ with probability $$\small \alpha=\min\left[1, \frac{p({\bf x}') q({\bf x}_{t}\|{\bf w}^{(t)},{\bm \mu}_{1:N}^{(t)}, {\bf C}_{1:N}^{(t)})}{p({\bf x}_t) q({\bf x}'\|{\bf w}^{(t)},{\bm \mu}_{1:N}^{(t)}, {\bf C}_{1:N}^{(t)})} \right], \label{AlphaEq}$$ otherwise set ${\bf x}_{t+1}={\bf x}_t$. 3. If $t<T_{stop}$, [update parameters of the proposal:]{} 1. Find the closest Gaussian to ${\bf x}_{t+1}$ (w.r.t. Euclidean distance), i.e., find the index $$j=\arg \min_{i} \| {\bm \mu}_i^{(t)}- {\bf x}_{t+1}\|^2.$$ 2. \[Step3b\] Set $m_j=m_j+1$ and update (adding a new column) the $j$-th auxiliary matrix $${\bf S}_j^{(t+1)}=[{\bf S}_j^{(t)}, {\bf s}_j^{(m_j)}={\bf x}_{t+1}],$$ whereas ${\bf S}_i^{(t+1)}={\bf S}_i^{(t)}$, for all $i\neq j$. 3. If $t>T_{train}$: update the parameters of $j$-th Gaussian, $$\label{EqUp1} {\bm \mu}_j^{(t+1)}=\frac{1}{m_j}\sum_{i=1}^{m_j}{\bf s}_j^{(i)},$$ and $$\begin{gathered} \small \begin{split} \label{EqUp2} {\bf C}_j^{(t+1)}=\frac{{\bf \tilde{S}}_j^{(t+1)} \cdot \big[{\bf \tilde{S}}_j^{(t+1)}\big]^T +(m_j-1) \epsilon {\bf I}_d}{m_j-1}, \end{split} \end{gathered}$$ whrere ${\bf \tilde{S}}_j^{(t+1)} = {\bf S}_j^{(t+1)} - {\bm \mu}_j^{(t+1)} \otimes {\bf 1}_{m_j}^{\top}$, with $\otimes$ denoting the Kronecker product [@VanLoan2000]. Morev,er set ${\bm \mu}_i^{(t+1)}={\bm \mu}_i^{(t)}$, ${\bf C}_i^{(t+1)}={\bf C}_i^{(t)}$, $\forall i\neq j$. Since $m_j$ has been incremented, then update also the weights $$\label{EqUp3} w_i^{(t+1)}=\frac{m_i}{\sum_{k=1}^{N} m_k}, \quad i=1,...,N,$$ so that ${\bf w}^{(t+1)}=[w_1^{(t+1)},...,w_N^{(t+1)}]^T$. 4. If $t<T_{tot}$ repeat from step 2. Observe that the proposal PDF is only updated for $T_{train}<t<T_{stop}$. Moreover, note that the matrix ${\bf S}_i^{(t)}$ (and ${\bf \tilde{S}}_i^{(t)}$), for some $i\in \{1,...,N\}$, has dimension $d\times m_i$, so that ${\bf C}_i^{(t)}$ has always dimension $d\times d$. The identity matrix $\epsilon {\bf I}_d$ is used just to avoid numerical problems (the matrix ${\bf C}_i^{(t)}$ must be positive definite), as in [@Haario01]. Efficient Recursive Update of the Parameters {#sec:recursive} ============================================ \[UpdateParameter\] To update the parameters of the selected ($j$-th) Gaussian PDF in the mixture, we can use recursive expressions. Indeed, Recalling that $m_j=m_j+1$ is already updated and ${\bf s}_j^{(m_j)}={\bf x}_{t+1}$ in step \[Step3b\] of the algorithm, can be rewritten as $$\label{EqUp1rec} {\bm \mu}_j^{(t+1)}=\frac{1}{m_j} {\bf x}_{t+1}+\frac{m_j-1}{m_j} {\bm \mu}_j^{(t)},$$ and the Eq. becomes $$\begin{gathered} \small \begin{split} \label{EqUp2rec} {\bf C}_j^{(t+1)}=\frac{1}{m_j-1} \Bigg[ &\frac{({\bf x}_{t+1}-{\bm \mu}_j^{(t+1)})({\bf x}_{t+1}-{\bm \mu}_j^{(t+1)})^T}{m_j}+\epsilon {\bf I}_d \Bigg] +\frac{m_j-2}{m_j-1} {\bf C}_j^{(t)}.\\ \end{split} \end{gathered}$$ Finally, note that $$\sum_{k=1}^{N} m_k=t+1+N,$$ for $T_{train}<t<T_{stop}$, so that $$\label{EqUp3rec} w_i^{(t+1)}=\frac{m_i}{t+N+1}, \quad i=1,...,N,$$ In this way, the novel technique becomes computationally efficient in high dimensional problems, as well. AGM-MH as black-box method {#sec:blackBox} ========================== The AGM-MH method shows sensitive dependence on the initial conditions. If some prior information about the target is available, it can be used to choose the initial parameters. If no prior informations is available, the AGM-MH can be applied as black-box algorithm in the following way: - Use a great number $N$ of Gaussians (typically the number must be greater if the dimension $d$ increases). - Select randomly the means ${\bm \mu_{1:N}^{(0)}}$ in order to cover as possible the support domain $\mathcal{D} \subseteq \mathbb{R}^d$. - Choose diagonal covariance matrices ${\bf C}_{1:N}^{(0)}=\sigma^2 {\bf I}_d$, with a big value of $\sigma^2$ in order to explore the space $\mathcal{D} \subseteq \mathbb{R}^d$ in the train period, $t\leq T_{train}$. - The parameter $T_{train}$ should be chosen bigger for greater dimensions $d$. Numerical results suggest that $T_{train}=100d$ could be a suitable choice. In general, for more complicated target distributions a greater $T_{train}$ could be needed. The use of a huge number of Gaussians does not generate computational problems since in the adaptation of the AGM-MH the weights of the irrelevant vanish quickly to zero. Therefore, the computational cost is controlled by the adaptation so that only the Gaussians located close to high probability regions survive. The useless Gaussian PDFs located faraway to the modes of the target are virtually discarded (the weights becomes quickly zero). Finally it is important to remark that, in general, the proposal is refined from the initial setting and the performance is improved. Simulations {#sec:results} =========== Example 1 --------- In this toy example, we apply the AGM-MH method to draw from a univariate bimodal target PDF defined as $$\begin{gathered} \begin{split} p_o(x)\propto p(x)&=\exp\left\{-\frac{(x^2-4)^2}{4}\right\}\\ &=\exp\left\{-\frac{x^4-4x^2+16}{4}\right\}, \end{split}\end{gathered}$$ that, clearly, has two modes at $x=\pm 2$. We set $N=2$, number of Gaussian PDFs in the proposal PDF, $w_{i}^{(0)}=0.5$, and $(\sigma_i^2)^{(0)}=10$ with $i=1,2$. The two initial means $\mu_1^{(0)}\sim \mathcal{U}([-4,0])$, $\mu_1^{(0)}\sim \mathcal{U}([0,4])$ are chosen uniformly in $[-4,0]$ and $[0,4]$, respectively. We draw $T_{tot}=5000$ iterations of the chain, and set $T_{train}=200$, $T_{stop}=T_{tot}$ (i.e., the adaptation is never stopped). The initial state is randomly choose as $x_{0}\sim N(x;0,1)$. We use [all]{} the generated samples to estimate the mean of the target (that is $0$ since $p(x)$ is symmetric). The mean square error (MSE) of the estimation (averaged over $2000$ runs) is $\approx 15 \cdot 10^{-4}$. The estimated linear correlation between contiguous samples is $\approx 0.18$. Without using any adaptation, namely using a standard MH, the correlation is $\approx 0.78$. Hence, we can observe as the correlation decreases using the AGM-MH algorithm. The final averaged locations of the means of the proposal are $\mu_1^{(T_{tot})}\approx -1.88$, $\mu_2^{(T_{tot})}\approx 1.88$. The final weights of the mixture are $w_{i}^{(T_{tot})}\approx 0.5$ and $\sigma_i^2\approx 0.16$, $i=1,2$. Example 2 --------- For the sake of simplicity, we consider again an univariate target density. However, now we consider that target distribution is itself a mixture of Gaussian PDFs.[^1] Specifically, the target PDF is formed by $M$-Gaussians, i.e., $$\begin{gathered} \begin{split} p_o(x)\propto p(x)=\sum_{i=1}^{M}a_i \mathcal{N}(x\| \eta_i, \rho_i^2), \end{split}\end{gathered}$$ where the weights are $a_i=1/M$ and the variances $\rho_i^2=4$, $i=1,...,M$. We consider different $3$ cases with $M=2,3,6$. The means are, for each case, $$\eta_i=-10,10, \mbox{ }\mbox{ }\mbox{ for }\mbox{ }\mbox{ } M=2,$$ $$\eta_i=-10,0,10, \mbox{ }\mbox{ }\mbox{ for }\mbox{ }\mbox{ } M=3,$$ $$\eta_i=-15,-10,-5,5,10,15 \mbox{ }\mbox{ }\mbox{ for }\mbox{ }\mbox{ } M=6,$$ with $i=1,...,M$. In the proposal we also use $N=M$ Gaussians and each initial mean is chosen uniformly in $[-20,20]$. All the initial variances are set $(\sigma_j^2)^{(0)}=10$ and the weights $w_j^{(0)}=1/N$, $j=1,...,N$. As in the first example, we draw $T_{tot}=5000$ iterations of the chain, set $T_{train}=200$ and $T_{stop}=T_{tot}$ (i.e., the adaptation is never stopped). The initial state of the chain is randomly choose as $x_{0}\sim N(x;0,1)$. We use all the generated samples to estimate the normalizing constant of the target. The mean square error (MSE) of the estimation (averaged over $1000$ runs) is $1.6 \cdot 10^{-4}$, $1.1 \cdot 10^{-4}$ and $2 \cdot 10^{-5}$ for $M=2,3,6$ respectively. The resulting correlation is $0.13$, $0.14$ and $0.16$ for $M=2,3,6$, respectively. With a standard MH (without adaptation) the correlation is $0.81$, $0.72$ and $0.46$, for $M=2,3,6$. Hence, another time, we remark as the adaptation of the AGM-MH reduces considerably the correlation among the generated samples. Finally, Fig. \[fig1\](a) depicts the averaged values of the acceptance function $\alpha$ in Eq. as function of $t$ and for different $M$. In this case, for $t>T_{train}$, the averaged values of $\alpha$ increase because of the proposal becomes closer to the target PDF owing to the adaptation. Example 3 --------- In this example, our goal is drawing from a bivariate target PDF using the AGM-MH as a black-box technique. Just for simplicity, we also consider a bivariate mixture of $M=2$ Gaussians as target distribution, with means ${\bm \eta}_1=[-2,-2]^{T}$, ${\bm \eta}_2=[0,4]^{T}$ and covariance matrices ${\bm \Sigma}_1=[0.3 \ \ 0.1; \ 0.1 \ \ 0.3]$, ${\bm \Sigma}_2=[0.8 \ \ -0.3; \ -0.3 \ \ 0.8]$. The weights are $a_i=0.5,0.5$, $i=1,2$. We set $T_{tot}=7000$, set $T_{train}=200$ and $T_{stop}=T_{tot}$ (i.e., the adaptation is never stopped). First, we use for the proposal $N=2$ Gaussian PDFs, $w_i^{(0)}=0.5$, ${\bf C}_i^{(0)}=10{\bf I}_d$ for $i=1,2$. The means are selected uniformly, ${\bm \mu}_1\sim \mathcal{U}([-5,5]\times[0,5])$ and ${\bm \mu}_2\sim \mathcal{U}([-5,5]\times[-5,0])$. In this case, all the parameters of the mixture in the proposal convergences always to the corresponding values of the mixture forming the target PDF. Moreover, we also consider the case with $N=10$. We choose the means ${\bm \mu}_i$ of the proposal uniformly in the square $[-5,5]\times [-5,5]$. In this situation, the AGM-MH refines the initial proposal PDF improving the parameters of the Gaussians in the mixture that are in good locations, whereas the weights of the unhelpful Gaussians decrease quickly to zero and their parameters remain invariant, as shown in Fig. \[fig1\]. Discussion {#sec:discussion} ========== We have proposed a novel adaptive independent MH algorithm (AGM-MH) to draw samples from arbitrary multi-modal and multi-dimensional targets. AGM-MH builds on the work of [@Haario01], extending it by using a Gaussian mixture proposal and updating also the means and the weights of the Gaussians. Compared to a previous extension provided by [@Giordani10], our approach is more efficient, updates the proposal at every iteration instead of only at a fixed number of iterations. [^1]: It is important to remark that we consider a mixture of Gaussians as a target PDF just to discuss the performance of the AGM-MH algorithm. More specifically, we desire to show as, in this case, the proposal convergences to real shape of the target, depending on the initial setting. However, clearly, the algorithm can be used to draw from any kind of target distribution.	ArXiv
--- abstract: \| The effect of narrow dibaryon resonances to nuclear matter and structure of neutron stars is investigated in the mean-field theory (MFT) and in the relativistic Hartree approximation (RHA). The existence of massive neutron stars imposes constraints to the coupling constants of the $\omega $- and $% \sigma $-mesons with dibaryons. We conclude that the experimental candidates to dibaryons d$_1$(1920) and d’(2060) if exist form in nuclear matter a Bose condensate stable against compression. This proves stability of the ground state for nuclear matter with a Bose condensate of the light dibaryons. author: - \| Amand Faessler$^1$, A. J. Buchmann$^1$, M. I. Krivoruchenko$^{1,2}$\ [$^1$[Institut für Theoretische Physik, Universität Tübingen, Auf der Morgenstelle 14 ]{}]{}\ \ [$^2$[Institute for Theoretical and Experimental Physics, B.Cheremushkinskaya 25]{}]{}\ title: 'Constraints to Coupling Constants of the $\omega $- and $\sigma $-Mesons with Dibaryons' --- =-1.3truecm 16 truecm The prospect to observe the long-lived H-particle predicted in 1977 by R. Jaffe [@Jaf] stimulated considerable activity in the experimental searches of dibaryons. It was proposed to examine the H-particle production in different reactions [@San]. The experiments [@Aok] did not give a positive sign for the H-particle, however, the existence of the H-particle remains an open question which must eventually be settled by experiment. The non-strange dibaryons with exotic quantum numbers, which have a small width due to zero coupling to the $NN$-channel, are promising candidates for experimental searches [@Mul]. The data on pion double charge exchange (DCE) reactions on nuclei [@Bil] exhibit a peculiar energy dependence, which can be interpreted [@Mar] as evidence for the existence of a narrow d’ dibaryon with quantum numbers $T=0$, $J^p=0^{-}$ and the total resonance energy of $2063$ MeV. Recent experiments at TRIUMPF (Vancouver) and CELSIUS (Uppsala) seem to support the existence of the d’ dibaryon [@Mey]. A method for searching narrow, exotic dibaryon resonances in the double proton-proton bremsstrahlung reaction is discussed in Ref. [@Ger]. Recently, some indications for a d$_1$(1920) dibaryon in this reaction have been found [@Khr]. When density of nuclear matter is increased beyond a critical value, production of dibaryons becomes energetically favorable. Dibaryons are Bose particles, so they condense in the ground state and form a Bose condensate [@Bal; @Kri]. An exactly solvable model for a one-dimensional Fermi-system of fermions interacting through a potential leading to a resonance in the two-fermion channel is analyzed in Ref. [@Buc]. The behavior of the system with increasing the density can be interpreted in terms of a Bose condensation of two-fermion resonances. The effect of narrow dibaryon resonances on nuclear matter in the mean field theory (MFT) is analyzed in Refs. [@Fae; @Faes]. In the limit of vanishing decay width, a dibaryon can be approximately described as an elementary field. Despite the dibaryon Bose condensate does not exist in ordinary nuclei, dibaryons affect properties of nuclear matter and the ordinary nuclei through a Casimir effect. Presence of the background $\sigma $-meson mean field inside of nuclei modifies the nucleon and dibaryon masses and in turn modifies the zero-point vacuum fluctuations of the nucleon and dibaryon fields. This effect contributes to the energy density and pressure. It can be evaluated within the relativistic Hartree approximation (RHA). For nucleons, this effect is well known [@Wal]. In the loop expansion of quantum hadrodynamics (QHD), MFT corresponds to the lowest approximation (no loops), while RHA corresponds to the one-loop approximation in a calculation of the equation of state for nuclear matter. At zero temperature, a uniformly distributed system of bosons with attractive potential is energetically unstable against compression and collapses [@Abr]. In such a case, the long wave excitations (sound in the medium) have imaginary dispersion law: The square of the sound velocity is negative $a_s^2<0$. The amplitude of these excitations increases with the time, providing instability of the system. It is necessary to analyze dispersion laws of other elementary excitations also. We shall see, however, that in MFT and RHA only sound waves can generate an instability. The ground state of nuclear matter with a Bose condensate of dibaryons is stable or unstable against small perturbations according as the repulsive $\omega $-meson exchange or the attractive $\sigma $-meson exchange is dominant between dibaryons. In this paper, we investigate the hypothesis that the dibaryon matter is unstable against compression. In such a case, formation of dibaryons in nuclear matter can be treated as a possible mechanism for a phase transition into the quark matter. If central density of a massive neutron stars exceeds a critical value for formation of dibaryons, the neutron star should convert into a quark star, a strange star, or a black hole. Some of the observed pulsars are identified quite reliably with ordinary neutron stars [@Sha]. From the requirement that the dibaryon formation is not energetically favorable at densities lower than the central density of neutron stars with a mass $1.3M_{\odot }$, we derive constraints to the coupling constants of the mesons and dibaryons d$_1$(1920) and d$^{\prime }$(2060) and conclude that narrow dibaryons in this mass range can form a Bose condensate stable against perturbations only. The effect of the dibaryons to stability and structure of neutron stars in different phenomenological models is analyzed in Refs. [@Kri; @Tam]. Constraints to the binding energy of strange matter [@Wit] from the existence of massive neutron stars are discussed in Ref. [@KrMa]. The dibaryonic extension of the Walecka model [@Wal] is obtained by including dibaryons to the Lagrangian density [@Fae; @Faes] $$\label{I} \begin{array}{c} {\cal L}=\bar \Psi (i\partial _\mu \gamma _\mu -m_N-g_\sigma \sigma -g_\omega \omega _\mu \gamma _\mu )\Psi +\frac 12(\partial _\mu \sigma )^2-\frac 12m_\sigma ^2\sigma ^2 \\ -\frac 14F_{\mu \nu }^2+\frac 12m_\omega ^2\omega _\mu ^2+(\partial _\mu -ih_\omega \omega _\mu )\varphi ^{}(\partial _\mu +ih_\omega \omega _\mu )\varphi -(m_D+h_\sigma \sigma )^2\varphi ^{}\varphi . \end{array}$$ Here, $\Psi $ is the nucleon field, $\omega _\mu $ and $\sigma $ are fields of the $\omega $- and $\sigma $-mesons, $F_{\mu \nu }=\partial _\nu \omega _\mu -\partial _\mu \omega _\nu $, $\varphi $ is the dibaryon isoscalar-scalar (or isoscalar-pseudoscalar) field. The values $m_\omega \ $ and $m_\sigma $ are the $\omega $- and $\sigma $-meson masses and the values $g_\omega $, $g_\sigma $, $h_\omega $, $h_\sigma $ are coupling constants of the $\omega $- and $\sigma $-mesons with nucleons ($g$) and dibaryons ($h$). The $\sigma $-meson mean field $\sigma _c$ determines the effective nucleon and dibaryon masses in the medium $$\label{II}m_N^{}=m_N+g_\sigma \sigma _c,$$ $$\label{III}m_D^{}=m_D+h_\sigma \sigma _c.$$ The nucleon scalar density in the RHA is defined by expression [@Wal] $$\label{IV}\rho _{NS}=<\bar \Psi (0)\Psi (0)>=\gamma \int \frac{d{\bf p}}{% (2\pi )^3}\frac{m_N^{}}{E^{}({\bf p})}\theta (p_F-\|{\bf p}\|)-4m_N^3\zeta (m_N^{}/m_N)$$ where $$4\pi ^2\zeta (x)=x^3lnx+1-x-\frac 52(1-x)^2+\frac{11}2(1-x)^3.$$ The last term in Eq.(4) occurs after the renormalization of the scalar density. Here, $\gamma =2$ for neutron matter and $\gamma =4$ for nuclear matter. We investigate here properties of the nuclear matter below the critical density for formation of dibaryons, so the dibaryon condensate is zero $% <\|\varphi (0)\|>=0$. The vacuum contribution to the scalar density of dibaryons can be found to be $$\label{V}2m_D^{}\rho _{DS}=2m_D^{}<\varphi (0)^{}\varphi (0)>=m_D^3\zeta (m_D^{}/m_D).$$ It differs from the nucleon term in the sign and in the statistical factor (one should replace $4$ ($=2_s\times 2_I)\rightarrow 1$). The self-consistency condition for the nucleon effective mass has the form $$\label{VI}m_N^{}=m_N-\frac{g_\sigma }{m_\sigma ^2}(g_\sigma \rho _{NS}+h_s2m_D^{}\rho _{DS}).$$ The renormalized vacuum contribution to the nucleon energy-momentum tensor has the form [@Wal] $$\label{VII}<T_{\mu \nu }^N(0)>_{vac}=-4g_{\mu \nu }m_N^4\eta (m_N^{}/m_N)$$ where$$16\pi ^2\eta (x)=x^4lnx+1-x-\frac 72(1-x)^2+\frac{13}3(1-x)^3-\frac{25}{12}% (1-x)^4.$$ For dibaryons, we get expression $$\label{VIII}<T_{\mu \nu }^D(0)>_{vac}=g_{\mu \nu }m_D^4\eta (m_D^{}/m_D).$$ The elementary excitations in nuclear matter with a Bose condensate of dibaryons correspond to nucleons and antinucleons, $\omega $-mesons, $\sigma $-mesons, and dibaryons and antidibaryons. The dispersion laws for these quasiparticles are found in Ref. [@Faes]. The nucleon and antinucleon dispersion laws have the same form as in the vacuum with a replacement $% m_N\rightarrow m_N^{}$ and therefore cannot generate an instability of the system. The dispersion laws of $\omega $-mesons, $\sigma $-mesons, and antidibaryons turn out to be real also. The possible source of the instability are the dibaryon quasiparticle excitations only, which are responsible for a long wave perturbations of the system and connected with existence of the sound in the medium. The square of the sound velocity has the form [@Faes] $$\label{IX}a_s^2=\frac \alpha {1+\alpha }$$ where $$\label{X}\alpha =2\rho _{DS}\frac{m_\sigma ^2}{\tilde m_\sigma ^2}(\frac{% h_\omega ^2}{m_\omega ^2}-\frac{h_\sigma ^2}{m_\sigma ^2})$$ and $\tilde m_\sigma ^2=m_\sigma ^2+2h_\sigma ^2\rho _{DS}$. We see that $% a_s^2>0$ for $$\label{XI}\frac{h_\omega ^2}{m_\omega ^2}>\frac{h_\sigma ^2}{m_\sigma ^2}.$$ The validity of inequality (11) is sufficient condition for stability of the ground state of nuclear matter with a Bose condensate of dibaryons. The physical meaning of the inequality (11) can be clarified by considering the interaction energy of a uniformly distributed dibaryon matter $\rho _{DV}({\bf x}_1)=\rho _{DV}=$ constant: $$\label{XII}W=\frac 12\int d{\bf x}_1d{\bf x}_2\rho _{DV}({\bf x}_1)\rho _{DV}({\bf x}_2)V(\|{\bf x}_1-{\bf x}_2\|).$$ The Yukawa potential $V(r)$ for two dibaryons has the form $$\label{XIII}V(r)=\frac{h_\omega ^2}{4\pi }\frac{e^{-m_\omega r}}r-\frac{% h_\sigma ^2}{4\pi }\frac{e^{-m_\sigma r}}r.$$ The integration gives $$\label{XIV}W=\frac 12N_D\rho _{DV}(\frac{h_\omega ^2}{m_{_\omega }^2}-\frac{% h_\sigma ^2}{m_{_\sigma }^2})$$ where $N_D$ is the number of dibaryons. When the dibaryon density $\rho _{DV} $ increases, the energy for $a_s^2>0$ increases also, the pressure is positive, and so the system is stable. At present, the coupling constants of the mesons with dibaryons are not known with precision good enough to draw a definite conclusion concerning the stability of the dibaryon matter. Here, we show that violation of the inequality (11) for light dibaryons is in contradiction with the existence of massive neutron stars. Given that the ratio $h_\sigma /(2g_\sigma )$ between the $\sigma $-meson couplings with dibaryons and nucleons is fixed, one can find the $\omega $- and $\sigma $-meson couplings with nucleons by fitting the nuclear matter binding energy $E/A-m_N=-15.75$ MeV at the empirical equilibrium density $% \rho _0=0.148$ fm$^{-3}$ determined from the density in the interior of $% ^{208}Pb$ [@Wal]. It corresponds to the equilibrium Fermi wavenumber $% k_F=1.3$ fm$^{-1}$. The dependence of the effective nucleon mass at the equilibrium density on the ratio $h_\sigma /(2g_\sigma )$ is shown on Fig.1 (a). In Figs.1 (b) and (c) we give dependence of the incompressibility $% K=9\rho _0(\partial ^2\varepsilon /\partial \rho ^2)\|_{\rho =\rho _0}$ and the asymmetry coefficient $a_4$ on the ratio $h_\sigma /(2g_\sigma )$. In Fig.1 (d), the values $C_s^2=g_\sigma ^2(m_N/m_\sigma )^2$ and $C_\omega ^2=g_\omega ^2(m_N/m_\omega )^2$ are plotted. Notice that $h_\sigma /(2g_\sigma )=0$ is equivalent to RHA with no dibaryons. For comparison, we give the results of MFT where the effect of dibaryons to the nuclear matter below the critical density for occurrence of a Bose condensate of dibaryons is absent. The results RHA for different dibaryons are not much different. In this work, we study the physical implications of the effective Lagrangian (1) describing nucleon and dibaryon degrees of freedom. Other baryons can be included in the QHD framework in a similar way. Their effect is quite small. We have checked that the Casimir effect caused by the inclusion of other octet baryons ($2 \times 8 = 16$ degrees of freedom) shift the critical value $h_{\sigma }$ only by about 25$\%$ if one assumes that the ${\sigma }$-meson is an SU(3)$_{f}$ singlet. The inclusion of octet and decuplet baryons ($2 \times 8 + 4 \times 10 = 56$ degrees of freedom) with a universal sigma-meson coupling constant increases the critical value of $h_{\sigma }/(2g_{\sigma })$ to 1.2. When the ratio $h_\sigma /(2g_\sigma )$ approaches a value $0.8$, the system of equations blows up and the empirical equilibrium properties of the nuclear matter can no longer be reproduced. When $x\rightarrow \ 0$ $\zeta (x)=O(x^4)$, so the zero-point contributions to the scalar density of nucleons and dibaryons, which have the opposite signs, are comparable for $% 4g_\sigma ^4/m_N\approx h_\sigma ^4/m_D$. The dibaryon effects become large for $h_\sigma /(2g_\sigma )\approx 0.5(4m_D/m_N)^{1/4}\approx 0.84$. The greater dibaryon mass, the greater the upper limit to the ratio $h_\sigma /(2g_\sigma )$. This effect is seen in Fig.1. The saturation curve is shown in Fig.2 and the effective nucleon mass dependence on the Fermi wavenumber $k_F$ is shown in Fig.3 for $h_\sigma /(2g_\sigma )=0.6$ in case of the H-particle. EOS in RHA is softer than in MFT. The contributions of the vacuum zero-point fluctuations of nucleons and dibaryons partially cancel each other, so the inclusion of the dibaryons makes the EOS stiffer. In Fig.2, we see that the dashed curves corresponding to RHA with dibaryons lie above the solid lined corresponding to the RHA with no dibaryons. The same effect is seen on Fig.3. In MFT, the nucleon effective mass decreases with the density faster then in RHA. Due to the partial compensation of the nucleon and dibaryon contributions to the vacuum scalar density, the dashed lines lie below the solid lines. In Fig.4 we show the critical densities for occurrence of dibaryons H(2220), d’(2060), and d$_1$(1920) in MFT and RHA in nuclear and neutron matter versus the ratio $h_\sigma /(2g_\sigma )$ for $h_\omega /h_\omega ^{max}=1$, $0.8$, and $0.6$ where $h_\omega ^{max}=h_\sigma m_\omega /m_\sigma $ is the maximum value for the $\omega $-meson coupling constant with dibaryons at which the inequality (11) is violated ($h_\omega /h_\omega ^{max}=1$ corresponds to $a_s^2=0$ and $h_\omega /h_\omega ^{max}=0.8$ and $0.6$ correspond to $a_s^2<0$). In RHA, dibaryons occur at higher densities. The coupling constant $h_\omega $ determines the energy of dibaryons in the positive $\omega $-meson mean field. The greater the $h_\omega $, the greater the density is required to make production of dibaryons energetically favorable. This effect is seen in Fig.4: The solid lines $% h_\omega /h_\omega ^{max}=1$ lie above the long-dashed and dashed lines $% h_\omega /h_\omega ^{max}=0.8$ and $0.6$, respectively. When the dibaryon matter is unstable against compression, production of dibaryons with increasing the density results to instability of neutron stars with subsequent phase transition into the quark matter and conversion of neutron stars into quark stars, strange stars, or black holes. In such a case, the maximum masses of neutron stars are determined by the mass and the coupling constants of the mesons with the lightest dibaryon. In Fig.5 we show the minimal neutron star masses in which dibaryons can occur. The MFT and RHA EOS for neutron matter at supranuclear densities are matched smoothly with the BBP EOS [@BBP] at densities $\rho _{drip}<\rho <0.8\rho _0$ where $\rho _{drip}=4.3$ $10^{11}$g/cm$^3$ and then with the BPS EOS [@BPS] at densities $\rho <\rho _{drip}$. The maximum neutron star masses are sensitive to the value of the equilibrium Fermi wavenumber. If we chose $k_F=1.42$ fm$^{-1}$ instead of $k_F=1.3$ fm$^{-1}$, the maximum masses in MFT (with no dibaryons) are reduced from $3M_{\odot }$ down to $% 2.6M_{\odot }$ [@Wal]. The choice $k_F=1.3$ fm$^{-1}$ provides less stringent (therefore more conservative) constraints to the meson-dibaryon coupling constants. We do not show the results for the d$_1$(1920) dibaryon, since its condensation starts at a density $\rho \approx $ $\rho _0$ providing conversion to quark stars of neutron stars with very low masses $% M<0.2M_{\odot }$. In Fig. 6 we show the parameter space for the coupling constants of dibaryons with the mesons. Our discussion and the validity of our arguments are restricted to the region $a_s^2<0$ in which the dibaryon matter is unstable against compression. The requirement of stability of the normal nuclear matter at the saturation density allows to get constraints to the coupling constants. The corresponding curves (straight lines in the MFT case) marked by arrows with the white ends restrict from below the parameter space of the coupling constants. The dotted line in the MFT case, which refers to the d$_1$(1920) dibaryon, is very close to the dashed-dotted line $% a_s^2=0$. For low values $h_\sigma $ the dotted line lies above the line $% a_s^2=0$. In such a case, the dibaryon matter unstable against compression cannot exist. The window in the parameter space for the unstable dibaryon matter for d$_1$(1920) is, however, much greater for RHA. If a conservative assumption is used, namely, that pulsars with a mass $% 1.3M_{\odot }$ are ordinary neutron stars, the constraints to the meson coupling constants with dibaryons can further be improved. The corresponding curves (straight lines in the MFT case) are shown on Fig.6. We see that the dotted and dashed lines lie above or very close to the line $a_s^2=0$. It means that for d$_1$(1920) and d’(2060) dibaryons, the dibaryon matter unstable against compression cannot exist (owing to a very small window for d’(2060) at higher values of the $h_\sigma $). For the H-particle, there is a window in the parameter space between the line $a_s^2=0$ and the solid curves marked by arrows with the black ends, which corresponds to the dibaryon matter unstable against compression. The H-particle interaction were studied in the non-relativistic quark cluster model [@Str; @Oka] successful in describing the $NN$-phase shifts. The coupling constants of the mesons with the H-particle can be fixed by fitting the depth and the position of the minimum of the HH-adiabatic potential [@Sak] to give $h_\omega /(2g_\omega )=0.89$ and $h_\sigma /(2g_\sigma )=0.80$. These values are marked on Fig.6 with a cross. These estimates are used in the MFT-calculations [@Fae; @Faes]. They correspond to the unstable dibaryon matter and are in the allowed region of the parameter space for the H-particle. The energetically favorable compression of the H-matter can lead to the formation of the absolutely stable strange mater [@Wit], producing conversion of neutron stars to strange stars [@Oli]. MFT and RHA EOS both are very stiff. In the soft models like the Reid one (for a review of the nuclear matter models see [@Sha]), the central density of neutron stars is much greater than in stiff models. Respectively, conditions for occurrence of new forms of nuclear matter are more favorable. From Figs. 5 and 6, we see that the softer RHA EOS produces lower upper limits to the neutron star masses and, respectively, more stringent constraints to the meson-dibaryon coupling constants as compared to the stiffer MFT EOS, despite in RHA dibaryons occur at higher densities (see Fig.4). One can assume that this effect is of the general validity and that softer EOS like the Reid one give more stringent constraints to the meson-dibaryon coupling constants. We consider therefore the constraints given in Fig.6 as the conservative ones. In the conclusion, we showed that the hypothesis of instability of the dibaryon matter against compression is in contradiction with the hypothesis that pulsars of a mass $1.3M_{\odot }$ are ordinary neutron stars for dibaryons d$_1$(1920) and d’(2060). This conclusion is valid for all narrow dibaryons with the same quantum numbers in the same mass range. The H-particle is sufficiently heavy, its condensation starts at higher densities, respectively, constraints to the meson-dibaryon coupling constants are not much stringent, allowing a possibility for the H-particle to form a condensate unstable against compression with subsequent formation of the strange matter. 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The solid, dashed, and dotted curves correspond to the dibaryons H(2220), d$^{\prime }$(2060), and d$_1$(1920). The reasonable description of the properties of the nuclear matter at the saturation density is possible for $h_\sigma /(2g_\sigma )<0.8$. [Fig.2]{} Saturation curve for nuclear matter in RHA with no dibaryons (solid line) and in RHA with inclusion of the H-dibaryon (dashed line) for $% h_\sigma /(2g_\sigma )=0.6$. [Fig.3]{} The effective nucleon mass versus the fermi momentum of nucleons in the nuclear matter (upper curves) and in the neutron matter (lower curves). The solid lines correspond to RHA with no dibaryons, the dashed lines correspond to RHA with inclusion of the H-dibaryon. [Fig.4]{} Critical densities for occurrence of the dibaryons H(2220), d’(2060), and d$_1$(1920) in nuclear and neutron matter in MFT and RHA versus the $\sigma $-meson coupling constant with dibaryons for $h_\omega /h_\omega ^{max}=1$, $0.8$, and $0.6$ (the solid, longed-dashed, and dashed lines, respectively), where $h_\omega ^{max}=h_\sigma m_\omega /m_\sigma $ is the maximum value for the $\omega $-meson coupling constant with dibaryons at which the dibaryon matter is unstable against compression (see the text). [Fig.5]{} The minimum neutron star masses, in which the dibaryon formation becomes energetically favorable, versus the $\sigma $-meson coupling constant with dibaryons for $h_\omega /h_\omega ^{max}=1$, $0.8$, and $0.6$ (the solid, longed-dashed, and dashed lines, respectively) where $h_\omega ^{max}$ is the maximum value for the $\omega $-meson coupling constant at which the dibaryon matter is unstable against compression ($a_s^2<0$). The results are given in MFT and RHA and for two dibaryons H(2220) and d’(2060). The minimum neutron star masses for the d$_1$(1920) are very small ($% <0.2M_{\odot }$). [Fig.6]{} Parameter space for the coupling constants of the $\sigma $- and $\omega $-mesons with dibaryons in MFT and RHA. The dashed-dotted line $% a_s^2=0$ divides the parameter space into two parts. The upper left part of the parameter space corresponds to the dibaryon matter stable against compression (square of the sound velocity is positive $a_s^2>0$), the lower right part corresponds to the nuclear matter unstable against compression ($% a_s^2>0$). The solid, dashed, and doted curves constrain from below the regions in which the dibaryon formation is energetically not favorable in ordinary nuclei (curves marked by arrows with white ends) and at the density equal to the central density of a $1.3M_{\odot }$ mass neutron star (curves marked by arrows with black ends). The cross refers to the H-particle coupling constants with the mesons, determined from the adiabatic potential [@Sak].	ArXiv
--- abstract: 'The purpose of this paper is to develop a general existence theory for constrained minimization problems for functionals defined on function spaces on metric measure spaces $(\mathcal M, d, \mu)$. We apply this theory to functionals defined on metric graphs $\mathcal G$, in particular $L^2$-constrained minimization problems of the form $$E(u) = \frac{1}{2} a(u,u) - \frac{1}{q}\int_{\mathcal K} \|u\|^q \, \mathrm dx,$$ where $q>2$ and $a(\cdot, \cdot)$ is a suitable symmetric, sesquilinear form on some function space on $\mathcal G$ and $\mathcal K \subseteq \mathcal G$ is given. We show how the existence of solutions can be obtained via decomposition methods using spectral properties of the operator $A$ associated with the form $a(\cdot, \cdot)$ and discuss the spectral quantities involved. An example that we consider is the higher-order variant of the stationary NLS (nonlinear Schrödinger) energy functional with potential $V\in L^2+ L^\infty(\mathcal G)$ $$E^{(k)}(u)= \frac{1}{2} \int_{\mathcal G} \|u^{(k)}\|^2+ V(x) \|u\|^2 \, \mathrm dx - \frac{1}{p} \int_{\mathcal K} \|u\|^q \, \mathrm dx$$ defined on a class of higher-order Sobolev spaces $H^k(\mathcal G)$ that we introduce. When $\mathcal K$ is a bounded subgraph, one has localized nonlinearities, which we treat as a special case. When $k=1$ we also consider metric graphs with infinite edge set as well as magnetic potentials. Then the operator $A$ associated to the linear form is a Schrödinger operator, and in the $L^2$-subcritical case $2<q<6$, we obtain generalizations of existence results for the NLS functional as for instance obtained by Adami, Serra and Tilli \[JFA 271 (2016), 201-223\], and Cacciapuoti, Finco and Noja \[Nonlinearity 30 (2017), 3271–3303\], among others.' address: - 'University of Lisbon, Portugal' - \| Grupo de Física Matemática\ Faculdade de Ciências da Universidade de Lisboa\ Campo Grande, Edifício C6\ P-1749-016 Lisboa, Portugal author: - Matthias Hofmann title: An existence theory for nonlinear equations on metric graphs via energy methods --- Introduction ============ In recent years, there has been a growth of interest in functionals on metric graphs $\mathcal G=(\mathcal V, \mathcal E)$ of the stationary NLS (Nonlinear Schrödinger) energy functional $$\label{eq:introminprobfunc} E_{\text{NLS}}(u, \mathcal G)= \frac{1}{2} \int_{\mathcal G} \|u'\|^2\, \mathrm dx - \frac{\mu}{q} \int_{\mathcal G} \|u\|^q \, \mathrm dx, \qquad u\in H^1(\mathcal G),\; \\|u\\|_{L^2}^2 =1,\; q>2, \; \mu >0$$ and associated ground states of the stationary NLS energy functional, i.e. minimizers for the constrained minimization problem $$\label{eq:introminprob} E_{\text{NLS}}(\mathcal G):= \inf_{\substack{u\in H^1(\mathcal G)\\ \\|u\\|_{L^2}^2=1}} E_{\text{NLS}}(u, \mathcal G), \qquad 2<q< 6.$$ Such minimizers are solutions to the stationary nonlinear Schrödinger equation on $\mathcal G$ given by $$\begin{cases} -u'' + \lambda u = \mu \|u\|^{q-2} u \qquad \text{edgewise,}\vspace{.5em}\\ \begin{gathered}u \text{ is continuous on }\mathcal G \text{ and satisfies the Kirchhoff condition}\\ \sum_{e\in \mathcal E:e\succ \mathsf v} \frac{\partial u}{\partial \nu} \Big \|_e(\mathsf v)=0, \qquad \forall \mathsf v\in \mathcal V, \end{gathered} \end{cases}$$ where we recall that $e\succ \mathsf v$ denotes the relation that the edge $e$ is adjacent to the vertex $\mathsf v\in \mathcal V$ and $\frac{\partial u}{\partial \nu}\|_e(\mathsf v)$ denotes the inward pointing derivative at $\mathsf v$ towards the interior of the edge $e$. While in the simplest case of the real line, existence of minimizers in can be deduced by standard techniques, on general noncompact graphs existence results are not as easy to obtain due to the lack of a concept of translation invariance. In [@adami2015nls] it was shown on the one hand that under certain topological configurations the problem does not admit a minimizer; on the other, in a later paper the same authors derive an existence principle based on a comparison inequality: \[thm:introast2016\] Let $\mathcal G$ be a noncompact metric graph with finitely many edges and $2<q<6$. Assume $$\label{eq:introadamiestimate} E_{\text{NLS}}(\mathcal G) < E_{\text{NLS}}(\mathbb R),$$ then there exists a minimizer for $E_{\text{NLS}}(\mathcal G)$. This result can be used to obtain existence results on concrete graphs $\mathcal G$ via construction of so called competitors, i.e. test functions $u\in H^1(\mathcal G)$ for which $E_{\text{NLS}}(u, \mathcal G) < E_{\text{NLS}}(\mathbb R)$. This allows to deduce existence of minimizers in certain situations as shown in [@tentarelli2016nls] and [@adami2017negative]. A variant of this problem with potential was considered in [@cacciapuoti2017ground] and [@cacciapuoti2018existence], where the energy functional was given by $$\label{eq:introfunctionaltocons} E_{\text{NLS}}^{V}(u) = \frac{1}{2} \int_{\mathcal G} \left \| u'\right \|^2+ V\|u\|^2 \, \mathrm dx -\frac{\mu}{q} \int_{\mathcal G} \|u\|^q \, \mathrm dx, \qquad \\|u\\|_{L^2}^2=1.$$ In [@cacciapuoti2017ground] the existence of minimizers of was related to the existence of eigenvalues of the Schrödinger operator $-\Delta+V$ below the essential spectrum: \[thm:introcfn2017\] Let $\mathcal G$ be a noncompact metric graph with finitely many edges and $V\in L^1+ L^\infty(\mathcal G)$ with $V_- \in L^r(\mathcal G)$ for $r\in[1,1+ \frac{2}{q-2}]$ and $2<q\le 6$. Assume $$\label{eq:introJameswants} \inf \sigma(-\Delta+V) < \inf \sigma_{\text{ess}}(-\Delta+V).$$ Then there exists $\mu^>0$ such that for $\mu \in (0, \mu^)$ the functional is bounded below and the associated constrained minimization problem $$E_{\text{NLS}}^{V}:= \inf_{\substack{u\in H^1(\mathcal G)\\ \\|u\\|_{L^2}^2=1}} E_{\text{NLS}}^V(u)$$ admits a minimizer. \[thm:introcac2018\] Let $\mathcal G$ be a noncompact metric graph with finitely many edges and $V\in L^1+L^\infty(\mathcal G)$ satisfying the assumptions in Theorem \[thm:introcfn2017\]. Let $$\Sigma_0 := \inf \sigma(-\Delta+V)<0, \qquad \gamma_q := \inf_{\substack{u\in H^1(\mathcal G)\\ \\|u\\|_{L^2}^2=1}}\frac{1}{2} \int_{\mathbb R} \left \| u'\right \|^2+ V\|u\|^2 \, \mathrm dx -\frac{1}{q} \int_{\mathcal G} \|u\|^q \, \mathrm dx<0.$$ Then we have existence of minimizers of $$E_{\text{NLS}}^{V}= \inf_{\substack{u\in H^1(\mathcal G)\\ \\|u\\|_{L^2}^2=1}} E_{\text{NLS}}^V(u)$$ for $0<\mu\le (\Sigma_0/\gamma_q)^{\frac{3}{2}-\frac{q}{4}}$. Our goal in this paper is threefold. Firstly, we develop a general existence theory in a far more abstract setting which can be applied to a variety of problems as for example $E_{\text{NLS}}$ and $E_{\text{NLS}}^{V}$ but which is not limited to metric graphs. For example, the existence theory may be also applied to functionals defined on function spaces on combinatorial graphs or general domains in $\mathbb R^n$. We then use this existence theory to obtain generalizations of the results in [@adami2016threshold] and [@cacciapuoti2017ground] by considering more general graphs, higher-order derivatives in the functionals, as well as different variants of the problems. We will also discuss the spectral quantities appearing in , which replaces the inequality in Theorem \[thm:introast2016\] for the existence results. Variants we will consider include the case of decaying potentials and localized nonlinearities, i.e. we replace the set of integration in the term corresponding to the nonlinearity by a bounded subgraph $\mathcal K \subset \mathcal G$. Unlike [@adami2016threshold] our proof of Theorem \[thm:introast2016\] does not involve rearrangement inequalities, which allows us to consider potentials and deduce a similar existence result in the case of decaying potentials. Let us be now more precise about the abstract setting we will consider. Let $(\mathcal M, d, \mu)$ be a nonempty metric measure space. Assume $p\in [1,\infty]$ and let $X(\mathcal M)\subset L^p(\mathcal M)$ be a Banach space continuously and locally compactly imbedded in $L^p(\mathcal M)$, i.e. for any relatively compact, connected subset $K$, the restriction $X(K)$ is compactly imbedded in $L^p(K)$. In the case of metric graphs a prototype would be $H^1(\mathcal G)$, but we will also apply this to higher-order Sobolev spaces $H^k(\mathcal G)$ with $k\in \mathbb N$. We will consider functionals $E\in C(X(\mathcal M), \mathbb R)$ with $E(0)=0$ for which the mapping $$t\mapsto E_t := \inf_{\substack{u\in X(\mathcal M)\\ \\|u\\|_p^p =t}}E(u)$$ is continuous for $t\ge 0$. It is easy to verify that these properties are satisfied by and . Our first result is a dichotomy result for a class of such functionals: \[thm:introdich\] Let $c>0$ and $(\mathcal M, d, \mu)$ be a metric measure space. Let $X(\mathcal M)$ and $E$ be as above. Assume additionally (1) $E$ is strictly subadditive, i.e. $$E_{t_1+t_2} < E_{t_1} + E_{t_2}, \quad \forall t_1,t_2 >0,$$ (2) $E$ is weak limit superadditive, i.e. for any weakly convergent sequence $u_n \rightharpoonup u$ in $X(\mathcal M)$ there exists a subsequence such that $$\limsup_{n\to \infty} E(u_n) \ge E(u) + \limsup_{n\to \infty} E(u_n-u)$$ If there exists a weakly convergent minimizing sequence $u_n \rightharpoonup u$ in $X(\mathcal M)$ for $$\label{eq:introdichminprob} E_c= \inf_{\substack{u\in X(\mathcal M)\\ \\|u\\|_p^p =c}}E(u),$$ then either (i) $u= 0$ or (ii) $u_n \to u$ in $L^p(\mathcal M)$ and $u\neq 0$ is a minimizer of $E_c$. Observe that under our assumptions, if $u_n \rightharpoonup 0$ in $X(\mathcal M)$, then $$u_n \to 0 \qquad \text{ on } L^p_{\text{loc}}(\mathcal G).$$ For functionals as considered in Theorem \[thm:introdich\] this means that either minimizing sequences of vanish to infinity, or converge towards minimizers in an appropriate sense. In the first case, we will say that minimizing sequences are vanishing. In this case one cannot deduce existence of minimizers without any additional information. This gives rise to our second result, which gives a sufficient condition for the existence of minimizers. Since the mass of vanishing sequences moves outside of any precompact set of $\mathcal M$ it is natural to separate the supports of the functions $u_n$ into an expanding part $O_n^{(1)}$ around some fixed precompact set and a part $O_n^{(2)}$ retreating to infinity via sequences of partitions of unity $$\{\Psi_{O_n^{(1)}}, \Psi_{O_n^{(2)}}\}.$$ If we then require that such a sequence of partitions of unity does not increase the energy functional $E$ under decomposition for vanishing sequences $u_n \rightharpoonup 0$ in Theorem \[thm:introdich\], i.e. $$\limsup_{n\to \infty} E(v_n) \ge \limsup_{n\to \infty} E(\Psi_{O_n^{(1)}} v_n)+ \limsup_{n\to \infty} E(\Psi_{O_n^{(2)}} v_n),$$ then we say $E$ is superadditive with respect to this vanishing-compatible sequence of partitions of unity and obtain the existence principle: \[thm:introextenceprinciple\] Let $c>0$ and $K$ be a precompact subset of $\mathcal M$. Assume $E\in C(X(\mathcal M), \mathbb R)$ is - strictly subadditive, - weak limit superadditive, - superadditive with respect to a vanishing-compatible sequence of partitions of unity, and $$E_c < \widetilde E_c:= \lim_{n\to \infty} \inf_{\substack{u\in X(\mathcal M)\\ \operatorname{supp} u\subset \mathcal M\setminus K_n, \; \\|u\\|_{L^p}^p=c}}E(u),$$ where $K_n:= \{x\in \mathcal G \| d(x, K)<n\}$ is the expanding ball around $K$. Assume there exists a weakly convergent minimizing sequence $u_n \rightharpoonup u$, then $u\neq 0$ is a minimizer of $E_c$. It will turn out that the functionals and as considered in [@adami2016threshold] and [@cacciapuoti2017ground] satisfy the prerequisites of this theory. In fact, we will apply Theorem \[thm:introextenceprinciple\] to a natural generalization of , namely the higher-order stationary NLS energy functional $$\label{eq:introkfunc} E^{(k)}(u) = \frac{1}{2} \int_{\mathcal G} \|u^{(k)}\|^2 + V\|u\|^2 \, \mathrm dx - \frac{\mu}{q} \int_{\mathcal G} \|u\|^q\, \mathrm dx, \qquad \begin{multlined}\mu>0,\quad 2<q<4k+2,\\ V\in L^2+L^\infty(\mathcal G)\end{multlined}$$ and consider the ground state problem $$\label{eq:introkground} E^{(k)} = \inf_{\substack{u\in H^k(\mathcal G)\\ \\|u\\|_{L^2}^2 =1}} \frac{1}{2} \int_{\mathcal G} \|u^{(k)}\|^2 + V\|u\|^2 \, \mathrm dx - \frac{\mu}{q} \int_{\mathcal G} \|u\|^q\, \mathrm dx.$$ Here, we write $V\in L^2+ L^\infty$ to mean that $V$ admits a decomposition $V=V_2+V_\infty$ such that $V_2\in L^2(\mathcal G)$ and $V_\infty\in L^\infty(\mathcal G)$. When $k=1$ the energy functional reduces to the stationary NLS energy functional and we derive conditions for which the theory is applicable. Minimizers of satisfy the stationary higher-order nonlinear Schrödinger equation $$\begin{cases} (-1)^k u_e^{(2k)} + \left ( V+\lambda\right ) u_e = \mu \|u_e\|^{q-1} u_e, \qquad \forall e \in \mathcal E\vspace{1em}\\ \begin{multlined} u^{(i)} \in C(\mathcal G)\quad \text{ for all } i\le 2k-1 \text{ even} \qquad \text{\it (Continuity)}\\ \qquad \land \quad\sum_{e:e\succ \mathsf v} u^{(k)}_e(\mathsf v) =0 \quad \forall i\le 2k-1 \text{ odd } \;\forall \mathsf v\in V\\ \text{\it (Kirchhoff condition)}. \end{multlined} \end{cases}$$ While to the best of our knowledge this functional has not yet been considered on metric graphs, the stationary higher-order nonlinear Schrödinger equation on the real line of 4^th^ order is for instance related to traveling wave solutions of the nonlinear higher-order Schrödinger equation for the pulse envelope with higher-order dispersion as shown in [@kruglov2019exact §II]. For combinatorial locally finite graphs a discussion on the existence of solutions of the nonlinear higher-order Schrödinger equation of 4^th^ order was for instance considered very recently in [@hanshaozhao2019]. A minor difficulty in defining is that one needs to define higher-order Sobolev spaces $H^k(\mathcal G)$, as to date no standard way to define these spaces has emerged. We will define them in such a way that the formal Polylaplacian $$\label{eq:operators1} \begin{gathered} A=(-\Delta)^k+V\\ D(A) = H^{2k}(\mathcal G) \end{gathered}$$ is a self-adjoint operator on $L^2(\mathcal G)$. We remark that the choice is not necessarily unique. A discussion of self-adjoint realizations for the Bilaplacian on metric graphs can be for instance found in [@gregorio2017bi]. One major result on the existence of minimizers in is as follows: \[thm:intromainnowcloser\] Let $\mathcal G$ be a noncompact metric graph with finitely many edges. Assume that either (i) there exists $V=V_2+V_\infty$ such that $V_2\in L^2(\mathcal G)$ and $V_\infty \in L^\infty (\mathcal G)$ and $$V_\infty(x)\to 0\qquad (x\to \infty)$$ on all edges of infinite length, or (ii) $A=(-\Delta)^k+V$ admits a ground state, i.e. $\inf \sigma(A)$ is an eigenvalue. Then $E^{(k)}$ is strictly subadditive, and if additionally $$\label{eq:intromainnowcloser}E^{(k)} < \widetilde{E^{(k)}} := \lim_{n\to \infty}\inf_{\substack{u\in H^k(\mathcal G)\\\\|u\\|_{L^2}^2=1,\; \operatorname{supp} u \subset \mathcal G\setminus K_n}} E^{(k)}(u),$$ then $E^{(k)}$ admits a minimizer. Analogously to and , we will refer to the minimizers as ground states. Theorem \[thm:intromainnowcloser\] generalizes Theorem \[thm:introast2016\] since satisfies the prerequisites of Theorem \[thm:intromainnowcloser\]. Indeed, one can show with a test function argument (see Example \[ex:NLSclassic\]) that if $\mathcal G$ is a metric graph with finitely many edges then $$\widetilde{E_{\text{NLS}}}(\mathcal G) := \lim_{n\to \infty}\inf_{\substack{u\in H^1(\mathcal G)\\\\|u\\|_{L^2}^2=1,\; \operatorname{supp} u \subset \mathcal G\setminus K_n}} E_{\text{NLS}}(u, \mathcal G)= E_{\text{NLS}}(\mathbb R)$$ and we recover Theorem \[thm:introast2016\]. Under the assumption that eigenvalues exist below the essential spectrum, i.e. $$\inf \sigma((-\Delta)^k+ V) <\inf \sigma_{\text{ess}}((-\Delta)^k+V),$$ by a perturbation argument one can ensure that is satisfied for small nonlinearities and deduce a generalization of Theorem \[thm:introcfn2017\]: \[thm:intromain1\] Let $\mathcal G$ be a noncompact metric graph with finite edge set. If $$\label{eq:introimportantinequality} \inf \sigma ( (-\Delta)^k +V) < \inf \sigma_{\text{ess}}((-\Delta)^k +V)$$ then admits a ground state for sufficiently small $\mu>0$. For $\mathcal G=\mathbb R$ and $V\equiv 0$, due to translation invariance the inequalities in Theorem \[thm:intromainnowcloser\] and Theorem \[thm:intromain1\] cannot be satisfied. However, here one can exploit the translation invariance to obtain a similar existence result, for which we are unaware of any reference in the literature: \[thm:intromain11\] Let $V\equiv 0$ and $\mathcal G=\mathbb R$, then admits a ground state for all $\mu >0$. In particular, there exists a solution $u\in C^\infty(\mathbb R)$ to the stationary higher-order NLS equation $$(-1)^k u^{(2k)} + \lambda u = \mu \|u_e\|^{q-1} u,$$ with $\\|u\\|_{L^2(\mathbb R)}^2=1$ for some $\lambda \in \mathbb R$. ![An illustration for the classes of graphs that are considered. To the left a finite graph, sometimes referred to as starlike, consisting of a core graph and attached rays and to the right an infinite tree graph as an example for a locally finite graph, i.e. finite on any precompact set.[]{data-label="fig:introstuff"}](graph_example.pdf "fig:") ![An illustration for the classes of graphs that are considered. To the left a finite graph, sometimes referred to as starlike, consisting of a core graph and attached rays and to the right an infinite tree graph as an example for a locally finite graph, i.e. finite on any precompact set.[]{data-label="fig:introstuff"}](infinite_tree_more.pdf "fig:") The results in Theorem \[thm:introast2016\] and Theorem \[thm:introcfn2017\] were shown for metric graphs with finitely many edges, which we refer to as finite graphs throughout the paper. Such graphs consist of a finite number (possibly zero) of edges of infinite length, i.e. half-lines, whch we call rays, and a complement, which is compact, and which we will call the core of the graph. [@cacciapuoti2017ground], [@cacciapuoti2018existence] call such graphs starlike graphs (see also Figure \[fig:introstuff\]). Our theory also allows us to handle more general graphs. For the next result we will consider a class of graphs with countable edge set, which is finite when restricted to any precompact subset. We will to refer to such graphs as locally finite graphs in the following. On locally finite graphs we consider the following variant of the NLS energy functional $$\label{eq:introNLSfunc} E_{\text{NLS}}^{(\mathcal K)}(u) = \frac{1}{2} \int_{\mathcal G} \left \|\left (i \frac{\mathrm d}{\mathrm dx} + M\right )u\right \|^2+ V\|u\|^2 \, \mathrm dx - \frac{\mu}{p} \int_{\mathcal K} \|u\|^q \, \mathrm dx,\qquad \\|u\\|_{L^2}^2=1,$$ where $\mathcal K\subseteq \mathcal G$ is a subgraph of $\mathcal G$. In this context, we consider the magnetic Schrödinger operator with external potential $$\label{eq:operators2} \begin{gathered} A^M= \left ( i \frac{\mathrm d}{\mathrm dx} + M\right )^2+ V\end{gathered}$$ with its natural domain of definition, which we describe in detail in §5.1. The following theorem is an analog of Theorem \[thm:intromain1\]. Interestingly, if one considers localized nonlinearities, i.e. $\mathcal K$ is a bounded subgraph of $\mathcal G$, then the existence result can be shown independent of the parameter $\mu>0$ in the nonlinearity: \[thm:intromain2\] Let $\mathcal G$ be a noncompact locally finite graph and $\mathcal K\subseteq \mathcal G$ a connected subgraph. Suppose $A^M=(i \frac{\mathrm d}{\mathrm dx}+ M)^2+V$ admits a ground state that does not vanish identically on $\mathcal K$. (i) If $\inf \sigma(A^M) < \inf \sigma_{\text{ess}}(A^M)$, then $$E_{\text{NLS}}^{(\mathcal K)}:= \inf_{\substack{u\in H^1(\mathcal G)\\ \\|u\\|_{L^2}^2=1}}\frac{1}{2} \int_{\mathcal G} \left \|\left (i \frac{\mathrm d}{\mathrm dx} + M\right )u\right \|^2+ V\|u\|^2 \, \mathrm dx - \frac{\mu}{q} \int_{\mathcal K} \|u\|^q \, \mathrm dx$$ admits a minimizer for sufficiently small $\mu>0$. (ii) If $\mathcal K$ is a bounded subgraph of $\mathcal G$, then minimizers exist for all $\mu >0$. In §7 we are going to show that for a tree graph $\mathcal G$ the ground states of Schrödinger operators with magnetic potential do not vanish anywhere on $\mathcal G$. Then, given a decaying potential $V\in L^2+ L^\infty(\mathcal G)$ with $V=V_2+V_\infty$, such that $V_2\in L^2(\mathcal G)$ and $V_\infty\in L^\infty(\mathcal G)$ satisfying $$\label{eq:introdecaying} \sup_{x\in \mathcal G\setminus K} \|V_\infty(x)\| \to 0 \qquad (n\to \infty),$$ we show: \[thm:introlastlastlast\] Let $\mathcal G$ be a noncompact locally finite tree graph with finitely many vertices of degree $1$ and suppose that $V\in L^2+ L^\infty$ satisfies . Then admits a minimizer if $$E_{\text{NLS}}^{(\mathcal K)} =\inf_{\substack{u\in H^1(\mathcal G)\\ \\|u\\|_{L^2}^2=1}}\frac{1}{2} \int_{\mathcal G} \left \|\left (i \frac{\mathrm d}{\mathrm dx} + M\right )u\right \|^2+ V\|u\|^2 \, \mathrm dx - \frac{\mu}{q} \int_{\mathcal K} \|u\|^q \, \mathrm dx< E_{\text{NLS}}(\mathbb R).$$ In particular, if $$\inf \sigma\left ( \left ( i \frac{\mathrm d}{\mathrm dx} +M\right )^2 +V\right )<0,$$ then we have existence of minimizers of $E_{\text{NLS}}^{(\mathcal K)}$ for $0 < \mu \le (\Sigma_0/\gamma_q)^{\frac{3}{2}-\frac{p}{4}}$ as in Theorem \[thm:introcac2018\]. This paper is organized as follows. In §2 we introduce higher-order Sobolev spaces and obtain inequalities on Sobolev spaces on metric graphs including variants of Sobolev and Gagliardo–Nirenberg inequalities. We also discuss basic properties of these spaces such as density results and a characterization of $W^{1,\infty}$ via uniformly bounded Lipschitz functions. In §3 we build the existence theory that is the foundation for all of our results. Theorem \[thm:introdich\] is shown in §3.1 and Theorem \[thm:introextenceprinciple\] is shown in §3.2. In §3.3 we develop an existence result for translation-invariant functionals. In §4 we discuss the application of this existence theory to $E^{(k)}$ on finite graphs and prove Theorem \[thm:intromainnowcloser\] and Theorem \[thm:intromain1\]. In §4.1 we formalize the problem and show basic properties of the functional. In §4.2 we construct suitable partitions of unity and prove a decomposition formula in §4.3, which we use to show that the existence theory is applicable. In §4.4 we prove Theorem \[thm:intromain11\] and show in §4.5 that the existence theory is applicable for decaying potentials. In §5 we discuss existence results for ground states of $E_{\text{NLS}}^{(\mathcal K)}$, where $\mathcal K\subseteq \mathcal G$, on locally finite graphs. In §6 we discuss the energy inequality that is essential for the existence theory and relate it to spectral estimates by developing a Persson type theory for the operators in and . This will also conclude the proof of Theorem \[thm:intromain2\]. In particular, we discuss sufficient conditions for the potential $V$ such that is satisfied. In §7 we finish the paper with an application of the existence results to infinite metric trees via reduction of the problem to one without magnetic potential and prove Theorem \[thm:introlastlastlast\]. Let us finish the introduction by mentioning a few other recent results on related topics. For a general reference on metric graphs we refer to [@berkolaiko2013introduction]. For a broad overview of spectral theory of operators we refer to [@reedmethods]. We refer to [@exner2018spectral] for a recent article on spectral theory for metric graphs with infinitely many edges. The stationary energy functional $$E_{\text{NLS}}^{(\mathcal K)} (u) = \frac{1}{2} \int_{\mathcal G} \|u'\|^2 \, \mathrm dx - \frac{\mu}{q} \int_{\mathcal K} \|u\|^q\, \mathrm dx, \qquad \\|u\\|_{L^2}^2 =1.$$ with $\mathcal K=\mathcal G$ was considered in [@adami2012stationary], [@adami2015nls], [@adami2016threshold], [@adami2017negative] among others. A variant of the problem with localized nonlinearities in the $L^2$-subcritical case was considered in [@tentarelli2016nls] and for the $L^2$-critical case extended in [@dovetta2018ground] and [@dovetta20182], where the domain of integration in the nonlinearity is taken to be a bounded subgraph $\mathcal K$. A very recent survey on results on the stationary NLS energy functional with localized nonlinearity can be found in [@borrelli2019overview]. The main tool for the existence theorems regarding existence of ground states of the stationary NLS energy functional is the constructions of so called competitors, i.e. test functions or a sequence of test functions that establish the energy inequality of Theorem \[thm:introast2016\]. Recently, classes of graphs that do not necessarily consist of finitely many edges have also been considered. For instance, [@dovetta2019nls] deals with a certain class of infinite tree graphs, which fall into the category of the locally finite graphs that we consider here. We would also like to mention the results obtained by [@Akduman2019] for the NLS energy functional with growing potentials for a class of general metric graphs satisfying certain volume growth assumptions using a generalized Nehari approach. Acknowledgments. I want to thank James Kennedy, who helped me shape the article and for all the helpful discussions with him. I thank Hugo Tavares, who suggested studying the NLS energy functional and provided me with references and overall knowledge on the topic and giving a few helpful suggestions. I thank Marcel Griesemer for helpful discussions as the approach on the real line is loosely based on discussions we had in the past in the case of the NLS energy functional on the real line. The work was supported by the Fundação para a Ciência e a Tecnologia, Portugal, via the program “Bolseiro de Investigação”, reference PD/BD/128412/2017, and FCT project UID/MAT/00208/2019. Sobolev spaces on graphs ======================== In this section we define metric graphs and define Sobolev spaces on metric graphs and prove Sobolev inequalities on these spaces. Metric graphs ------------- Let $\mathcal G=(\mathcal V,\mathcal E)$ be a metric graph, where each edge $e\in \mathcal E$ is associated with an interval $I_e$ of length $l_e\in (0,\infty]$, where $I_e=[0,\infty)$ if $l_e=\infty$ and $I_e=[0, l_e]$ otherwise. We assume every vertex to be at least of degree one. For every $e\in {{\mathcal E}}$ joining two vertices we associate each of the vertices to $0$ and $l_e<\infty$ respectively on the interval $I_e$. However, we always assume that the half-line $I_e=[0,\infty)$, which we also call a ray, is attached to the remaining part of the graph at $x_e=0$, and the vertex of the graph corresponding to $x_e=\infty$ is called a vertex at infinity. In particular, there are no edges between vertices at infinity. We denote by $\mathcal E_\infty\subset \mathcal E$ the set of all rays. A connected metric graph $\mathcal G$ admits a natural structure of a metric space. The metric is the shortest distance measured along the edges of the graph. A more detailed introduction to metric graphs can be found for instance in [@berkolaiko2013introduction]. We consider two classes of noncompact graphs: \[df:localfin\] Let $\mathcal G$ be a connected metric graph. Then we say (1) $\mathcal G$ is a finite graph if there are at most finitely many edges. (2) $\mathcal G$ is a locally finite graph if ${{\mathcal E}}$ is a countable set such that any bounded subset of the graph intersects at most finitely many edges We will always take our graphs to be connected, locally finite noncompact metric graphs. Note that a finite graph $\mathcal G$ is compact if and only if the graph $\mathcal G$ does not admit any rays, that is, there are no edges of $\mathcal G$ that are half-lines. Note that a locally finite graph is compact, if and only if the graph is bounded. We see immediately that all compact locally finite metric graphs are finite, and all finite graphs are locally finite. In particular for compact graphs the notions in Definition \[df:localfin\] coincide. First-order Sobolev spaces -------------------------- Let $\mathcal G$ be any locally finite graph. Here and in the rest of the article denote $$\label{eq:short} u_e:=u\big \|_e.$$ We denote by $C(\mathcal G)$ the set of continuous complex-valued functions in $\mathcal G$ and define for $1\le p <\infty$ $$\begin{aligned} L^p(\mathcal G) &= \left \{u\in \bigoplus_{e\in {{\mathcal E}}} L^p(I_e)\bigg \| \\|u\\|_p^p:=\sum_{e\in {{\mathcal E}}} \left \\|u_e \right \\|_{p}^p<\infty\right \}, \\ W^{1,p}(\mathcal G)&=\left \{u\in C(\mathcal G) \bigg\| u_e \in W^{1,p}(I_e)\; \land\; \\|u\\|_{W^{1,p}}^p:=\sum_{e\in {{\mathcal E}}} \\|u_e \\|_{W^{1,p}(I_e)}^p <\infty \right \}. \end{aligned}$$ Then we set $H^1(\mathcal G) = W^{1,2}(\mathcal G)$ as usual. For $q=\infty$ we need to adapt the definition above slightly: $$\begin{aligned} L^\infty(\mathcal G) &= \left \{u\in \bigoplus_{e\in {{\mathcal E}}} L^\infty(I_e)\bigg \| \\|u\\|_\infty:=\max_{e\in {{\mathcal E}}} \left \\|u_e \right \\|_{\infty}<\infty\right \}, \\ W^{1,\infty}(\mathcal G)&=\bigg \{u\in C(\mathcal G) \| u_e \in W^{1,\infty}(I_e)\; \\ &\qquad\qquad \land\; \\|u\\|_{W^{1,\infty}}:=\max_{e\in {{\mathcal E}}} \\|u_e \\|_{W^{1,\infty}(I_e)} <\infty \bigg \}. \end{aligned}$$ Gagliardo–Nirenberg inequality with magnetic potential ------------------------------------------------------ The goal of this section is to show a Gagliardo–Nirenberg inequality for locally finite, connected metric graphs. With future applications in mind we actually consider a modified Gagliardo–Nirenberg inequality: \[cor:Gagliardo-nirenberg\] Let $\mathcal G$ be a locally finite, connected metric graph and $M\in C(\mathcal G)$. For $p\in [2,\infty)$ there exists a constant $C>0$ independent of $M$ such that $$\label{eq:Gagliardo-nirenberg} \\|u\\|_{p}^p \le C \left \\|\left ( i \frac{\mathrm d}{\mathrm dx}+M\right )u\right \\|_{2}^{\frac{p-2}{2}} \\| u\\|_{2}^{\frac{p+2}{2}},$$ for all $u\in H^1(\mathcal G)$. The inequality reduces to the usual Gagliardo–Nirenberg inequality when $M\equiv 0$. Suppose $\mathcal G$ is a tree graph at first. Then using the unitary gauge transform $G: H^1(\mathcal G) \to H^1(\mathcal G)$ (see also §7 for details) we deduce that is equivalent to $$\label{eq:Gagliardo-nirenberg2} \\|u\\|_{p}^p \le C \left \\|u'\right \\|_{2}^{\frac{p-2}{2}} \\| u\\|_{2}^{\frac{p+2}{2}},$$ which can be shown via symmetrization methods as shown[^1] in [@adami2015nls]. In particular, the constant $C>0$ can be chosen independent of $M$. Cutting the graph at a discrete set of points on the metric graph, i.e. we can find a tree graph $\widetilde{\mathcal G}$ such that identifying a discrete set of points on the graph results in a graph isometric isomorph to $\mathcal G$. Moreover, since $H^1(\mathcal G)$ imbeds isometrically into $H^1(\overline{\mathcal G})$, also holds for $H^1(\widetilde{\mathcal G})$ and the constant $C>0$ can be chosen independent of $M\in C(\mathcal G)$. Similarly, one can also show the following Sobolev inequality. Let $\mathcal G$ be a locally finite, connected metric graph and $M\in C(\mathcal G)$. Let $p\in [2,\infty]$ then there exists a constant $C>0$ independent of $M$ such that $$\label{eq:Sobolevinequality} \\|u\\|_{p} \le C \left (\int_{\mathcal G} \left \|\left ( i \frac{\mathrm d}{\mathrm dx} + M\right ) u\right \|^2\, \mathrm dx+\int_{\mathcal G} \|u\|^2\, \mathrm dx\right )^{1/2},$$ for all $u\in H^1(\mathcal G)$. The aproach is similar as before, indeeed we can use the known result in absence of $M$ and use a gauge transform to show that holds with a constant $C>0$ independent of the potential $M\in C(\mathcal G)$. Higher-order Sobolev spaces --------------------------- In this section we introduce the notion of higher-order Sobolev spaces on graphs for $p\in [1,\infty)$. Let $\mathcal G$ be a locally finite metric graph. One naive way of doing so is simply defining it analogously as in $W^{1,p}(\mathcal G)$ $$\widetilde {W^{k,p}}(\mathcal G) := \left \{u\in C(\mathcal G) \bigg \| u_e \in W^{k,p} (I_e)\quad \forall e\in E\; \land\; \\|u \\|_{W^{k,p}}^p:=\sum_{e\in E} \\|u_e \\|_{W^{k,p}(I_e)}^p <\infty \right \}$$ Then for $u\in \widetilde{W^{k,p}}(\mathcal G)$ we always have $u_e \in C^{k-1}(I_e)$ for all $e\in \mathcal E$. However, we also want to specify a condition on the higher-order derivatives. We define $$\begin{gathered} W^{k,p}(\mathcal G) = \{u\in \widetilde{W^{k,p}}(\mathcal G)\| u^{(j)} \in C(\mathcal G)\quad \forall j\le k-1 \text{ even}\\ \land \quad\sum_{e:e\succ \mathsf v} \frac{\partial^j}{\partial\nu^j}u_e(\mathsf v) =0 \quad \forall j\le k-1 \text{ odd }\, \forall \mathsf v\in V\},\end{gathered}$$ where $e:e\succ \mathsf v$ denotes the set of edges $e$ adjacent to a vertex $\mathsf v$. This definition is natural in the sense that if we consider a dummy vertex $\hat v$ of degree $2$, i.e. subdividing an edge $e\in \mathcal E$ connecting two vertices $v_1, v_2$ into two edges $e_1, e_2$ connecting $v_1, \hat v$ and $\hat v, v_2$ respectively such that the total length of the graph is preserved, then the Kirchhoff condition simply reduces to a continuity statement of the derivatives. As usual we define $\widetilde{H^k}(\mathcal G)= \widetilde{W^{k,2}}(\mathcal G)$ and $H^k(\mathcal G)= W^{k,2}(\mathcal G)$. While the Sobolev spaces as defined here are domains of self-adjoint realizations of differential operators on $L^2(\mathcal G)$, the definitions are not necessarily canonical. We refer to [@gregorio2017bi] for a discussion on self-adjoint extension of the Bilaplacian, and a discussion for $W^{2,p}$ spaces on graphs. In this context, we are going to define some useful related spaces: $$\label{eq:sobolev0} \begin{gathered} \widetilde{W^{k,p}_{0}}(\mathcal G) := \{u\in \widetilde{W^{k,p}}(\mathcal G)\| u^{(l)}(\mathsf v)=0, \quad \forall 1\le l\le k-1, \quad \forall \mathsf v\in \mathcal V\}\\ \widetilde{W^{k,p}_{c}} (\mathcal G):= \left \{u\in \widetilde{W^{k,p}_0}(\mathcal G)\| \operatorname{supp}(u) \text{ compact}\right \}. \end{gathered}$$ Of special importantance will be the following test function spaces: $$\begin{split} \widetilde{C^\infty(\mathcal G)} &:= \bigcap_{k\in \mathbb N}\widetilde{W^{k, \infty}}(\mathcal G)\\ \widetilde{C^\infty_b(\mathcal G)} &:= \bigcap_{k\in \mathbb N}\widetilde{W_0^{k, \infty}}(\mathcal G)\\ \widetilde{C^\infty_c(\mathcal G)} &:= \bigcap_{k\in \mathbb N}\widetilde{W_c^{k, \infty}}(\mathcal G). \end{split}$$ Higher-order Gagliardo–Nirenberg inequality for finite metric graphs -------------------------------------------------------------------- Let $\mathcal G$ be a finite metric graph. Consider the norm on $H^k(\mathcal G)$ defined as $$\|u\|_{H^k} := \left (\int_{\mathcal G} \|u^{(k)}\|^2+ \|u\|^2\, \mathrm dx \right )^{1/2}.$$ Then due to the Gagliardo–Nirenberg interpolation inequality on intervals (see e.g. [@leoni2017first Theorem 7.41]) applied edgewise $$\label{eq:ineedthis} \|u\|_{H^k}^2 \le \\|u\\|_{H^k}^2\le C \|u\|_{H^k}^2$$ and we conclude that $\\|\cdot\\|_{H^k}$ and $\|\cdot\|_{H^k}$ are equivalent norms in $H^k(\mathcal G)$. \[prop:higherordergag\] Let $\mathcal G$ be a finite metric graph. Then $$\label{eq:propgagk} \\|u\\|_{p}^p \le C \\|u\\|_2^{\frac{(2k-1)p+2}{2k}} \|u\|_{H^k}^{\frac{p-2}{2k}}$$ From the Gagliardo–Nirenberg inequality on metric graphs and Gagliardo–Nirenberg interpolation inequality on intervals we compute $$\begin{aligned} \\|u\\|_{p}^p &\le C_1 \\|u\\|_{2}^{\frac{p}{2}+1}\\|u'\\|_{2}^{\frac{p}{2}-1}\\ &\le C_k \left \\|u\right \\|_{2}^{\frac{(2k-1)p+2}{2k}} \big \|u\big \|_{H^{k}}^{\frac{p-2}{2k}}. \end{aligned}$$ On the density of Sobolev spaces -------------------------------- \[prop:density\] Let $\mathcal G$ be a finite, connected metric graph and $p\in [1,\infty)$, then $W^{m,p}(\mathcal G)$ is dense in $W^{n,p}(\mathcal G)$ for $m\ge n\ge 1$. For $n=0$ this corresponds to the fact that $W^{m,p}$ is dense in $L^p$ for $m\ge 1$. It suffices to prove that $W^{k+1,p}(\mathcal G)$ is dense in $W^{k,p}(\mathcal G)$. To this end, let $u\in W^{k}(\mathcal G)$ arbitrary and $u_n$ be an edgewise approximating sequence in $\oplus_{e\in \mathcal E} C^\infty(I_e)\cap W^{k+1,p}(I_e)$ such that $$\label{eq:ineedthisnow} \sum_{e\in \mathcal E} \left \\|u_n -u\big \|_e\right \\|_{W^{k,p}} \le \frac{1}{2^n}$$ for all $n\in \mathbb N$. The general idea is to construct sequences $v_n \in \oplus_{e\in \mathcal E} W^{k+1,p}(I_e)$ such that $u_n + v_n \in W^{k+1,p}$ and $$u_n + v_n \stackrel{W^{k,p}}\longrightarrow u\qquad (n\to \infty).$$ For fixed $\mathsf v\in \mathcal V$ and for $n$ satisfying $$\frac{2}{n} \le \min_{e\in \mathcal E} \|I_e\|$$ for all $e\succ \mathsf v$ and $\hat k \in \{0, \ldots, k\}$ we define $$\label{eq:testfunctions} v_{n,\hat k, \mathsf v}(x)\big \|_e=\begin{cases} \frac{c_{n, \hat k, \mathsf v}}{\hat k!} x_{\mathsf v}^{\hat k} \left (1-nx_{\mathsf v}\right )^{k+1}, &\quad \text{for }x\in e \text{ with } x_{\mathsf v}:=\operatorname{dist}(x,\mathsf v)\le \frac{1}{n} \\ 0, &\quad \text{otherwise}. \end{cases}$$ where $c_{n, \hat k, \mathsf v}$ is given by $$\begin{aligned} \text{for }\hat k=0:\qquad c_{n, 0, \mathsf v}&= u-u_n \big \|_e (0_{\mathsf v})\\ \text{for } 1\le \hat k \le k-1:\qquad c_{n, \hat k, \mathsf v} &= u^{(\hat k)}-u_n^{(\hat k)} \big \|_e(0_{\mathsf v}) - \sum_{\ell=0}^{\hat k-1} (k+1)_{\ell} (-n)^{\ell} c_{n,\ell, \mathsf v}. \end{aligned}$$ We can extend the functions $v_{n,\hat k, \mathsf v}$ by zero on the rest of the graph. With the Leibniz rule for $1\le \ell \le k+1$ we compute $$\label{eq:leibnizcomp} v_{n,\hat k, v}^{(\ell)} (x)\big \|_e= \chi_{\{x_{\mathsf v}\le \frac{1}{n}\}}\sum_{m=0}^\ell \frac 1{\hat k!}{\binom{\ell}{m}} c_{n, \hat k, \mathsf v} (-n)^{\ell-m}(\hat k)_m (k)_{\ell-m} x_{\mathsf v}^{\hat k - m}(1-n x_{\mathsf v})^{k+m+1-\ell}$$ Then $$\tilde v_n := \sum_{\ell=0}^{k-1}\sum_{\mathsf v\in \mathcal V} v_{n,\ell, \mathsf v}$$ satisfies $\tilde v_n \in \oplus_{e\in \mathcal E} W^{k+1,p}(I_e)$ and $u_n+\tilde v_n \in W^{k,p}(\mathcal G)$ since $u_n+\tilde v_n$ coincides in all $k-1$ derivatives with $u$ by construction. Indeed, the restrictions of the $k$^th^ derivatives at the vertices are of rank $\le 2\|\mathcal E\|$. Then we can find $c_{n,k,\mathsf v}$ for all $\mathsf v\in \mathcal V$ $$v_{n,k,\mathsf v}\big \|_e (x) =\begin{cases} \frac{c_{n, k, \mathsf v}}{k!} x_{\mathsf v}^{k} \left (1-\max\{n, c_{n,k, \mathsf v}^{2}\}x_{\mathsf v}\right )^{k+1}, &\quad \text{for }x\in e \text{ with } \\ &x_{\mathsf v}\le \min\{n^{-1}, c_{n,k, \mathsf v}^{-2}\} \\ 0, &\quad \text{otherwise}. \end{cases}$$ such that $$u_n + \tilde v_n + \sum_{\mathsf v\in \mathcal V} v_{n,k,\mathsf v}\in W^{k+1,p}(\mathcal G).$$ By assumption we deduce by applying the Sobolev imbedding edgewise $$\label{eq:additionalinequ} \sum_{e\in \mathcal E} \left \\|u_n-u\big \|_e\right \\|_{C^{k-1}} \le \frac{C}{2^n}$$ for all $n\in \mathbb N$ and some $C>0$ and satisfies by construction $$\label{eq:trickher} (-n)^\ell c_{n,\hat k, \mathsf v} \to 0 \qquad (n\to \infty)$$ for all $1\le \ell\le k$ and $\mathsf v\in \mathcal V$. By a change of variables we then compute for $0\le m \le \ell \le k$ $$\begin{gathered} n^{\ell-m} \int_{I_e} x_{\mathsf v}^{\hat k-m} (1-n x_{\mathsf v})^{k+m+1-\ell}\, \mathrm dx_{\mathsf v}= n^{\ell-1 -\hat k} \int_0^1 t^{\hat k-m} (1-t)^{k+m+1-\ell}\, \mathrm dt\\ \begin{multlined} c_{n,k, \mathsf v}\max\{n, c_{n,k, \mathsf v}^2\}^{\ell-m} \int_{I_e} x_{\mathsf v}^{k-m} (1-\max\{n, c_{n,k, \mathsf v}^2\} x_{\mathsf v})^{k+m+1-\ell}\, \mathrm dx_{\mathsf v}\\ = c_{n,k, \mathsf v}\min\{n^{-1}, c_{n,k, \mathsf v}^{-2}\}^{k+1-\ell} \int_0^1 t^{k-m} (1-t)^{k+m+1-\ell}\, \mathrm dt \longrightarrow 0 \\ (n\to\infty) \end{multlined}\end{gathered}$$ and with and we conclude $$\begin{aligned} &\left \\|u- \left [u_n + \tilde v_n + \sum_{\mathsf v\in \mathcal V} v_{n,k,\mathsf v} \right ]\right \\|_{W^{k,p}}\\ &\qquad \qquad\le \left \\|u- u_n\right \\|_{W^{k,p}} + \left \\|\tilde v_n + \sum_{\mathsf v\in \mathcal V} v_{n,k,\mathsf v}\right \\|_{W^{k,p}}\longrightarrow 0 \qquad (n\to \infty). \end{aligned}$$ \[prop:densitycompact\] Let $\mathcal G$ be a locally finite, connected metric graph and $p\in [1,\infty)$. Then $$W^{1,p}_c(\mathcal G)= \{u\in W^{1,p}(\mathcal G)\| \operatorname{supp} u \text{ is bounded}\}$$ is dense in $W^{1,p}(\mathcal G)$. Let $K$ be a bounded, connected subgraph of $\mathcal G$. For $R>0$ set $$K_R :=\{x\in \mathcal G\| \operatorname{dist}(x, K)<R\}.$$ We define the cut-off functions $\psi_n$ via $$\widetilde{\psi_n} := \frac{1}{n}\max \{n, \operatorname{dist}(x,K_n)\},\quad \psi_n:= 1- \widetilde{\psi_n}$$ For all $u\in W^{1,p}(\mathcal G)$ one then computes $$\begin{aligned} &\limsup_{n\to \infty}\\|u- \psi_n u\\|_{W^{1,p}}^p = \limsup_{n\to \infty}\left [\int_{\mathcal G} \left \|\frac{\mathrm d}{\mathrm dx} \widetilde{\psi_n} u\right \|^p\, \mathrm dx+ \int_{\mathcal G} \left \|\widetilde {\psi_n} u\right \|^p\, \mathrm dx\right ]\\ &\qquad\le \limsup_{n\to \infty} \left [\frac{2^p}{n^p} \int_{\mathcal G\setminus K_n} \|u\|^p\, \mathrm dx +2^p \int_{\mathcal G\setminus K_n} \left \|\widetilde \psi_n u \right \|^p\, \mathrm dx + \int_{\mathcal G \setminus K_n} \left \|\widetilde \psi_n u\right \|^p\, \mathrm dx\right]=0, \end{aligned}$$ where in the equation we used $$\int_{\mathcal G\setminus K_n} \|\widetilde{\psi_n} u\|^p \, \mathrm dx \le \int_{\mathcal G\setminus K_n} \|u\|^p \, \mathrm dx \to 0 \qquad (n\to \infty).$$ As such $\psi_n u \to u$ in $W^{1,p}(\mathcal G)$ as $n\to \infty$. \[prop:densitylocfinite\] Let $\mathcal G$ be a locally finite, connected metric graph and $p\in [1,\infty)$. Then $$W^{2,p}_c(\mathcal G)= \{u\in W^{2,p}(\mathcal G)\| \operatorname{supp} u \text{ is bounded}\}$$ is dense in $W^{1,p}$. Let $u\in W^{1,p}(\mathcal G)$. By Lemma \[prop:densitycompact\] we can find a sequence $u_n\in W^{1,p}_c(\mathcal G)$ with $u_n \to u$ in $W^{1,p}$. Then by Proposition \[prop:density\] for each $n$ we find a sequence $u_{n,m} \in W^{2,p}(\mathcal G)$, after extending by zero on the whole graph, converging towards $u_n$ in $W^{1,p}(\mathcal G)$ as $m\to \infty$. Then one can construct a sequence in $W^{2,p}_c(\mathcal G)$ converging to $u$ in $W^{1,p}$ by a diagonal argument. Characterization of $W^{1,\infty}$ ---------------------------------- We give a characterization of $W^{1, \infty}$ on locally finite, connected metric graphs in the following: \[prop:lipschitz\] Let $\mathcal G$ be a locally finite, connected metric graph. Then $W^{1,\infty}(\mathcal G)= C^{0,1}_b(\mathcal G)$ is the set of uniformly bounded, Lipschitz continuous functions. Assume $u\in W^{1,\infty}(\mathcal G)$. Let $x,y\in \mathcal G$ and $\gamma$ be a path of length $L(\gamma)$ connecting $x,y$, parametrized by arc length. In the first step let us assume $u\in C^1$ edgewise, then using the continuity of $u$ we have $$\|u(x)-u(y)\| \le \int_0^{L(\gamma)} \|u'(\gamma)\|\, \mathrm d\|\gamma\| \le \max_{t} \|u'(\gamma(t)\| L(\gamma).$$ Due to density this holds also for $W^{1,\infty}(\mathcal G)$. Taking the infimum over all paths connecting $x,y$ we conclude $$\|u(x)-u(y)\|\le \\|u'\\|_{\infty} \operatorname{dist}(x,y)$$ and thus $u\in C^{0,1}_b(\mathcal G)$. On the other hand, let $f\in C^{0,1}_b(\mathcal G)$, then $$\frac{\|u(x)-u(y)\|}{\operatorname{dist}(x,y)} \le L$$ for some constant $L>0$. On each edge $e\in \mathcal E$ then $u\in W^{1,\infty}(I_e)$ and $u'$ exists a.e. and $$\\|u'\\|_{\infty} \le L.$$ We conclude $u\in W^{1,\infty}(\mathcal G)$ since $u$ is also uniformly bounded by assumption. A general Existence Principle ============================= In this section we derive an existence theory for ground states of functionals as in and . To do so, we derive a more general existence principle for functionals on function spaces on metric measure spaces, which we will apply later to the functionals introduced before to discuss the existence of minimizers. A Dichotomy Result ------------------ In the following we work with a slightly more abstract space $X(\mathcal M)$, namely a function space on a metric measure space $\mathcal M$. \[as:assumption1\] Let $p\in [1,\infty)$. Let $(\mathcal M, d)$ be a metric space with a locally finite Borel measure $\mu$ on $\mathcal M$. Assume $X= X(\mathcal M)\subset L^p(\mathcal M)$ is a nontrivial Banach function space continuously and locally compactly imbedded in $L^p(\mathcal M)$, i.e. $\mathcal M$ restricted to $$K_R(y) := \{x\in \mathcal M\|\operatorname{dist}(x,y) \le R\}$$ is compactly imbedded in $L^p(K_R(y))$ for all $R>0$ and $y\in \mathcal M$. Our prototype to satisfy Assumption \[as:assumption1\] is $X(\mathcal G)= H^1(\mathcal G)$ where $\mathcal G$ is a connected, locally finite metric graph. However, it is for instance also satisfied by $X(\Omega) = H^1(\Omega)$ for a domain $\Omega \subset \mathbb R^n$ with $n\in \mathbb N$. Let $p\ge 2$ and let $\mathcal M$ and $X=X(\mathcal M)$ be as in Assumption \[as:assumption1\]. Let $E \in C(X(\mathcal M), \mathbb R)$ and $$E_t := \inf_{\substack{u\in X(\mathcal M)\\ \\|u\\|_{p}^p=t}} E(u)$$ be bounded from below for each $t\ge 0$ and $E(0)=0$. We say: (1) $t\mapsto E_t$ is strictly subadditive if $$E_{t_1+t_2} < E_{t_1} + E_{t_2}, \qquad \forall t_1, t_2 >0.$$ (2) $E$ is weak limit superadditive in $X$ if for all $c>0$ any weakly convergent minimizing sequence $u_n \rightharpoonup u$ in $X(\mathcal M)$ of $E_c$ satisfies up to a subsequence $$\limsup_{n\to \infty} E(u_n) \ge E(u) + \limsup_{n\to \infty} E(u_n -u).$$ \[thm:main1\] Let $p\in [2,\infty)$, $c>0$ and let $\mathcal M$, $X=X(\mathcal M)$ be as in Assumption \[as:assumption1\]. Let $E\in C(X(\mathcal M), \mathbb R)$ be a weak limit superadditive functional in $X$. Let $$t\mapsto E_t = \inf_{\substack{u\in X(\mathcal M)\\ \\|u\\|_{p}^p=t}} E(u)$$ be a strictly subadditive, continuous function of $t\in [0,c]$. Let $u_n$ be a minimizing sequence of $E_c$, and assume there exists $u\in X$ such that up to a subsequence $u_n \rightharpoonup u$. Then either $u\equiv 0$ or $u_n \to u$ in $L^p(\mathcal M)$ and $u\neq 0$ is a minimizer. Theorem \[thm:main1\] gives rise to a dichotomy. If the requirements of Theorem \[thm:main1\] are satisfied, then a minimizing sequence satisfies either $u_n \rightharpoonup 0$ in $X$ or there exists an $L^p$ convergent subsequence converging to a minimizer of $E_c$. Up to a subsequence, this implies pointwise convergence almost everywhere. On the other hand, $u_n \rightharpoonup 0$ in $X(\mathcal M)$ implies $\\|u_n\\|_{L^p(K)}\to 0$ on any bounded subset $K$ of $\mathcal G$, but since $\\|u_n\\|_{p}^p=c$ for all $n\in \mathbb N$ the mass needs to move outside any compact set. In virtue of Theorem \[thm:main1\] we say a minimizing sequence of $E_c$ is vanishing if $u_n \rightharpoonup 0$ in $X$ and non-vanishing otherwise. Let $u_n \rightharpoonup u$ in $X$ with $u\neq 0$. Then since $u_n \to u$ in $L^{p}_{\text{loc}}$ we deduce $u\neq 0$ and $$c\ge \\|u\\|_{p}^p>0.$$ Up to a subsequence $u_n \to u$ pointwise almost everywhere, and from the Brezis-Lieb Lemma (see [@brezis1983relation]) we conclude $$\\|u\\|_p^p + \limsup_{n\to \infty} \\|u_n-u\\|_p^p =c.$$ By weak limit superadditivity and strict subadditivity of $t\mapsto E_t$ we deduce that up to a subsequence $$\begin{aligned} E_c &\ge E(u) + \limsup_{n\to \infty} E(u-u_n)\\ &\ge E_{\\|u\\|_p^p} + \limsup_{n\to \infty} E_{\\|u-u_n\\|_p^p}\\ &\ge E_{\\|u\\|_p^p} + E_{\limsup_{n\to \infty} \\|u_n-u\\|_p^p} \ge E_c. \end{aligned}$$ where equality is only attained when $\\|u\\|_p^p=c$ and $\limsup_{n\to \infty} \\|u_n-u\\|_p^p=0$. Thus $\\|u\\|_p^p=c$ and we conclude $$E_c = E(u)$$ and $u$ is a minimizer of $E_c$. Vanishing sequences and Ionization Energies ------------------------------------------- As in the previous section we consider $\mathcal M$ to be a metric measure space and $X(\mathcal M)\subset L^p(\mathcal M)$ to be a Banach space which is locally compactly imbedded in $L^p(\mathcal M)$. In the following we want to introduce partitions of unity and therefore assume the following: \[as:assumption2\] Let $(\mathcal M, d)$ be a metric space with locally finite Borel measure $\mu$ on $\mathcal M$. We assume $X(\mathcal M)$ to be a subspace of the space of $\mu$-measurable functions on $\mathcal M$ and $Y(\mathcal M)$ to be a set of real-valued functions on $\mathcal M$ such that $X(\mathcal M)$ is invariant with respect to multiplication of elements in $Y(\mathcal M)$. Let $\mathcal G$ be a locally finite metric graph. If $X(\mathcal G)= H^1(\mathcal G)$ and $Y=W^{1,\infty}(\mathcal G)$, then $X(\mathcal G), Y(\mathcal G)$ are imbedded in $C(\mathcal G)$ and $X(\mathcal G)$ is invariant by multiplications of $Y(\mathcal G)$, i.e. elements in $X(\mathcal G)$ are invariant by pointwise multiplication of elements in $Y(\mathcal G)$, and for $f\in W^{1,\infty}(\mathcal G)$ and $g\in H^1(\mathcal G)$ $$(fg)'= f' g+ g'f.$$ In this section we show that the existence of vanishing sequences gives a bound from below on the ground state energy $E_c$, which allows us, under stronger assumptions, to deduce an existence result from Theorem \[thm:main1\]. Let us first introduce partitions of unity on metric spaces. Let $Y$ be as in Assumption \[as:assumption2\]. Assume $\cup_{O\in \mathcal O} O=\mathcal G$ is a locally finite open covering $\mathcal O$ of $\mathcal M$. Then we say a family of nonnegative functions $\psi_O\in Y(\mathcal G)$ is a partition of unity subordinate to $\mathcal O$ if $$\operatorname{supp} \psi_O \subset O, \quad \forall O\in \mathcal O \quad\land \quad \bigcup_{O\in \mathcal O} \operatorname{supp} \psi_O = \mathcal G \quad \land \quad 0 \le \psi_O \le 1$$ and $\sum_{O\in \mathcal O} \psi_O(x)\neq 0$ for all $x\in \mathcal M$ and $$\Psi_O(x)= 1, \qquad \forall x\in \operatorname{supp} \Psi_O \setminus \bigcup_{\widehat O\in \mathcal O \setminus \{O\}} \operatorname{supp} \Psi_{\widehat{O}}.$$ In the following we define for $R> 0$ the open and closed $R$-neighborhoods of a subset $K\subset \mathcal M$ by $$\label{eq:core} \begin{aligned} K_{R} &:= \{x\in \mathcal M\| \operatorname{dist} (x, K)< R\}\\ \overline{K_{R}} &:= \{x\in \mathcal M\|\operatorname{dist} (x, K) \le R\}. \end{aligned}$$ \[ex:first\] Let $\mathcal G$ be a locally finite graph. Let $K$ be some compact, connected subgraph. A simple example for a partition of unity in $W^{1,\infty}(\mathcal G)= C^{0,1}_b(\mathcal G)$ on a locally finite metric graph subordinate to $K_2$, $\mathcal G\setminus K_1$ is given by $$\psi(x) = \max\{\operatorname{dist}(\mathcal G\setminus K_{2}, x), 1\}, \qquad \widetilde{\psi}(x) = 1-\psi.$$ In the following we will characterize a property of functionals with regards to partitions of unity. Let $k \in \mathbb N$ and let $\mathcal M$ be a metric space. Let $K$ be a bounded subset of $\mathcal M$ and $K_n$ be defined by for $n\in \mathbb N$. We say a sequence of open coverings $\mathcal O_n=\{O_n^{(1)}, \ldots, O_n^{(k)}\}$ consisting of $k$ open subsets (not necessarily connected) is vanishing-compatible, if $$K_n\cap O_n^{(i)}= \emptyset, \qquad \forall i\in \{2,\ldots, k\}$$ and $O_n^{(1)}$ is bounded for all $n$. In particular, $K_n \subset O_n^{(1)}$. That is, for a sequence of open coverings $\mathcal O_n=\{O_n^{(1)}, \ldots, O_n^{(k)}\}$ all its members except $O_n^{(1)}$ move away from $K$. Furthermore, this notion does not depend on the choice of $K$, i.e. up to a subsequence any sequence of open coverings is vanishing-compatible for any other $K$. Let $k\in \mathbb N$ and $\mathcal O_n=\{O_n^{(1)}, \ldots, O_n^{(k)}\}$ be a vanishing-compatible sequence of open coverings. Then we say $E\in C(X(\mathcal M), \mathbb R)$ is $k$-superadditive with respect to a sequence of partitions of unity $$\{\psi_{O}\}_{O\in \mathcal O_n}=\left \{\psi_{O_n^{(1)}},\ldots, \psi_{O_n^{(k)}}\right \}$$ if for any vanishing sequence $(v_n)$ up to a subsequence $$\limsup_{n\to \infty} E(v_n) \ge \sum_{i=1}^k \limsup_{n\to \infty} E(\psi_{O_{n}^{(i)}} v_n).$$ \[ex:unity\] Let $\mathcal G$ be a locally finite metric graph and let $K$ be a compact subgraph of $\mathcal G$. Recall on $\mathcal G$ the open covering $\mathcal O$ as in Example \[ex:first\] with partition of unity $\psi, \widetilde{\psi}$. Similarly, we define the sequence of partitions of unity $$\psi_n= \frac{1}{n} \max \{\operatorname{dist}(\mathcal G \setminus K_{2n}, x), n\}, \quad \widetilde{\psi_n}= 1- \psi_n$$ and gives a sequence of partitions of unity with regards to a vanishing-compatible sequence of open coverings. This gives rise to our second main result: \[thm:main2\] Let $p\in [2,\infty)$, $c>0$ and let $(\mathcal M, \mu)$, $X(\mathcal M)$ and $Y(\mathcal M)$ satisfy Assumption \[as:assumption1\] and Assumption \[as:assumption2\]. Let $K$ be a bounded, connected, nonempty set in $\mathcal G$. Let $E\in C(X(\mathcal M), \mathbb R)$, such that $$t\mapsto E_t = \inf_{\substack{u\in X(\mathcal M)\\ \\|u\\|_{p}^p=t}} E(u)$$ is continuous and assume $E$ to be $2$-superadditive with respect to a sequence of partitions of unity $\{\psi_{O}\}_{O\in \mathcal O_n}$ in $Y(\mathcal M)$ subordinate to a vanishing-compatible sequence of open coverings $\mathcal O_n=(O_1^{(n)}, O_2^{(n)})$. If there exists a minimizing sequence which is vanishing, then $$E_c = \lim_{R\to \infty} \inf_{\substack{u\in X(\mathcal M), \\|u\\|_{p}^p=c\\ \operatorname{supp} u \subset \mathcal M \setminus K_R}} E(u) =: \widetilde{E_c}.$$ Let $u_n$ be a vanishing sequence. Assume $(O_n^{(1)}, O_n^{(2)})$ to be such that $$K \subset O_n^{(1)}$$ and $O_n^{(1)}$ is bounded. For each fixed $m\in \mathbb N$ we have $$\int_{O_m^{(1)}} \|u_n\|^p\, \mathrm d\mu \to 0 \qquad (n\to \infty).$$ Then for any $m\in \mathbb N$ we find an $n_m$, such that for $n>n_m$ $$\int_{O_m^{(1)}} \|u_n\|^p \, \mathrm d\mu \le \frac{1}{m}.$$ Using a diagonal argument we deduce the existence of a subsequence of $u_n$, still denoted by $u_n$, such that $$\int_{O_n^{(1)}} \|u_n\|^p \, \mathrm d\mu \to 0 \qquad (n\to \infty).$$ In particular, $$\begin{aligned} 0\le \int_{O_n^{(1)}} \|\psi_{O_n^{(1)}}u_n\|^p\, \mathrm d\mu &\le \int_{O_n^{(1)}} \|u_n\|^p\, \mathrm d\mu\\ c -\int_{O_n^{(1)}} \|u_n\|^p\, \mathrm d\mu \le \int_{O_n^{(2)}} \|\psi_{O_n^{(2)}}u_n\|^p \, \mathrm d\mu&\le \int_{\mathcal M} \|u_n\|^p\, \mathrm d\mu =c \end{aligned}$$ and we obtain $$\begin{aligned} \int_{O_n^{(1)}} \|\psi_{O_n^{(1)}}u_n\|^p \, \mathrm d\mu &\to 0 \qquad (n\to \infty)\\ \int_{O_n^{(2)}} \|\psi_{O_n^{(2)}}u_n\|^p \, \mathrm d\mu &\to c \qquad (n\to \infty). \end{aligned}$$ Then by superadditivity we have $$\begin{aligned} E_c &= \lim_{n\to \infty} E(u_n) \\ &\ge \limsup_{n\to \infty} E\left (\psi_{O_{n}^{(2)}} u_n\right )\ge \widetilde {E_c}. \end{aligned}$$ This concludes the inequality $E_c \ge \widetilde{E_c}$. The reverse inequality is trivial since $$E_c \le \inf_{\substack{u\in X(\mathcal M), \\|u\\|_{p}^p=c\\ \operatorname{supp} u \subset \mathcal M \setminus K_R}} E(u)$$ for all $R>0$. \[cor:existence\] Suppose the assumptions in Theorem \[thm:main1\] and Theorem \[thm:main2\] are satisfied and $$E_c <\widetilde{E_c},$$ then a minimizer of $E_c$ exists, and any minimizing sequence for $E_c$ admits a subsequence converging in $L^p$ towards a minimizer of $E_c$. Throughout the rest of the paper given a functional $E\in C(X(\mathcal G), \mathbb R)$ we define the corresponding threshold energy $$\label{eq:defion} \widetilde{E_c} := \lim_{R\to \infty} \inf_{\substack{u\in X(\mathcal G), \, \\|u\\|_{p}^p=c\\ \operatorname{supp} u \subset \mathcal G\setminus K_R}}E(u).$$ In the case of many-body quantum particle systems, this quantity refers to the ionization energy (see [@griesemer2004exponential] and [@simon1983semiclassical]). For this reason, throughout of the rest of the paper we are going to refer to the quantity in also as the ionization threshold or ionization energy. \[ex:NLSclassic\] Let $\mathcal G$ be a locally finite, connected noncompact metric graph with at least one ray. Consider the NLS energy functional as considered in [@adami2016threshold] $$E_{\text{NLS}}(u,\mathcal G) = \int_{\mathcal G} \|u'\|^2\, \mathrm dx - \frac{\mu}{p} \int_{\mathcal G} \|u\|^p\, \mathrm dx.$$ Consider the minimization problem $$\label{eq:exeq} E_{\text{NLS}}(\mathcal G) := \inf_{\substack{u\in H^1(\mathcal G) \\\\|u\\|_2^2=1}} E_{\text{NLS}}(u, \mathcal G).$$ Then in the context of decaying potentials, we will see in §4.5 that the functional $E_{\text{NLS}}$ satisfies the prerequisites of Theorem \[thm:main1\] and Theorem \[thm:main2\]. Moreover, we will characterize the ionization threshold of $\widetilde{E_{\text{NLS}}}(\mathcal G)$. As discussed in Remark \[rmk:decayingpotential\] we will show $$\label{eq:excomp} \widetilde{E_{\text{NLS}}}(\mathcal G):=\lim_{n\to \infty}\inf_{\substack{u\in H^1(\mathcal G) \\\\|u\\|_2^2=1,\; \operatorname{supp} u \subset \mathcal G\setminus K_n}} E_{\text{NLS}}(u, \mathcal G) \le E_{\text{NLS}}(\mathbb R).$$ If $\mathcal G$ is a finite graph we have equality in due to Lemma \[lem:decayingpotential\]. In particular Corollary \[cor:existence\] gives a generalization of the result in [@adami2016threshold] where it was shown that if $\mathcal G$ is finite then minimizers of exist if $$\label{eq:inequality} E_{\text{NLS}}(\mathcal G) < E_{\text{NLS}}(\mathbb R).$$ Since does not guarantee existence by Corollary \[cor:existence\] under the assumption one cannot necessarily extend this result to general locally finite graphs. But as shown in Example \[ex:unrootedtrees\], for a class of infinite tree graphs, one can show the reverse inequality $$\label{eq:excomp2} \widetilde{E_{\text{NLS}}}(\mathcal G) \ge E_{\text{NLS}}(\mathbb R).$$ In particular one can derive for such graphs satisfying existence of minimizers of $E_{\text{NLS}}(\mathcal G)$ under assumption . An existence result on the real line for translation-invariant functionals -------------------------------------------------------------------------- Assume $X(\mathbb R)\subset L^p(\mathbb R)$ is a Banach space on $\mathbb R$. Let $E \in C(X, \mathbb R)$ be a translation invariant functional, i.e. if $T_{\lambda} u(x)= u(x-\lambda)$ then $$E(u)= E(T_\lambda u)$$ for all $\lambda \in \mathbb R$. The space $\mathbb R$ can be understood as a graph consisting of two half-lines. In this context we consider $K=\{0\}$ to be the compact core of this particular graph. \[thm:main3\] Let $p\in [2,\infty)$, $c>0$, let $E\in C(X(\mathbb R), \mathbb R)$ be translation invariant and satisfy Assumption \[as:assumption1\] and Assumption \[as:assumption2\]. Let $$t\mapsto E_t = \inf_{\substack{u\in X(\mathbb R)\\ \\|u\\|_p^p =t}} E(u)$$ be a strictly subadditive functional in $X(\mathbb R)$. Assume $E$ to be superadditive with respect to a sequence of $3$-partitions of unity $\{\psi_O\}_{O\in \mathcal O_n}$ subordinate to the vanishing-compatible sequence of open coverings $$\mathcal O_n = \{(-2n,2n), (n, \infty), (-\infty, -n)\},$$ then it admits a minimizer. By Theorem \[thm:main2\] we only need to construct non-vanishing minimizing sequences. Assume $u_n$ to be a minimizing sequence of the functional $E_c$. Then we may construct such a sequence by using the translation invariance of the functional. Indeed, we may assume up to translation invariance $$\begin{aligned} \int_{0}^\infty \|u_n\|^p\, \mathrm dx &\to \frac{c}{2} \quad (n\to \infty)\\ \int_{-\infty}^0 \|u_n\|^p\, \mathrm dx &\to \frac{c}{2} \quad (n\to \infty). \end{aligned}$$ Assume $u_n$ is vanishing. Then since $u_n \to 0$ in $L^\infty_{\text{loc}}$ (up to a subsequence) due to a diagonal argument, we have that$$\begin{aligned} \int_{\mathbb R} \|\psi_{(-2n,2n)} u_n\|^p \, \mathrm dx &\to 0 \qquad (n\to \infty) \\ \int_{\mathbb R} \|\psi_{(n, \infty)} u_n\|^p \, \mathrm dx &\to \frac{c}{2}\qquad (n\to \infty) \\ \int_{\mathbb R} \|\psi_{(-n, -\infty)} u_n\|^p \, \mathrm dx &\to \frac{c}{2} \qquad (n\to \infty) \end{aligned} $$ Then using the subadditivity of the functional and the strict subadditivity of $t\mapsto E_t$ we conclude $$\begin{aligned} E_c &= \lim_{n\to \infty} E(u_n) \\ &\ge \limsup_{n\to \infty} E\left ( \psi_{(-\infty, -n)} u_n\right ) + \limsup_{n\to \infty} E\left (\psi_{(n, \infty)} u_n\right ) \\ &\ge E_{c/2} + E_{c/2} > E_c. \end{aligned}$$ By contradiction after translating the $u_n$ if necessary we can find a non-vanishing subsequence. Passing to a further subsequence there exists a weakly convergent subsequence in $H^1$ that converges up to a further subsequence to a minimizer by Theorem \[thm:main1\]. \[ex:NLSclassic2\] Recall the NLS energy functional from Example \[ex:NLSclassic\] on the real line $$E_{\text{NLS}}(\mathbb R) = \inf_{\substack{u\in H^1(\mathbb R) \\\\|u\\|_2^2=1}} E_{\text{NLS}}(u, \mathbb R).$$ Then as discussed in Example \[ex:NLSclassic\], we have $$\widetilde{E_{\text{NLS}}}(\mathbb R) = E_{\text{NLS}}(\mathbb R).$$ In particular inequality cannot be satisfied and we cannot apply Corollary \[cor:existence\]. On the other hand, the prerequisites of Theorem \[thm:main3\] are satisfied. In particular this ensures existence of ground states for all $\mu >0$. More explicitly, one can show that ground states of the NLS energy functional on the real line are unique up to translation invariance of the functional and are the well-known soliton solutions. Existence of Ground states of a class of Nonlinear Equations with Polylaplacian on Finite Graphs ================================================================================================ In this section, we give a first application of the results derived in §3 on finite graphs. In this context we show a decomposition formula for the Polylaplacian. Formulation of the problem {#sec:existence} -------------------------- Let $\mathcal G$ be a connected, finite metric graph and let $K$ be a connected subgraph of $\mathcal G$. For $k\in \mathbb N$ consider the energy functional $$E^{(k)} = \frac{1}{2}\int_{\mathcal G} \|u^{(k)}\|^2 + V \|u\|^2 \, \mathrm dx - \frac{\mu}{p} \int_{\mathcal G} \|u\|^p\, \mathrm dx,$$ with $2<p< 4k+2$ and $V\in L^2+ L^\infty$ and for $c>0$ consider the minimization problem $$\label{eq:minimizationeq} E_{c}^{(k)} := \inf_{\substack{u\in H^k\\ \\|u\\|_{2}^2=c}} E^{(k)}(u).$$ \[lem:energyestimateneu\] Let $\mathcal G$ be a finite connected metric graph. The functional $E^{(k)}$ under the $L^2$-constraint $\\|\cdot\\|_{L^2}^2=c$ is bounded below for $2<p<4k+2$ and $c>0$. Moreover, for each $0<\varepsilon <1$ there exists a $C_\varepsilon>0$, such that $$E^{(k)}(u) \ge \frac{1-\varepsilon}{2} \int_{\mathcal G} \|u^{(k)}\|^2 + V \|u\|^2 \, \mathrm dx - C_\varepsilon.$$ Let $\varepsilon>0$. Consider a decomposition of $V\in L^2+ L^\infty$ such that $$V = V_2+ V_\infty, \quad \\|V_2\\|_{2}\le\varepsilon.$$ Then $$\begin{gathered} \int_{\mathcal G} \left \|u^{(k)}\right \|^2- \left \\|V_\infty\right \\|_\infty \|u\|^2\, \mathrm dx- \varepsilon \\|u\\|_{4}^2\\ \le \int_{\mathcal G} \left \|u^{(k)}\right \|^2+ V \|u\|^2\, \mathrm dx\\ \le \int_{\mathcal G} \left \|u^{(k)}\right \|^2+ \left \\|V_\infty\right \\|_\infty \|u\|^2\, \mathrm dx+ \varepsilon \\|u\\|_{4}^2.\end{gathered}$$ By the Sobolev inequality we infer $$\\|u\\|_4^2 \le C_1 \\|u\\|_{H^k}^2 \le C_2 \left (\left \|u^{(k)}\right \|_2^2+ \left \|u\right \|_2^2 \right ).$$ Adding a constant to the potential if necessary we have that $$\\|u\\| :=\left (\int_{\mathcal G} \left \|u^{(k)}\right \|^2+ V \|u\|^2\, \mathrm dx\right )^{1/2}$$ defines an equivalent norm on $H^k$. From Proposition \[prop:higherordergag\] we have $$\\|u\\|_{L^p}^p\le C \left \\|u\right \\|_{L^2(\mathcal G)}^{\frac{(2k-1)p+2}{2k}} \big \\|u\big \\|^{\frac{p-2}{2k}}$$ for some $C>0$. Let $0<\varepsilon <1$, then with Young’s inequality we infer for all $u\in H^k(\mathcal G)$ with $\\|u\\|_2^2=c$$$\frac{\mu}{p}\\|u\\|_{L^p}^p \le \frac{\varepsilon}{2} \\|u\\|^2 + C_{\varepsilon,c}$$ for some $C_{\varepsilon, c} >0$ and we obtain $$E^{(k)} \ge \frac{1-\varepsilon}{2} \int_{\mathcal G} \|u^{(k)}\|^2 + V \|u\|^2 \, \mathrm dx - C_{\varepsilon,c}$$ for $2< p<4k+2$. \[prop:NLSmult\]Let $\mathcal G$ be a finite, connected metric graph. Assume $u\in H^k$ is a minimizer of $E^{(k)}_c$, then $u\in H^{2k}$ and there exists $\lambda\in \mathbb R$ such that $$\label{eq:NLSELG} (-1)^k u_e^{(2k)} + \left ( V+\lambda\right ) u_e = \mu \|u_e\|^{p-1} u_e$$ for all $e\in {{\mathcal E}}$. Since $E^{(k)}\in C^1(H^k, \mathbb R)$ and the $L^2$ constraint is also $C^1$, and $u$ is a constrained critical point we can compute the Gâteaux derivative $$\int_{\mathcal G} \left ( u^{(k)} \eta^{(k)}- u \|u\|^{p-2} \eta\right )\, \mathrm dx+ \int_{\mathcal G} \left ( V+\lambda\right ) u\eta\, \mathrm dx=0, \qquad \forall \eta\in H^k(\mathcal G)$$ where $\lambda$ is a Lagrange multiplier. Fixing an edge $e$, then with $\eta \in C_c^\infty(I_e)$ and integration by parts we deduce for each $e\in {{\mathcal E}}$ and by elliptic regularity $u\in \widetilde{H^{2k}}$. Fixing now $\mathsf v\in V$ and taking $\eta\in H^k$ to be locally supported near $\mathsf v$ and not supported at any other vertex, then by integration by parts we deduce $$\sum_{j=1}^{k}(-1)^{j} \sum_{e\succ \mathsf v}\tfrac{\partial^{(k+j-1)}}{\partial^{(k+j-1)}\nu}u_e \tfrac{\partial^{(k-j)}}{\partial^{(k-j)}\nu}\eta_e(\mathsf v)=0.$$ Since the choice $\eta\in H^{k}$ is arbitrary we deduce $$\begin{cases} \sum_{e\succ \mathsf v} \frac{\partial^\ell}{\partial \nu^\ell}u_e(\mathsf v)=0, \qquad \forall k\le \ell \le 2k-1 \text{ odd},\\ u_{e_1}^{(\ell)}(\mathsf v) = u_{e_2}^{(\ell)}(\mathsf v), \qquad \forall k \le \ell \le 2k-1 \text{ even and } \forall e_1, e_2 \text{ adjacent at } \mathsf v \end{cases}$$ for all $\mathsf v\in \mathcal V$. Partitions of unity in $\widetilde{C_b^\infty}$ ----------------------------------------------- Let $\mathcal G$ be any locally finite, connected graph and $\mathcal O$ be a finite covering of $\mathcal O$. We construct a partition of unity in $\widetilde{C_b^\infty}(\mathcal G)$ by choosing appropriate partitions of unities subordinate to the covering. One rather different “normalization” of the usual partition of unity will be especially useful in applications: \[lem:unity2\] Let $\mathcal G$ be a metric graph. Consider any finite open covering $\mathcal O$ of $\mathcal G$. There exists a partition of unity subordinate to $\mathcal O$ in $\widetilde {C_b^\infty}$ satisfying $$\label{eq:normalizationt} \sum_{O\in \mathcal O} \Psi_O^2 \equiv 1.$$ Consider any smooth partition of unity $\{\psi_O\}_{O\in \mathcal O}$ on the graph subordinate to the open covering $\mathcal O$ satisfying $$\sum_{O\in \mathcal O} \Psi_O \equiv 1.$$ Then we may define $$\Psi_O:= \frac{\psi_O}{\sqrt{\sum_{O\in \mathcal O} \psi_O^2}}$$ for all $O\in \mathcal O$, which is smooth restricted as functions on all edges since $\sum_{O\in \mathcal O} \psi_O^2(y)\neq 0$ for all $y\in\mathcal G$. Furthermore, it is constant in a neighborhood of any vertex and we infer $\Psi_O\in \widetilde{C_b^\infty}$. By construction we conclude $$\sum_{O\in \mathcal O} \Psi_O^2 \equiv 1.$$ The normalization in replaces in this context the typical normalization, where one assumes $$\sum_{O\in \mathcal O } \psi_O \equiv 1.$$ Throughout the rest of the paper we will only work with partitions of unity that satisfy the normalization . \[ex:unity2\] Let $\mathcal G$ be a finite, connected metric graph with core $K= \mathcal G\setminus \mathcal E_\infty$. Consider on $\mathcal G$ the open covering $ \mathcal O$ given by $K_2$ and $\mathcal G\setminus K_1$, where $K_1$ and $K_2$ are the neighborhoods of $K$ given as in , such that $\mathcal G\setminus K_1$ only consists of disjoint rays. Consider the partition of unity subordinate to $\mathcal O$ from Lemma \[lem:unity2\] given by $\psi_K, \{\psi_e\}_{e\in \mathcal E_\infty}$ respective to $K_2$ and $\mathcal G\setminus K_1$, then we define slight modifications $$\begin{aligned} \psi_{K,R}(x) &=\begin{cases} 1, &\qquad \text{on K}\\ \psi_{K}(x/R) &\qquad \text{on all rays } e\in \mathcal E_\infty; \end{cases}\\ \psi_{e,R}(x) &=\begin{cases} 0, &\qquad \text{on }\mathcal G\setminus \{e\}\\ \psi_{e}(x/R) &\qquad \text{on } e\in \mathcal E_\infty. \end{cases}\end{aligned}$$ Then by definition, $\{\psi_{O,n}\}_{O\in \mathcal O}$ is a vanishing-compatible sequence of partitions of unity subordinate to the open coverings given by $K_{2n}$ and $\mathcal G\setminus K_{n}$. By Lemma \[lem:unity2\] there exists a sequence of partitions of unity $$\Psi_n:=\Psi_{K,n},\qquad \widetilde{\Psi_n}:=\sum_{e\in \mathcal E_\infty}\Psi_{e,n}$$ in $\widetilde{C_b^\infty}$ satisfying $$\Psi_{n}^2+ \widetilde{\Psi_n}^2\equiv 1.$$ A decomposition formula {#sec:decompositionpoly} ----------------------- In the following we identify a given function $f\in \widetilde{C_b^\infty}(\mathcal G)$ with its corresponding multiplication operator $\mathcal M_f \phi:= f\phi$. Let $A$ be an operator such that $fD(A) \subset D(A)$, then we can define the commutator $[A,f]=Af-fA$ and $$\begin{aligned} fAf&= f^2 A + f [A, f]\\ fAf&= Af^2 + [A,f]f.\end{aligned}$$ Averaging the two preceding equations we conclude $$\label{eq:decompositionn} fAf = \frac{1}{2} ( f^2 A + A f^2) + \frac{1}{2} ( f [A,f] - [A,f]f).$$ \[lem:IMSunity\] Let $\mathcal G$ be a locally finite, connected metric graph. Assume $\{\psi_k\}_{k=1}^N$ is a family of function in $\widetilde{C_b^\infty}(\mathcal G)$ with $0 \le \psi_k\le 1$ for all $k\in \{1,\ldots, k\}$ and $$\sum_{k=1}^N \psi_k^2 \equiv 1.$$ Assume $D(A)$ is invariant under multiplication by elements in $\widetilde{C_b^\infty}(\mathcal G)$, then $$\label{eq:decompositionadv} A= \sum_{k=1}^N \psi_k A \psi_k -\frac{1}{2} ( \psi_k [A,\psi_k] - [A,\psi_k]\psi_k).$$ Follows immediately with . We refer to as a decomposition formula for $A$. In the following, we develop a decomposition formula for the Polylaplacian $A=(-\Delta)^k$. Let $\mathcal G$ be a finite metric graph and let $k\in \mathbb N$. In the following, we define the Polylaplacian $A=(-\Delta)^k$ on $\mathcal G$ as an operator $A: D(A) \subset L^2(\mathcal G) \to L^2(\mathcal G)$ given by $$\begin{gathered} \left ((-\Delta)^k u\right )_e := (-\Delta)^k u_e:= (-1)^k u_e^{(2k)}\\ D(A) = H^{2k}(\mathcal G).\end{gathered}$$ \[lem:decomposition2\] Let $\mathcal G$ be a locally finite connected graph. Let $A=(-\Delta)^k$ with $D(A)= H^{2k}$, then (i) $fD(A)\subset D(A)$ for all $f\in \widetilde{C_b^\infty}(\mathcal G)$. (ii) Let $f\in \widetilde{C_b^\infty}(\mathcal G)$, then the operator $fAf$ is given by $$\label{eq:decomposition2} \begin{aligned} \left (fAf\right ) \phi &= \frac{1}{2} ( f^2 A + A f^2)\phi \\ &\qquad+ \frac{(-1)^{k+1}}{2} \sum_{m=1}^{2k-1} \sum_{n=1}^{2k-m} \frac{(2k)_{m+n}}{m! n!} f^{(m)} f^{(n)} \phi^{(2k-m-n)} \end{aligned}$$ for all $\phi \in D(A)$. We apply Leibniz’ formula and compute $$\begin{aligned} [A,f]\phi &= (-\Delta)^k f \phi - f(-\Delta)^k \phi\\ &= (-1)^k \sum_{m=1}^{2k} \binom{2k}{l} f^{(m)} \phi^{(2k-m)}. \end{aligned}$$ Then we apply Leibniz’ formula again and compute $$\begin{aligned} \left ([A,f]f\right )\phi &= (-1)^k \sum_{m=1}^{2k}\sum_{n=0}^{2k-m} \binom{2k}{m}\binom{2k-m}{n} f^{(m)} f^{(n)} \phi^{2k-m-n}\\ &= (-1)^k \sum_{m=1}^{2k} \sum_{n=0}^{2k-m} \frac{(2k)_{m+n}}{m! n!} f^{(m)} f^{(n)} \phi^{(2k-m-n)} \end{aligned}$$ and we conclude $$\frac{1}{2} ( f [A,f] - [A,f]f)\phi=\frac{(-1)^{k+1}}{2} \sum_{m=1}^{2k-1} \sum_{n=1}^{2k-m} \frac{(2k)_{m+n}}{m! n!} f^{(m)} f^{(n)} \phi^{(2k-m-n)}.$$ The statement follows upon combining this with . Given the core $K= \mathcal G\setminus {{\mathcal E}}_\infty$ of $\mathcal G$ and $R>0$ we define $$\label{eq:numerateR} \begin{aligned} D_R&:= \{\phi \in D(A) \|\operatorname{supp}(\phi) \subset \mathcal G \setminus K_R\}\\ \Sigma_R&:= \inf\{\langle \phi, A\phi\rangle\|\phi \in D_R, \\|\phi\\|_{2}^2=1\}, \end{aligned}$$ where $K_R$ was defined in . For $R=0$ we set $$\label{eq:numerate0} \begin{aligned} D_0&:= D(A)\\ \Sigma_0&:= \inf\{\langle \phi, A\phi \rangle \|\phi \in D(A), \\|\phi\\|_{2}^2=1\}. \end{aligned}$$ The last relevant quantity, which we will discuss later in §\[sec:onthreshold\] is $$\label{eq:numerate1} \Sigma := \lim_{R\to \infty} \Sigma_R=\sup_{R>0} \Sigma_R.$$ \[lem:preconditions\] Let $\mathcal G$ be a finite, connected metric graph and let $V\in L^2+ L^\infty(\mathcal G)$. $E^{(k)}$ is weak limit superadditive, superadditive with respect to the partition of unity in Example \[ex:unity2\]. Assume $A= (-\Delta)^k + V$ admits a ground state, then $t\mapsto E_t$ as defined in is strictly subadditive. Weak limit superadditivity. We showed in the proof of Lemma \[lem:energyestimateneu\] that $$\\|u\\| =\left (\int_{\mathcal G} \left \|u^{(k)}\right \|^2+ V \|u\|^2\, \mathrm dx\right )^{1/2}$$ defines an equivalent norm on $H^k$ upon adding a constant to $V$. Assume $u_n \rightharpoonup u$ in $H^k$ weakly, then up to a subsequence, which we still denote by $u_n$, by the Brezis–Lieb Lemma and weak convergence $$\begin{gathered} \limsup_{n\to \infty} \\|u_n\\|^2 = \\|u\\|^2 + \limsup_{n\to \infty} \\|u-u_n\\|,\\ \limsup_{n\to \infty} \int_{\mathcal G} \|u_n\|^p \, \mathrm dx= \int_{\mathcal G} \|u\|^p\, \mathrm dx + \limsup_{n\to \infty}\int_{\mathcal G} \|u-u_n\|^p \, \mathrm dx. \end{gathered}$$ Then $$\limsup_{n\to \infty} E^{(k)} (u_n) =E^{(k)} (u)+ \limsup_{n\to \infty} E^{(k)} (u-u_n)$$ and $E^{(k)}$ is weak limit superadditive. Superadditivity with respect to a sequence of partitions of unity. Finally we need to show superadditivity with respect to the partition of unity $\{\Psi_n, \widetilde {\Psi_n}\}$ in Example \[ex:unity2\]. Let $u_n\rightharpoonup 0$ be a vanishing sequence with $\\|u_n\\|_{L^2}^2=c$, then $$\\|\Psi_n u_n\\|_{L^2}^2 +\\|\widetilde{\Psi_n} u_n\\|_{L^2}^2= c$$ and up to a subsequence, still denoted by $u_n$, $$\begin{gathered} \lim_{n\to \infty} \int_{K_{2n}} \|u_n\|^p\, \mathrm dx =0, \qquad \lim_{n\to \infty} \int_{K_{2n}} \|u_n\|^2\, \mathrm dx = 0\\ \lim_{n\to \infty} \int_{\mathcal G} \|u_n\|^p=\liminf_{n\to \infty} \int_{\mathcal G} \|u_n\|^p = \limsup_{n\to \infty} \int_{\mathcal G} \|u_n\|^p \, \mathrm dx. $$ Then from $$\begin{aligned} 0&\le \liminf_{n\to \infty} \int_{K_{2n}} \|\Psi_n u_n\|^p\, \mathrm dx \le \limsup_{n\to \infty} \int_{K_{2n}} \|\Psi_n u_n\|^p\, \mathrm dx \le \lim_{n\to \infty} \int_{K_{2n}} \|u_n\|^p\, \mathrm dx\\ 0&\le \liminf_{n\to \infty} \int_{K_{2n}} \|\widetilde{\Psi_n} u_n\|^p\, \mathrm dx \le \limsup_{n\to \infty} \int_{K_{2n}} \|\widetilde{\Psi_n} u_n\|^p\, \mathrm dx \le \lim_{n\to \infty} \int_{K_{2n}} \|u_n\|^p\, \mathrm dx \end{aligned}$$ we deduce $$\begin{gathered} \lim_{n\to \infty} \int_{\mathcal G} \|\Psi_n u_n\|^p\, \mathrm dx=\lim_{n\to \infty} \int_{\mathcal K_{2n}} \|\Psi_n u_n\|^p =0\\ \lim_{n\to \infty} \int_{K_{2n}} \|\widetilde{\Psi_n} u_n\|^p\, \mathrm dx=0.\end{gathered}$$ and in particular $$\lim_{n\to \infty} \int_{\mathcal G} \|u_n\|^p{\mathrm dx} = \lim_{n \to \infty} \int_{\mathcal G\setminus K_{2n}} \|\widetilde{\Psi_n} u_n\|^p \, \mathrm dx = \lim_{n\to \infty} \int_{\mathcal G} \|\widetilde{\Psi_n} u_n\|^p.$$ $H^{2k}$ is dense in $H^k$ by Proposition \[prop:density\] and we may assume that there exists a minimizing sequence in $H^{2k}$. Let $u_n$ be a minimizing sequence in $H^{2k}$, then by Lemma \[lem:decomposition2\] we conclude passing to a subsequence, still denoted by $u_n$, using integration by parts and Young’s inequality $$\begin{aligned} &\sum_{i=1}^{2k-1} \sum_{j=1}^{2k-m} \frac{(2k)_{m+n}}{i! j!} \left \|\left \langle \Psi_n^{(i)} \Psi_n^{(j)} u_n^{(2k-i-j)}, \phi\right \rangle_{L^2} \right \| \\ &\qquad= \sum_{i=1}^{2k-1} \sum_{j=1}^{2k-m} \frac{(2k)_{m+n}}{i! j!} \left \|\left\langle \frac{\mathrm d^{k-j}}{\mathrm dx^{k-j}} \Psi_n^{(i)} \Psi_n^{(j)}u_n, u_n^{(k-i)}\right \rangle_{L^2}\right \|\\ &\qquad\le \sum_{i=1}^{2k-1} \sum_{j=1}^{2k-m} \frac{(2k)_{m+n}}{i! j!} \left \\| \frac{\mathrm d^{k-j}}{\mathrm dx^{k-j}} \Psi_n^{(i)} \Psi_n^{(j)}u_n \right \\|_{L^2} \left \\|u_n^{(k-i)}\right \\|_{L^2} \\ &\qquad \le \frac{C }{n^2}\sum_{i=1}^{2k-1} \sum_{j=1}^{2k-m} \frac{(2k)_{m+n}}{i! j!} \\|u_n\\|_{H^k}\to 0 \qquad (n\to \infty), \end{aligned}$$ and we infer $$\begin{aligned} E^{(k)}&= \lim_{n\to \infty} \frac{1}{2}\langle u_n, Au_n\rangle + \frac{\mu}{p}\\|u_n\\|_{p}^p\\ &= \limsup_{n\to \infty} \frac{1}{2}\langle \Psi_n u_n, A \Psi_nu_n \rangle + \frac{\mu}{p}\\|\Psi_n u_n \\|_p^p \\ &\qquad + \limsup_{n\to \infty}\frac{1}{2}\langle \widetilde{\Psi_n} u_n, A \widetilde{\Psi_n} u_n \rangle+ \frac{\mu}{p}\\|\widetilde{\Psi_n} u_n \\|_p^p\\ &= \limsup_{n\to \infty} E^{(k)} (\Psi_n u_n) + \limsup_{n\to \infty} E^{(k)} (\widetilde{\Psi_n} u_n) \end{aligned}$$ and $E^{(k)}$ is (super-)additive with respect to the partition of unity $\{\Psi_n, \widetilde{\Psi_n}\}$ in Example \[ex:unity2\]. Subadditivity. To show the subadditivity, note that $$\label{eq:scalingarg2} E_t^{(k)} = t \inf_{\substack{u\in H^1\\ \\|u\\|_{L^2}^2=1}} \left \{\frac{1}{2}\int_{\mathcal G} \left \|u^{(k)}\right \|^2+ V \|u\|^2\, \mathrm dx - t^{\frac{p-2}{2}} \frac{\mu}{p} \int_{\mathcal G} \|u\|^p\, \mathrm dx \right \}.$$ We deduce the property by showing that $t\mapsto E_t^{(k)}$ is a concave function. Indeed, the scaling defines a concave function and passing to the limit we deduce concavity of the functional. Hence, $$\label{eq:concavityarg2} E_t^{(k)} \ge t E_1^{(k)}, \qquad t\in [0,1],$$ so that $$E_t^{(k)} + E_{1-t}^{(k)} \ge E_1^{(k)}, \qquad t\in [0,1].$$ For the strictness in the inequality it suffices to show strictness in the inequality . Assume $$E_t^{(k)} = t E_1^{(k)}$$ for some $t\in (0,1)$ and let $u_n$ be a minimizing sequence for $E_t^{(k)}$, then in particular due to $$\label{eq:recipe1} \int_{\mathcal G} \|u_n\|^p\, \mathrm dx \to 0 \qquad (n\to \infty).$$ By density we may assume $u_n \in D(A)$ and we deduce $$\begin{aligned} E_t^{(k)} &= \lim_{n\to \infty} \frac{1}{2} \langle A u_n, u_n\rangle - \frac{\mu}{p} \int_{\mathcal G} \|u\|^p\, \mathrm dx \\ &= \lim_{n\to \infty} \frac{1}{2} \langle A u_n, u_n \rangle \ge \frac{\Sigma_0 t}{2}. \end{aligned}$$ Now suppose that a ground state to $E^{(k)}$ exists, i.e. there exists $u\in D(A)$ with $\\|u\\|_2^2=t$ and $\langle Au, u\rangle = \Sigma_0 t$, then $$\label{eq:recipe2} E^{(k)}_t \le \frac{1}{2}\langle Au, u\rangle - \frac{\mu}{p}\int_{\mathcal G} \|u\|^p\, \mathrm dx < \frac{\Sigma_0 t}{2}.$$ This is a contradiction, and we conclude strict subadditivity in this case. \[prop:second\] Assume $\mathcal G$ is a finite, connected metric graph and $\Sigma_0 < \Sigma$ as defined in and . Then there exists $\hat \mu >0$, such that for all $\mu \in (0, \hat \mu)$ $$\widetilde \Sigma_0^{(\mu,k)} := \inf_{\substack{u \in D(A)\\ \\|\phi\\|_2^2=1}} E^{(k)}(u) < \inf_{\substack{u \in D_R(A)\\\\|\phi\\|_2^2=1}} E^{(k)}(u) =: \widetilde \Sigma^{(\mu,k)}$$ for some $R>0$. Without loss of generality we may assume $\Sigma_0>0$ as we otherwise can simply add a constant to the functional. In particular, $$\\|u\\|= \left ( \int_{\mathcal G} \|u^{(k)}\|^2 + V \|u\|^2\, \mathrm dx \right )^{1/2}$$ defines an equivalent norm on $H^k$ as shown in Lemma \[lem:energyestimateneu\]. Let $\varepsilon>0$. By Proposition \[prop:higherordergag\] $$\\|u\\|_{L^p}^p \le C \\|u\\|_{L^2(\mathcal G)}^{\frac{(2k-1)p+2}{2k}} \\|u\\|^{\frac{p-2}{2k}}$$ for some $C>0$. In particular, for sufficiently small $\mu$ with Young’s inequality we have $$\frac{\mu}{p} \\|u\\|_{L^p}^p\le \frac{\varepsilon}{2} \int_{\mathcal G} \|u^{(k)}\|^2+ V\|u\|^2\, \mathrm dx + \frac{\widetilde{C} \varepsilon}{2}$$ for sufficiently small $\mu>0$ we deduce $$E^{(k)}(u) \ge \frac{1- \varepsilon}{2} \left ( \int_{\mathcal G} \|u^{(k)}\|^2+ V\|u\|^2\, \mathrm dx\right )- \frac{\widetilde{C} \varepsilon}{2}.$$ Hence, $$\widetilde{\Sigma}^{(\mu,k)} -\widetilde{\Sigma}_0^{(\mu,k)} \ge \frac{1-\varepsilon}{2} \Sigma-\frac{\varepsilon}{2} - \frac{1}{2}\Sigma_0 = \frac{1}{2} \left (\Sigma - \Sigma_0\right ) -\frac{\varepsilon}{2} \left (\widetilde{C}+ \Sigma\right ).$$ Since $\varepsilon$ can be chosen arbitrarily small and $\Sigma > \Sigma_0>0$, we have for sufficiently small $\mu$ $$\widetilde \Sigma^{(\mu,k)} > \widetilde \Sigma^{(\mu,k)}_0.$$ \[thm:bigresult2\] Let $\mathcal G$ be a finite, connected graph and let $c>0$. Assume $\Sigma_0 < \Sigma$ as defined in and , then $E_c^{(k)}(\mathcal G)$ admits a minimizer for $\mu \in (0, \hat \mu)$ as in Proposition \[prop:second\] and let $\hat \mu$ be as in Proposition \[prop:second\]. Then $E_c^{(k)}$ admits a minimizer for any $\mu \in (0,\hat \mu)$. By Lemma \[lem:energyestimateneu\] any minimizing sequence admits a weakly convergent subsequence. $E^{(k)}$ satisfies the prerequisites of Theorem \[thm:main1\] and Theorem \[thm:main2\] with $X(\mathcal G)= H^k(\mathcal G)$ and $Y(\mathcal G) = \widetilde{C_b^\infty}(\mathcal G)$ by Lemma \[lem:preconditions\]. Then due to Proposition \[prop:second\] the energy inequality in Corollary \[cor:existence\] is satisfied. In particular we deduce existence of a minimizer of $E_c^{(k)}(\mathcal G)$ under the assumptions of the statement. The case $V\equiv 0$ on the real line ------------------------------------- Consider the infimization problem on the real line $$E^{(k)}(\mathbb R) = \inf_{\substack{u\in H^1(\mathbb R)\\ \\|u\\|_2^2=1}} \frac{1}{2} \int_{\mathbb R} \|u^{(k)}\|^2\, \mathrm dx - \frac{\mu}{p} \int_{\mathbb R} \|u\|^p \, \mathrm dx.$$ This is the special case when $V\equiv 0$ and $\mathcal G=\mathbb R$ in . In this case $E_c$ admits a minimizer due to Theorem \[thm:main3\]: \[thm:bigresult3\] Let $V\equiv 0$ and $\mathcal G=\mathbb R$. The minimization problem $$E^{(k)}(\mathbb R) = \inf_{\substack{u\in H^1(\mathbb R)\\ \left \\|u\right \\|_2^2=1}} E^{(k)}(u)$$ admits a minimizer for all $\mu >0$. Furthermore, any minimizer of $E_c(\mathbb R)$ is $C^\infty_b$ and satisfies the Euler–Lagrange equation $$(-1)^k u^{(2k)} + \lambda u = \mu \|u\|^{p-2}u,$$ where $\lambda \in \mathbb R$ is a Lagrange multiplier. Due to Lemma \[lem:preconditions\] it suffices to show that $$t\mapsto E_{t}^{(k)}= \inf_{\substack{u\in H^1(\mathbb R)\\ \\|u\\|_2^2=t}}\frac{1}{2} \int_{\mathbb R} \|u^{(k)}\|^2\, \mathrm dx - \frac{\mu}{p} \int_{\mathbb R} \|u\|^p \, \mathrm dx.$$ is strictly subadditive, then the prerequisites of Theorem \[thm:main3\] are satisfied. The Euler–Lagrange equation is satisfied because of Proposition \[prop:NLSmult\]. The regularity is due to elliptic regularity and a bootstrap argument. For a contradiction, assume as in the proof of Lemma \[lem:preconditions\] $$E_t = t E_1$$ for some $t\in [0,c]$. Assume $u_n$ is a minimizing sequence for $E_t$, then as in the proof of Lemma \[lem:preconditions\] we have $$\label{eq:rcase} \lim_{n\to \infty} \int_{\mathbb R} \|u_n\|^p\, \mathrm dx = 0.$$ Given a test function $\phi \in C_c^\infty(\mathbb R)$ with $\\|\phi\\|_{L^2}^2=1$, we define the rescaling for $\lambda >0$ $$\phi_{\lambda} := \lambda^{1/2} \phi( \lambda x).$$ Then $\\|\phi_{\lambda}\\|_{L^2}^2=1$ for all $\lambda >0$. Then $$E^{(k)}(\phi_{\lambda}) = \lambda^{2k} \int_{\mathbb R} \|\phi^{(k)}\|^2\, \mathrm dx - \lambda^{\frac{p}{2}-1} \int_{\mathbb R} \|\phi\|^p \, \mathrm dx.$$ In particular for sufficiently small $\lambda>0$ $$E^{(k)} \le E^{(k)}(\phi_{\lambda}) <0.$$ On the other hand with $$E^{(k)} = \lim_{n\to \infty} E^{(k)}(u_n) = \lim_{n\to \infty} \int_{\mathcal G} \|u_n^{(k)}\|^2\, \mathrm dx\ge 0$$ and we conclude subadditivity due to the contradiction. This concludes the proof. \[cor:mainresult3\] Let $V\equiv 0$. Then $$E^{(k)}(\mathbb R)<0.$$ Decaying potentials ------------------- In the following we study the minimization problem on finite metric graphs $\mathcal G$ under the assumption that $V=V_2+V_\infty$ with $V_2\in L^2(\mathcal G), V_\infty\in L^\infty(\mathcal G)$ such that $$\label{eq:decayingpotential} V_\infty(x) \to 0 \qquad (x\to \infty)$$ on all of the rays. Consider the quantitities $$\begin{aligned} \widetilde{\Sigma}_0^{(\mu,k)} &= \inf_{\substack{\phi \in D(A)\\ \\|\phi\\|_{L^2}^2 =1}} E^{(k)}(u)\\ \widetilde{\Sigma}^{(\mu,k)} &= \lim_{R\to \infty} \inf_{\substack{\phi \in D_R(A)\\ \\|\phi\\|_{L^2}^2 =1}} E^{(k)}(u)\\ E^{(k)}(\mathbb R) &= \inf_{\substack{u\in H^1(\mathbb R)\\ \\|u\\|_{L^2(\mathbb R)}^2=1}} \frac{1}{2} \int_{\mathbb R} \|u^{(k)}\|^2\, \mathrm dx - \frac{\mu}{p} \int_{\mathbb R} \|u\|^p \, \mathrm dx. \end{aligned}$$ \[lem:decayingpotential\] Let $\mathcal G$ be a finite metric graph and assume that $V\in L^2+L^\infty(\mathcal G)$ satisfies . Then $$\widetilde{\Sigma}^{(\mu,k)} = E^{(k)}(\mathbb R).$$ Assume $\phi$ is a minimizer of $E^{(k)}(\mathbb R)$, which exists due to Theorem \[thm:bigresult3\]. Due to density, we can consider a minimizing sequence $u_n$ for $E^{(k)}(\mathbb R)$ in $C_c^\infty(\mathbb R)$ satisfying $\\|u_n\\|_{2}^2=1$, such that $u_n \to \phi$ strongly in $H^1$ as $n\to \infty$. Now by translation invariance we may assume that $u_n$ is supported in $[n,\infty)$ for $n\in \mathbb N$. identifying the half-line with one of the rays of $\mathcal G$, we may consider $u_n$ as a function in $H^1(\mathcal G)$. Then $$\begin{aligned} \left \|\int_{\mathcal G} V \|u_n\|^2\, \mathrm dx \right \| &= \left \|\int_{\mathcal G\setminus K_n} V \|u_n\|^2\, \mathrm dx \right \|\\ &\le C \left (\sup_{x\in \mathcal G\setminus K_n} \|V_\infty(x)\|^2 +\int_{\mathcal G\setminus K_n} \|V_2\|^2\, \mathrm dx\right )\to 0 \qquad (n\to \infty) \end{aligned}$$ and we compute $$\begin{aligned} E^{(k)}(\mathbb R)&=\lim_{n\to \infty}E^{(k)}(u_n)\\ &= \lim_{n\to \infty} \frac{1}{2}\int_{\mathcal G} \|u^{(k)}_n\|^2 \, \mathrm dx - \frac{\mu}{p} \int_{\mathcal G} \|u\|^p\, \mathrm dx\\ &= \lim_{n\to \infty} E^{(k)}(u_n)\ge \widetilde{\Sigma}^{(\mu,k)}. \end{aligned}$$ On the other hand given a minimizing sequence $u_n$ for $\widetilde{\Sigma}^{(\mu,k)}$, such that $\operatorname{supp} u_n \subset \mathcal G \setminus K_n$ then the functions in the sequence are supported on each of the rays and $$\label{eq:goesto0} \begin{aligned} &\left \|\int_{\mathcal G} V \|u_n\|^2\, \mathrm dx \right \| = \left \|\int_{\mathcal G\setminus K_n} V \|u_n\|^2\, \mathrm dx \right \|\\ &\qquad\le C \left (\sup_{x\in \mathcal G\setminus K_n} \|V_\infty(x)\|+\left (\int_{\mathcal G\setminus K_n} \|V_2\|^2\, \mathrm dx\right )^{1/2}\right )\to 0 \qquad (n\to \infty). \end{aligned}$$ By density we can consider a collection of sequences $u_n^{(1)}, \ldots, u_n^{(\|\mathcal E_\infty\|)}$ in $C_c^\infty(\mathbb R)$ and choose them to have disjoint supports. Then if we define $$\widetilde{u}_n := \sum_{i=1}^{\|\mathcal E_\infty\|} u_n^{(i)}.$$ Then with we compute $$\begin{aligned} \widetilde{\Sigma}^{(\mu,k)} &= \lim_{n\to \infty} \sum_{i=1}^{\mathcal E_\infty} E^{(k)}(u_n^{(i)})\\ &= \lim_{n\to \infty} E^{(k)}\left ( \widetilde{u}_n \right ) \ge E^{(k)}(\mathbb R). \\ \end{aligned}$$ \[rmk:decayingpotential\] Suppose $\mathcal G$ is a locally finite metric graph with at least one ray. Then the inequality $$\widetilde{\Sigma}^{(\mu,k)}\le E^{(k)}(\mathbb R)$$ can still be shown as in the proof of Lemma \[lem:decayingpotential\] using the test function argument on the half-line. \[thm:decayingpotential\] Let $\mathcal G$ be a finite metric graph. Assume $V\in L^2+L^\infty(\mathcal G)$ satisfies , then $E^{(k)}$ is strictly subadditive and if $$\widetilde{\Sigma}_0^{(\mu,k)} < E^{(k)}(\mathbb R)$$ then there exists a minimizer to the minimization problem. By Lemma \[lem:energyestimateneu\] any minimizing sequence admits a weakly convergent subsequence. By Lemma \[lem:preconditions\] and Lemma \[lem:decayingpotential\] it suffices to prove the strict subadditivity of $E^{(k)}$. As in Lemma \[lem:preconditions\] we can argue by contradiction. Assume namely that $$E_t^{(k)}= t E_1^{(k)}$$ for some $t\in (0,1)$ and let $u_n$ be a minimizing sequence for $E_t^{(k)}$, then in particular $$\int_{\mathcal G} \|u_n\|^p \, \mathrm dx \to 0 \qquad (n\to \infty).$$ But then $u_n$ is a vanishing sequence and passing to a subsequence still denoted by $u_n$, we deduce with superaddditivity with respect to a sequence of partitions of unity as defined in Example \[ex:unity2\] $$\label{eq:reipse2} \begin{aligned} E_t^{(k)} &= \limsup_{n\to \infty} E^{(k)}(\Psi_n u_n) +\limsup_{n\to \infty} E^{(k)}(\widetilde{\Psi_n} u_n) \\ &\ge \frac{1}{2}\lim_{n\to \infty}\inf_{\substack{\phi\in H^1\\ \\|u\\|_{L^p}^p=t, \operatorname{supp} u \subset \mathcal G\setminus K_n}}\langle A u, u\rangle\\ &\ge -\frac{C}{2} \lim_{n\to \infty} \left (\sup_{x\in \mathcal G\setminus K_n} \|V_\infty(x)\|+ \left (\int_{\mathcal G\setminus K_n} \|V_2\|^2\, \mathrm dx\right )^{1/2}\right ) =0, \end{aligned}$$ since $\\|u_n\\|_{H^1}\le C$ for some $C>0$ by Lemma \[lem:energyestimateneu\]. On the other hand, by Lemma \[lem:decayingpotential\] and Corollary \[cor:mainresult3\] $$\widetilde{\Sigma}^{(\mu,k)} = E^{(k)}(\mathbb R)<0$$ and by contradiction we deduce strict subadditivity. Hence, the prerequisites of Theorem \[thm:main1\] and Theorem \[thm:main2\] are satisfied with $X(\mathcal G)= H^k(\mathcal G)$ and $Y(\mathcal G) = \widetilde{C_b^\infty}(\mathcal G)$. Then due to Proposition \[prop:second\] the energy inequality in Corollary \[cor:existence\] is satisfied. In particular we deduce existence of a minimizer of $E^{(k)}(\mathcal G)$ under the stated assumptions. \[ex:shouldbeputinintroduction\] Let $\mathcal G$ be a finite metric graph and let $V\in L^2+L^\infty$ satisfy . Similarly as in Lemma \[lem:decayingpotential\] we can show $$\Sigma=\lim_{R\to \infty} \inf_{\substack{\phi \in D_R(A)\\ \\|\phi\\|_{L^2}^2 =1}} \langle \phi, A\phi\rangle_{L^2}=0.$$ In particular if $\Sigma_0<0$, then by Theorem \[thm:bigresult2\] there exists $\hat \mu>0$, such that for $\mu \in (0,\hat\mu]$ there exists a minimizer to $E^{(1)}$. As in [@cacciapuoti2018existence] one can show due to scaling properties that $$\Sigma^{(\mu,1)}_0 < \Sigma_0 \le \gamma_p \mu^{\frac{4}{6-p}}= E^{(1)}(\mathbb R)$$ for some $\gamma_p<0$ and $0<\mu\le (\Sigma_0/\gamma_p)^{\frac{3}{2}-\frac{p}{4}}$. In particular, we can deduce existence of minimizers for $E^{(1)}$ and $0<\mu\le(\Sigma_0/\gamma_p)^{\frac{6-p}{4}}$ by Theorem \[thm:decayingpotential\]. Locally Finite Graphs ===================== In this section, we study the NLS energy functional with potentials on more general graphs. We show a decomposition formula for the form associated with the magnetic Schrödinger operator and adapt previous arguments by introducing a suitable sequence of partitions of unity in the case of locally finite graphs. Formulation of the problem {#formulation-of-the-problem} -------------------------- Consider the Schödinger operator with potentials $M\in H^1+W^{1,\infty}(\mathcal G)$ and $V\in L^2+L^\infty(\mathcal G)$ satisfying natural vertex conditions on $\mathcal G$: $$\label{eq:schroedinger2} \begin{aligned} A&=\left (i \frac{\mathrm d}{\mathrm dx}+M\right )^2+V \\ D(A)&= \bigg \{u\in C(\mathcal G)\bigg\| u_e\in H^2(e), \qquad\forall e\in {{\mathcal E}}\\ &\qquad\qquad\qquad \land\qquad \sum_{e \succ \mathsf v} \left ( i \frac{\partial}{\partial\nu} + M \right ) u_e (\mathsf v)=0\bigg \}. \end{aligned}$$ Consider the NLS functional $$E_\text{NLS}^{(\mathcal K)} (u) := \frac{1}{2}\int_{\mathcal G} \left \|\left (i \frac{\mathrm d}{\mathrm dx}+M\right ) u\right \|^2 + V \|u\|^2\, \mathrm dx- \frac{\mu}{p} \int_{\mathcal K}\|u\|^p\, \mathrm dx$$ where $V\in L^2+ L^\infty$ and $\mathcal K$ is a not necessarily bounded subgraph of $\mathcal G$. Define the corresponding minimization problem $$E_{\text{NLS}}^{(\mathcal K)}:=\inf_{\substack{u\in H^1(\mathcal G)\\ \\|u\\|_2^2=1}} E_{\text{NLS}}^{(\mathcal K)} (u)$$ similarly as in Section \[sec:existence\]. ses: - The localized case, when $\mathcal K$ is a bounded subgraph of $\mathcal G$; - The global case, when $\mathcal K= \mathcal G$ is the whole graph. In this case, we drop the argument and simply define $$E_\text{NLS} (u) := \frac{1}{2}\int_{\mathcal G} \left \|\left (i \frac{\mathrm d}{\mathrm dx}+M\right ) u\right \|^2 + V \|u\|^2\, \mathrm dx- \frac{\mu}{p} \int_{\mathcal G}\|u\|^p\, \mathrm dx$$ and $$E_{\text{NLS}}:=\inf_{\substack{u\in H^1(\mathcal G)\\ \\|u\\|_2^2=1}} E_{\text{NLS}} (u).$$ We define quantities analogous to , and . Given a bounded subgraph of $\mathcal G$ and $R>0$ we define $$\begin{aligned}\label{eq:numerateR2} D_R&:= \{\phi \in D(A) \|\operatorname{supp}(\phi) \subset \mathcal G \setminus K_R\}\\ \Sigma_R&:= \inf\{\langle \phi, A\phi\rangle\|\phi \in D_R, \\|\phi\\|_{2}^2=1\}, \end{aligned}$$ where $K_R$ was defined in , $$\label{eq:numerate02} \begin{aligned} D_0&:= D(A)\\ \Sigma_0&:= \inf\{\langle \phi, A\phi \rangle \|\phi \in D(A), \\|\phi\\|_{2}^2=1\} \end{aligned}$$ and $$\label{eq:numerate12} \Sigma := \lim_{R\to \infty} \Sigma_R=\sup_{R>0} \Sigma_R.$$ Most results can be extended simply to this case following the previous proofs. The principal difficulty lies in establishing superadditivity with respect to a suitable sequence of partitions of unity. We give a construction of such a sequence of partitions of unity in the following. Partitions of unity in $W^{1,\infty}(\mathcal G)$ ------------------------------------------------- Here we give an important example for a partition of unity in $W^{1,\infty}(\mathcal G)= C^{0,1}(\mathcal G)$. Given any partition of unity in $W^{1,\infty}(\mathcal G)$ one can always find a renormalization as in Lemma \[lem:unity2\]: \[lem:unity\] Let $\mathcal G$ be a connected, locally finite metric graph. Consider any finite open covering $\mathcal O$ of $\mathcal G$. Then there exists a partition of unity in $W^{1,\infty}(\mathcal G)$ subordinate to $\mathcal O$ satisfying $$\sum_{O\in \mathcal O} \Psi^2_O \equiv 1.$$ Consider a partition of unity $\{\psi_O\}_{O\in \mathcal O}$ on the graph subordinate to the open covering $\mathcal O$. Then we define $$\Psi_O := \frac{\psi_O}{\sqrt{\sum_{O\in \mathcal O} \psi_O^2}}$$ for all $O\in \mathcal O$. As a product of uniformly bounded Lipschitz continuous functions, $\Psi_O$ is also one; and by Proposition \[prop:lipschitz\] we conclude $\psi_O\in W^{1,\infty}(\mathcal G)$. Moreover, $\sum_{O\in \mathcal O} \psi_O^2 \equiv 1$ by construction. \[ex:first2\] Let $\mathcal G$ be a locally finite graph and let $K$ be some bounded, connected subgraph. Let $X(\mathcal G)= H^1(\mathcal G)$ and $Y(\mathcal G)= W^{1,\infty}(\mathcal G)$. Then $X(\mathcal G),Y(\mathcal G)$ satisfy Assumption \[as:assumption1\] and Assumption \[as:assumption2\]. Recall the partition of unity in $W^{1,\infty}(\mathcal G)$ in Example \[ex:first\] $$\psi(x) = \max\{\operatorname{dist}(\mathcal G\setminus K_{2}, x), 1\}, \qquad \widetilde {\psi}(x) = 1-\psi.$$ We construct a sequence of partitions of unity via $$\psi_n(x)= \frac{1}{n} \max\{\operatorname{dist}(\mathcal G\setminus K_{2n}, x), n\}, \qquad \widetilde {\psi_n}(x) = 1-\psi_n.$$ By Lemma \[lem:unity\] we can rescale them in such a way that $$\Psi_n^2 + \widetilde{\Psi_n}^2 \equiv 1.$$ By definition the partitions $K_{2n}, \mathcal K\setminus K_n$ are vanishing-compatible and $\Psi_n, \widetilde{\Psi_n}$ is a vanishing-compatible sequence of partitions of unity. \[df:kirchhoff\] Let $f\in C^{0,1}(\mathcal G)$. We call a point $x\in \mathcal G$ a Kirchhoff point of $f$ if one of the following holds: (1) $x\in \mathcal V$ is a vertex of degree $d_x\neq 2$, the derivatives $f_e'(x)$ exist for all $e\succ x$, and $f$ satisfies the Kirchhoff condition $$\sum_{e\succ x} \tfrac{\partial}{\partial\nu}f_e(x) =0,$$ (2) $x\in\mathcal G$ is an interior point of an edge (equivalently, a dummy vertex of degree $2$), and $f$ is differentiable at $x$. We call the set $$\mathcal N_f = \mathcal G\setminus \{x\in \mathcal G: x \text{ is a Kirchhoff point of }f\}$$ the non-Kirchoff set of $f$. \[rmk:important\] The sequence constructed in Example \[ex:unity2\] do not work here, since the functions are not in $W^{1,\infty}(\mathcal G)$. We are going to consider the sequence of partitions of unity in Example \[ex:first2\] instead. This concrete sequence has some interesting properties, such that for all $n\in \mathbb N$ $$\label{eq:interestingprop1} \\|\psi_n'\\|_{L^\infty} = \frac{1}{n} \qquad \\|\widetilde{\psi_n}'\\|_{L^\infty} = \frac{1}{n}$$ and in particular $$\label{eq:interestingprop2} \\|\Psi_n'\\|_{L^\infty} \le \frac{C}{n} \qquad \\|\widetilde{\Psi_n}'\\|_{L^\infty} \le \frac{C}{n}$$ for a $C=C(\mathcal G)$ only dependent on the graph. A decomposition formula {#a-decomposition-formula} ----------------------- For the Schrödinger operator with magnetic potential $$\label{eq:test} \begin{gathered} \widetilde{A}=\left (i \frac{\mathrm d}{\mathrm dx}+M\right )^{2}\\ D(\widetilde A)= \widetilde{H^2} \end{gathered}$$ one can show as in Section \[sec:decompositionpoly\], see Lemma \[lem:decomposition2\]: \[lem:decompositionmagnetic\] Let $\mathcal G$ be a locally finite connected metric graph. Let $$\begin{gathered} \widetilde A:=\left (i \frac{\mathrm d}{\mathrm dx}+M\right )^{2}\\ D(\widetilde A):= \widetilde{H^2}(\mathcal G) \end{gathered}$$ edgewise defined, i.e. $$\left (\widetilde A \phi\right )_e = \widetilde A \phi_e.$$ Then $\widetilde A$ defines an unbounded operator on $L^2(\mathcal G)$ and satisfies (i) $fD(\widetilde A)\subset D(\widetilde A)$ for all $f\in \widetilde{C^\infty}(\mathcal G)$. (ii) Let $f\in \widetilde{C^\infty}(\mathcal G)$, then the operator $f\widetilde Af$ is given by $$\label{eq:formulareduced} f\widetilde A f= \frac{1}{2} \left (f^2\widetilde A + \widetilde A f^2\right ) + \|f'\|^2$$ The proof is analogous to the one in Lemma \[lem:decomposition2\]. does not uniquely determine an operator. Indeed is the special case of when $k=1$. In particular, formula in the case $k=1$ holds for all self-adjoint realizations of the magnetic Schrödinger operators (e.g. ) and independent of the choice of $M\in H^1+ W^{1,\infty}(\mathcal G)$. We will be interested in a decomposition lemma on the form associated to $A$ as given in . \[lem:lemma5.7\] Let $\mathcal G$ be a locally finite, connected metric graph and $a(\cdot, \cdot)$ be the symmetric bilinearform given by $$a(u,v):= \int_{\mathcal G} \overline{\left ( i \frac{\mathrm d}{\mathrm dx}+M\right )u} \left ( i \frac{\mathrm d}{\mathrm dx}+M\right ) v \, \mathrm dx$$ for $u,v\in H^1(\mathcal G)$ Then for $f\in W^{1,\infty}(\mathcal G)\cap \widetilde{C^\infty}(\mathcal G)$ we have $$\label{eq:IMSformulapre} a(fu, fv) = \frac{1}{2} \left (a_{\mathcal G}(u, f^2 v)+ a(f^2 u, v)\right )+\langle \|f'\|^2 u, v\rangle_{L^2(\mathcal G)}$$ By Proposition \[prop:densitylocfinite\] we may assume $u,v\in H^2_{c}(\mathcal G)$ and $fu, fv\in \widetilde{H^2}(\mathcal G)\cap H^1_c(\mathcal G)$. Integrating by parts on an arbitrary bounded subgraph $K$ containing $\operatorname{supp}u$ and $\operatorname{supp}{v}$ we compute $$\begin{aligned} a(fu,fv)&= -\int_{K} \overline {\left (fAf\right ) u} v \, \mathrm dx + \sum_{\mathsf v\in \mathcal N_f\cap K} \sum_{e\succ \mathsf v} \left [\overline{\left ( i \frac{\mathrm d}{\mathrm dx} + M\right ) fu}\right ]_e fv(\mathsf v)\\ &=-\int_{K} \left (\overline{\frac{1}{2}\left (f^2\widetilde A + \widetilde A f^2\right )u + \|f'\|^2 u}\right ) v\, \mathrm dx\\ &\qquad \qquad\quad \qquad\qquad + \sum_{\mathsf v\in \mathcal N_f\cap K} \sum_{e\succ \mathsf v} \left [\overline{\left ( i \frac{\mathrm d}{\mathrm dx} + M\right ) fu}\right ]_e fv(\mathsf v)\\&= \frac{1}{2} \left (a(u, f^2 v)+ a(f^2 u, v)\right )+ \int_{\mathcal G} \|f'\|^2 \overline u v\, \mathrm dx\\ &\qquad\qquad - \sum_{\mathsf v\in \mathcal N_f\cap K} \sum_{e\succ \mathsf v} \frac{1}{2}\left [\overline{\left ( i \frac{\mathrm d}{\mathrm dx} + M\right ) f^2 u}\right ]_e v(\mathsf v)\\ &\qquad \qquad \qquad-\sum_{\mathsf v\in \mathcal N_f\cap K} \sum_{e\succ \mathsf v} \frac{1}{2}\left [\overline{\left ( i \frac{\mathrm d}{\mathrm dx} + M\right ) u}\right ]_e f^2v(\mathsf v)\\ &\qquad \qquad \qquad \qquad + \sum_{\mathsf v\in \mathcal N_f\cap K} \sum_{e\succ \mathsf v} \left [\overline{\left ( i \frac{\mathrm d}{\mathrm dx} + M\right ) fu}\right ]_e fv(\mathsf v)\\ &= \frac{1}{2} \left (a(u, f^2 v)+ a(f^2 u, v)\right )+ \int_{\mathcal G} \|f'\|^2 \overline u v\, \mathrm dx \end{aligned}$$ and the statement follows by density. Existence of NLS ground state for a class of Schrödinger operators ------------------------------------------------------------------ ### The localized setting In the following we study the localized case. We also remark that some of the lemmas will also apply to the global case. For $t>0$ we define $$\label{eq:minimizationdefloc} E_t^{(\mathcal K)}:= \inf_{\substack{u\in H^1(\mathcal G)\\\\|u\\|_{L^2}^2=t}} E_{\text{NLS}}^{(\mathcal K)}.$$ \[eq:NLSboundedbelow\] Let $\mathcal G$ be a connected locally finite metric graph. Let $\mathcal K$ be a not necessarily bounded subset of $\mathcal G$. The functional $E_{\text{NLS}}^{(\mathcal K)}$ under $L^2$-constraint $\\|\cdot\\|_{L^2}^2=1$ is bounded below for $2<p<6$. From the Gagliardo–Nirenberg inequality we have $$\begin{aligned} \int_{\mathcal K} \|u\|^p\, \mathrm dx &\le \int_{\mathcal G} \|u\|^p\, \mathrm dx \\ &\le \varepsilon \int_{\mathcal G} \left \|\left ( i \frac{\mathrm d}{\mathrm dx} + M \right )u\right \|^2+ V \|u\|^2\, \mathrm dx + C_{\varepsilon} \int_{\mathcal G} \|u\|^2\, \mathrm dx \end{aligned}$$ and therefore $$E_{\text{NLS}}^{(\mathcal K)}(u) \ge(1-\varepsilon) \int_{\mathcal G} \left \|\left ( i \frac{\mathrm d}{\mathrm dx} + M \right )u\right \|^2+ V \|u\|^2\, \mathrm dx - C_{\varepsilon} \ge -C_{\varepsilon}.$$ for all $u\in H^1(\mathcal G)$ satisfying $\\|u\\|_2^2=1$. \[lem:energyinequality\] Let $\mathcal G$ be a locally finite, connected metric graph. Assume $A= \left ( i \frac{\mathrm d}{\mathrm dx} + M\right )^2 + V$ admits a ground state, then $$E_t^{(\mathcal K)}=\inf_{\substack{u\in H^1(\mathcal G)\\\\|u\\|_{L^2}^2=t}} E_{\text{NLS}}^{(\mathcal K)} \le \frac{\Sigma_0t}{2}.$$ The inequality is strict if the ground state does not vanish identically on $\mathcal K$ Assume $u$ is a ground state of $A= \left ( i \frac{\mathrm d}{\mathrm dx} + M\right )^2 + V$ with $\\|u\\|_{L^2}^2=t$, then $$E_{\text{NLS}}^{(\mathcal K)}(u) = \frac{\Sigma_0t}{2} - \frac{\mu}{p} \int_{\mathcal K} \|u\|^{p}\, \mathrm dx\le \frac{\Sigma_0}{2} t$$ and the inequality is strict if $u$ is not identically vanishing on $\mathcal K$. In particular $$\inf_{\substack{u\in H^1\\\\|u\\|_{L^2}^2=t}} E_{\text{NLS}}^{(\mathcal K)} \le \frac{\Sigma_0t}{2}$$ with strictness in the inequality if there exists a ground state, which is not identically vanishing on $\mathcal K$. \[lem:prework\] Let $\mathcal G$ be a locally finite, connected metric graph and let $\mathcal K$ be any subgraph. Assume $A= \left ( i \frac{\mathrm d}{\mathrm dx} + M\right )^2 + V$ admits a ground state that is not identically vanishing on $\mathcal K$, then the functional $E_{\text{NLS}}^{(\mathcal K)}$ is weak limit superadditive, superadditive with respect to the partition of unity in Example \[ex:first2\] and $t\mapsto E_t$ as defined in is strictly subadditive. With the Minkowski inequality we have $$\begin{aligned} &\left ( \int_{\mathcal G}\|u'\|^2\mathrm dx\right )^{1/2}- \left (\int_{\mathcal G} \|M\|^2 \|u\|^2\, \mathrm dx\right )^{1/2}\\ &\qquad \qquad \le \left (\int_{\mathcal G} \left \|\left ( i \frac{\mathrm d}{\mathrm dx} + M \right )u\right \|^2\, \mathrm dx\right )^{1/2}\\ &\qquad \qquad \qquad \qquad\le \left ( \int_{\mathcal G}\|u'\|^2\mathrm dx\right )^{1/2}+ \left (\int_{\mathcal G} \|M\|^2 \|u\|^2\, \mathrm dx\right )^{1/2}. \end{aligned}$$ Adding a constant to the potential similarly as in the proof of Lemma \[lem:preconditions\] we may assume $$\\|u\\|_{2, M,V} =\left (\int_{\mathcal G} \left \|\left ( i \frac{\mathrm d}{\mathrm dx} + M \right )u\right \|^2+ V \|u\|^2\, \mathrm dx\right )^{1/2}$$ to define an equivalent norm on $H^1$. Weak limit superadditivity. Assume $u_n \rightharpoonup u$ weakly in $H^1$, then up to a subsequence by the Brezis–Lieb Lemma and weak limit superadditivity (similarly as in the proof of Lemma \[lem:preconditions\]) $$\limsup_{n\to \infty} E_{\text{NLS}}^{(\mathcal K)} (u_n) =E_{\text{NLS}}^{(\mathcal K)}(u)+ \limsup_{n\to \infty} E_{\text{NLS}}^{(\mathcal K)} (u-u_n)$$ and $E_{\text{NLS}}$ is weak limit superadditive. Superaddivity with respect to a sequence of partitions of unity. For the superadditivity, since $u_n$ is vanishing, up to a subsequence $$\\|\widetilde{\Psi_n} u_n\\|_p^p- \\|u_n\\|_p^p\to 0 \qquad (n\to \infty).$$ Then using the decomposition formula we compute, similarly as in the proof of Lemma \[lem:preconditions\]: $$\begin{aligned} \limsup_{n\to \infty}E_{\text{NLS}}^{(\mathcal K)}(u_n) &= \limsup_{n\to \infty} \frac{1}{2} a(u_n, u_n) - \frac{\mu}{p} \\|u_n\\|_p^p\\ &\ge \limsup_{n\to \infty} a(\widetilde{\Psi_n} u_n, \widetilde{\Psi_n}u_n) - \frac{\mu}{p} \\|\widetilde{\Psi_n} u_n\\|_p^p\\ &\qquad \qquad \qquad + a(\Psi_n u_n, u_n) - \frac{\mu}{p} \\|\Psi_n u_n\\|_p^p\\ &= \limsup_{n\to \infty} E_{\text{NLS}}^{(\mathcal K)}(\widetilde{\Psi_n}u_n)+ E_{\text{NLS}}^{(\mathcal K)}(\Psi_n u_n). \end{aligned}$$ Subadditivity. To show the subadditivity, note that $$\label{eq:scalingarg} E_t^{(\mathcal K)} = t \inf_{\substack{u\in H^1\\ \\|u\\|_{L^2}^2=1}} \left \{\frac{1}{2}\int_{\mathcal G} \left \|\left ( i \frac{\mathrm d}{\mathrm dx} + M \right )u\right \|^2+ V \|u\|^2\, \mathrm dx - t^{\frac{p-2}{2}} \frac{\mu}{p} \int_{K} \|u\|^p\, \mathrm dx \right \}.$$ We deduce the property by showing that $t\mapsto E_t^{(\mathcal K)}$ is a concave function. Indeed, the scaling defines a concave function and in the limit we deduce concavity of the functional. In particular $$\label{eq:concavityarg} E_t^{(\mathcal K)} \ge t E_1^{(\mathcal K)}, \qquad t\in [0,1].$$ Then $$E_t^{(\mathcal K)} + E_{1-t}^{(\mathcal K)} \ge E_1^{(\mathcal K)}, \qquad t\in [0,1].$$ For the strictness in the inequality it suffices to show strictness in the inequality . Assume $$E_t^{(\mathcal K)} = t E_1^{(\mathcal K)}$$ for some $t\in (0,1)$ and let $u_n$ be a minimizing sequence for $E_t$, then in particular due to $$\int_{\mathcal K} \|u_n\|^p\, \mathrm dx \to 0 \qquad (n\to \infty).$$ Then by density we may assume $u_n \in D(A)$ and we infer $$\begin{aligned} E_t^{(\mathcal K)} &= \lim_{n\to \infty} E_{NLS}^{(\mathcal K)} (u_n)\\ &\ge \frac{1}{2} \limsup_{n\to \infty} \langle Au_n, u_n\rangle \ge \frac{\Sigma_0 t}{2}, \end{aligned}$$ which is a contradiction to the inequality in Lemma \[lem:energyinequality\]. \[thm:bigresulttt\] Let $\mathcal G$ be a connected, locally finite metric graph. Assume $A= \left ( i \frac{\mathrm d}{\mathrm dx} + M\right )^2 + V$ admits a ground state, which is not identically vanishing on $\mathcal K$, then $E_{\text{NLS}}^{(\mathcal K)}$ admits a minimizer for all $\mu >0$. For $R>0$ sufficiently large since $\mathcal K$ is considered to be bounded $$\begin{aligned} \inf_{\substack{u\in D_R(A) \\ \\|u\\|_{L^2}^2=1}} E_{\text{NLS}}^{(\mathcal K)}(u)&= \inf_{\substack{u\in D_R(A)\\\\|u\\|_{L^2}^2=1}} \frac{1}{2}\langle Au, u\rangle\\ &\ge \inf_{\substack{u\in D(A) \\ \\|u\\|_{L^2}^2=1}} \frac{1}{2}\langle Au, u\rangle= \frac{\Sigma_0}{2} \end{aligned}$$ In particular with Lemma \[lem:energyinequality\] we have $$E_{\text{NLS}}^{(\mathcal K)} < \lim_{R\to \infty} \inf_{\substack{u\in D_R(A) \\ \\|u\\|_{L^2}^2=1}} E_{\text{NLS}}^{(\mathcal K)}(u)=: \widetilde{E_{\text{NLS}}^{(\mathcal K)}}.$$ Due to Lemma \[lem:prework\] the requirements of Theorem \[thm:main1\] and \[thm:main2\] are satisfied and up to a subsequence any minimizing sequence admits a strong limit in $L^p$ such that the limit achieves the minimum in $E_{\text{NLS}}^{(\mathcal K)}$. \[rem:importantvanishing\] If $M\equiv 0$ then we can assume that a ground state of $A$ is nonnegative and in fact by Hopf’s maximum principle positive everywhere. In particular, any ground state of $A$ is not identically vanishing on any subset of $\mathcal G$. We will see in §6 that $\Sigma_0<\Sigma$ implies the existence of ground states of $A$. In particular, this condition becomes obsolete in this case. ### The global setting $\mathcal K=\mathcal G$ Consider now the global case, where we consider the functional $$E_\text{NLS} (\phi) = \frac{1}{2}\int_{\mathcal G} \left \|\left ( i \frac{\mathrm d}{\mathrm dx} + M \right )\phi\right \|^2+ V \|\phi\|^2\, \mathrm dx- \frac{\mu}{p} \int_{\mathcal G}\|\phi\|^p\, \mathrm dx, \qquad \\|\phi\\|_{L^2}=1.$$ In the global case Lemma \[lem:prework\] applies since any ground state of the magnetic Schrödinger operator $A=\left ( i \frac{\mathrm d}{\mathrm dx}+M\right )^2+V$ is not identically zero. In the following we give a criterion for existence of ground states with regards to these quantities. \[prop:first\] Assume $\mathcal G$ is a locally finite, connected metric graph and $\Sigma_0 < \Sigma$. Then there exists $\hat \mu >0$, such that for all $\mu \in (0, \hat \mu)$. $$\widetilde \Sigma_0^{(\mu)} := \inf_{\phi \in D(A)} E_{\text{NLS}}(\phi) < \lim_{R\to \infty}\inf_{\phi \in D_R(A)} E_{\text{NLS}}(\phi) =: \widetilde \Sigma^{(\mu)}.$$ W.l.o.g. $\Sigma_0>0$; otherwise we simply add a constant to the potential $V$. Let $0<\varepsilon<1$ arbitrary, which we will only fix later. With Proposition \[cor:Gagliardo-nirenberg\] we deduce (similarly as in Proposition \[prop:second\]) that for sufficiently small $\mu>0$ $$E_\text{NLS}(\phi) \ge \frac{1- \varepsilon}{2} \left ( \int_{\mathcal G} \left \|\left ( i \frac{\mathrm d}{\mathrm dx} + M \right )\phi\right \|^2+ V\|\phi\|^2\, \mathrm dx\right )- \frac{C\varepsilon}{2}.$$ Then $$\widetilde{\Sigma}^{(\mu)} -\widetilde{\Sigma}_0^{(\mu)} \ge \frac{1-\varepsilon}{2} \Sigma-\frac{\varepsilon}{2} - \frac{1}{2}\Sigma_0 = \frac{1}{2} \left (\Sigma - \Sigma_0\right ) -\frac{\varepsilon}{2} \left (\widetilde{C}+ \Sigma\right ).$$ Since $\varepsilon$ can be chosen arbitrarily small, we have for sufficiently small $\mu$ $$\widetilde \Sigma^{(\mu)} > \widetilde \Sigma^{(\mu)}_0.$$ \[lem:energyinequality2\] Let $\mathcal G$ be a locally finite, connected metric graph. Assume $A= \left ( i \frac{\mathrm d}{\mathrm dx} + M\right )^2 + V$ admits a ground state, then $$E_t=\inf_{\substack{u\in H^1(\mathcal G)\\\\|u\\|_{L^2}^2=t}} E_{\text{NLS}}(u) < \frac{\Sigma_0t}{2}.$$ In the nonlocalized case, we can proceed analogously to before. Given a ground state $u\in H^2$ we simply compute analogously as in Lemma \[lem:energyinequality\] $$E_t < \frac{\Sigma_0 t}{2}.$$ \[lem:essential\] Assume $\mathcal G$ is a locally finite, connected metric graph and $\Sigma_0 < \Sigma$. Then $ E_{\text{NLS}}$ is weak limit superadditive, superadditive with respect to the sequence of partitions of unity in Example \[ex:unity2\] and $t\mapsto E_t$ defines a strictly subadditive functional. The proof is analogous to the one in Lemma \[lem:prework\] by simply replacing $\mathcal K$ with the whole graph. $\Sigma_0 < \Sigma$, as we will see later, implies by Theorem \[thm:persson2\] $$\inf \sigma\left ( \left ( i \frac{\mathrm d}{\mathrm dx} +M\right )^2 +V\right ) < \inf \sigma_{\text{ess}}\left ( \left ( i \frac{\mathrm d}{\mathrm dx} +M\right )^2 +V\right ).$$ In particular, there exist discrete eigenvalues below the essential spectrum and $A$ admits a ground state. \[thm:bigresultt\] Let $\mathcal G$ be a locally finite, connected metric graph. Assume $\Sigma_0 < \Sigma$, then for $\mu \in (0, \hat \mu)$ as in Proposition \[prop:second\] $$E_\text{NLS} (\phi) = \frac{1}{2}\int_{\mathcal G} \left \|\left ( i \frac{\mathrm d}{\mathrm dx} + M \right )\phi\right \|^2+ V \|\phi\|^2\, \mathrm dx- \frac{\mu}{p} \int_{\mathcal G}\|\phi\|^p\, \mathrm dx, \qquad \\|\phi\\|_2=1.$$ admits a minimizer. By Lemma \[lem:essential\] the requirements of Theorem \[thm:main2\] are satisfied. Furthermore the energy inequality in Corollary \[cor:existence\] is satisfied by Proposition \[prop:first\] and we infer the statement. On the Threshold condition {#sec:onthreshold} ========================== In this section we study the quantities $\Sigma_0$ and $\Sigma$ that appeared in the applications of the previous sections. Relation to essential spectrum ------------------------------ For details on the definitions and characterizations of the essential spectrum we refer to [@reedmethods §VII]. ### Polylaplacians Let $\mathcal G= (\mathcal V, \mathcal E)$ be a connected metric finite graph in this first part of the section. In particular $\mathcal G$ consists of a compact core $K$ with rays $\mathcal E_\infty\subset \mathcal E$ attached to $K$. Consider the Polylaplacian on $\mathcal G$ defined edgewise on $H^{2k}(\mathcal G)$: $$A= (-\Delta)^k,\qquad D(A) =H^{2k}(\mathcal G)$$ Combining Lemma \[lem:IMSunity\] and the abstract decomposition formula in Lemma \[lem:decomposition2\] we have the decomposition formula for the Polylaplacian: \[lem:decomposition3\] Let $\mathcal G= (V, \mathcal E)$ be a connected finite graph and assume $\{\Psi_1, \ldots, \Psi_N\}$ to be a partition of unity subordinate to an open covering $\mathcal O=\{O_1, \ldots, O_n\}$ satisfying $$\sum_{k=1}^N \Psi_k^2 \equiv 1.$$ Then $$\label{eq:decomposition3} \begin{aligned} A\phi&= \sum_{j=1}^k \Psi_j A \Psi_j\phi+ \frac{(-1)^k}{2} \sum_{m=1}^{2k} \sum_{n=1}^{2k-m} \frac{(2k)_{m+n}}{m! n!} \Psi_j^{(m)} \Psi_j^{(n)} \phi^{(2k-m-n)} \end{aligned}$$ for all $\phi \in D(A)$. We recall that $K=\mathcal G\setminus \mathcal E_\infty$ is the core of the graph and that for $R>0$ $$\begin{aligned} D_R&= \{\phi \in D(A) \|\operatorname{supp}(\phi) \subset \mathcal G \setminus K_R\}\\ \Sigma_R&= \inf\{\langle \phi, A\phi\rangle\|\phi \in D_R, \\|\phi\\|_{2}^2=1\}.\end{aligned}$$ Since $D(A)$ is nontrivial and invariant under multiplication by test functions in $\widetilde{C_c^\infty}$ the set $D_R$ is nonempty. For $R=0$ we set $$\begin{aligned} D_0&= D(A)\\ \Sigma_0&= \inf\{\langle \phi, A\phi \rangle \|\phi \in D(A), \\|\phi\\|_{2}^2=1\}\end{aligned}$$ and recall that $$\Sigma = \lim_{R\to \infty} \Sigma_R=\sup_{R>0} \Sigma_R.$$ In the following we characterize the quantities that were central to the existence theorems in the existence results before. Since $A$ is self-adjoint one can show $$\Sigma_0 = \inf \sigma(A).$$ \[thm:persson\] Assume $\mathcal G$ is a finite metric graph. Let $A$ be a self-adjoint, nonnegative operator on $L^2(\mathcal G)$ that satisfies the decomposition formula . Additionally let $f(A+i)^{-1}$ be compact for all $f\in\widetilde{C_c^\infty}(\mathcal G)$. Then $$\Sigma= \inf \sigma_{\text{ess}} (A).$$ $\inf \sigma_{\text{ess}}(A)\ge \Sigma$. Let $\lambda \in \sigma_{\text{ess}}(A)$ and let $(\phi_n)$ be an associated Weyl sequence satisfying $\\|\phi_n\\|_2^2=1$. Consider the vanishing-compatible sequence of partitions of unity $\Psi_{n}, \widetilde{\Psi_n}$ from Example \[ex:unity2\]. Since $\Psi_R^2 (A+i)^{-1}$ is compact for all $R>0$, and since $(A+i)\phi_n\rightharpoonup 0$ as $n\to \infty$ we deduce that $$\\|\Psi_R \phi_n\\|_2= \\|\Psi_R (A+i)^{-1} (A+i) \phi_n\\|_2\to 0\qquad (n\to \infty)$$ and passing to a subsequence, still denoted by $\phi_n$, we may assume $$\\|\Psi_n \phi_n\\|_2= \\|\Psi_n (A+i)^{-1} (A+i) \phi_n\\|_2\to 0.$$ Furthermore, with we deduce that $$\\|\phi_n\\|_{H^{2k}} \le C \|\phi_n\|_{H^{2k}}= C \left ( \\|A \phi_n\\|_2^2+ \\|\phi_n\\|_2^2\right )^{1/2}$$ is uniformly bounded. Since $\phi_n$ is a Weyl sequence for $\lambda \in \sigma_{\text{ess}}(A)$ with the decomposition formula in Lemma \[lem:decomposition3\] we then compute $$\begin{aligned} \lambda &= \lim_{n\to \infty} \langle \phi_n, A \phi_n\rangle_{L^2} \\ &= \lim_{n\to \infty} \langle \Psi_n\phi_n, A \Psi_n \phi_n\rangle_{L^2} + \langle \widetilde{\Psi_n} \phi_n, A \widetilde{\Psi_n} \phi_n\rangle_{L^2} + O\left (\frac{1}{n^2}\right ) \\ &\ge \lim_{n\to \infty} \sum_{e\in \mathcal E_\infty} \langle \widetilde{\Psi_n} \phi_n, A\widetilde{\Psi_n} \phi_n\rangle_{L^2}\ge \lim_{n\to \infty} \Sigma_n = \Sigma. \end{aligned}$$ Since $\lambda \in \sigma_{\text{ess}}(A)$ was arbitrary, we conclude $\inf \sigma_{\text{ess}}(A)\ge \Sigma$. $\inf \sigma_{\text{ess}}(A)\le \Sigma$. Assume for a contradiction that $\inf \sigma_{\text{ess}}(A)\ge \Sigma+ 3\varepsilon$ with $\varepsilon >0$. Then $\sigma(A) \cap (-\infty, \Sigma+ 2\varepsilon]$ is discrete and since $A$ is bounded from below, the spectral projector $P_\Sigma:= P_{(-\infty, \Sigma+2\varepsilon]}$ is of finite rank. Assume $\phi_n \in D_n(A)$ is a sequence such that $$\langle \phi_n, A\phi_n\rangle \le \Sigma + \varepsilon$$ and $\phi_n \rightharpoonup 0$ in $L^2$. Then since $\left (A +\Sigma + 2\varepsilon\right )P_\Sigma $ is a compact operator and $$\left (A +\Sigma + 2\varepsilon\right )P_\Sigma \phi_n\to 0 \qquad (n\to \infty).$$ Hence $$\begin{aligned} \langle \phi_n, A\phi_n\rangle_{L^2} &= \langle \phi_n, A(1-P_{\Sigma})\phi_n\rangle + \langle \phi_n, A P_\Sigma \phi_n\rangle_{L^2}\\ &\ge (\Sigma + 2\varepsilon) \langle \phi_n, (1-P_{\Sigma})\phi_n\rangle_{L^2} + \langle \phi_n, AP_\Sigma \phi_n\rangle_{L^2} \\ &\ge \Sigma+2\varepsilon + \left \langle \phi_n, \left (A +\Sigma + 2\varepsilon\right )P_\Sigma \phi_n \right \rangle_{L^2}. \end{aligned}$$ Passing to the limit we conclude $$\liminf_{n\to \infty} \langle \phi_n, A\phi_n\rangle_{L^2} \ge \Sigma+2\varepsilon$$ and the statement follows by contradiction. ### Schrödinger operators and IMS formula on locally finite graphs Let $\mathcal G=(\mathcal V, \mathcal E)$ be a locally finite graph throughout the rest of the section. Consider the Schrödinger operators with potentials $M$ and $V$ satisfying natural vertex conditions on $\mathcal G$: $$\label{eq:schroedinger} \begin{aligned} A&=\left (i \frac{\mathrm d}{\mathrm dx}+M\right )^2+V \\ D(A)&= \bigg \{u\in C(\mathcal G)\bigg\| u_e\in H^2(e), \qquad\forall e\in {{\mathcal E}}\\ &\qquad\qquad\qquad \land\qquad \sum_{e \succ \mathsf v} \left ( i \frac{\mathrm d}{\mathrm dx} + M \right ) u_e (\mathsf v)=0,\qquad \forall \mathsf v \in \mathcal V\bigg \}. \end{aligned}$$ In the case of magnetic Schrödinger operators $$A= \left ( i \frac{\mathrm d}{\mathrm dx}+M\right )^2 + V$$ on domains $\Omega \subset \mathbb R^N$ the decomposition formula can be obtained via the IMS formula[^2] $$A = \sum_{j=1}^k \Psi_k A \Psi_k + \|\Psi_k'\|^2 \quad \text{where}\quad \sum_{j=1}^k \Psi_k^2 \equiv 1.$$ When considering locally finite graphs, we may not use the approach as before because general locally finite graphs do not necessarily contain a core and we need to adapt the theory using sequences of partitions of unity as in Example \[ex:first2\]. \[df:IMS\] Let $\mathcal G$ be a locally finite, connected metric graph. Let $A: D(A) \subset L^2(\mathcal G) \to L^2(\mathcal G)$ be a densely defined, self-adjoint operator and assume $a(\cdot, \cdot)$ is the associated symmetric, sesquilinear form, defined on $H^1(\mathcal G)$. We say $A$ satisfies the IMS formula if for all $f\in W^{1,\infty}(\mathcal G)\cap \widetilde{C^\infty}(\mathcal G)$ $$\begin{gathered} \label{eq:IMSformula} a(fu,fv) = \frac{1}{2} \left (a(u, f^2 v)+ a(f^2 u, v)\right )+\langle \|f'\|^2 u, v\rangle_{L^2}, \qquad \forall u,v\in D(A).\end{gathered}$$ We showed in Lemma \[lem:lemma5.7\] that the magnetic Schrödinger operator in satisfies the IMS formula ; and in particular the following result applies: \[thm:persson2\] Assume $\mathcal G$ is a locally finite, connected metric graph. Let $A$ be a self-adjoint, nonnegative operator on $L^2(\mathcal G)$ that satisfies the IMS formula . Additionally let $f(A+i)^{-1}$ be compact for all $f\in C_c^{0,1}\cap \widetilde{C^\infty}$ then $$\Sigma= \inf \sigma_{\text{ess}} (A).$$ $\inf \sigma_{\text{ess}}(A)\le \Sigma$ follows by an abstract argument analogous to the argument in Theorem \[thm:persson\]. To establish $$\inf \sigma_{\text{ess}}(A)\le \Sigma$$ consider $\lambda \in \sigma_{\text{ess}}(A)$ and let $(\phi_n)$ be an associated Weyl sequence satisfying $$\\|\phi_n\\|_2^2=1.$$ Consider the vanishing-compatible sequence of partitions of unity $\Psi_{n}, \widetilde{\Psi_n}$ from Example \[ex:first2\]. Since $\Psi_R^2 (A+i)^{-1}$ is compact for all $R>0$, and since $(A+i)\phi_n\rightharpoonup 0$ as $n\to \infty$ we deduce $$\\|\Psi_R \phi_n\\|_2= \\|\Psi_R (A+i)^{-1} (A+i) \phi_n\\|_2\to 0\qquad (n\to \infty)$$ and passing to a subsequence, still denoted by $\phi_n$, we may assume $$\\|\Psi_n \phi_n\\|_2= \\|\Psi_n (A+i)^{-1} (A+i) \phi_n\\|_2\to 0.$$ Since $\phi_n$ is a Weyl sequence for $\lambda \in \sigma_{\text{ess}}(A)$, with the decomposition formula in Lemma \[lem:decomposition3\] we then compute $$\begin{aligned} \lambda &= \lim_{n\to \infty} \langle \phi_n, A \phi_n\rangle_{L^2} \\ &= \lim_{n\to \infty} a(\Psi_n \phi_n, \Psi_n \phi_n) + a(\widetilde{\Psi_n} \phi_n, \widetilde{\Psi_n}\phi_n) + O\left (\frac{1}{n^2}\right ) \\ &\ge \lim_{n\to \infty} \sum_{e\in \mathcal E_\infty} a(\widetilde{\Psi_n}\phi_n, \widetilde{\Psi_n}\phi_n)\ge \lim_{n\to \infty} \Sigma_n = \Sigma. \end{aligned}$$ Since $\lambda \in \sigma_{\text{ess}}(A)$ was arbitrary, we conclude $\inf \sigma_{\text{ess}}(A)\ge \Sigma$. Sufficient conditions for the threshold condition for the Polylaplacian {#sec:subsecsuff} ----------------------------------------------------------------------- In this section we obtain criteria for the threshold condition for the operator $$\begin{gathered} A= (-\Delta)^k + V\\ D(A)=H^{2k} \end{gathered}$$ We start with the case $k=1$ for general locally finite graphs satisfying a volume growth assumption. For $k\ge 2$ we restrict ourselves to finite graphs. \[thm:lastresultt1\] Let $\mathcal G$ be a locally finite, connected metric graph and let $K$ be a connected, precompact subgraph. We suppose additionally the volume assumption $$\label{eq:volumegrowthassumpt} \left \|K_{2n} \setminus K_{n}\right \|=o(n^2) \qquad (n\to \infty).$$ Assume $\sigma_\text{ess}(-\Delta+V) \subset [0, \infty)$ and assume additionally either (i) $V\in L^1(\mathcal G)\cap L^2(\mathcal G)$ and $$\int_{\mathcal G} V \, \mathrm dx <0$$ (ii) or $V<0$ on $\mathcal G$. Then $\Sigma_0 < \Sigma$ (as defined in and ) and there exists $\hat \mu>0$ such that the minimization problem $$\label{eq:minimizerlast} E^{(1)}=\inf_{\substack{u\in H^1\\\\|u\\|_{L^2}^2=1}} E^{(1)}(u)$$ admits a minimizer for $\mu \in (0,\hat \mu)$. Consider as a test function $\Psi_n$ as defined in Example \[ex:first2\], then we only need to show that for $n$ sufficiently high, the Rayleigh quotient $$\mathcal R[\Psi_n] := \frac{\int_{\mathcal G} \|\Psi'_n\|^2 + V \|\Psi_n\|^2\, \mathrm dx }{\int_{\mathcal G} \|\Psi_n\|^2\, \mathrm dx}<0.$$ Indeed, since $\\|\Psi'_n\\|_{\infty}^2\le O(\frac{1}{n^2})$ as $n\to \infty$ we deduce $$\\|\Psi'_n\\|_2^2\le \\|\Psi'_n\\|_{\infty}^2 \|K_{2n}\setminus K_n\|\to 0 \qquad (n\to \infty).$$ If $V<0$ then for sufficiently large $n$ and $\varepsilon>0$ sufficiently small $$\int_{\mathcal G} V \|\Psi_n\|^2\, \mathrm dx\le -\left \|\left \{x\in \mathcal G: V(x)\le -\varepsilon\right \}\right \| \varepsilon<0.$$ If $\int_{\mathcal G} V \, \mathrm dx <0$, then $$\liminf_{n\to \infty}\int_{\mathcal G} V \|\Psi_n\|^2 \, \mathrm dx= \int_{\mathcal G} V\, \mathrm dx<0$$ by dominated convergence. In particular for $n$ large enough $$\int_{\mathcal G} V\|\Psi_n\|^2 \le \frac{1}{2} \int_{\mathcal G} V\, \mathrm dx <0.$$ We deduce $R[\Psi_n]<0$ and thus $\inf \sigma((-\Delta)^k+V)<0$. Then $\Sigma_0 < \Sigma$ and we conclude the existence of minimizers of by Theorem \[thm:bigresultt\]. Similarly, for $k\ge 2$ we have: \[thm:lastresultt2\] Let $\mathcal G$ be a finite, connected metric graph and $k\ge 1$. Assume $\sigma_\text{ess}((-\Delta)^k+V) \subset [0,\infty)$ and assume additionally either (i) $V\in L^1(\mathcal G)\cap L^2(\mathcal G)$ and $$\int_{\mathcal G} V \, \mathrm dx <0$$ (ii) or $V<0$ on $\mathcal G$. Then $\Sigma_0 < \Sigma$ (as defined in and ) and there exists $\hat \mu>0$, such that the minimization problem $$\label{eq:minimizerklast} E^{(k)}=\inf_{\substack{u\in H^1\\\\|u\\|_{L^2}^2=1}} E^{(k)}(u)$$ admits a minimizer for $\mu \in (0,\hat \mu)$. The proof is analagous to the proof in Proposition \[thm:lastresultt1\]. We only need to replace the test functions $\Psi_n$ with the ones in Example \[ex:unity2\]. Then $$\\|\Psi_n^{(k)}\\|_\infty \le \frac{1}{n^{2k}}C$$ with $C$ independent of $n$. We infer the result as in the proof of Proposition \[thm:lastresultt1\]. If $V$ is a relativly compact perturbation of $(-\Delta)^k$, i.e. $$V\left ((-\Delta)^k+i\right )^{-1}$$ is compact, then $\inf \sigma_\text{ess}\left ((-\Delta)^k+V\right ) =0$ and we deduce $$\inf \sigma_\text{ess}\left ((-\Delta)^k+V\right )\subset [0,\infty).$$ We finish the section by giving a criterion for the potential $V$ such that $$\Sigma = \lim_{n\to \infty} \inf_{\substack{u\in D(A) \\ \\|u\\|_2^2=1, \; \operatorname{supp} u\subset \mathcal G\setminus K_n}} \langle u, Au\rangle \ge 0.$$ In the situations considered in §6.1 this in particular implies $$\sigma_\text{ess}((-\Delta)^k+V) \subset [0,\infty).$$ Consider decaying potentials $V=V_2+ V_\infty$ with $V_2\in L^2(\mathcal G)$ and $V_\infty\in L^\infty(\mathcal G)$ such that $$\label{eq:newdecayingpotentials} \sup_{x\in \mathcal G\setminus K_n} \|V_\infty(x)\|\to 0 \qquad (n\to \infty).$$ \[prop:Iwantthistoend\] Let $\mathcal G$ be a locally finite metric graph. Assume $V\in L^2+ L^\infty$ satisfying . Let $A=(-\Delta)^k+V$, then $$\Sigma = \lim_{n\to \infty} \inf_{\substack{u\in H^k(\mathcal G)\\ \\|u\\|_2^2=1, \; \operatorname{supp} u\subset \mathcal G\setminus K_n}} \langle u, Au\rangle_{L^2} \ge 0.$$ Assume $u_n$ is a minimizing sequence, such that $\\|u_n\\|_{L^2}^2$, $\operatorname{supp} u\subset \mathcal G\setminus K_n$ and $$\langle u_n, Au_n\rangle_{L^2}\to \Sigma.$$ With we deduce that $$\label{eq:ineedthis2} \\|u_n\\|_{H^k} \le C \left (\langle u_n, Au_n\rangle_{L^2}^2 + \\|u_n\\|_2^2\right )$$ is uniformly bounded. Integrating by parts and using we infer $$\begin{aligned} \int_{\mathcal G} \left \|u^{(k)}_n\right \|^2 + V\|u_n\|^2\, \mathrm dx&\ge \int_{\mathcal G} \left \|u^{(k)}_n\right \|^2\, \mathrm dx \\ &\qquad - \widetilde{C} \left ( \left (\int_{\mathcal G\setminus K_n} \|V\|^2\, \mathrm dx\right )^{1/2} + \sup_{x\in \mathcal G\setminus K_n} \|V_\infty(x)\|\right ). \end{aligned}$$ We have $$\left ( \left (\int_{\mathcal G\setminus K_n} \|V\|^2\, \mathrm dx\right )^{1/2} + \sup_{x\in \mathcal G\setminus K_n} \|V_\infty(x)\|\right )\to 0 \qquad (n\to \infty).$$ Thus, $$\Sigma = \lim_{n\to \infty} \langle u_n, Au_n\rangle_{L^2} \ge 0.\qedhere$$ Application: Schrödinger operators with magnetic potentials on infinite tree graphs =================================================================================== In certain cases as discussed in [@berkolaiko2013introduction §2.6] the gauge transform $G$, as defined below, unitarily transforms the Schrödinger operator with magnetic potential into a Schrödinger operator without magnetic potential, and the NLS functional under gauge transform reduces to a problem without magnetic potential. We may use the results from Section \[sec:subsecsuff\] to show existence of minimizers of the NLS functional with magnetic potential. For infinite tree graphs in the context of locally finite, connected metric graphs it is particularly easy to see this. In this context, let $\mathcal G$ be an infinite tree graph. Given a vertex $\mathsf v$ we can define the gauge transform $G$ radially. For any $x\in \mathcal G$, let $\gamma$ be a simple path from $\mathsf v$ to $x$ parametrized by arc length, then $$G:u(x)\mapsto e^{i \int_{\operatorname{im} \gamma} M\, \mathrm d\gamma} u(x).$$ Assume $A^{M}= (i \frac{\mathrm d}{\mathrm dx} + M)^2+V$ admits a ground state. In this particular case since $$G^{-1} A^{M} G = -\Delta + V= A^{0},$$ this is equivalent to the assertion that $A^{0}$ admits a ground state. Indeed, let $u_M$ be a ground state to $A^{M}$, then $$A^{0} G^{-1} u_0 =G^{-1} A^{M} u_M = \Sigma G^{-1} u_M$$ and $G^{-1} u_M$ is a ground state of $A^{0}$. Then we may assume $u_0>0$ by phase invariance and the maximum principle. Then $u_M$ does not vanish anywhere. In particular independent of $M\in H^1+W^{1,\infty}(\mathcal G)$ $$\label{eq:unitaryequivalence} \begin{aligned} \Sigma_0^{M} &= \inf_{\substack{u\in D(A^{M})\\\\|u\\|_{2}^2=1}}\left \langle A^{M}u, u\right \rangle= \inf_{\substack{u\in D(A^{0})\\\\|u\\|_{2}^2=1}}\left \langle A^{0}u, u\right \rangle= \Sigma_0\\ \Sigma_{R}^{M} &= \inf_{\substack{u\in D_R(A^{M})\\\\|u\\|_{2}^2=1}}\left \langle A^{M}u, u\right \rangle= \inf_{\substack{u\in D_R(A^{0})\\\\|u\\|_{2}^2=1}}\left \langle A^{0}u, u\right \rangle= \Sigma_R\\ \Sigma^{M}&= \lim_{R\to \infty} \Sigma_R^M = \lim_{R\to \infty} \Sigma_R= \Sigma \end{aligned}$$ and in Section \[sec:subsecsuff\] we gave sufficient conditions for $\Sigma_0 < \Sigma$. \[prop:app1\] Assume $\mathcal G$ is an infinite tree graph, connected and locally finite. Assume $\mathcal K$ is a bounded subgraph of $\mathcal G$ and $-\Delta+V$ admits a ground state, then the infimization problem $$E_{\text{NLS}}^{(\mathcal K)}=\inf_{\substack{u\in H^1(\mathcal G)\\ \\|u\\|_2^2=1}} \frac{1}{2}\int_{\mathcal G} \left \|\left ( i \frac{\mathrm d}{\mathrm dx} + M \right )u\right \|^2+ V \|u\|^2\, \mathrm dx - \frac{\mu}{p} \int_{\mathcal K} \|u\|^p\, \mathrm dx$$ admits a minimizer for all $\mu \in \mathbb R$. This follows immediately from Theorem \[thm:bigresulttt\] and the unitary equivalence of the problem in absence of a magnetic potential under the gauge transform. \[prop:app2\] Assume $\mathcal G$ is a infinite tree graph, locally finite and connected. Assume $\mathcal K$ is any unbounded subgraph and $\Sigma_0 < \Sigma$ then there exists $\hat \mu>0$, such that the infimization problem $$E_{\text{NLS}}=\inf_{\substack{\phi\in H^1(\mathcal G)\\ \\|\phi \\|_2^2=1}} \frac{1}{2}\int_{\mathcal G} \left \|\left ( i \frac{\mathrm d}{\mathrm dx} + M \right )\phi\right \|^2+ V \|\phi\|^2\, \mathrm dx- \int_{\mathcal K} \frac{\mu}{p} \|\phi\|^p\, \mathrm dx.$$ admits a minimizer for all $\mu\in (0,\hat \mu)$. This follows immediately from Theorem \[thm:bigresultt\] and the unitary equivalence of the problem in absence of a magnetic potential under the gauge transform For decaying potentials in §6.2 we discussed criteria such that $\Sigma_0< \Sigma$ is satisfied. Indeed, for any given locally finite graph, one can construct decaying potentials in the following way: Let $\mathcal G$ be a locally finite, connected graph and $K$ a bounded, connected subgraph. Consider the higher-order Schrödinger operator $A=(-\Delta)^k+V$ with potential $V$. We define a potential $V$ a.e. via $$\begin{aligned} V\bigg \|_{K_1} &\equiv -\frac{1}{2}\\ V\bigg \|_{K_{2n}\setminus K_n} &\equiv -\frac{1}{2^n \|K_{2n} \setminus K_n\|}, \quad n\ge 2 \end{aligned}$$ on each “annulus” $K_{2n}\setminus K_n$. Then $V\in L^2 \cap L^1(\mathcal G)$, $$\int_{\mathcal G} V\, \mathrm d\mu= -\sum_{n=0}^\infty \frac{1}{2^n} <0$$ and by Proposition \[prop:Iwantthistoend\] we infer $\inf \sigma_{\text{ess}}(A)\ge 0$. In particular, if $\mathcal G$ is an infinite tree graph satisfying the volume growth assumption , then the prerequisites in Proposition \[thm:lastresultt1\] are satisfied as well and we have $$\Sigma_0 < \Sigma.$$ In particular Proposition \[prop:app1\] and Proposition \[prop:app2\] are applicable and there exists $\hat \mu >0$ such that $$E_{\text{NLS}}^{(\mathcal K)}=\inf_{\substack{u\in H^1(\mathcal G)\\ \\|u\\|_2^2=1}} \frac{1}{2}\int_{\mathcal G} \left \|\left ( i \frac{\mathrm d}{\mathrm dx} + M \right )u\right \|^2+ V \|u\|^2\, \mathrm dx - \frac{\mu}{p} \int_{\mathcal K} \|u\|^p\, \mathrm dx$$ admits a minimizer for $\mu \in (0, \hat \mu)$. If $\mathcal K\subset \mathcal G$ is precompact, then minimizers exist for all $\mu>0$. For a certain class of infinite tree graphs we can in a similar way as in Example \[ex:shouldbeputinintroduction\] give an explicit $\hat \mu$ such that for $\mu\in (0,\hat \mu]$ the minimization problem $E_{NLS}$ admits a minimizer. \[ex:unrootedtrees\] Consider an unrooted tree graph $\mathcal G$ as considered for instance in [@dovetta2019nls], i.e. there are no vertices of degree $1$ apart of vertices at infinity. Such trees in particular satisfy the (H)-condition formulated in [@adami2015nls] in the special case of finite graphs: - For every point $x\in \mathcal G$, there exist two injective curves $\gamma_1, \gamma_2:[0,+\infty) \to \mathcal G$ parametrized by arc length, with disjoint images except on a discrete set of points, and such that $\gamma_1(0)=\gamma_2(0)=x$. and by rearrangement methods as in [@dovetta2019nls] one can show for decaying potentials $V= V_2+V_\infty$ with $V_2\in L^2(\mathcal G)$ and $V_\infty\in L^\infty(\mathcal G)$ satisfying $$\sup_{x\in \mathcal G\setminus K_n} \|V_\infty(x)\|\to 0 \qquad (n\to \infty)$$ that $$\begin{aligned} \widetilde{\Sigma}^{(\mu)} &= \lim_{n\to \infty} \inf_{\substack{u\in H^1(\mathcal G\\\\|u\\|_2^2=1, \, \operatorname{supp} u\subset \mathcal G\setminus K_n}} E_{\text{NLS}}^V(u)\\ &\ge \lim_{n\to \infty} \inf_{\substack{u\in H^1(\mathcal G\\\\|u\\|_2^2=1, \, \operatorname{supp} u\subset \mathcal G\setminus K_n}} E_{\text{NLS}}^0(u) \ge E_{\text{NLS}}(\mathbb R), \end{aligned}$$ where by Remark \[rmk:decayingpotential\] one has equality if $\mathcal G$ contains a half line. When $V\equiv 0$, by strictness in the rearrangement inequality one can prove nonexistence results similarly as in [@adami2015nls]. On the other hand, under the assumption $$\Sigma_0 = \inf \sigma(-\Delta +V) <0,$$ as discussed in Example \[ex:shouldbeputinintroduction\] we have thus the existence of minimizers of $E_{\text{NLS}}$ for $$\mu \in \left [ 0, \left ( \frac{\Sigma_0}{\gamma_p} \right )^{\frac{6-p}{4}} \right ].$$ \[rmk:unrooted\] The arguments in Example \[ex:unrootedtrees\] can be applied to all graphs that satisfy the (H)-condition. One can even consider more general graphs as long they satisfy the following weaker version of the (H)-condition: - There exists a precompact set $K\subset \mathcal G$, such that for every point $x\in \mathcal G\setminus K$, there exist two injective curves $\gamma_1, \gamma_2:[0,+\infty) \to \mathcal G$ parametrized by arc length, with disjoint images except on a discrete set of points, and such that $\gamma_1(0)=\gamma_2(0)=x$. Consider the graph consisting of two half-lines and a pendant edge joined at a single vertex (see also Figure \[fig:donedone\]), then the graph satisfies the (H)-condition but not the (H)-condition and the existence result from Example \[ex:unrootedtrees\] as discussed in Remark \[rmk:unrooted\] is still applicable. ![The graph consisting of two half-lines and a pendant edge as an example of a graph that satisfies the the (H)-condition but not the (H)-condition.[]{data-label="fig:donedone"}](simple3.pdf) We finish this section by proving Theorem \[thm:introlastlastlast\]: Let $\mathcal G$ be a locally finite metric tree graph that contains at most finitely many vertices of degree $1$. Then there exists a connected, precompact set $K\subset \mathcal G$ that contains all vertices of degree $1$ by assumption. Consider the set $\overline{\mathcal G}$ of points $x\in \mathcal G$, such that there exist two injective curves $\gamma_1, \gamma_2:[0,+\infty)\to \mathcal G$ parametrized by arc length, with disjoint images except on a discrete set of points, and such that $\gamma_1(0)= \gamma_2(0)=x$. In particular, if $x\in \overline{\mathcal G}$, then $$\operatorname{im} \gamma_1,\; \operatorname{im} \gamma_2\subset \overline{\mathcal G}.$$ 1st Case: $\overline{\mathcal G}\neq \emptyset$. Then by assumption $\mathcal G\setminus \overline{\mathcal G}$ contains at most finitely many connected components. Moreover of the connected components is precompact. Otherwise one could construct an injective curve $\gamma_1:[0, +\infty)\to \mathcal G\setminus \overline{\mathcal G}$ for all $x\in \mathcal G\setminus \overline{\mathcal G}$ and since we assumed $\overline{\mathcal G}\neq \emptyset$, we can construct $\gamma_2:[0,+\infty)\to \mathcal G\setminus \overline{\mathcal G}$. This would then imply that $\mathcal G\setminus \overline{\mathcal G}$ is necessarily precompact. Since $\mathcal G$ is a tree graph, this also implies that each connected component of $\mathcal G\setminus \overline{\mathcal G}$ contains necessarily a vertex of degree $1$. In particular, $\mathcal G\setminus \overline{\mathcal G}$ admits at most finitely many connected components and is precompact. By construction, $\overline{\mathcal G}$ satisfies the (H)-condition and hence $\mathcal G$ satisfies the (H)-condition. Then as in Example \[ex:unrootedtrees\] we have $$\begin{aligned} \widetilde{\Sigma}^{(\mu)} &= \lim_{n\to \infty} \inf_{\substack{u\in H^1(\mathcal G\\\\|u\\|_2^2=1, \, \operatorname{supp} u\subset \mathcal G\setminus K_n}} E_{\text{NLS}}^V(u)\\ &\ge \lim_{n\to \infty} \inf_{\substack{u\in H^1(\mathcal G\\\\|u\\|_2^2=1, \, \operatorname{supp} u\subset \mathcal G\setminus K_n}} E_{\text{NLS}}^0(u) \ge E_{\text{NLS}}(\mathbb R) \end{aligned}$$ and we obtain existence of minimizers of $E_{\text{NLS}}$ for $$\mu \in \left [ 0, \left ( \frac{\Sigma_0}{\gamma_p} \right )^{\frac{6-p}{4}} \right ].$$ 2nd Case: $\overline{\mathcal G}=\emptyset$. In particular for each $x\in \mathcal G$ there exists only one connected component of $\mathcal G$ that contains a vertex at infinity. Assume $K$ is a precompact set that contains all vertices of degree $1$, then by assumption for any $x\in \mathcal G\setminus K$ the connected components of $\mathcal G\setminus \{x\}$ consist of a compact core graph containing all vertices of degree $1$ and a half-line. In particular, $\mathcal G$ is a finite graph and Example \[ex:shouldbeputinintroduction\] yields the existence of minimizers of $E_{\text{NLS}}$ for $$\mu \in \left [ 0, \left ( \frac{\Sigma_0}{\gamma_p} \right )^{\frac{6-p}{4}} \right ].$$ [EKMN18]{} Riccardo Adami, Claudio Cacciapuoti, Domenico Finco, and Diego Noja. Stationary states of [NLS]{} on star graphs. , 100(1):10003, 2012. Setenay Akduman and Alexander Pankov. Nonlinear schr[ö]{}dinger equation with growing potential on infinite metric graphs. , 184, 2019. Riccardo Adami, Enrico Serra, and Paolo Tilli. ground states on graphs. , 54(1):743–761, 2015. Riccardo Adami, Enrico Serra, and Paolo Tilli. 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--- abstract: 'When a large collection of objects (e.g., robots, sensors, etc.) has to be deployed in a given environment, it is often required to plan a coordinated motion of the objects from their initial position to a final configuration enjoying some global property. In such a scenario, the problem of minimizing some function of the distance travelled, and therefore energy consumption, is of vital importance. In this paper we study several motion planning problems that arise when the objects must be moved on a graph, in order to reach certain goals which are of interest for several network applications. Among the others, these goals include broadcasting messages and forming connected or interference-free networks. We study these problems with the aim of minimizing a number of natural measures such as the average/overall distance travelled, the maximum distance travelled, or the number of objects that need to be moved. To this respect, we provide several approximability and inapproximability results, most of which are tight.' author: - Davide Bilò - Luciano Gualà - Stefano Leucci - Guido Proietti date: 'Received: date / Accepted: date' title: 'Exact and approximate algorithms for movement problems on (special classes of) graphs [^1] [^2] ' --- Introduction ============ In many practical applications a number of centrally controlled objects need to be moved in a given environment in order to complete some task. Problems of this kind often occur in robot motion planning where we seek to move a set of robots from their starting position to a set of ending positions such that a certain property is satisfied. For example, if the robots are equipped with a short range communication device we might want to move them so that a message originating from one of the robots can be routed to all the others. If the robots’ goal is to monitor a certain area we might want to move them so that they are not too close to each other. Other interesting problems include gathering (placing robots next to each other), monitoring of traffic between two locations, building interference-free networks, and so on. To make things harder, objects to be moved are often equipped with a limited supply of energy. Preserving energy is a critical problem in ad-hoc networking, and movements are expensive. To prolong the lifetime of the objects we seek to minimize the energy consumed during movements and thus the distance travelled. Sometimes, instead, movements are cheap but before and/or after an object moves it needs to perform expensive operations. In this scenario we might be interested in moving the minimum number of objects needed to reach the goal. In this paper, we assume the underlying environment is actually a network, which can be modelled as an undirected graph $G$, and the moving objects are centrally controlled pebbles that are initially placed on vertices of $G$, and that can be moved to other vertices by traversing the graph edges. To this respect, we study several movement planning problems that arise by various combinations of final positioning goals and movement optimization measures. In particular, we focus our study on the scenarios where we want the pebbles to be moved to a connected subgraph ([$\textsc{Con}$]{}), an independent set ([$\textsc{Ind}$]{}), or a clique ([$\textsc{Clique}$]{}) of $G$, while minimizing either the overall movement ([$\textsc{Sum}$]{}), the maximum movement ([$\textsc{Max}$]{}), or the number of moved pebbles ([$\textsc{Num}$]{}). We also give some preliminary results on the problem of moving the pebbles to an s-t-cut, i.e., a set of vertices whose removal makes two given vertices $s,t$ disconnected ([$s\mbox{-}t\mbox{-}\textsc{Cut}$]{}) while minimizing the above measures. We will denote each of the above problems with $\psi$-$c$, where $\psi$ represents the goal to be achieved and $c$ the measure to be minimized. For a more rigorous definition of the problems we refer the reader to Section \[sec:formal\_definition\]. #### Related work. Although movement problems were deeply investigated in a distributed setting (see [@PS06] for a survey), quite surprisingly the centralized counterpart has received attention from the scientific community only in the last few years. The first paper which defines and studies these problems in this latter setting is [@demaine2007minimizing]. In their work, the authors study the problem of moving the pebbles on a graph $G$ of $n$ vertices so that their final positions form a connected component, a path (directed or undirected) between two specified nodes, an independent set, or a matching (two pebbles are matched together if their distance is exactly $1$). Regarding connectivity problems, in [@demaine2007minimizing] the authors show that all the variants are hard and that the approximation ratio of ${\ensuremath{\textsc{Con}}}$-${\ensuremath{\textsc{Max}}}$ is between $2$ and ${\ensuremath{O}}(1+\sqrt{k/c^})$, where $k$ is the number of pebbles and $c^$ denotes the measure of an optimal solution. This result has been improved in [@berman2011], where the authors show that ${\ensuremath{\textsc{Con}}}$-${\ensuremath{\textsc{Max}}}$ can be approximated within a constant factor. In [@demaine2007minimizing] it is also shown that ${\ensuremath{\textsc{Con}}}$-${\ensuremath{\textsc{Sum}}}$ and ${\ensuremath{\textsc{Con}}}$-${\ensuremath{\textsc{Num}}}$ are not approximable within ${\ensuremath{O}}(n^{1-\epsilon})$ (for any positive $\epsilon$) and $o(\log n)$, respectively, while they admit approximation algorithms with ratios of ${\ensuremath{O}}(\min\{n \log n, k\})$ and ${\ensuremath{O}}(k^\epsilon)$, respectively. Moreover, the authors also provide an exact polynomial-time algorithm for ${\ensuremath{\textsc{Con}}}$-${\ensuremath{\textsc{Max}}}$ on trees. Concerning independency problems, in [@demaine2007minimizing] the authors remark that it is [$\textsf{\upshape NP}$]{}-hard even to find any feasible solution on general graphs since it would require to find an independent set of size at least $k$. This clearly holds for all three objective functions. For this reason, they study an Euclidean variant of these problems where pebbles have to be moved on a plane so that their pairwise distances are strictly greater than $1$. In this case, the authors provide an approximation algorithm that guarantees an additive error of at most $1+1/\sqrt{3}$ for ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Max}}}$, and a polynomial time approximation scheme for ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Num}}}$. More recently, in [@friggstad2011minimizing], a variant of the classical facility location problem has been studied. This variant, called mobile facility location, can be modelled as a movement problem and is approximable within $(3+\epsilon)$ (for any constant $\epsilon>0$) if we seek to minimize the total movement [@ahmadian2013local], while the variant where the maximum movement has to be minimized admits a tight $2$-approximation [@demaine2007minimizing; @friggstad2011minimizing]. Moreover, as it is frequent in the practice to have a small number of pebbles compared to the size of the environment (i.e., the vertices of the graph), the authors of [@demaine2009FPT] turn to study fixed-parameter tractability. They show a relation between the complexity of the problems and their minimal configurations (sets of final positions of the pebbles that correspond to feasible solutions, such that any removal of an edge makes them unacceptable). Finally, we mention that in [@BDGMPW13] it was considered a set of vertex-to-vertex motion planning problems in a simple polygon, with the aim of forming final configurations enjoying some sort of visual connectivity among the pebbles. #### Our results. We start by studying connectivity motions problems in the case where pebbles move on a tree, and we devise two polynomial-time dynamic programming algorithms for ${\ensuremath{\textsc{Con}}}$-${\ensuremath{\textsc{Sum}}}$ and ${\ensuremath{\textsc{Con}}}$-${\ensuremath{\textsc{Num}}}$. These algorithms complement the already known polynomial-time algorithm for ${\ensuremath{\textsc{Con}}}$-${\ensuremath{\textsc{Max}}}$ on trees shown in [@demaine2007minimizing]. Then, we study independency motion problems on graphs where a maximum independent set (and thus a feasible solution for the corresponding motion problem) can be computed in polynomial time. This class of graphs includes, for example, perfect and claw-free graphs. More precisely, we show that ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Max}}}$ and ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Sum}}}$ are [$\textsf{\upshape NP}$]{}-hard even on bipartite graphs (which are known to be perfect graphs [@bollobas1998modern]). Moreover, we devise three exact polynomial-time algorithms: one for solving ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Max}}}$ on paths, and the other two for solving ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Sum}}}$ and ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Num}}}$ on trees, respectively. Moreover, we devise a polynomial-time approximation algorithm for ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Max}}}$ which is optimal unless an additive term of $1$ (this is clearly tight). Concerning the problem of moving pebbles towards a clique of a general graph, we prove that all the three variants are [$\textsf{\upshape NP}$]{}-hard. Then, we provide an approximation algorithm for ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Max}}}$ which is optimal unless an additive term of $1$ (this result is clearly tight). Moreover, we show that both ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Sum}}}$ and ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Num}}}$ are approximable within a factor of $2$, but they are not approximable within a factor better than $10\sqrt{5}-21 > 1.3606$, unless ${\ensuremath{\textsf{\upshape P}}}={\ensuremath{\textsf{\upshape NP}}}$. If the unique game conjecture [@khot2002power] is true, then both problems are not approximable within a factor better than $2$ and the provided approximation algorithms are tight. These results are obtained by showing a non-trivial relation with the minimum vertex cover problem. We also show that an exact solution for ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Num}}}$ can be computed in polynomial time on every class of graphs for which finding a maximum-weight clique requires polynomial time (these classes of graphs also include perfect and claw-free graphs). Finally, we present a strong inapproximability results of $\Omega(n^{1-\epsilon})$ (for any $\epsilon>0$) for ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Max}}}$ and ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Sum}}}$, unless ${\ensuremath{\textsf{\upshape P}}}={\ensuremath{\textsf{\upshape NP}}}$, along with two approximation algorithms. The approximation algorithm for ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Max}}}$ is essentially tight, while we show that any constant-factor approximation for ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Num}}}$ would imply a tight approximation for ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Sum}}}$. The paper is organized as follows: in Section \[sec:formal\_definition\] we provide a formal definition of our problems, while in Sections \[sec:con\]–\[sec:stcut\] we give our results for [$\textsc{Con}$]{}, [$\textsc{Ind}$]{}, [$\textsc{Clique}$]{}, and [$s\mbox{-}t\mbox{-}\textsc{Cut}$]{}, respectively (for a summary of the state of the art of the studied problems, along with the results presented in this paper, see Table \[table:results\]). Finally, Section \[sec:concl\] concludes the paper. [\|c\|M\|L\|M\|]{} & ${\ensuremath{\textsc{Max}}}$ & ${\ensuremath{\textsc{Sum}}}$ & ${\ensuremath{\textsc{Num}}}$\ ${\ensuremath{\textsc{Con}}}$ & G: $2 \le \rho = {\ensuremath{O}}(1)$ [@demaine2007minimizing; @berman2011] T: polynomial [@demaine2007minimizing] & G: $\rho = \Omega(n^{1-\epsilon})$ [@demaine2007minimizing] $\rho={\ensuremath{O}}(\min\{n \log n,k\})\!$ [@demaine2007minimizing] T: polynomial & G: $\rho=\Omega(\log n)$ [@demaine2007minimizing] $\rho={\ensuremath{O}}(k^\epsilon)$ [@demaine2007minimizing] T: polynomial\ ${\ensuremath{\textsc{Ind}}}$ & G: [$\textsf{\upshape NP}$]{}-hard [@demaine2007minimizing] IS: [$c^* + 1$]{}, [$\rho \le 2$]{} B: [$\rho \ge 2$]{} P: polynomial & G: [$\textsf{\upshape NP}$]{}-hard [@demaine2007minimizing] B: [$\textsf{\upshape NP}$]{}-hard T: polynomial & G: [$\textsf{\upshape NP}$]{}-hard [@demaine2007minimizing] T: polynomial\ ${\ensuremath{\textsc{Clique}}}$ & G: [$\textsf{\upshape NP}$]{}-hard & G: [$10\sqrt{5}-21 \le \rho \le 2$]{} & G: [$10\sqrt{5}-21 \le \rho \le 2$]{} MWC: polynomial\ ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$ & G: [$\rho = \Omega(n^{1-\epsilon})$]{} & G: [$\rho = \Omega(n^{1-\epsilon})$]{} & G: $\rho$-apx $\Longrightarrow (\rho \cdot d)$-apx for ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Sum}}}$\ \ \ \[table:results\] Formal definitions {#sec:formal_definition} ================== A pebble motion problem, denoted by $\psi$-$c$, is an optimization problem whose instances consist of a loop-free connected undirected graph $G=(V(G),E(G))$ on $n$ nodes, a set $P=[k]=\{1,\dots, k\}$ of pebbles, a function $\sigma : P \to V(G)$ that assigns each pebble to a start vertex of $G$, and a boolean predicate $\psi : 2^{V(G)} \to \{ {\ensuremath{\texttt{True}}},{\ensuremath{\texttt{False}}}\}$ that assigns a truth value to every possible subset of vertices of $G$. A (feasible) solution is a function $\mu : P \to V(G)$ that maps each pebble to an end vertex of $G$ (in other words, moves a pebble from its start to its end position) such that $\psi(\mu[P])$ is true, where $\mu[P]$ denotes the image of $P$ under $\mu$. Notice that, in general, it is not required for $\sigma$ or $\mu$ to be injective and thus we allow more than one pebble to be placed on the same vertex. In the rest of the paper, we will assume that a pebble moving from a vertex $u$ to a vertex $v$ always uses a shortest path in $G$ between $u$ and $v$, say $\pi_G(u,v)$. Moreover we denote by $d_G(u,v)$ the length of such a path. Finally, $c(\mu) \in \mathbb{N}_0$ is a measure function that assigns a non-negative integer to each feasible solution (i.e., to each set of moves). A solution $\mu^$ that minimizes $c$ is said to be optimal. In the following, we will study some of the movement problems that arise from the different choices of predicates and measures. In particular, we will consider the following predicates: Connectivity:* : ${\ensuremath{\textsc{Con}}}(U)$ is true if and only if the subgraph of $G$ induced by the set of vertices $U \subseteq V(G)$ is connected; Independency: : ${\ensuremath{\textsc{Ind}}}(U)$ is true if and only if $U \subseteq V(G)$ is an independent set of $G$ of size $k$, i.e., there is at most one pebble per vertex and no two pebbles are on adjacent vertices; Clique: : ${\ensuremath{\textsc{Clique}}}(U)$ is true if and only if $U \subseteq V(G)$ induces a clique in $G$, i.e., for each pair $u,v$ of distinct vertices in $U$ there exists the edge $(u,v) \in E(G)$; $s$-$t$-Cut: : Given $s,t \in V(G)$ with $s \not= t$, then ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}(U)$ is true if and only if $s \not \in U$, $t \not\in U$ and $U \subseteq V(G)$ is an s-t-cut (i.e., there exists no path between $s$ and $t$ in the graph induced by the vertices in $V(G) \setminus U$); and the following measures: Overall movement: : The sum of the distances travelled by pebbles has to be minimized: every pebble $p \in P$ moves from its starting vertex $\sigma(p)$ to his end vertex $\mu(p)$, so the overall distance is ${\ensuremath{\textsc{Sum}}}(\mu) = \sum_{p\in P} d_G( \sigma(p), \mu(p) )$; Maximum movement: : We want to minimize the maximum distance travelled by a pebble, i.e., the measure ${\ensuremath{\textsc{Max}}}(\mu)= \max_{p \in P} d_G( \sigma(p), \mu(p) )$; Number of moved pebbles: : We aim to minimize the number of pebbles that need to be moved from their starting positions. The associated measure is ${\ensuremath{\textsc{Num}}}(\mu) = \| \{ p \in P : \sigma(p) \not= \mu(p)\} \|$. Connectivity motion problems {#sec:con} ============================ In this section we describe two polynomial-time algorithms for solving on trees ${\ensuremath{\textsc{Con}}}$-${\ensuremath{\textsc{Sum}}}$ and ${\ensuremath{\textsc{Con}}}$-${\ensuremath{\textsc{Num}}}$, respectively. In this way we complement the result provided in [@demaine2007minimizing] for ${\ensuremath{\textsc{Con}}}$-${\ensuremath{\textsc{Max}}}$ on trees. Solving ${\ensuremath{\textsc{Con}}}$-${\ensuremath{\textsc{Sum}}}$ on trees {#sec:con_sum_trees} ---------------------------------------------------------------------------- Our dynamic-programming algorithm relies on the following property of optimal solutions: \[lemma:sum\_tree\_no\_crossing\] In any optimal solution $\mu^$ for an instance of ${\ensuremath{\textsc{Con}}}$-${\ensuremath{\textsc{Sum}}}$ on trees, there exists no edge that is traversed in opposite directions by pebbles. Let $\pi_G(u,v)$ denote a shortest path in $G$ between the vertices $u$ and $v$. Suppose by contradiction that there exists an optimal solution $\mu^$, an edge $(x,y) \in E(G)$, and two pebbles $p, q \in P$ such that $p$ moves through the path $\pi_G(\sigma(p), x) \cup \{ (x,y) \} \cup \pi_G(y, \mu^(p))$ and $q$ moves through the path $\pi_G(\sigma(q), y) \cup \{ (y,x) \} \cup \pi_G(x, \mu^(q))$. Consider the solution $\mu^\prime$ obtained from $\mu^$ by swapping the final positions for $p$ and $q$, i.e., $\mu^\prime(p)=\mu^(q)$ and $\mu^\prime(q)=\mu^(p)$. Clearly $\mu^\prime[P]=\mu^[P]$, therefore $\mu^\prime$ is feasible. Moreover $c(\mu^\prime) < c(\mu^)$ as we can move $p$ through the path $\pi_G(\sigma(p), x) \cup \pi_G(x,\mu^\prime(p))$ and $q$ through the path $\pi_G(\sigma(q), y) \cup \pi_G(y, \mu^\prime(q))$, thus saving $2$. The algorithm first guesses a vertex $r \in V(G)$ such that there exists an optimal solution that places a pebble on $r$, then roots the tree $G$ at $r$ to obtain a rooted tree $G_r$, and finally considers all the subtrees of $G_r$ in a bottom-up fashion. For a given subtree $T_u$ of $G_r$ rooted at the vertex $u$, let us denote by $\eta(u)$ the number of pebbles placed on $V(T_u)$ w.r.t. $\sigma$. When the subtree $T_u$ is examined, we consider an auxiliary problem. In this problem we want to place exactly $j \le k$ of the pebbles on the vertices of $T_u$ in order to satisfy the following properties: 1. \[it:con\_P1\] the subgraph of $T_u$ induced by the final positions of the pebbles must be connected; 2. \[it:con\_P2\] if $j \ge 1$ then at least one pebble must be placed on $u$. Moreover, if $j < \eta(u)$ we want to move the $\eta(u)-j$ exceeding pebbles to the parent of $u$ (and thus outside $T_u$). In a similar manner, if $j > \eta(u)$ then the $j-\eta(u)$ missing pebbles are to be moved into $T_u$ from the parent of $u$, where we assume they are initially placed. We point out that, by Lemma \[lemma:sum\_tree\_no\_crossing\], we do not need to consider the case where some pebbles move into $T_u$ while others move out of $T_u$. ![Left: an instance of ${\ensuremath{\textsc{Con}}}$-${\ensuremath{\textsc{Sum}}}$ and its optimal solution. A single pebble is placed on each black vertex while the optimal movements are denoted by arrows. Right: the auxiliary problems corresponding to the optimal solution which are considered by the dynamic-programming algorithm.[]{data-label="fig:mov_con_sum"}](Fig1.eps) We will denote by ${\ensuremath{\texttt{Opt}}}[u,j]$ the cost of the optimal movement for this auxiliary problem. Notice that, in ${\ensuremath{\texttt{Opt}}}[u,j]$, we are accounting for the cost of traversing all the edges of $T_u$ plus the edge from $u$ to its parent. To solve the original problem we need to find a solution corresponding to ${\ensuremath{\texttt{Opt}}}[r,k]$. Clearly as $k=\eta(r)$ we do not have exceeding or missing pebbles, in this case. We now show how to combine these auxiliary problems. If $u$ is a leaf of $G_r$ then ${\ensuremath{\texttt{Opt}}}[u,j]=\|\eta(u) - j\|$. Otherwise, if $u$ is not a leaf, we can distinguish two cases: $j=0$ and $j > 0$. If $j=0$ then no pebble can be placed on $u$ or in any descendant of $u$, therefore all the pebbles must first be moved towards $u$ and then to the parent of $u$. Let $v_1, \dots, v_\ell$ be the set of children of $u$ in $G_r$, we have: ${\ensuremath{\texttt{Opt}}}[u,0] = \eta(u) + \sum_{i=1}^\ell {\ensuremath{\texttt{Opt}}}[v_i, 0]$. Otherwise, if $j > 0$, we can move any number of pebbles between $1$ and $j$ to $u$, and place the remaining pebbles on the subtrees rooted at the children of $u$. Therefore we have: $${\ensuremath{\texttt{Opt}}}[u,j] = \|\eta(u)-j\| + \min_{ \substack{0 \le j_1, \dots, j_\ell < j \\ \sum_{i=1}^{\ell} j_i < j } } \left\{ \sum_{i=1}^{\ell} {\ensuremath{\texttt{Opt}}}[v_i, j_i] \right\} \mbox{.}$$ Notice that the minimum considers all the possible ways for distributing less than $j$ pebbles on the subtrees, i.e., all the vectors $(j_i)_i$ of $\ell$ elements whose sum is less than $j$. We now argue on the fact that, despite the number of such vectors can be exponential on $j$, the minimum can be found in polynomial time. This can again be done by using dynamic programming: let ${\ensuremath{\texttt{Min}}}[i,h]$ denote the minimum cost of placing $h$ pebbles in the first $i$ subtrees. Clearly when $i=1$ we have ${\ensuremath{\texttt{Min}}}[1, h] = {\ensuremath{\texttt{Opt}}}[v_1, h]$, while for $i > 1$ the following holds: $${\ensuremath{\texttt{Min}}}[i, h] = \min_{0 \le z \le h} \left\{ {\ensuremath{\texttt{Min}}}[i-1, z] + {\ensuremath{\texttt{Opt}}}[v_i, h-z] \right\} \mbox{.}$$ Therefore, the equation for ${\ensuremath{\texttt{Opt}}}[u,j]$ can be rewritten as: $${\ensuremath{\texttt{Opt}}}[u,j] = \|\eta(u)-j\| + \min_{0 \le i < j} {\ensuremath{\texttt{Min}}}[\ell, i] \mbox{.}$$ Notice how this way of distributing the pebbles is general and does not depend on the specific movement problem: in fact, it can be used every time we are interested in minimizing the cost of distributing a number of items in a set of bins if, for each bin, we incur a cost that depends on the number of items placed therein. An example of an optimal decomposition into subproblems along with the corresponding optimal solution is shown in Figure \[fig:mov\_con\_sum\]. Regarding the complexity of the algorithm, the time required to compute a specific ${\ensuremath{\texttt{Opt}}}[u,j]$ is $O(\ell \cdot j) = O(\ell \cdot k)$. As the sum of the $\ell$-values over all the vertices is $n-1$, the time needed to compute ${\ensuremath{\texttt{Opt}}}[v,j]$ for a fixed $j$ and all $v \in V$ is $O(n \cdot k)$. It follows that all the possible subproblems can be solved in time $O(n \cdot k^2)$. As we have to guess the vertex $r$, a naïve strategy would be repeating the above procedure $n$ times, one for each vertex of $G$. This would require an overall time of $O(n^2 \cdot k^2)$. We can do better by using a more sophisticated approach: consider a centroid[^3] $v$ of $G$, and notice that either there exists an optimal solution that places a pebble on $v$, or every optimal solution places all the pebbles on a single connected component of $G-v$. We first apply the above algorithm using $v$ as the root and then we proceed recursively on the trees of the forest $G-v$ (each of which has at most half of the vertices). More precisely, for every subtree $T$ of $G-v$, rooted at $v^\prime$, we recursively solve an instance consisting of the tree $T$ where all the pebbles in $V(T)$ are left unmoved and all pebbles not in $V(T)$ have been moved to $v^\prime$. This movement cost, i.e., $\sum_{p \in P : \sigma(p) \not\in V(T)} d(\sigma(p), v^\prime)$, is then added to the measure of the solution returned by the recursive call. Among all the computed solutions we choose the cheapest one. By doing so, we are able to reduce the computational complexity to $O(n \cdot k^2 \log n)$. Indeed, the recurrence relation describing the running time of the algorithm is $T(n) = \sum_{n_j} T(n_j) + O(n \cdot k^2)$, where $n_j \le \frac{n}{2}$ denotes the number of vertices of the $j$-th subtree of $G-v$. Clearly, the depth of the recursion is $O(\log n)$ while the amount of work on each level of the recursion-tree is $O(n \cdot k^2)$. Once the value of the optimal solution $\mu^$ has been found, it is not too hard to see that the optimal solution itself can be reconstructed by proceeding in a bottom-up fashion, while keeping track of both the pebbles that move out of each subtree and the position where missing pebbles are to be placed. To summarize, we have the following: \[thm:con\_sum\_trees\] ${\ensuremath{\textsc{Con}}}$-${\ensuremath{\textsc{Sum}}}$ on trees can be solved in $O(n \cdot k^2 \log n)$ time. Solving ${\ensuremath{\textsc{Con}}}$-${\ensuremath{\textsc{Num}}}$ on trees ---------------------------------------------------------------------------- The algorithm is similar to the one for ${\ensuremath{\textsc{Con}}}$-${\ensuremath{\textsc{Sum}}}$: we guess a vertex $r \in V(G)$ such that there exists an optimal solution that places a pebble on $r$, then we root the tree $G$ at $r$ (call $G_r$ the rooted tree) and we consider all the subtrees of $G_r$ in a bottom-up fashion. Let $\varphi(u)=\|\{ p \in P : \sigma(p)=u \}\|$ be the number of pebbles whose initial position is the vertex $u \in V(G)$. As before, when the subtree $T_u$ (rooted at the vertex $u$) of $G_r$ is examined, we consider an auxiliary problem where we want to place exactly $j \le k$ pebbles on the vertices of $T_u$ in order to satisfy the properties \[it:con\_P1\] and \[it:con\_P2\]. We will measure the cost of a solution for this auxiliary problem by examining the number of pebbles placed on each vertex of $T_u$. Removing pebbles from a vertex costs nothing, while placing a pebble on a vertex costs $1$ if it comes from a different vertex. A way to visualize this auxiliary problem is to imagine the tree $T_u$ where no pebbles have been placed and a pool of $j$ pebbles to be distributed on its vertices. Each vertex $v$ of $T_u$ can hold up to $\varphi(v)$ pebbles for free, while each additional pebble placed on $v$ increases the overall cost by $1$. We will denote by ${\ensuremath{\texttt{Opt}}}[u,j]$ the cost of the optimal movement for this auxiliary problem. To solve the original problem we need to find a solution corresponding to ${\ensuremath{\texttt{Opt}}}[r,k]$. We now show how to combine these auxiliary problems. If $u$ is a leaf of $G_r$ then we have: $${\ensuremath{\texttt{Opt}}}[u,j] = \begin{cases} 0 & \mbox{if } j \le \varphi(u); \\ j-\varphi(u) & \mbox{if } j > \varphi(u). \end{cases}$$ Otherwise, if $u$ is not a leaf, we can either place some pebbles on $u$ and the others on the subtrees rooted at its children ($j \ge 1$), or place no pebble at all in the whole subtree rooted at $u$ ($j=0$). We call $z$ the number of pebbles that are to be placed on $u$. If $j=0$ we have ${\ensuremath{\texttt{Opt}}}[u,j] = 0$, otherwise: $${\ensuremath{\texttt{Opt}}}[u,j] = \min_{1 \le z \le j}\left\{ \max \{z-\varphi(u), 0\} + \min_{\substack{0 \le j_1, \dots, j_{\ell} \le j-z \\ \sum_{i=1}^{\ell} j_i = j-z }} \left\{ \sum_{i=1}^{\ell} {\ensuremath{\texttt{Opt}}}[v_i, j_i] \right\} \right\}$$ where $v_1, \dots, v_\ell$ are the children of $u$ in $G_r$. As before, using the already shown dynamic-programming approach to optimally distributing the pebbles on the subtrees, we can find the values of ${\ensuremath{\texttt{Opt}}}[u,j]$ for a fixed $j$ and all $v \in V$, in $O(n \cdot k)$ time. Therefore the time required to compute all ${\ensuremath{\texttt{Opt}}}$ values for a single root $r$ is $O(n \cdot k^2)$ and the measure of the best solution is found in ${\ensuremath{\texttt{Opt}}}[r, k]$. As for ${\ensuremath{\textsc{Con}}}$-${\ensuremath{\textsc{Sum}}}$, it is not necessary to run the algorithm for all roots $r \in V(G)$ but we can choose a centroid $v$ of $G$ as starting root and then proceed recursively on the trees of $G-v$. To summarize, we have the following: \[thm:con\_num\_trees\] ${\ensuremath{\textsc{Con}}}$-${\ensuremath{\textsc{Num}}}$ on trees can be solved in $O(n \cdot k^2 \log n)$ time. Independency motion problems {#sec:ind} ============================ In this section we focus on independency motion problems. First, we give a better characterization of the hardness of ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Max}}}$ and ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Sum}}}$ (depending on the input graph), and then we show some positive results for our considered variants on paths and trees. Since if $k \geq n$ there is no feasible solution, we will consider only instances where $k < n$. Hardness of ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Max}}}$ and ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Sum}}}$ on bipartite graphs ----------------------------------------------------------------------------------------------------------------------------------------------------------- As we already pointed out, for independency problems on general graphs it is [$\textsf{\upshape NP}$]{}-hard even to find any feasible solution since it would require to find an independent set of size at least $k$. Nevertheless, one may wonder whether independency motion problems are tractable on instances on which a maximum independent set can be found in polynomial time. We provide a negative answer to this question, at least for ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Max}}}$ and ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Sum}}}$, by showing the following ![Instance of ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Max}}}$ and of ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Sum}}}$ corresponding to the formula $(\bar{x}_1 \vee \bar{x}_2 \vee x_3) \wedge (x_1 \vee x_2 \vee \bar{x}_3)$. Pebbles are placed on black vertices.[]{data-label="fig:mov_ind_max"}](Fig2.eps) \[thm:ind\_max\_sum\_hard\_bipartite\] ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Max}}}$ and ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Sum}}}$ are [$\textsf{\upshape NP}$]{}-hard on bipartite graphs. We will show a polynomial reduction from $\textsc{3-Sat}$ to the decisional versions of ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Max}}}$ and ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Sum}}}$. Recall that $\textsc{3-Sat}$ is the problem of deciding whether a formula $f$ in conjunctive normal form with three literals per clause is satisfiable. Let $X = \{ x_1, \dots, x_\tau \}$ be the set containing the variables of $f$, and let $m$ be the number of clauses. We will denote the $i$-th literal of the $j$-th clause with $\ell^i_j$. Given an instance $f$ for $\textsc{3-Sat}$ we construct an instance for ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Max}}}$ and for ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Sum}}}$ in the following manner: - For each variable $x_i \in X$ create a star with $3$ leaves labelled $u_i, x_i, \bar{x}_i$ and label the internal node $v_i$. Place one pebble on $u_i$ and one on $v_i$. - For each clause $(\ell^1_j \vee \ell^2_j \vee \ell^3_j)$ create a star with $4$ leaves labelled $\ell^1_j, \ell^2_j, \ell^3_j, w_j$ and label the internal node $z_j$. Place one pebble on $w_j$ and one on $z_j$. - For each literal $\ell^i_j$ of $f$ let $x_s$ be the corresponding variable; then, if $\ell^i_j$ is asserted add an edge between the two nodes labelled $\ell^i_j$ and $\bar{x}_s$, otherwise add an edge between the two nodes labelled $\ell^i_j$ and $x_s$. Let $G$ be the resulting graph, $P$ the set containing the $2 \cdot (\tau+m)$ placed pebbles, and $\sigma$ the function that maps each pebble to its starting position (see Figure \[fig:mov\_ind\_max\]). Notice that $G$ is bipartite as we can partition the vertices into two sets $A=\{v_i : 1 \le i \le \tau\} \cup \{ \ell^i_j : 1 \le i \le 3 \wedge 1 \le j \le m \} \cup \{ w_j : 1 \le j \le m \}$ and $B=V(G) \setminus A$ such that no edge of $G$ has both its endpoints in the same set. We claim that there exists an assignment that satisfies $f$ if and only if the optimal solution $\mu$ for the instance of ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Max}}}$ (resp., ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Sum}}}$) has measure at most $1$ (resp., $\tau + m$). Suppose the existence of an assignment $\theta^* : X \to \{ {\ensuremath{\texttt{True}}}, {\ensuremath{\texttt{False}}}\}$ that satisfies $f$. Then we move a pebble starting on vertex $v_i$ to the vertex labelled $x_i$ if $\theta^(x_i)={\ensuremath{\texttt{True}}}$, or to the vertex labelled $\bar{x}_i$ if $\theta^(x_i)={\ensuremath{\texttt{False}}}$. Moreover, for each clause $(\ell^1_j \vee \ell^2_j \vee \ell^3_j)$ of $f$ at least one literal $\ell^i_j$ must be true w.r.t. $\theta^$. This implies that the vertex labelled $\ell^i_j$ is adjacent only to $z_j$ and to a vertex $x_s$ where no pebble has been placed. We then move the pebble initially placed on vertex $z_j$ to the vertex $\ell^i_j$. The resulting configuration of pebbles is an independent set for $G$ and each pebble has been moved to a node adjacent to its starting position. This implies that the maximum movement is $1$ and that the overall distance travelled by pebbles is $\tau + m$. Conversely, suppose the existence of an optimal solution $\mu^$ for the instance of ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Max}}}$ (resp., ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Sum}}}$) that has measure equal to $1$ (resp., $\tau+m$). Notice that for every pebble $p$ initially placed on a vertex labelled $v_i$, $\mu^(p) \in \{ x_i, \bar{x}_i \}$ must hold, otherwise either $\mu^$ would be unfeasible or $c(\mu^)$ would be greater than $1$ (resp., $\tau+m$, since $\tau+m$ pebbles need to be moved). Similarly for every pebble $p$ initially placed on a vertex labelled $z_j$, $\mu^(p) \in \{ \ell^j_1, \ell^j_2, \ell^j_3 \}$ must hold. We construct an assignment $\theta$ for the variables in the following manner: if there is a pebble on the vertex labelled $x_i$ we set $\theta(x_i)= \mathrm{{\ensuremath{\texttt{True}}}}$, otherwise there must be a pebble on $\bar{x}_i$ and we set $\theta(x_i) = {\ensuremath{\texttt{False}}}$. We now show that $\theta$ is, indeed, an assignment that satisfies $f$. For each clause $(\ell^1_j \vee \ell^2_j \vee \ell^3_j)$ the pebble placed on $z_j$ has been moved to a vertex $\ell^i_j$ corresponding to one of the three literals. This implies that no pebble has been moved to the unique vertex in $\cup_{i=1}^t\{x_i, \bar{x}_i\}$ that is adjacent to vertex $\ell^i_j$. The above implies that the variable corresponding to literal $\ell^j_i$ has been set to the value that satisfies $\ell^j_i$ and thus the whole clause is satisfied. Apparently, the above technique cannot be straightforwardly adapted to ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Num}}}$, and so we leave this as an interesting open problem. Approximability of ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Max}}}$ ------------------------------------------------------------------------------ Actually, as shown in Theorem \[thm:ind\_max\_sum\_hard\_bipartite\], ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Max}}}$ is hard already when the cost of an optimal solution is 1. This immediately implies the following: \[cor:ind\_max\_2-inapx\] ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Max}}}$ on bipartite graphs is not approximable in polynomial time within a factor of $2-\epsilon$ for any positive $\epsilon$, unless ${\ensuremath{\textsf{\upshape P}}}={\ensuremath{\textsf{\upshape NP}}}$. We now show that this bound is tight, by providing a polynomial-time solution, which is optimal unless an additive term of $1$, to ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Max}}}$ on any class of graphs where a maximum independent set can be found in polynomial time, e.g., perfect graphs (which include bipartite graphs), interval graphs, and claw-free graphs. Given a graph $H$ and a subset of vertices $A \subseteq V(H)$ we will denote the open neighbourhood of $A$ by $N_H(A)=\left\{ v \in V(H) : \exists u \in A \mbox{ s.t. } (u,v) \in E(H) \right\}$. Moreover we will denote the closed neighbourhood of $A$ by $N_H[A]=A \cup N_H(A)$. Let $U^$ be a maximum independent set of $G$, the following lemma holds: \[lemma:independent\_set\_neighborhood\] For each independent set $U$ of $G$ it is true that $\|U^ \cap N_G[U]\| \ge \|U\|$. By contradiction, let $\|U^* \cap N_G[U]\| < \|U\|$ then $U^\prime = \left( U^* \setminus N_G[U] \right) \cup U = \left( U^* \setminus \left( U^* \cap N_G[U] \right) \right) \cup U$ is an independent set of $G$ and $\|U^\prime\| > \|U^\|$. To prove the next lemma we use the following well known result: \[thm:hall\_s\_matching\] Let $H=(V_1 \cup V_2, E)$ be a bipartite graph. There exists a matching of size $\|V_1\|$ on $H$ iff $\|A\| \le \|N_H(A)\|, \linebreak \forall A \subseteq V_1$. \[lemma:injective\_function\] For each independent set $U$ of $G$, there exists an injective function $f : U \to U^$ such that $d_G(u,f(u)) \le 1$. Construct the bipartite graph $H=(U \cup U^, E)$ where all vertices of $U$ are considered to be distinct from the ones in $U^$ and $E=\{(u,v) \in U \times U^* : v \in N_G[\{u\}] \}$. Notice that, by construction, if two vertices $u,v$ are adjacent in $H$ either they are the same vertex or they are adjacent in $G$, i.e., $d_G(u,v)\le 1$. Now, Lemma \[lemma:independent\_set\_neighborhood\] implies that, for every $A \subseteq U$, we have $\|N(A)\| = \|U^* \cap N_G[A]\| \ge \|A\|$. Hence, from Hall’s Matching Theorem, there is a matching of size $\|U\|$ on $H$ (and thus the function $f$ exists). We are now ready to prove: \[thm:mov\_ind\_max\_opt\_plus\_one\] There exists a polynomial-time algorithm for ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Max}}}$ which, for every class of graphs where the maximum independent set can be found in polynomial time, computes a solution $\tilde{\mu}$ such that $c(\tilde{\mu}) \le c^* + 1$ where $c^$ is the measure of an optimal solution. The algorithm computes a maximum independent set $U^$ of $G$ then, if $k > \|U^\|$ it reports infeasibility, otherwise it optimally moves the pebbles towards (a subset of) the vertices of $U^$. To do that, it proceeds as follows: for every value of $z$ from $0$ to $n-1$, it computes a solution $S_z$ for the maximum matching problem on the auxiliary bipartite graph $H=(A \cup U^,E)$, where each vertex in $A$ is associated with a pebble, and $(p \in A,v \in U^) \in E$ if and only if $d_G(\sigma(p), v) \le z$. Let $\tilde{z}$ be the first value of $z$ such that $\|S_z\|=k$, i.e., all the pebbles have been matched. For every pebble $p \in P$ set $\tilde{\mu}(p)=v$, where $v$ is the only vertex such that $(p,v)\in S_{\tilde{z}}$, and return $\tilde{\mu}$. Clearly $c(\tilde{\mu})=\tilde{z}$. Let $\mu^$ be an optimal solution to ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Max}}}$ and let $U=\mu^[P]$. By Lemma \[lemma:injective\_function\] there exists an injective function $f$ that maps every vertex of the independent set $U$ to an adjacent vertex of $U^$. Thus, for every $p \in P$ we have $d_G(\sigma(p), \mu^(p)) + d_G(\mu^(p), f(\mu^(p))) \le c^* + 1$, and therefore there exists a way to place all the pebbles on vertices of $U^$ while travelling a maximum distance of at most $c^+1$. This implies $\tilde{z} \le c^+1$. Independency motion problems on trees and paths ----------------------------------------------- Concerning independency motion problems on trees, we are able to devise two dynamic-programming algorithms for ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Sum}}}$ and ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Num}}}$, respectively. We note that it remains open to establish whether ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Max}}}$ on trees can be solved in polynomial time. We will, however, devise an efficient algorithm for solving ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Max}}}$ on paths. The details of these algorithm are presented in the following subsections. ### Solving ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Sum}}}$ on trees Our dynamic-programming algorithm relies on the property of optimal solutions shown by Lemma \[lemma:sum\_tree\_no\_crossing\] that is valid also for ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Sum}}}$. The algorithm first roots the tree $G$ at an arbitrary vertex $r \in V(G)$ to obtain the rooted tree $G_r$, then considers all the subtrees of $G_r$ in a bottom-up fashion. Here, the auxiliary problem we consider when a subtree $T_u$ of $G$ rooted at vertex $u$ is examined is that of placing exactly $j \le k$ pebbles on the vertices of $T_u$ such that their final positions induce an independent set of $G$ (and each pebble is placed on different vertex). Remind that $\eta(u)$ denotes the number of pebbles placed on $V(T_u)$. As for the connectivity problems, if $j < \eta(u)$ we want to move the $\eta(u)-j$ exceeding pebbles to the parent of $u$ (and thus outside $T_u$). In a similar manner, if $j > \eta(u)$, then the $j-\eta(u)$ missing pebbles are to be moved into $T_u$ from the parent of $u$, where we assume they are initially placed. We will denote by ${\ensuremath{\texttt{Opt}}}[u,j]$ the cost of the optimal movement for this auxiliary problem. Notice that, in ${\ensuremath{\texttt{Opt}}}[u,j]$, we are accounting for the cost of traversing all the edges of $T_u$ plus the edge from $u$ to its parent. To solve the original problem we need to find a solution corresponding to ${\ensuremath{\texttt{Opt}}}[r,k]$. Clearly as $k=\eta(r)$ we do not have exceeding or missing pebbles, in this case. We now show how to combine these auxiliary problems. Let ${\ensuremath{\texttt{Opt}}}^+[u,j]$ (resp., ${\ensuremath{\texttt{Opt}}}^-[u,j]$) with $u \in V(G)$ and $0 \le j \le n$ be the value of an optimal solution to the auxiliary problem where exactly one pebble must be placed on $u$ (resp., no pebble can be placed on $u$). Notice that ${\ensuremath{\texttt{Opt}}}[u,j] = \min \{ {\ensuremath{\texttt{Opt}}}^+[u,j], {\ensuremath{\texttt{Opt}}}^-[u,j] \}$. We will say that infeasible solutions have cost $+\infty$. If $u$ is a leaf we clearly have: $${\ensuremath{\texttt{Opt}}}^+[u,j] = \begin{cases} \|\eta(u) - 1\| & \mbox{if } j = 1; \\ +\infty & \mbox{if } j=0 \mbox{ or } j \ge 2. \\ \end{cases} \quad \enskip {\ensuremath{\texttt{Opt}}}^-[u,j] = \begin{cases} \eta(u) & \mbox{if } j = 0; \\ +\infty & \mbox{if } j \ge 1. \end{cases}$$ If $u$ is not a leaf then we can either place a pebble on $u$ or not. If we do place it, the corresponding optimal value is: $${\ensuremath{\texttt{Opt}}}^+[u,j] = \begin{cases} +\infty & \mbox{if } j=0; \\ \|\eta(u) - j\| + {\displaystyle \min_{\substack{0 \le j_1, \dots, j_{\ell} < j \\ \sum_{i=1}^{\ell} j_i = j-1}}} \left\{ \sum_{i=1}^{\ell} {\ensuremath{\texttt{Opt}}}^-[v_i, j_i] \right\} & \mbox{if } j \ge 1, \end{cases}$$ where $v_1, \dots, v_\ell$ is the set of children of $u$ in $G_r$. If $j$ is at least $1$ we place a pebble on $u$ and then consider all the possible ways of placing a total of $j-1$ pebbles on the subtrees rooted at the children of $u$, but not on the children themselves. If we do not place a pebble on $u$, then the optimal value is: $${\ensuremath{\texttt{Opt}}}^-[u,j] = \|\eta(u) - j\| + {\displaystyle \min_{\substack{0 \le j_1, \dots, j_{\ell} \le j \\ \sum_{i=1}^{\ell} j_i = j }}} \left\{ \sum_{i=1}^{\ell} {\ensuremath{\texttt{Opt}}}[v_i, j_i] \right\}.$$ This corresponds to the optimal way of placing $j$ pebbles on the subtrees rooted at the children of $u$. As we have previously shown, the optimal assignment to $j_1, \dots, j_{\ell}$ can be found in $O(\ell \cdot k)$ time by using dynamic programming. The value of the optimal solution is stored in ${\ensuremath{\texttt{Opt}}}[r, k]$ and the solution itself can be reconstructed by proceeding backwards as shown in Section \[sec:con\_sum\_trees\] for ${\ensuremath{\textsc{Con}}}$-${\ensuremath{\textsc{Sum}}}$. As we can compute the values ${\ensuremath{\texttt{Opt}}}[u,j]$, for a fixed $j$ and all $v \in V$, in $O(n \cdot k)$ time (the $\ell$-values sum up to $n-1$), the total time required by the algorithm is $O(n \cdot k^2)$. The above discussion, immediately leads to the following: \[thm:ind\_sum\_trees\] ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Sum}}}$ on trees can be solved in ${\ensuremath{O}}(n \cdot k^2)$ time. ### Solving ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Num}}}$ on trees The algorithm is similar to the one for ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Sum}}}$: we root the tree $G$ at an arbitrary vertex $r \in V(G)$ to obtain the rooted tree $G_r$, then we consider all the subtrees of $G_r$ in a bottom-up fashion. Let $\gamma(u)$ be $1$ if at least one pebble is initially placed on the vertex $u \in V(G)$, and $0$ otherwise. As before, when the subtree $T_u$, rooted at the vertex $u$, of $G_r$ is examined, we consider an auxiliary problem where we want to place exactly $j \le k$ pebbles on the vertices of $T_u$ such that their final positions induce an independent set of $G$ (and each pebble is placed on a different vertex). We will measure the cost of a solution for this auxiliary problem by examining the number of pebbles placed on each vertex of $T_u$. Removing pebbles from a vertex costs nothing, while placing a pebble on a vertex costs $1$ if it comes from a different vertex. A way to visualize this auxiliary problem is to imagine the tree $T_u$ where no pebbles have been placed, and a pool of $j$ pebbles to be distributed on its vertices. If a vertex $v$ is such that $\gamma(v)=1$ then it can hold a single pebble for free, otherwise placing a pebble on $v$ will increase the overall cost by $1$. We will denote by ${\ensuremath{\texttt{Opt}}}[u,j]$ the cost of the optimal movement for this auxiliary problem. To solve the original problem we need to find a solution corresponding to ${\ensuremath{\texttt{Opt}}}[r,k]$. We now show how to combine these auxiliary problems. Let ${\ensuremath{\texttt{Opt}}}^+[u,j]$ (resp., ${\ensuremath{\texttt{Opt}}}^-[u,j]$) with $u \in V(G)$ and $0 \le j \le n$ be the value of an optimal solution to the auxiliary problem where exactly one pebble must be placed on $u$ (resp., no pebbles can be placed on $u$). Notice that ${\ensuremath{\texttt{Opt}}}[u,j] = \min \{ {\ensuremath{\texttt{Opt}}}^+[u,j], {\ensuremath{\texttt{Opt}}}^-[u,j] \}$. We will say that infeasible solutions have cost $+\infty$. If $u$ is a leaf of $G_r$ then we have: $${\ensuremath{\texttt{Opt}}}^+[u,j] = \begin{cases} +\infty & \mbox{if } j=0 \mbox{ or } j \ge 2; \\ 1-\gamma(u) & \mbox{if } j=1. \end{cases} \quad \quad {\ensuremath{\texttt{Opt}}}^-[u,j] = \begin{cases} 0 & \mbox{if } j=0; \\ +\infty & \mbox{if } j \ge 1. \end{cases}$$ If $u$ is not a leaf then, with a reasoning similar to that we did for ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Sum}}}$ we have: $${\ensuremath{\texttt{Opt}}}^+[u,j] = \begin{cases} +\infty & \mbox{if } j=0; \\ 1-\gamma(u) + {\displaystyle \min_{\substack{0 \le j_1, \dots, j_{\ell} < j \\ \sum_{i=1}^{\ell} j_i = j-1}}} \left\{ \sum_{i=1}^{\ell} {\ensuremath{\texttt{Opt}}}^-[v_i, j_i] \right\} & \mbox{if } j \ge 1, \end{cases}$$ and $${\ensuremath{\texttt{Opt}}}^-[u,j] = \min_{ \substack{0 \le j_1, \dots, j_{\ell} \le j \\ \sum_{i=1}^{\ell} j_i = j} } \left\{ \sum_{i=1}^{\ell} {\ensuremath{\texttt{Opt}}}[v_i, j_i] \right\},$$ where $v_1, \dots, v_{\ell}$ are the children of $u$ in $G_r$. Again, the optimal assignment to $j_1, \dots, j_\ell$ can be found in $O(\ell \cdot k)$ time by using dynamic programming. The value of the optimal solution is stored in ${\ensuremath{\texttt{Opt}}}[r, k]$ and the solution itself can be reconstructed by proceeding backwards. As we can compute the values ${\ensuremath{\texttt{Opt}}}[u,j]$, for a fixed $j$ and all $v \in V$, in $O(n \cdot k)$ time (the $\ell$-values sum up to $n-1$), the total time required by the algorithm is $O(n \cdot k^2)$. The above discussion immediately leads to the following: \[thm:ind\_num\_trees\] ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Num}}}$ on trees can be solved in ${\ensuremath{O}}(n \cdot k^2)$ time. ### Solving ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Max}}}$ on paths In this section we concentrate on ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Max}}}$, and we devise an efficient algorithm for solving ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Max}}}$ on the special case where $G$ is a path. We start by proving the following: \[lemma:swap\] Let $s_1, s_2, e_1, e_2 \in \mathbb{N}$ where $s_1 \le s_2$ and $e_2 \le e_1$, we have $\max\left\{ \| s_1 - e_1\|, \|s_2 - e_2\| \right\} \ge \max\left\{ \| s_1 - e_2\|, \|s_2 - e_1\| \right\}$. Suppose that $\| s_1 - e_2\| \ge \|s_2 - e_1\|$. If $s_1 \ge e_2$ then $\|s_2 - e_2\| \ge \|s_1 - e_2\|$, else if $s_1 < e_2$ then $\|s_1 - e_1\| = e_1 - s_1 \ge e_2 -s_1 = \|s_1 - e_2\|$. In a similar manner suppose $\|s_1 - e_2\| \le \|s_2 - e_1\|$. If $s_2 \ge e_1$ then $\|s_2 - e_2\| \ge \|s_2 - e_1\|$, else if $s_2 < e_1$ then $\|s_1 - e_1\| = e_1 - s_1 \ge e_1 - s_2 = \|s_2 - e_1\|$. Let $r$ be any endpoint of $G$, with a little abuse of notation we will identify the vertices of $G$ by their distance from $r$: the vertex $i$ with $0 \le i < n$ will be the unique vertex $u$ such that $d_G(r, u)=i$. Let $p_1, \dots, p_k$ be an ordering of the pebbles such that $\sigma(p_i) \le \sigma(p_{i+1})$ for $i=1,\dots,k-1$. The following lemma shows that there exists an optimal solution where the ordering of pebbles is preserved, i.e., no edge of $G$ is traversed by two pebbles going in opposite directions.[^4] \[lemma:no\_crossing\] If there exists a feasible solution $\mu$ for ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Max}}}$ on paths then there also exists a solution $\mu^\prime$ such that $\mu^\prime(p_i) < \mu^\prime(p_{i+1})$ for every $i=1,\dots,k-1$ and $c(\mu^\prime) \le c(\mu)$. Let $\mu$ be a feasible solution. We will show that whenever there are two pebbles $p_i$ and $p_{i+1}$ such that $\mu(p_i) > \mu(p_{i+1})$, they can be swapped without increasing the cost of $\mu$ to obtain another feasible solution.[^5] If needed, the procedure can be repeated until all the pebbles are placed in the right order. In order to prove the above, consider a new solution $\mu^\prime$ where $\mu^\prime(p_i)=\mu(p_{i+1})$, $\mu^\prime(p_{i+1})=\mu(p_i)$, and $\mu^\prime(p_j) = \mu(p_j)$ for every $j \not\in \{i,i+1\}$. Clearly $\mu^\prime$ is feasible (as we only swapped two pebbles) and requires at most the same maximum movement as $\mu$, because: $$\begin{gathered} \max\left\{ d_G(\sigma(p_i), \mu(p_i)), d_G(\sigma(p_{i+1}), \mu(p_{i+1})) \right\} \ge \\ \max\left\{ d_G(\sigma(p_i), \mu(p_{i+1})), d_G(\sigma(p_{i+1}), \mu(p_{i})) \right\}, \end{gathered}$$ where every distance $d_G(u,v)$ between two vertices $u$ and $v$ can be rewritten as $\|u - v\|$ and Lemma \[lemma:swap\] holds. The algorithm takes as input an instance of ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Max}}}$ plus a non-negative integer $z$ and exploits the previous property to compute a solution of cost at most $z$, if it exists. The idea is simple: the pebbles are moved towards the endpoint $r$ of the path as most as possible while preserving the feasibility of the solution and the constraint on the cost. The pseudocode is given below. $p_1, \dots, p_k \gets$ pebbles sorted by distance from $r$ $next \gets 0$ $\mu$ \[lemma:alg1\_correctness\] Algorithm \[alg:IndMax\] correctly returns a solution with maximum movement at most $z$, if such a solution exists, or reports infeasibility if there is no such solution. First notice that all solutions returned by the algorithm are feasible as no two pebbles can be placed on adjacent vertices and, by construction, no pebble travels a distance greater than $z$. This implies that if there is not a feasible solution, the algorithm correctly reports infeasibility. Now we will prove that the algorithm correctly computes a solution when this exists. Let $\mu^$ be a solution with maximum movement at most $z$ that preserves the ordering of the pebbles (by Lemma \[lemma:no\_crossing\] such a solution always exists). We will prove by induction that at the end of the $j$-th loop the first $j$ pebbles are correctly placed (they are on an independent set), and that for the $j$-th pebble $\mu(p_j) \le \mu^(p_j)$ holds. Base case: the first pebble is placed on the first reachable node from $r$, thus $\mu(p_1) \le \mu^(p_1)$ must hold. Inductive step: suppose that the property is true for the first $j \ge 1$ pebbles, we will prove that it is also true also for the $(j+1)$-th pebble. If the pebble $p_{j+1}$ is placed on vertex $\sigma(p_{j+1})-z$ then the property holds, since this is the vertex with the smallest distance from $r$ that is reachable by $p_{j+1}$. Otherwise $\mu(p_{j+1}) = \mu(p_j)+2$, and we have $\mu^(p_{j+1}) \ge \mu^(p_j) + 2 \ge \mu(p_j) + 2=\mu(p_{j+1})$. We are ready to give the following: \[thm:ind\_max\_paths\] ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Max}}}$ on paths can be solved in ${\ensuremath{O}}(n + k \log n)$ time. From Lemma \[lemma:alg1\_correctness\] it follows that ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Max}}}$ on paths can be solved as follows: if $k > n$ then there is no solution, otherwise we can perform a binary search using Algorithm \[alg:IndMax\] to find the optimal value of $k$ (or report infeasibility). As far as the time complexity is concerned, each invocation of the algorithm requires time $O(k)$ and at most $O(\log n)$ invocations are required. For Algorithm \[alg:IndMax\] to work, vertices and pebbles must be sorted w.r.t. their distance from $r$, and this needs to be done only once and requires $O(n + k)$ time. The whole procedure then takes $O(n + k \log n)$ time. Clique motion problems {#sec:clique} ====================== In this section we prove that the problems ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Max}}}$, ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Sum}}}$, and ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Num}}}$ are [$\textsf{\upshape NP}$]{}-hard. Then, we give a tight approximation algorithm for ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Max}}}$ that computes a solution that costs at most one more than the optimal solution. As of ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Sum}}}$ and ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Num}}}$, we show that the problems are not approximable within any factor smaller than $10\sqrt{5}-21$, unless ${\ensuremath{\textsf{\upshape P}}}={\ensuremath{\textsf{\upshape NP}}}$, and we devise two $2$-approximation algorithms. Actually, we will show that any approximation for ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Sum}}}$ or ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Num}}}$ implies an approximation with the same ratio for the minimum vertex cover problem, which is known to be not approximable under $10\sqrt{5}-21$, unless ${\ensuremath{\textsf{\upshape P}}}={\ensuremath{\textsf{\upshape NP}}}$ [@dinur2005hardness]. Moreover, if the unique game conjecture [@khot2002power] is true, then both approximation algorithms are tight as the corresponding problems are not approximable within any constant factor better than $2$ [@khot2008vertex]. Finally, we show that for classes of graphs where we can find a maximum-weight clique in polynomial time, we can also solve ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Num}}}$ in polynomial time. Approximability of ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Max}}}$ --------------------------------------------------------------------------------- We prove the following: ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Max}}}$ is [$\textsf{\upshape NP}$]{}-hard. We show a reduction from the problem of determining if there exists a dominating clique in a graph $H$, which is known to be [$\textsf{\upshape NP}$]{}-complete [@haynes1998domination]. A dominating clique is a subset of vertices $C \subseteq V(H)$ such that $C$ is both a clique and a dominating set for $H$. We construct an instance of ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Max}}}$ by setting $G=H$ and placing a pebble on each vertex of $G$. We claim that there exists a dominating clique in $H$ if and only if the optimal solution for ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Max}}}$ has measure $c^$ at most $1$. Suppose that there exists a dominating clique $C$ in $H$. By definition, $C$ is also a dominating set of $G$. We define $\mu$ so that every pebble initially placed on a vertex $u \not\in C$ is moved to a vertex $v \in C$ such that $(u,v) \in E(G)$ (notice that such a vertex always exists). After their movement the pebbles are placed on a clique of $G$ and each pebble has travelled a distance of at most $1$. Now suppose that there exists a solution $\mu$ for ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Max}}}$ such that $c(\mu) \le 1$. Clearly $\mu[P]$ is a clique, and it is also a dominating set, since for each vertex $u \not\in \mu[P]$, there exists a vertex $v \in \mu[P]$ such that $(u,v) \in G$. On the positive side, we have: It is possible to compute a solution $\tilde{\mu}$ for ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Max}}}$ such that $c(\tilde{\mu}) \le c^ + 1$ in polynomial time, where $c^$ is the measure of an optimal solution. Consider the following algorithm: for each vertex $u \in V(G)$ construct a solution $\mu_u$ that moves all the pebbles to $u$ (i.e., set $\mu_u(p)=u, \; \forall p \in P$) and compute $c(\mu)$. Among the $n$ possible solutions choose the one of minimum cost and call it $\tilde{\mu}$. Now, let $\mu^$ be an optimal solution and recall that, by definition, $c(\mu^) = \max_{p \in P} \left\{ d_G(\sigma(p), \mu^(p)) \right\}$. Let $C=\mu^[P]$ be the clique where pebbles have been placed by $\mu^$, and notice that when the above algorithm considers a node $u \in C$ we have: $ d_G(\sigma(p), \mu_u(p)) \le d_G(\sigma(p), \mu^(p)) + 1 \; \forall p \in P $, thus $c(\tilde{\mu}) \le c(\mu_u) \le c(\mu^)+1$. The claim follows. Approximability of ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Num}}}$ --------------------------------------------------------------------------------- We prove the following: \[thm:2-apx-mov-clique-num\] ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Num}}}$ is $2$-approximable and is not approximable within any constant factor smaller than $10\sqrt{5}-21$, unless ${\ensuremath{\textsf{\upshape P}}}={\ensuremath{\textsf{\upshape NP}}}$. Let $\langle G, P, \sigma \rangle$ be an instance of ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Num}}}$ and let $\varphi(u) = \|\{p \in P : \sigma(p)=u \}\|$ be the number of pebbles that are initially placed on vertex $u \in V$. Let us assume that $\varphi(u) \le 1$ for every $u \in V$, i.e., no two pebbles are placed on the same vertex. We will show later that this assumption is not restrictive. Call $H$ the graph induced by the vertices $u$ with $\varphi(u)=1$. Let $\bar{H}$ be the complement graph of $H$ w.r.t. the edge set, that is the graph such that $V(\bar{H})=V(H)$ and $E(\bar{H}) = \{ (u,v) : u,v \in V(H) \wedge u \not= v \wedge (u,v) \not\in E(H) \}$. We will show that there exists a vertex cover $\mathcal{C}$ for $\bar{H}$ if and only if there exists a solution for the instance $\langle G, P, \sigma \rangle$ of ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Num}}}$ of cost $\|\mathcal{C}\|$. Let $\mathcal{C}$ be a vertex cover for $\bar{H}$, this implies that $Q=V(H)-\mathcal{C}$ is an independent set for $\bar{H}$, and therefore a clique for $H$ and $G$. We construct a solution $\mu$ for ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Num}}}$ by moving the $\|\mathcal{C}\|$ pebbles that are not yet placed on vertices in $Q$ to one of such vertices. Now let $\mu$ be a solution for the instance $\langle G, P, \sigma \rangle$ of ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Num}}}$ and let $Q=\{u \in V(H) : \exists p \in P \mbox{ s.t. } \sigma(p)=\mu(p)=u\}$. Notice that the cost of $\mu$ is exactly $k-\|Q\|=\|V(H)-Q\|$. Clearly $Q \subseteq \mu[P]$ is a clique for $G$ and $H$, therefore it is also an independent set for $\bar{H}$. This implies that $\mathcal{C}=V(H)-Q$ is a vertex cover for $\bar{H}$. From the above it follows that the cost of an optimal solution is equal to the size of the minimum vertex cover for $\bar{H}$. To approximate ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Num}}}$ we construct the graph $\bar{H}$, we compute a $2$-approximate minimum vertex cover $\mathcal{\tilde{C}}$, and then we reconstruct the solution $\mu$ as shown. If the previous assumption is not met, i.e., there exists at least a vertex on which two or more pebbles are placed by $\sigma$, a slight modification to the instance is needed before we can apply the previous approach. We modify the graph $G$ by replacing each vertex $u$ such that $\varphi(u)>1$ with a clique of size $\varphi(u)$. Each edge $e$ incident to $u$ is replaced by $\varphi(u)$ edges connecting every vertex of the clique to the other endpoint of $e$. Then, we modify the function $\sigma$ so that the $\varphi(u)$ pebbles that were placed on $u$ are assigned to each of the $\varphi(u)$ vertices of the corresponding clique. After the modifications, the cost of an optimal solution has not changed, moreover every solution for the modified instance can be easily reconverted to a solution for the original instance without increasing its cost. Hence, we have designed a $2$-approximation algorithm for ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Num}}}$. Concerning the inapproximability result, we prove it by contradiction: we show that if there exists an algorithm that approximates ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Num}}}$ with a ratio better than $10\sqrt{5}-21$, then this would allow to approximate minimum vertex cover with the same approximation ratio. Indeed, let $\bar{H}$ be the instance of minimum vertex cover. Now, add an isolated vertex $u$ and call $G$ the complement of such a graph w.r.t. the edge set. Then, let $P=[\|V(G)\|-1]$, and let $\sigma$ be a function that places a single pebble on each vertex of $G$ except $u$. We can now compute an approximate solution for the instance $\langle G, P, \sigma \rangle$ of ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Num}}}$ and reconstruct a solution with the same cost for minimum vertex cover, as shown before.[^6] We close this section by proving that ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Num}}}$ can be solved in polynomial time whenever we are able to solve the maximum-weight clique problem on a weighted variant of $G$. We remark that this is known to be the case of several classes of graphs, including perfect graphs. We refer the interested reader to [@bomze1999maximum] for a survey. An exact solution to ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Num}}}$ can be found in polynomial time on every class of graphs where a maximum-weight clique can be found in polynomial time. For every $v \in V$ let $\varphi(v)$ be the number of pebbles starting on vertex $v$, i.e., $\varphi(v)=\|\{p \in P : \sigma(p)=v\}\|$. We consider a graph $H=(V,E)$ that is a copy of $G$ where each vertex $v \in V$ has a weight $\varphi(v)$. Then, we compute a maximum-weight clique $Q$ of $H$ and construct a solution $\mu$ for ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Num}}}$ by moving every pebble starting on a vertex in $V \setminus Q$ to an arbitrary vertex of $Q$.[^7] It is now easy to see that this is an optimal solution, since for every other solution $\mu^\prime$, we have: $$c(\mu^\prime) = \sum_{v \not\in \mu^\prime[P]} \varphi(v) = k-\sum_{v \in \mu^\prime[P]} \varphi(v) \ge k-\sum_{v \in Q} \varphi(v) = \sum_{v \not\in \mu[P]} \varphi(v) = c(\mu).$$ Approximability of ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Sum}}}$ --------------------------------------------------------------------------------- We prove the following: ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Sum}}}$ is $2$-approximable and is not approximable within any constant factor better than $10\sqrt{5}-21$, unless ${\ensuremath{\textsf{\upshape P}}}={\ensuremath{\textsf{\upshape NP}}}$. Let $\mu^$ be an optimal solution to ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Sum}}}$. If $\mu^$ moves all the pebbles (i.e., $\sigma(p) \not= \mu^(p), \; \forall p \in P$) then we can compute a $2$-approximate solution $\tilde{\mu}$ by guessing a vertex $u \in \mu^[P]$ and moving all the pebbles to $u$ (i.e., setting $\tilde{\mu}(p)=u, \; \forall p \in P$). Indeed, we have: $$\frac{c(\tilde{\mu})}{c(\mu^)} = \frac{\sum_{p \in P}d_G(\sigma(p), \tilde{\mu}(p))}{\sum_{p \in P}d_G(\sigma(p), \mu^(p))} \le \frac{\|P\| + \sum_{p \in P}d_G(\sigma(p), \mu^(p))}{\sum_{p \in P}d_G(\sigma(p), \mu^(p))} \le 2$$ where we used the fact that $d_G(\sigma(p), \tilde{\mu}(p)) \le d_G(\sigma(p), \mu^(p))+1$, and that $d_G(\sigma(p), \mu^(p)) \ge 1$, for every pebble $p \in P$. On the other hand, if there exists at least one pebble $p^\prime \in P$ such that $\sigma(p^\prime)=\mu^(p^\prime)$, then we guess its starting vertex $u=\sigma(p^\prime)$. We call $P_0$ the set of pebbles whose starting vertex is $u$, $P_1$ the set of pebbles whose starting vertex is adjacent to $u$, and $P_{2}$ the set of pebbles that are initially placed on a vertex at distance $2$ or more from $u$. We then set $\tilde{\mu}(p)=u$ if $p \in P_0$ or $p \in P_{2}$. With a reasoning similar to the one of the previous case we can show that: $$\sum_{p \in P_{2}} d_G(\sigma(p), \tilde{\mu}(p)) \le \|P_{2}\| + \sum_{p \in P_{2}} d_G(\sigma(p), \mu^(p)) \le 2 \sum_{p \in P_{2}} d_G(\sigma(p), \mu^(p)) \mbox{.}$$ Concerning $P_1$, assume $P_1\not=\emptyset$, and so we need to compute $\tilde{\mu}$ for the pebbles in $P_1$. To do that, consider the instance $\langle H, P_1, \sigma \rangle$ of ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Num}}}$ where $H$ is the subgraph of $G$ induced by the vertices initially occupied by pebbles in $P_1 \cup \{ p^\prime \}$, and compute a $2$-approximate solution $\mu^\prime$ as shown in Theorem \[thm:2-apx-mov-clique-num\]. Set $\tilde{\mu}(p)=\sigma(p)$ for every pebble $p \in P_1$ such that $\mu^\prime(p)=\sigma(p)$, and set $\tilde{\mu}(p)=u$ for the remaining pebbles in $P_1$. Clearly $\tilde{\mu}[P]$ is a clique for $G$ as the vertices in $\mu^\prime[P_1]$ are a clique for $G$, $u$ is adjacent to every vertex in $\mu^\prime[P_1]$, and $\tilde{\mu}[P] \subseteq \mu^\prime[P_1] \cup \{ u \}$. Notice that the cost of moving the pebbles in $P_1$ w.r.t. $\mu^$ is greater than or equal to the cost of the optimal solution for the instance $\langle H, P_1, \sigma \rangle$ of ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Num}}}$. Moreover, the cost of moving the pebbles in $P_1$ w.r.t. $\tilde{\mu}$ is equal to the cost of $\mu^\prime$. From the above it follows that: $$\sum_{p \in P_1} d_G(\sigma(p), \tilde{\mu}(p)) \le 2 \sum_{p \in P_1} d_G(\sigma(p), \mu^(p)) \mbox{.}$$ Therefore, the overall cost of this approximated solution is: $$c(\tilde{\mu}) \le 2 \sum_{p \in P_1} d_G(\sigma(p), \mu^(p)) + 2 \sum_{p \in P_{2}} d_G(\sigma(p), \mu^(p)) \le 2 \,c(\mu^) \mbox{.}$$ Concerning the inapproximability result, take a graph $H=(V, E)$ and construct the graph $G$ by complementing $H$ w.r.t. the edge set and adding an additional vertex $v_0$ adjacent to every other vertex. Let $P=[\|V(H)\|]$ and let $\sigma$ be a function that places one pebble on each vertex of $G$ except $v_0$. We will show that, given any solution for ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Sum}}}$, it is possible to construct a vertex cover of $H$ having the same cost, and vice versa. This implies that any approximation algorithm for ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Sum}}}$ converts into an approximation algorithm for minimum vertex cover with the same approximation ratio, therefore no approximation algorithm with an approximation ratio less than $10\sqrt{5}-21$ can exist for ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Sum}}}$ [@dinur2005hardness], unless ${\ensuremath{\textsf{\upshape P}}}={\ensuremath{\textsf{\upshape NP}}}$. Let $\mathcal{C}$ be a vertex cover for $H$, then $V(H)-\mathcal{C}$ is an independent set for $H$ and a clique for $G$. The solution that moves all the pebble of $\mathcal{C}$ to $v_0$ and leaves the others on their starting position is feasible and has a cost of $\mathcal{C}$. Now let $\mu$ be a solution for the instance of ${\ensuremath{\textsc{Clique}}}$-${\ensuremath{\textsc{Sum}}}$. We modify $\mu$ in such a way that every pebble that moves at least by $1$ is now moving to $v_0$. This modification cannot increase the cost of $\mu$. Then, let $Q= \mu[P] \setminus \{ v_0 \}$, and let $\mathcal{C}=V(H) \setminus Q$. The cost of $\mu$ is $\|V(H)\|-\|Q\|=\|\mathcal{C}\|$, and $Q \subseteq \mu[P]$ is a clique for $G$. Therefore $Q$ is also an independent set for $H$, and $\mathcal{C}$ is a vertex cover for $H$. ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$ motion problems {#sec:stcut} =============================================================== In this section we discuss (in)approximability results for ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Max}}}$ and ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Sum}}}$. Among the others, we provide an essentially tight approximation algorithm for ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Max}}}$. Regarding ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Num}}}$, establishing its tractability remains open, but we will show that approximating such a problem can be useful to approximate ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Sum}}}$, as well. We start by proving the following: ![Instance of ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Max}}}$ corresponding to the formula $(x_1 \vee x_2 \vee x_3) \wedge (\bar{x}_1 \vee \bar{x}_2 \vee x_3)$. Pebbles are placed on black vertices. All the edges not incident to a black vertex represent paths of length $h$ between their endpoints.[]{data-label="fig:cut_max_hard"}](Fig3.eps) \[thm:cut\_max\_sum\_hard\_bipartite\] ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Max}}}$ and ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Sum}}}$ are [$\textsf{\upshape NP}$]{}-hard even if $G$ is a bipartite graph. We will show a reduction from the decision version of $\textsc{3-Sat}$. Let $X=\{x_1, \dots, x_\tau\}$ be the set of variables of the given formula $f$, and let $m$ be the number of clauses. Start with an empty graph $G$ and a set $P$ of $\tau + 2m$ pebbles, then construct an instance for ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Max}}}$ and ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Sum}}}$ as follows (see Figure \[fig:cut\_max\_hard\] for an example): - Add the two vertices $s$ and $t$. - For each variable $x_i \in X$ add a path of three vertices to $G$. Label the middle node $u_i$ and the two endpoints $x_i$ and $\bar{x}_i$, respectively. Place a pebble on $u_i$. - For each clause $(\ell^1_j \vee \ell^2_j \vee \ell^3_j)$ create a path of five vertices labelled, from one endpoint to another, $\ell^j_1, v_j, \ell^j_2, v_j^\prime, \ell^j_3$. Place a pebble on $v_j$ and one on $v_j^\prime$. - For each literal $\ell^i_j$ of $f$, let $x_s$ be the corresponding variable. If $\ell^i_j$ is asserted add a new “long” path of length $h> k = t + 2 m$ between the vertices labelled $\ell^i_j$ and $x_s$ (so that $d_G(\ell^i_j,x_s)=h$). Otherwise add a new “long” path between the vertices labelled $\ell^i_j$ and $\bar{x}_s$. - Connect $s$ to every vertex $x_i$ and to every vertex $\bar{x}_i$ with a new “long” path of length $h$. - Connect $t$ to every vertex $\ell^j_i$ with a new “long” path of length $h$. Notice that $G$ is bipartite as every cycle in $G$ must pass trough an even number of long paths (paths of length $h$) and an even number of other edges. We will show that there exists a truth assignment satisfying $f$ if and only if an optimal solution for ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Max}}}$ (resp., ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Sum}}}$) has cost at most $1$ (resp., $k$). Suppose that there exists a truth assignment $\theta^* : X \to \{ {\ensuremath{\texttt{True}}},{\ensuremath{\texttt{False}}}\}$ that satisfies $f$. Then, for each $x_i \in X$ we move the pebble placed on $u_i$ to $x_i$ if $\theta^(x_i)={\ensuremath{\texttt{True}}}$, and to $\bar{x}_i$ if $\theta^(x_i)={\ensuremath{\texttt{False}}}$. Moreover, for each clause $(\ell^1_j \vee \ell^2_j \vee \ell^3_j)$ there are at most $2$ literals whose truth values are false w.r.t. $\theta^$. We move one or two of the pebbles placed on $v_j$ and $v^\prime_j$ to the nodes of $G$ corresponding to the those literals. Notice that in this way each pebble travels at most a distance of $1$ (so the sum of the distances is at most $k$). To prove that the new positions of the pebbles induce an s-t-cut in $G$, consider any path $\pi(s,t)$ in $G$ between $s$ and $t$. We will show that $\pi(s,t)$ has been “blocked”, i.e, there exists at least one pebble that has been moved on some vertex of $\pi(s,t)$. The path $\pi(s,t)$ must contain (as subpath) at least one path of length $h$ that connects a vertex $x \in \{x_i, \bar{x}_i\}$ for (some $i$) to a vertex $\ell \in \{\ell^1_j, \ell^2_j, \ell^3_j\}$ (for some $j$). If a pebble has been placed on $x$ then $\pi(s,t)$ is blocked. Otherwise the literal represented by $\ell$ must be false and, by construction, a pebble has been placed on $\ell$. Now suppose that an optimal solution for ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Max}}}$ (resp., ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Sum}}}$) has cost at most $1$ (resp., $k$). Notice that every pebble must have travelled a distance of at most $1$. This is trivial for ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Max}}}$, while for ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Sum}}}$ it suffices to note that placing a pebble on an internal vertex of the paths of length $h$ blocks at most the same set of paths that are blocked by placing it on one of the endpoints. Moreover no pebble can traverse a whole path as $h > k$. We define $\theta(x_i)={\ensuremath{\texttt{True}}}$ if a pebble has been placed on $x_i$, and $\theta(x_i)={\ensuremath{\texttt{False}}}$ otherwise. For each clause $(\ell^1_j \vee \ell^2_j \vee \ell^3_j)$ there exists at least one vertex $\ell \in \{\ell^1_j, \ell^2_j, \ell^3_j \}$ such that no pebble has been moved to $\ell$. Let $x_i$ be the variable associated with $\ell_i$, and let $x$ be the vertex labelled $x_i$ if $\ell = x_i$, or the vertex labelled $\bar{x}_i$ if $\ell = \bar{x}_i$. The pebble placed on $u_i$ must have been moved to $x$, otherwise there would exist a path from $s$ to $t$ passing through $x$ and $\ell$, and therefore the clause is satisfied. We now show that ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Max}}}$ and ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Sum}}}$ are actually very hard to approximate: ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Max}}}$ and ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Sum}}}$ are not approximable within a factor of $n^{1-\epsilon}$ for every $\epsilon>0$, unless ${\ensuremath{\textsf{\upshape P}}}={\ensuremath{\textsf{\upshape NP}}}$. This also holds for bipartite graphs. As shown in the proof of Theorem \[thm:cut\_max\_sum\_hard\_bipartite\], it is possible to construct instances of ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Max}}}$ (and ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Sum}}}$) such that the optimal solution has measure at most $1$ (resp., $k=\tau + 2m$) if and only if a boolean formula in conjunctive normal form with three literals per clause, $\tau$ variables, and $m$ clauses is satisfiable. Moreover, any solution with cost $z$ such that $\tau+2m < z < h$ (recall that $h$ is the length of the “long” paths) can be easily transformed into a solution of cost at most $\tau + 2m$ by moving every pebble $p$ that has been placed on one long path to an appropriate endpoint (i.e., the one adjacent to $\sigma(p)$). This implies that if $f$ is satisfiable, the measure $z^$ of an optimal solution is at most $k$ for both the problems, while if $f$ is not satisfiable, $z^$ is at least $h$. Then, given a formula $f$, construct an instance for ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Max}}}$ (and ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Sum}}}$) as in the proof of Theorem \[thm:cut\_max\_sum\_hard\_bipartite\], and suppose that there exists a polynomial-time algorithm that approximates the optimal solution within a factor of $n^{1-\epsilon}$, for some positive $\epsilon \le 1$. Let $\tilde{c}$ be the measure of such an approximate solution. Observe that it is possible to upper bound the number of vertices $n$ of the instance with the quantity $2 + 3\tau + 5m + 2\tau h + 6mh \le 11h(\tau+m) \le 11hk$. If $f$ is satisfiable, we have $\tilde{c} \le k n^{1-\epsilon}$, while if $f$ is not satisfiable we have $\tilde{c} \ge h$. If $k n^{1-\epsilon} < h$ holds then it is possible to decide $\textsc{3-Sat}$ by running the approximation algorithm and looking the measure of the approximate solution. This can be guaranteed by choosing $h > (11k)^{2/\epsilon}$, as we have $h^\epsilon > 11 k^2 \ge 11^{1-\epsilon} k^{2-\epsilon} = k (11 k)^{1-\epsilon}$, which implies $h > k (11 h k)^{1-\epsilon} \ge k n^{1-\epsilon}$. By moving all the pebbles on a minimum $s$-$t$-cut of $G$ we can show that the inapproximability result provided above is essentially tight for ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Max}}}$: ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Max}}}$ is $d$-approximable in polynomial time, where $d < n$ is the diameter of $G$, while ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Sum}}}$ is $(k \cdot d)$-approximable in polynomial time. The approximation algorithm is as follows: if the initial position of the pebbles already makes $s$ and $t$ disconnected, then we are done. Otherwise, compute (in linear time) a minimum $s$-$t$-cut of $G$, say $C$. If $\|C\| > k$ this implies that both problems have no solution, otherwise move the pebbles on the vertices of $C$. The optimal measure must be at least $1$, while each of the $k$ pebbles travels a distance of at most $d$. We close this section proving a theorem which is useful in linking the approximability of ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Num}}}$ to that of ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Sum}}}$: If there exists a $\rho$-approximation algorithm for ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Num}}}$ then there exists a $(\rho\cdot d)$-approximation algorithm for ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Sum}}}$, where $d < n$ is the diameter of $G$. Consider an instance of ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Sum}}}$ an let $c^$ be the measure of an optimal solution. If $c^\prime$ is the measure of the optimal solution on ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Num}}}$, then $c^* \ge c^\prime$. If $c^\prime = 0$ the claim is trivial, therefore we consider $c^\prime > 0$. Then, a $\rho$-approximate solution for ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Num}}}$ is a $\rho \cdot d$-approximate solution for ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Sum}}}$, as each pebble travels a distance of at most $d$, and $\frac{\rho\cdot d \cdot c^\prime}{c^} \le \frac{\rho\cdot d \cdot c^\prime}{c^\prime} = \rho \cdot d$. Conclusions {#sec:concl} =========== In this paper we have been concerned with the emerging field of (centralized) pebble motion problems. In particular, we have provided approximability and inapproximability results –most of which were tight– for several relevant variants studied in the literature. Among the issues we left open, the most prominent are those of establishing whether ${\ensuremath{\textsc{Ind}}}$-${\ensuremath{\textsc{Max}}}$ on trees can be solved in polynomial time, and of settling the computational complexity of ${\ensuremath{s\mbox{-}t\mbox{-}\textsc{Cut}}}$-${\ensuremath{\textsc{Num}}}$. Motion planning of devices in a constrained environment deserves a further deep investigation in several respects. Here we have limited our attention to vertex-to-vertex motion on unweighted, undirected graphs, and with the objective of achieving very basic configurations, but it is easy to imagine more challenging scenarios. For instance, pebbles could be deployed on a 2-dimensional environment, or even on a terrain. In this case, the planning task could become substantially more difficult, because of the setting in the continuum. On the other hand, a simplifying yet very interesting scenario is that in which the given graph is a 2-dimensional grid. At an intermediate stage, an intriguing variant is that in which the underlying graph is weighted. We also plan to look at other goal configurations, e.g., reaching a set of vertices inducing a 2-edge-connected subgraph, or connected subgraph with a bounded diameter, just to mention few. Acknowledgements {#acknowledgements .unnumbered} ================ We would like to thank Feliciano Colella and Simone Galanti for insightful discussions on computational tractability of some of the problems studied in this paper. [99]{} Ahmadian, S., Friggstad, Z., Swamy, C.: Local-search based approximation algorithms for mobile facility location problems. Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’13), SIAM, 1607–1621, 2013. Bilò, D., Disser, Y., Gualà, L., Mihal’ak, M., Proietti, G., Widmayer, P.: Polygon-constrained motion planning problems. Proceedings of the 9th International Symposium on Algorithms and Experiments for Sensor Systems, Wireless Networks and Distributed Robotics (ALGOSENSORS’13), LNCS 8243, Springer, 67–82, 2013. Berman, P., Demaine, E.D., Zadimoghaddam, M.: $O(1)$-approximations for maximum movement problems. Proocedings of the 14th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems and of the the 15th International Workshop on Randomization and Computation (APPROX-RANDOM’11), LNCS 6845, Springer, 62–74, 2011. Bollob[á]{}s, B.: Modern Graph Theory 184, Springer, 1998. Bomze, I.M., Budinich, M., Pelillo, M., Rossi, C.: Annealed replication: a new heuristic for the maximum clique problem. Discrete Applied Mathematics* 121(1-3):27–49, 2002. Demaine, E.D., Hajiaghayi, M.T., Mahini, H., Sayedi-Roshkhar, A.S., Gharan, S.O., Zadimoghaddam, M.: Minimizing movement. ACM Transactions on Algorithms 5(3), article 30, 2009. Demaine, E.D., Hajiaghayi, M., Marx, D.: Minimizing movement: fixed-parameter tractability. Proceedings of the 17th Annual European Symposium on Algorithms (ESA’09), LNCS 5757, Springer, 718–729, 2009. Dinur, I., Safra, S.: On the hardness of approximating minimum vertex cover. Annals of Mathematics, 439–485, 2005. Friggstad, Z., Salavatipour, M.R.: Minimizing movement in mobile facility location problems. ACM Transactions on Algorithms 7(3), article 28, 2011. Hall, P.: On representatives of subsets. Journal of the London Mathematical Society 10(1):26–30, 1935. Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Domination in graphs: advanced topics 40, Marcel Dekker, New York, 1998. Khot, S.: On the power of unique 2-prover 1-round games. Proceedings on 34th Annual ACM Symposium on Theory of Computing (STOC’02), ACM, 767–775, 2002. Khot, S., Regev, O.: Vertex cover might be hard to approximate to within 2-epsilon. Journal of Computer and System Sciences 74(3):335–349, 2008. Prencipe, G., Santoro, N.: Distributed algorithms for autonomous mobile robots. Proceedings of the 4th International Conference on Theoretical Computer Science (IFIP-TCS’06), IFIP 209, Springer, 47–62, 2006. [^1]: Research partially supported by the Research Grant PRIN 2010 “ARS TechnoMedia” funded by the Italian Ministry of University and Research. [^2]: A preliminary version of this work appeared in the Proceedings of the 20th Colloquium on Structural Information and Communication Complexity (SIROCCO’13), LNCS 8179, Springer, 322–333, 2013. DOI: http://dx.doi.org/10.1007/978-3-319-03578-9\_27 [^3]: A centroid of a tree is a vertex whose removal minimizes the maximum number of nodes over all the trees of the resulting forest. Notice that each tree of the forest has at most half of the vertices of the original tree, and that a centroid can be easily found in linear time. [^4]: This is not true if $G$ is a tree: take the tree $T$ with vertex set $\{1, 2, \dots, 11\}$ and edge set $\{ (i,i+1) : 1 \le i \le 4 \} \cup \{ (2,6), (2,7), (3,8),(3,9), (4,10), (4,11) \}$, place $2$ pebbles on vertex $1$ and one pebble on each vertex in $\{6, 7, 8, 9, 10, 11\}$. It is easy to see that any optimal solution has cost $3$ and that at least one edge must be traversed in opposite directions. [^5]: The case $\mu(p_i) = \mu(p_{i+1})$ is clearly impossible. [^6]: Notice that the additional vertex $u$ is only instrumental to guarantee that $G$ is connected. [^7]: W.l.o.g. we assume that $Q$ contains no vertices of weight $0$.	ArXiv
--- abstract: 'This paper provides a sample of a LaTeX document which conforms, somewhat loosely, to the formatting guidelines for ACM SIG Proceedings.[^1]' author: - Ben Trovato - 'G.K.M. Tobin' - 'Lars Th[ø]{}rv[ä]{}ld' - 'Lawrence P. Leipuner' - Sean Fogarty - Charles Palmer - John Smith - 'Julius P. Kumquat' bibliography: - 'sample-bibliography.bib' subtitle: Extended Abstract title: SIG Proceedings Paper in LaTeX Format --- <ccs2012> <concept> <concept\_id>10010520.10010553.10010562</concept\_id> <concept\_desc>Computer systems organization Embedded systems</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10010520.10010575.10010755</concept\_id> <concept\_desc>Computer systems organization Redundancy</concept\_desc> <concept\_significance>300</concept\_significance> </concept> <concept> <concept\_id>10010520.10010553.10010554</concept\_id> <concept\_desc>Computer systems organization Robotics</concept\_desc> <concept\_significance>100</concept\_significance> </concept> <concept> <concept\_id>10003033.10003083.10003095</concept\_id> <concept\_desc>Networks Network reliability</concept\_desc> <concept\_significance>100</concept\_significance> </concept> </ccs2012> [^1]: This is an abstract footnote	ArXiv
--- abstract: 'The International Linear Collider (ILC) is the next large scale project in accelerator particle physics. Colliding electrons with positrons at energies from 0.3 TeV up to about 1 TeV, the ILC is expected to provide the accuracy needed to complement the LHC data and extend the sensitivity to new phenomena at the high energy frontier and answer some of the fundamental questions in particle physics and in its relation to Cosmology. This paper reviews some highlights of the ILC physics program and of the major challenges for the accelerator and detector design.' address: \| Department of Physics, University of California at Berkeley and\ Lawrence Berkeley National Laboratory\ Berkeley, CA 94720, USA\ [email protected] author: - Marco Battaglia --- The International Linear Collider ================================= Introduction {#sec0} ------------ Accelerator particle physics is completing a successful cycle of precision tests of the Standard Model of electro-weak interactions (SM). After the discovery of the $W$ and $Z$ bosons at the $Sp\bar{p}S$ hadron collider at CERN, the concurrent operation of hadron and $e^+e^-$ colliders has provided a large set of precision data and new observations. Two $e^+e^-$ colliders, the SLAC Linear Collider (SLC) at the Stanford Linear Accelerator Center (SLAC) and the Large Electron Positron (LEP) collider at the European Organization for Nuclear Research (CERN), operated throughout the 1990’s and enabled the study of the properties of the $Z$ boson in great detail. Operation at LEP up to 209 GeV, the highest collision energy ever achieved in electron-positron collisions, provided detailed information on the properties of $W$ bosons and the strongest lower bounds on the mass of the Higgs boson and of several supersymmetric particles. The collision of point-like, elementary particles at a well-defined and tunable energy offers advantages for precision measurements, as those conducted at LEP and SLC, over proton colliders. On the other hand experiments at hadron machines, such as the Tevatron $p \bar p$ collider at Fermilab, have enjoyed higher constituent energies. The CDF and D0 experiments eventually observed the direct production of top quarks, whose mass had been predicted on the basis of precision data obtained at LEP and SLC. While we await the commissioning and operation of the LHC $pp$ collider at CERN, the next stage in experimentation at lepton colliders is actively under study. For more than two decades, studies for a high-luminosity accelerator, able to collide electrons with positrons at energies of the order of 1 TeV, are being carried out world-wide. The path towards the ILC {#sec1} ------------------------ The concept of an $e^+e^-$ linear collider dates back to a paper by Maury Tigner [@Tigner:1965] published in 1965, when the physics potential of $e^+e^-$ collisions had not yet been appreciated in full. This seminal paper envisaged collisions at 3-4 GeV with a luminosity competitive with that of the SPEAR ring at SLAC, i.e. $3 \times 10^{30}$ cm$^{-2}$ s$^{-1}$. [A possible scheme to obtain $e^-e^-$ and $e^+e^-$ collisions at energies of hundreds of GeV]{} is the title of a paper [@Amaldi:1976] by Ugo Amaldi published a decade later in 1976, which sketches the linear collider concept with a design close to that now developed for the ILC. The parameters for a linear collider, clearly recognised as the successors of $e^+e^-$ storage rings on the way to high energies, were discussed by Burt Richter at the IEEE conference in San Francisco in 1979 [@Richter:1979cq] and soon after came the proposal for the [Single Pass Collider Project]{} which would become SLC at SLAC. From 1985, the CERN Long Range Planning Committee considered an $e^+e^-$ linear collider, based on the CLIC [@Schnell:1986ig] design, able to deliver collisions at 2 TeV with $10^{33}$ cm$^{-2}$ s$^{-1}$ luminosity, [vis-a-vis]{} a hadron collider, with proton-proton collisions at 16 TeV and luminosity of $1.4 \times 10^{33}$ cm$^{-2}$ s$^{-1}$, as a candidate for the new CERN project after LEP. That review process eventually led to the decision to build the LHC, but it marked an important step to establish the potential of a high energy $e^+e^-$ collider. It is important to note that it was through the contributions of several theorists, including John Ellis, Michael Peskin, Gordon Kane and others, that the requirements in terms of energy and luminosity for a linear collider became clearer in the mid 1980’s [@Ahn:1988vj]. The SLC project gave an important proof of principle for a high energy linear collider and the experience gained has shaped the subsequent designs in quite a significant way. After a decade marked by important progress in the R&D of the basic components and the setup of advanced test facilities, designs of four different concepts emerged: TESLA, based on superconducting RF cavities, the NLC/JLC-X, based on high frequency (11.4 GHz) room-temperature copper cavities, JLC-C, based on lower frequency (5.7 GHz) conventional cavities and CLIC, a multi-TeV collider based on a different beam acceleration technique, the two-beam scheme with transfer structures operating at 30 GHz. Accelerator R&D had reached the maturity to assess the technical feasibility of a linear collider project and take an informed choice of the most advantageous RF technology. The designs were considered by the International Linear Collider Technical Review Committee (ILC-TRC), originally formed in 1994 and re-convened by the International Committee for Future Accelerators (ICFA) in 2001 under the chairmanship of Greg A. Loew. The ILC-TRC assessed their status using common criteria, identified outstanding items needing R&D effort and suggested areas of collaboration. The TRC report was released in February 2003 [@trc] and the committee found that there were [no insurmountable show-stoppers to build TESLA, NLC/JLC-X or JLC-C in the next few years and CLIC in a more distant future, given enough resources]{}. Nonetheless, significant R&D remained to be done. At this stage, it became clear that, to make further progress, the international effort towards a linear collider should be focused on a single design. ICFA gave mandate to an International Technology Recommendation Panel (ITRP), chaired by Barry Barish, to make a definite recommendation for a RF technology that would be the basis of a global project. In August 2004 the ITRP made the recommendation in favour of superconducting RF cavities [@itrp]. The technology choice, which was promptly accepted by all laboratories and groups involved in the R&D process, is regarded as a major step towards the realization of the linear collider project. Soon after it, a truly world-wide, centrally managed design effort, the Global Design Effort (GDE) [@gde], a team of more than 60 persons, started, with the aim to produce an ILC Reference Design Report by beginning of 2007 and an ILC Technical Design Report by end of 2008. The GDE responsibility now covers the detailed design concept, performance assessments, reliable international costing, industrialization plan, siting analysis, as well as detector concepts and scope. A further important step has been achieved with release of the Reference Design Report in February 2007 [@rdr]. This report includes a preliminary value estimate of the cost for the ILC in its present design and at the present level of engineering and industrialisation. The value estimate is structured in three parts: 1.78 Billion ILC Value Units for site-related costs, such as those of tunneling in a specific region, 4.87 Billion ILC Value Units for the value of the high technology and conventional components and 13,000 person-years for the required supporting manpower. For this estimate the conversion factor is 1 ILC Value Unit = 1 US Dollar = 0.83 Euro = 117 Yen. This estimate, which is comparable to the LHC cost, when the pre-existing facilities, such as the LEP tunnel, are included, provides guidance for optimisation of both the design and the R&D to be done during the engineering phase, due to start in Fall 2007. Technical progress was paralleled by increasing support for the ILC in the scientific community. At the 2001 APS workshop [The Future of Physics]{} held in Snowmass, CO, a consensus emerged for the ILC as the right project for the next large scale facility in particle physics. This consensus resonated and expanded in a number of statements by highly influential scientific advisory panels world-wide. The ILC role in the future of scientific research was recognised by the OECD Consultative Group on High Energy Physics [@oecd], while the DOE Office of Science ranked the ILC as its top mid-term project. More recently the EPP 2010 panel of the US National Academy of Sciences, in a report titled [Elementary Particle Physics in the 21$^{st}$ Century]{} has endorsed the ILC as the next major experimental facility to be built and its role in elucidating the physics at the high energy frontier, independently from the LHC findings [@epp2010]. Nowadays, the ILC is broadly regarded as the highest priority for a future large facility in particle physics, needed to extend and complement the LHC discoveries with the accuracy which is crucial to understand the nature of New Physics, test fundamental properties at the high energy scale and establish their relation to other fields in physical sciences, such as Cosmology. A matching program of physics studies and detector R&D efforts has been in place for the past decade and it is now developing new, accurate and cost effective detector designs from proof of concepts towards that stage of engineering readiness, needed for being adopted in the ILC experiments. ILC Accelerator Parameters {#sec2} -------------------------- ### ILC Energy {#sec2.1} The first question which emerges in defining the ILC parameters is the required centre-of-mass energy $\sqrt{s}$. It is here where we most need physics guidance to define the next thresholds at, and beyond, the electro-weak scale. The only threshold which, at present, is well defined numerically is that of top-quark pair production at $\sqrt{s} \simeq$ 350 GeV. Beyond it, there is a strong prejudice, supported by precision electro-weak and other data, that the Higgs boson should be light and new physics thresholds may exist between the electro-weak scale and approximately 1 TeV. If indeed the SM Higgs boson exists and the electro-weak data is not affected by new physics, its mass $M_H$ is expected to be below 200 GeV as discussed in section \[sec3.1\]. Taking into account that the Higgs main production process is in association with a $Z^0$ boson, the maximum of the $e^+e^- \to H^0 Z^0$ cross section varies from $\sqrt{s}$ = 240 GeV to 350 GeV for 120 GeV $< M_H <$ 200 GeV. On the other hand, we know that the current SM needs to be extended by some New Physics. Models of electroweak symmetry breaking contain new particles in the energy domain below 1 TeV. More specifically, if Supersymmetry exists and it is responsible for the dark matter observed in the Universe, we expect that a significant fraction of the supersymmetric spectrum would be accessible at $\sqrt{s}$ = 0.5-1.0 TeV. In particular, the ILC should be able to study in detail those particles determining the dark matter relic density in the Universe by operating at energies not exceeding 1 TeV, as discussed in section \[sec3.2\]. Another useful perspective on the ILC energy is an analysis of the mass scale sensitivity for new physics vs. the $\sqrt{s}$ energy for lepton and hadron colliders in view of their synergy. The study of electro-weak processes at the highest available energy offers a window on mass scales well beyond its kinematic reach. A comparison of the mass-scale sensitivity for various new physics scenarios as a function of the centre-of-mass energy for $e^+e^-$ and $pp$ collisions is given in section \[sec3.3\]. These and similar considerations, emerged in the course of the world-wide studies on physics at the ILC, motivate the choice of $\sqrt{s}$ = 0.5 TeV as the reference energy parameter, but requiring the ILC to be able to operate, with substantial luminosity, at 0.3 TeV as well and to be upgradable up to approximately 1 TeV. It is useful to consider these energies in an historical perspective. In 1954 Enrico Fermi gave a talk at the American Physical Society, of which he was chair, titled [What can we learn with high energy accelerators ?]{}. In that talk Fermi considered a proton accelerator with a radius equal to that of Earth and 2 T bending magnets, thus reaching a beam energy of $5 \times 10^{15}$ eV [@Maiani:2001wi]. Stanley Livingstone, who had built with Ernest O. Lawrence the first circular accelerator at Berkeley in 1930, had formulated an empirical linear scaling law for the available centre-of-mass energy vs. the construction year and cost. Using Livingstone curve, Fermi predicted that such an accelerator could be built in 1994 at a cost of 170 billion \$. We have learned that, not only such accelerator could not be built, but accelerator physics has irrevocably fallen off the Livingstone curve, even in its revised version, which includes data up to the 1980’s. As horizons expanded, each step has involved more and more technical challenges and has required more resources. The future promises to be along this same path. This underlines the need of coherent and responsible long term planning while sustaining a rich R&D program in both accelerator and detector techniques. The accelerator envisaged by Enrico Fermi was a circular machine, as the almost totality of machines operating at the high energy frontier still are. Now, as it is well known, charged particles undergoing a centripetal acceleration $a = v^2/R$ radiate at rate $P = \frac{1}{6 \pi \epsilon_0} \frac{e^2 a^2}{c^3} \gamma^4$. If the radius $R$ is kept constant, the energy loss is the above rate $P$ times $t = 2 \pi R/v$, the time spent in the bending section of the accelerator. The energy loss for electrons is $W = 8.85 \times 10^{-5} \frac{E^4 \mathrm{(GeV^4)}}{R \mathrm{(km)}}$ MeV per turn while for protons is $W = 7.8 \times 10^{-3} \frac{E^4 \mathrm{(TeV^4)}}{R \mathrm{(km)}}$ keV per turn. Since the energy transferred per turn by the RF cavities to the beam is constant, $G \times 2 R \times F$, where $G$ is the cavity gradient and $F$ the tunnel fill factor, for each value of the accelerator ring radius $R$ there exists a maximum energy $E_{max}$ beyond which the energy loss exceeds the energy transferred. In practice, before this value of $E_{max}$ is reached, the real energy limit is set by the power dumped by the beam as synchrotron radiation. To make a quantitative example, in the case of the LEP ring, with a radius $R$ =4.3 km, a beam of energy $E_{beam}$=250 GeV, would lose 80 GeV/turn. Gunther Voss is thought to be the author of a plot comparing the guessed cost of a storage ring and a linear collider as a function of the $e^+e^-$ centre-of-mass energy. A $\sqrt{s}$=500 GeV storage ring, which would have costed an estimated 14 billions CHF in 1970’s is aptly labelled as the [Crazytron]{} [@Treille:2002iu]. LEP filled the last window of opportunity for a storage ring at the high energy frontier. Beyond LEP-2 energies the design must be a linear collider, where no bending is applied to the accelerated particles. Still the accelerator length is limited by a number of constraints which include costs, alignment and siting. Therefore, technology still defines the maximum reachable energy at the ILC. The ILC design is based on superconducting (s.c. ) radio-frequency (RF) cavities. While s.c. cavities had been considered already in the 1960’s, it was Ugo Amaldi to first propose a fully s.c. linear collider in 1975 [@Amaldi:1976]. By the early 1990’s, s.c. cavities equipped already one accelerator, TRISTAN at KEK in Japan, while two further projects were in progress, CEBAF at Cornell and the LEP-2 upgrade at CERN. LEP-2 employed a total of 288 s.c. RF cavities, providing an average gradient of 7.2 MV/m. It was the visionary effort of Bjorn Wijk to promote, from 1990, the TESLA collaboration, with the aim to develop s.c. RF cavities pushing the gradient higher by a factor of five and the production costs down by a factor of four, thus reducing the cost per MV by a factor of twenty. Such reduction in cost was absolutely necessary to make a high energy collider, based on s.c. cavities, feasible. Within less than a decade 1.3 GHz, pure niobium cavities achieved gradients in excess to 35 MV/m. This opened the way to their application to a $e^+e^-$ linear collider, able to reach centre-of-mass energies of the order of 1 TeV, as presented in detail in the TESLA proposal published in 2001 [@Brinkmann:2001qn] and recommended for the ILC by the ITRP in 2004 [@itrp]. Today, the ILC baseline design aims at matching technical feasibility to cost optimisation. One of the major goals of the current effort in the ILC design is to understand enough about its costs to provide a reliable indication of the scale of funding required to carry out the ILC project. Preparing a reliable cost estimate for a project to be carried out as a truly world-wide effort at the stage of a conceptual design that still lacks much of the detailed engineering designs as well as agreements for responsibility and cost sharing between the partners and a precise industrialisation plan is a great challenge. Still having good cost information as soon as possible, to initiate negotiations with the funding agencies is of great importance. An interesting example of the details entering in this process is the optimisation of the cost vs. cavity gradient for a 0.5 TeV collider. The site length scales inversely with the gradient $G$ while the cost of the cryogenics scales as $G^2/Q_0$ resulting in a minimum cost for a gradient of 40 MW/m, corresponding to a tunnel length of 40 km, and a fractional cost increase of 10 % for gradients of 25 MV/m or 57 MV/m. The chosen gradient of 35 MV/m, which is matched by the average performance of the most recent prototypes after electro-polishing, gives a total tunnel length of 44 km with a cost increment from the minimum of just 1 %. Beyond 1 TeV, the extension of conventional RF technology is more speculative. In order to attain collisions at energies in excess of about 1 TeV, with high luminosity, significantly higher gradients are necessary. As the gradient of s.c. cavities is limited below $\sim$ 50 MV/m, other avenues should be explored. The CLIC technology [@Assmann:2000hg], currently being developed at CERN and elsewhere, may offer gradients of the order of 150 MV/m [@wuensch], allowing collision energies in the range 3-5 TeV with a luminosity of $10^{35}$ cm$^{-2}$ s$^{-1}$, which would support a compelling physics program [@Battaglia:2004mw]. While RF cavities are limited to accelerating fields of order of 100-200 MV/m, or below, laser-wakefield accelerators are capable, in principle, of producing fields of 10-100 GV/m. Recently a 1 GeV $e^-$ beam has been accelerated over just 3.3 cm using a 40 TW peak-power laser pulse [@Leemans:2006], thus opening a possible path towards ultra-high energies in $e^+e^-$ collisions in some more distant future. ### ILC Luminosity {#sec2.2} The choice of a linear collider, rather than a circular storage ring, while solving the problem of the maximum reachable energy, introduces the challenge of achieving collisions with the required luminosity. The luminosity, $\cal{L}$, defined as the proportionality factor between the number of events produced and the process cross section $\sigma$, has requirements which depend on the typical values of s-channel cross sections and so scale as $1/s$. First luminosity requirements were already outlined in the 1980s [@Richter:1981uu; @Amaldi:1987xt] as ${\cal{L}} \simeq \frac{2 E_{beam}}{\mathrm{TeV}} \times 10^{33}$ cm$^{-2}$ s$^{-1}$, based on the estimated discovery potential. But in the present vision of the ILC role in probing the high energy frontier new requirements must be considered. One example is the precision study of electro-weak processes to look for deviations from the SM predictions, due to effect of new physics at high scales. The $e^+e^- \to b \bar b$ cross section at 1 TeV is just 96 fb, so this would corresponds to less than $10^3$ events per year at $10^{33}$ cm$^{-2}$ s$^{-1}$, which is certainly insufficient for the kind of precision measurements which we expect from the ILC. Another example is offered by one of the reactions most unique to the ILC: the double-Higgs production $e^+e^- \to HHZ$ sensitive to the Higgs self-coupling, which has a cross section of order of only 0.2 fb at 0.5 TeV. Therefore a luminosity of $10^{34}$ cm$^{-2}$ s$^{-1}$ or more is required as baseline parameter. The luminosity can be expressed as a function of the accelerator parameters as: $${\cal{L}} = f_{rep} n_b \frac{N^2}{4 \pi \sigma_x \sigma_y}.$$ Now, since in a linear machine the beams are collided only once and then dumped, the collision frequency, $f_{rep}$, is small and high luminosity should be achieved by increasing the number of particles in a bunch $N$, the number of bunches $n_b$ and decreasing the transverse beam size $\sigma$. Viable values for $N$ are limited by wake-field effects and the ILC parameters have the same number of electrons in a bunch as LEP had, though it aims at a luminosity three orders of magnitude higher. Therefore, the increase must come from a larger number of bunches and a smaller transverse beam size. The generation of beams of small transverse size, their preservation during acceleration and their focusing to spots of nanometer size at the interaction region presents powerful challenges which the ILC design must solve. A small beam size also induces beam-beam interactions. On one hand the beam self-focusing, due to the electrostatic attraction of particles of opposite charges enhances the luminosity. But beam-beam interactions also result in an increase of beamstrahlung with a larger energy spread of the colliding particles, a degraded luminosity spectrum and higher backgrounds. Beamstrahlung is energy loss due to particle radiation triggered by the trajectory bending in the interactions with the charged particles in the incoming bunch [@Noble:1986yz]. The mean beamstrahlung energy loss, which has to be minimised, is given by: $$\delta_{BS} \simeq 0.86 \frac{e r_e^3}{2 m_0 c^2} \frac{E_{cm}}{\sigma_z} \frac{N_b^2}{(\sigma_x+\sigma_y)^2}.$$ Since the luminosity scales as $\frac{1}{\sigma_x \sigma_y}$, while the beamstrahlung energy loss scales as $\frac{1}{\sigma_x + \sigma_y}$, it is advantageous to choose a large beam aspect ratio, with the vertical beam size much smaller than the horizontal component. The parameter optimisation for luminosity can be further understood by expressing the luminosity in terms of beam power $P = f_{rep} N E_{cm}$ = $\eta P_{AC}$ and beamstrahlung energy loss as: $${\cal{L}} \propto \frac{\eta P_{AC}}{E_{cm}} \sqrt{\frac{\delta_{BS}}{\epsilon_y}}H_D$$ which highlights the dependence on the cavity efficiency $\eta$ and the total power $P_{AC}$. The $H_D$ term is the pinch enhancement factor, that accounts for the bunch attraction in the collisions of oppositely charged beams. In summary, since the amount of available power is necessarily limited, the main handles on luminosity are $\eta$ and $\epsilon_y$. The efficiency for transferring power from the plug to the beam is naturally higher for s.c. than for conventional copper cavities, so more relaxed collision parameters can be adopted for a s.c. linear collider delivering the same luminosity. The main beam parameters for the ILC baseline design are given in Table \[tab:params\]. \[tab:params\] ILC Physics Highlights {#sec3} ---------------------- The ILC physics program, as we can anticipate it at present, is broad and diverse, compelling and challenging. The ILC is being designed for operation at 0.5 TeV with the potential to span the largest range of collision energies, from the $Z^0$ peak at 0.091 TeV up to 1 TeV, collide electrons with positrons, but optionally also electrons with electrons, photons with photons and photons with electrons, and combine various polarization states of the electron and positron beams. Various reports discussing the linear collider physics case, including results of detailed physics studies, have been published in the last few years [@Ahn:1988vj; @Murayama:1996ec; @Aguilar-Saavedra:2001rg; @Abe:2001wn; @Abe:2001gc; @Dawson:2004xz; @Battaglia:2004mw]. Here, I shall focus on three of the main ILC physics themes: the detailed study of the Higgs boson profile, the determination of neutralino dark matter density in the Universe from accelerator data, and the sensitivity to new phenomena beyond the ILC kinematic reach, through the analysis of two-fermion production, at the highest $\sqrt{s}$ energy. Results discussed in the following have been obtained mostly using realistic, yet parametric simulation of the detector response. Only few analyses have been carried out which include the full set of physics and machine-induced backgrounds on fully simulated and reconstructed events. With the progress of the activities of detector concepts and the definition of well-defined benchmark processes, this is becoming one of the priorities for the continuation of physics and detector studies. ### The Higgs Profile at the ILC {#sec3.1} Explaining the origin of mass is one of the great scientific quests of our time. The SM addresses this question by the Higgs mechanism [@Higgs]. The first direct manifestation of the Higgs mechanism through the Higgs sector will be the existence of at least one Higgs boson. The observation of a new spin-0 particle would represent a first sign that the Higgs mechanism of mass generation is indeed realised in Nature. This has motivated a large experimental effort, from LEP-2 to the Tevatron and, soon, the LHC, actively backed-up by new and more accurate theoretical predictions. After a Higgs discovery, which we anticipate will be possible at the LHC, full validation of the Higgs mechanism can only be established by an accurate study of the Higgs boson production and decay properties. It is here where the ILC potential in precision physics will be crucial for the validation of the Higgs mechanism, through a detailed study of the Higgs profile [@Heinemeyer:2005gs]. The details of this study depend on the Higgs boson mass, $M_H$. In the SM, $M_H = \sqrt{2 \lambda} v$ where the Higgs field expectation value $v$ is determined as $(\sqrt{2}G_F)^{-1/2} \approx 246$ GeV, while the Higgs self-coupling $\lambda$ is not specified, leaving the mass as a free parameter. However, we have strong indications that $M_H$ must be light. The Higgs self-coupling behaviour at high energies [@triv], the Higgs field contribution to precision electro-weak data [@ewwg:2005di] and the results of direct searches at LEP-2 [@Barate:2003sz] at $\sqrt{s} \ge$ 206 GeV, all point towards a light Higgs boson. In particular, the study of precision electro-weak data, which are sensitive to the Higgs mass logarithmic contribution to radiative corrections, is based on several independent observables, including masses ($m_{top}$, $M_W$, $M_Z$), lepton and quark asymmetries at the $Z^0$ pole, $Z^0$ lineshape and partial decay widths. The fit to eighteen observables results in a 95% C.L. upper limit for the Higgs mass of 166 GeV, which becomes 199 GeV when the lower limit from the direct searches at LEP-2, $M_H >$ 114.4 GeV, is included. As a result, current data indicates that the Higgs boson mass should be in the range 114 GeV $< M_H <$ 199 GeV. It is encouraging to observe that if the same fit is repeated, but excluding this time $m_{top}$ or $M_W$, the results for their values, 178$^{+12}_{-9}$ GeV and 80.361$\pm$0.020 GeV respectively, are in very good agreement with the those obtained the direct determinations, $m_{top}$ = 171.4$\pm$2.1 GeV and $M_W$ = 80.392$\pm$0.029 GeV. At the ILC the Higgs boson can be observed in the Higgs-strahlung production process $e^+e^- \rightarrow HZ$ with $Z \rightarrow \ell^+\ell^-$, independent of its decay mode, by the distinctive peak in the di-lepton recoil mass distribution. A data set of 500 fb$^{-1}$ at $\sqrt{s}$ = 350 GeV, corresponding to four years of ILC running, provides a sample of 3500-2200 Higgs particles produced in the di-lepton $HZ$ channel, for $M_H$ = 120-200 GeV. Taking into account the SM backgrounds, dominated by $e^+e^- \rightarrow Z^0Z^0$ and $W^+W^-$ production, the Higgs boson observability is guaranteed up to its production kinematical limit, independent of its decays. This sets the ILC aside from the LHC, since the ILC sensitivity to the Higgs boson does not depend on its detailed properties. After observation of a new particle with properties compatible with those of the Higgs boson, a significant experimental and theoretical effort will be needed to verify that this is indeed the boson of the scalar field responsible for the electro-weak symmetry breaking and the generation of mass. Outlining the Higgs boson profile, through the determination of its mass, width, quantum numbers, couplings to gauge bosons and fermions and the reconstruction of the Higgs potential, stands as a most challenging, yet compelling, physics program. The ILC, with its large data sets at different centre-of-mass energies and beam polarisation conditions, the high resolution detectors providing unprecedented accuracy on the reconstruction of the event properties and the use of advanced analysis techniques, developed from those successfully adopted at LEP and SLC, promises to promote Higgs physics into the domain of precision measurements. Since the Higgs mass $M_H$ is not predicted by theory, it is of great interest to measure it precisely. Once this mass, and thus $\lambda$, is fixed, the profile of the Higgs particle is uniquely determined in the SM. In most scenarios we expect the LHC to determine the Higgs mass with a good accuracy. At the ILC, this measurement can be refined by exploiting the kinematical characteristics of the Higgs-strahlung production process $e^+e^- \rightarrow Z^* \rightarrow H^0 Z^0$ where the $Z^0$ can be reconstructed in both its leptonic and hadronic decay modes. The $\ell^+\ell^-$ recoil mass for leptonic $Z^0$ decays yields an accuracy of 110 MeV for 500 fb$^{-1}$ of data, without any requirement on the nature of the Higgs decays. Further improvement can be obtained by explicitly selecting $H \rightarrow b \bar b$ ($WW$) for $M_H \le$($>$) 140 GeV. Here a kinematical 5-C fit, imposing energy and momentum conservation and the mass of a jet pair to correspond to $M_Z$, achieves an accuracy of 40 to 90 MeV for 120$< M_H <$ 180 GeV [@hmass1]. The total decay width of the Higgs boson is predicted to be too narrow to be resolved experimentally for Higgs boson masses below the $ZZ$ threshold. On the contrary, above $\simeq$ 200 GeV, the total width can be measured directly from the reconstructed width of the recoil mass peak, as discussed below. For the lower mass range, indirect methods must be applied. In general, the total width is given by $\Gamma_{tot}=\Gamma_X/\mathrm{BR}(H\to X)$. Whenever $\Gamma_X$ can be determined independently of the corresponding branching fraction, a measurement of $\Gamma_{tot}$ can be carried out. The most convenient choice is the extraction of $\Gamma_{H}$ from the measurements of the $WW$ fusion cross section and the $H\to WW^$ decay branching fraction . A relative precision of 6% to 13% on the width of the Higgs boson can be obtained at the ILC with this technique, for masses between 120 GeV and 160 GeV. The spin, parity and charge-conjugation quantum numbers $J^{PC}$ of Higgs bosons can be determined at the ILC in a model-independent way. Already the observation of either $\gamma \gamma \rightarrow H$ production or $H \rightarrow \gamma\gamma$ decay sets $J \ne 1$ and $C=+$. The angular dependence $\frac{d \sigma_{ZH}}{d \theta} \propto \sin^2 \theta$ and the rise of the Higgs-strahlung cross section: $$\sigma_{ZH} \propto~\beta \sim~\sqrt{s-(M_H+M_Z)^2}$$ allows to determine $J^P = 0^+$ and distinguish the SM Higgs from a $CP$-odd $0^{-+}$ state $A^0$, or a $CP$-violating mixture of the two [@Miller:2001bi; @Schumacher:2001ax]. But where the ILC has a most unique potential is in verifying that the Higgs boson does its job of providing gauge bosons, quarks and leptons with their masses. This requires to precisely test the relation $g_{HXX} \propto {m_X}$ between the Yukawa couplings, $g_{HXX}$, and the corresponding particle masses, $m_X$. In fact, the SM Higgs couplings to fermion pairs $g_{Hff} = m_f/v$ are fully determined by the fermion mass $m_f$. The corresponding decay partial widths only depend on these couplings and on the Higgs boson mass, QCD corrections do not represent a significant source of uncertainty [@Djouadi:1995gt]. Therefore, their accurate determination will represent a comprehensive test of the Higgs mechanism of mass generation [@Hildreth:1993dx]. Further, observing deviations of the measured values from the SM predictions will probe the structure of the Higgs sector and may reveal a non-minimal implementation of the Higgs model or the effect of new physics inducing a shift of the Higgs couplings [@Carena:2001bg; @Desch:2004cu; @Battaglia:2004js]. The accuracy of these measurements relies on the performances of jet flavour tagging and thus mostly on the Vertex Tracker, making this analysis an important benchmark for optimising the detector design. It is important to ensure that the ILC sensitivity extends over a wide range of Higgs boson masses and that a significant accuracy is achieved for most particle species. Here, the ILC adds the precision which establishes the key elements of the Higgs mechanism. It is important to point out that these tests are becoming more stringent now that the $B$-factories have greatly improved the determination of the $b$- and $c$-quark masses. When one of these studies was first presented in 1999 [@Battaglia:1999re], the $b$ quark mass was known to $\pm 0.11$ GeV and the charm mass to $\pm 0.13$ GeV, with the expectation that $e^+e^-$ $B$-factory and LHC data could reduce these uncertainties by a factor of two by the time the ILC data would be analysed. Today, the analysis of a fraction of the BaBar data [@Aubert:2004aw] has already brought these uncertainties down to 0.07 GeV for $m_b$ and, more importantly, 0.09 GeV for $m_c$, using the spectral moments technique in semi-leptonic $B$ decays, which had been pioneered on CLEO [@Bauer:2002sh] and DELPHI data [@Battaglia:2002tm]. Extrapolating to the anticipated total statistics to be collected at PEP-II and KEKB, we can now confidently expect that the $b$ quark mass should be known to better than $\pm 0.05$ GeV and the charm mass to better than $\pm 0.06$ GeV. This translates into less than $\pm 0.4$ % and $\pm 6.5$ % relative uncertainty in computing the Higgs SM couplings to $b$ and $c$ quarks, respectively, and motivates enhanced experimental precision in the determination of these couplings at the ILC. Detailed simulation shows that these accuracy can be matched by the ILC [@Kuhl:2004ri; @Barklow:2003hz]. While much of the emphasis on the ILC capabilities in the study of the Higgs profile is for a light Higgs scenario, preferred by the current electro-weak data and richer in decay modes, the ILC has also the potential of precisely mapping out the Higgs boson properties for heavier masses. If the Higgs boson turns out to weigh of order 200 GeV, the 95% C.L. upper limit indicated by electro-weak fits, or even heavier, the analysis of the recoil mass in $e^+e^- \to HZ$ at $\sqrt{s}$ = 0.5 TeV allows to precisely determine $M_H$, $\Gamma_H$ and the Higgs-strahlung cross section. Even for $M_H$ = 240 GeV, the mass can be determined to a 10$^{-3}$ accuracy and, more importantly, the total width measured about 10% accuracy. Decays of Higgs bosons produced in $e^+e^- \to H \nu \bar \nu$ give access to the Higgs couplings. The importance of the $WW$-fusion process $e^+e^- \to H^0 \nu \bar \nu$ to probe rare Higgs decays at higher energies, emerged in the physics study for a multi-TeV linear collider [@Battaglia:2002gq]. Since this cross section increases as $log \frac{s}{M_H^2}$, it becomes dominant around $\sqrt{s}$ = 1 TeV. Detailed studies have been performed and show that 1 ab$^{-1}$ of data at $\sqrt{s}$ = 1 TeV, corresponding to three to four years of ILC running, can significantly improve the determination of the Higgs couplings, especially for the larger values of $M_H$ [@Battaglia:2002av; @Barklow:2003hz]. $WW$ and $ZZ$ couplings can be determined with relative accuracies of 3 % and 5 % respectively, while the coupling to $b \bar b$ pairs, a rare decay with a branching fraction of just $2 \times 10^{-3}$ at such large masses, can be determined to 4 % to 14 % for 180 GeV $< M_H <$ 220 GeV. This measurement is of great importance, since it would offer the only opportunity to learn about the fermion couplings of such an heavy Higgs boson, and it is unique to a linear collider. A most distinctive feature of the Higgs mechanism is the shape of the Higgs potential: $$V(\Phi) = - \frac{\mu^2}{2} \Phi^2 + \frac{\lambda}{4} \Phi^4$$ with $v = \sqrt{\frac{\mu^2}{\lambda}}$. In the SM, the triple Higgs coupling, $g_{HHH} = 3 \lambda v$, is related to the Higgs mass, $M_H$, through the relation $$g_{HHH} = \frac{3}{2} \frac{M_H^2}{v}.$$ By determining $g_{HHH}$, the above relation can be tested. The ILC has access to the triple Higgs coupling through the double Higgs production processes $e^+e^- \to H H Z$ and $e^+e^- \to H H \nu\nu$ [@Djouadi:1999gv]. Deviations from the SM relation for the strength of the Higgs self-coupling arise in models with an extended Higgs sector [@Kanemura:2004mg]. The extraction of $g_{HHH}$ is made difficult by their tiny cross sections and by the dilution effect, due to diagrams leading to the same double Higgs final states, but not sensitive to the triple Higgs vertex. This makes the determination of $g_{HHH}$ a genuine experimental [tour de force]{}. Other modes, such as $e^+e^- \to H H b \bar b$, have also been recently proposed [@Gutierrez-Rodriguez:2006qk] but signal yields are too small to provide any precise data. Operating at $\sqrt{s}$ = 0.5 TeV the ILC can measure the $HHZ$ production cross section to about 15% accuracy, if the Higgs boson mass is 120 GeV, corresponding to a fractional accuracy on $g_{HHH}$ of 23% [@Castanier:2001sf]. Improvements can be obtained first by introducing observables sensitive to the presence of the triple Higgs vertex and then by performing the analysis at higher energies where the $HH \nu \bar \nu$ channel contributes [@Battaglia:2001nn]. In the $HHZ$ process events from diagrams containing the $HHH$ vertex exhibit a lower invariant mass of the $HH$ system compared to double-Higgstrahlung events. When the $M_{HH}$ spectrum is fitted, a relative statistical accuracy of $\pm 0.20$ can be obtained with 1 ab$^{-1}$ at $\sqrt{s}$ = 0.5 TeV. The availability of beam polarization increases the $HHZ$ cross section by a factor of two and that for $HH \nu \bar \nu$ by a factor of four, thus offering a further possible significant improvement to the final accuracy. The ILC and, possibly, a multi-TeV $e^+e^-$ collider represent a unique opportunity for carrying out this fundamental measurement. In fact, preliminary studies show that, the analysis of double Higgs production at the LHC is only possible after a luminosity upgrade and, even then, beyond the observation of double Higgs production, it would provide only a very limited information on the triple-Higgs coupling [@Baur:2002qd; @Baur:2003gp]. \[tab:summary\] ### Understanding Dark Matter at the ILC {#sec3.2} The search for new physics beyond the Standard Model has a central role in the science program of future colliders. It is instructive to contrast the LHC and the ILC in terms of their potential in such searches. Running at $\sqrt{s} \le$ 1 TeV the ILC might appear to be limited in reach, somewhere within the energy domain being probed by the Tevatron and that to be accessed by LHC. And yet its potential for fully understanding the new physics, which the LHC might have manifested, and for probing the high energy frontier beyond the boundaries explored in hadron collisions is of paramount importance. There are several examples of how the ILC will be essential for understanding new physics. They address scenarios where signals of physics beyond the SM, as observed at the LHC, may be insufficient to decide on the nature of the new phenomena. One such example, which has been studied in some details, is the case of Supersymmetry and Universal Extra Dimensions (UED), two very different models of new physics leading to the very same experimental signature: fermion pairs plus missing energy. Here, the limited analytical power of the LHC may leave us undecided [@Datta:2005zs; @Smillie:2005ar], while a single spin measurement performed at the ILC precisely identifies the nature of the observed particles [@Battaglia:2005zf]. But the ILC capability to fully understand the implications of new physics, through fundamental measurements performed with high accuracy, is manifested also in scenarios where the LHC could observe a significant fraction of the new particle spectrum. An especially compelling example, which can be studied quantitatively, is offered by Supersymmetry in relation to Dark Matter (DM). Dark Matter has been established as a major component of the Universe. We know from several independent observations, including the cosmic microwave background (CMB), supernovas (SNs) and galaxy clusters, that DM is responsible for approximately 20 % of the energy density of the universe. Yet, none of the SM particles can be responsible for it and the observation of DM is likely the first direct signal of new physics beyond the SM. Several particles and objects have been nominated as candidates for DM. They span a wide range of masses, from 10$^{-5}$ eV, in the case of axions, to 10$^{-5}$ solar masses, for primordial black holes. Cosmology tells us that a significant fraction of the Universe mass consists of DM, but does not provide clues on its nature. Particle physics tells us that New Physics must exist at, or just beyond, the EW scale and new symmetries may result in new, stable particles. Establishing the inter-relations between physics at the microscopic scale and phenomena at cosmological scale will represent a major theme for physics in the next decades. The ILC will be able to play a key role in elucidating these inter-relations. Out of these many possibilities, there is a class of models which is especially attractive since its existence is independently motivated and DM, at about the observed density, arises naturally. These are extensions of the SM, which include an extra symmetry protecting the lightest particle in the new sector from decaying into ordinary SM states. The lightest particle becomes stable and can be chosen to be neutral. Such a particle is called a weakly interacting massive particle (WIMP) and arises in Supersymmetry with conserved R-parity (SUSY) but also in Extra Dimensions with KK-parity (UED) [@Kong:2005hn]. Current cosmological data, mostly through the WMAP satellite measurements of the CMB, determine the DM density in the Universe with a 6 % relative accuracy [@wmap]. By the next decade, the PLANCK satellite will push this uncertainty to $\simeq$ 1 %, or below [@planck]. Additional astrophysical data manifest a possible evidence of DM annihilation. The EGRET data show excess of $\gamma$ emission in the inner galaxy, which has been interpreted as due to DM [@deBoer:2005tm] and the WMAP data itself may show a signal of synchrotron emission in the Galactic center [@Finkbeiner:2004us]. These data, if confirmed, may be used to further constrain the DM properties. Ground-based DM searches are also approaching the stage where their sensitivity is at the level predicted by Supersymmetry for some combinations of parameters [@Akerib:2005kh]. The next decades promise to be a time when accelerator experiments will provide new breakthroughs and highly accurate data to gain new insights, not only on fundamental questions in particle physics, but also in cosmology, when studied alongside the observations from satellites and other experiments. The questions on the nature and the origin of DM offer a prime example of the synergies of new experiments at hadron and lepton colliders, at satellites and ground-based DM experiments. It is essential to study, in well defined, yet general enough, models, which are the properties of the new physics sector, such as masses and couplings, most important to determine the resulting relic density of the DM particles. Models exist which allow to link the microscopic particle properties to the present DM density in the Universe, with mild assumptions. If DM consists of WIMPs, they are abundantly produced in the very early Universe when $T \simeq (t \mathrm{(sec)})^{-1/2} > $ 100 GeV and their interaction cross section is large enough that they were in thermal equilibrium for some period in the early universe. The DM relic density can be determined by solving the Boltzmann equation governing the evolution of their phase space number density [@Scherrer:1985zt]. It can be shown that, by taking the WMAP result for the DM relic density in units of the Universe critical density,$\Omega_{DM} h^2$, the thermal averaged DM annihilation cross section times the co-moving velocity, $<\sigma v>$, should be $\simeq$ 0.9. From this result, the mass of the DM candidate can be estimated as: $$M_{DM} = \sqrt{\frac{\pi \alpha^2}{8 <\sigma v>}} \simeq 100~\mathrm{GeV}.$$ A particle with mass $M = {\cal{O}}$(100 GeV) and weak cross section would naturally give the measured DM density. It is quite suggestive that new physics, responsible for the breaking of electro-weak symmetry, also introduce a WIMP of about that mass. In fact, in essentially every model of electroweak symmetry breaking, it is possible to add a discrete symmetry that makes the lightest new particle stable. Often, this discrete symmetry is required for other reasons. For example, in Supersymmetry, the conserved $R$ parity is needed to eliminate rapid proton decay. In other cases, such as models with TeV-scale extra dimensions, the discrete symmetry is a natural consequence of the underlying geometry. Data on DM density already set rather stringent constraints on the parameters of Supersymmetry, if the lightest neutralino $\chi^0_1$ is indeed responsible for saturating the amount of DM observed in the Universe. It is useful to discuss the different scenarios, where neutralino DM density is compatible with the WMAP result, in terms of parameter choices in the context of the constrained MSSM (cMSSM), to understand how the measurements that the ILC provides can establish the relation between new physics and DM. The cMSSM reduces the number of free parameters to just five: the common scalar mass, $m_0$, the common gaugino mass, $m_{1/2}$, the ratio of the vacuum expectation values of the two Higgs fields, $\tan\beta$, the sign of the Higgsino mass parameter, $\mu$, and the common trilinear coupling, $A_0$. It is a remarkable feature of this model that, as these parameters, defined at the unification scale, are evolved down to lower energies, the electroweak symmetry is broken spontaneously and masses for the $W^{\pm}$ and $Z^0$ bosons generated automatically. As this model is simple and defined by a small number of parameters, it is well suited for phenomenological studies. The cosmologically interesting regions in the $m_0$ - $m_{1/2}$ parameter plane are shown in Figure \[fig10\]. As we move away from the bulk region, at small values of $m_0$ and $m_{1/2}$, which is already severely constrained by LEP-2 data, the masses of supersymmetric particles increase and so does the dark matter density. It is therefore necessary to have an annihilation process, which could efficiently remove neutralinos in the early universe, to restore the DM density to the value measured by WMAP. Different processes define three main regions: i) the focus point region, where the $\chi^0_1$ contains an admixture of the supersymmetric partner of a neutral Higgs boson and annihilates to $W^+W^-$ and $Z^0Z^0$, ii) the co-annihilation region, where the lightest slepton has a mass very close to $M_{\chi^0_1}$, iii) the $A$ annihilation funnel, where $M(\chi^0_1)$ is approximately half that of the heavy $A^0$ Higgs boson, providing efficient $s$-channel annihilation, $\chi \chi \to A$. In each of these regions, researchers at the ILC will be confronted with several different measurements and significantly different event signatures. \[tab:bench\] It is interesting to observe that the DM constraint, reduces the dimensionality of the cMSSM plane, by one unit, since the allowed regions are tiny lines in the $m_0$ - $m_{1/2}$ plane, evolve with $\tan \beta$ and depend only very weakly on $A_0$ [@Battaglia:2003ab]. Representative benchmark points have been defined and their parameters are summarised in Table \[tab:bench\]. Even though these points have been defined in a specific supersymmetric model, their phenomenology is common to the more general supersymmetric solutions and we shall soon discuss the extension of results derived in this constrained model to the general MSSM. There are several features which are common to all these regions. First, the relic density depends on the mass of the lightest neutralino and of few additional particles, close in mass to it. The heavier part of the SUSY spectrum decouples from the value of $\Omega_{\chi} h^2$. This is of particular importance for the ILC. Running at $\sqrt{s} \le$ 1 TeV, the ILC will not be able to study supersymmetric particles exceeding $\simeq$450-490 GeV, in particular scalar quarks and heavy Higgs bosons in some regions of the parameter phase space. But, independently of the LHC results, the ILC will either observe and measure these particles if they may be relevant to determine the relic DM density, or it will set bounds that ensure their decoupling. A second important observation is that $\Omega_{\chi} h^2$ typically depends on SUSY parameters which can be fixed by accurate measurements of particle masses, particle mass splittings, decay branching fractions and production cross sections. In some instances the availability of polarised beams is advantageous. The LHC can often make precise measurements of some particles, but it is difficult for the LHC experiments to assemble the complete set of parameters needed to reconstruct annihilation cross section. It is also typical of supersymmetry spectra to contain light particles that may be very difficult to observe in the hadron collider environment. The ILC, in contrast, provides just the right setting to obtain both types of measurements. Again, it is not necessary for the ILC to match the energy of the LHC, only that it provides enough energy to see the lightest charged particles of the new sector. Rather detailed ILC analyses of the relevant channels for each benchmark point have been performed, [@Weiglein:2004hn; @Gray:2005ci; @Khotilovich:2005gb; @Battaglia:2004gk] based on parametric simulation, which includes realistic detector performances and effects of the ILC beam characteristics. It has been assumed that the ILC will be able to provide collisions at centre-of-mass energies from 0.3 TeV to 0.5 TeV with an integrated luminosity of 500 fb$^{-1}$ in a first phase of operation and then its collision energy can raised to 1 TeV to provide an additional data set of 1 ab$^{-1}$, corresponding to an additional three to four years of running. Results are summarised in terms of the estimated accuracies on masses and mass differences in Table \[tab:constraints\]. \[tab:constraints\] In order to estimate the implications of these ILC measurements on the estimation of neutralino dark DM density $\Omega_{\chi} h^2$, broad scans of the multi-parameter supersymmetric phase space need to be performed. For each benchmark point, the soft parameters (masses and couplings) at the electroweak scale can be computed with the full 2-loop renormalization group equations and threshold corrections using [Isajet 7.69]{} [@ISAJET]. Supersymmetric loop corrections to the Yukawa couplings can also be included. The electroweak-scale MSSM parameters are extracted from the high scale cMSSM parameters. The dark matter density $\Omega_{\chi} h^2$ can be estimated using the [DarkSUSY]{} [@Gondolo:2004sc] and [Micromegas]{} [@Belanger:2006is] programs. These programs use the same [Isajet]{} code to determine the particle spectrum and couplings, including the running Yukawa couplings, and compute the thermally averaged cross section for neutralino annihilation, including co-annihilation and solve the equation describing the evolution of the number density for the DM candidate. While the assumptions of the cMSSM are quite helpful for defining a set of benchmark points, the cMSSM is not representative of the generic MSSM, since it implies several mass relations, and its assumptions have no strong physics justification. Therefore, in studying the accuracy on $\Omega_{\chi} h^2$, the full set of MSSM parameters must be scanned in an uncorrelated way and the mass spectrum evaluated for each parameter set. A detailed study has recently been performed [@Baltz:2006fm]. I summarise here some of the findings, Table \[tab:dmsummary\] gives results for the neutralino relic density estimates in MSSM for the LHC, the ILC at 0.5 TeV and the ILC at 1 TeV. The LCC1 point is in the bulk region and the model contains light sleptons, with masses just above that of the lightest neutralino. The most important annihilation reactions are those with t-channel slepton exchange. At the LHC, many of the SUSY spectrum parameters can be determined from kinematic constraints. At the ILC masses can be determined both by the two-body decay kinematics of the pair-produced SUSY particles and by dedicated threshold scans. Let us consider the two body decay of a scalar quark $\tilde q \to q \chi^0_1$. If the scalar quarks are pair produced $e^+e^- \to \tilde q \tilde q$, $E_{\tilde q} = E_{beam}$ and the $\chi^0_1$ escapes undetected, only the $q$ (and the $\bar q$) are observed in the detector. In a 1994 paper, J. Feng and D. Finnell [@Feng:1993sd] pointed out that the minimum and maximum energy of production for the quark can be related to the mass difference between the scalar quark $\tilde q$ and the $\chi^0_1$: $$E_{max,~min} = \frac{E_{beam}}{2} \big( 1 \pm \sqrt{1 - \frac{m_{\tilde q}^2}{E_{beam}^2}} \big) \big( 1 - \frac{m_{\chi}^2}{m_{\tilde q}^2} \big).$$ The method can also be extended to slepton decays $\tilde \ell \to \ell \chi^0_1$, which share the same topology, and allows to determine slepton mass once that of the neutralino is known or determine a relation between the masses and get $m_{\chi^0_1}$ if that of the slepton can be independently measured. The measurement requires a precise determination of the endpoint energies of the lepton momentum spectrum, $E_{min}$ and $E_{max}$. It can be shown that accuracy is limited by beamstrahlung, affecting the knowledge of $E_{beam}$ in the equation above, more than by the finite momentum resolution, $\delta p/p$ of the detector. The ILC has a second, and even more precise, method for mass measurements. The possibility to precisely tune the collision energy allows to perform scans of the onset of the cross section for a specific SUSY particle pair production process. The particle mass and width can be extracted from a fit to the signal event yield as function of $\sqrt{s}$. The accuracy depends rather weakly on the number of points, $N$, adopted in the scan and it appears that concentrating the total luminosity at two or three different energies close to the threshold is optimal [@Martyn:2000; @Blair:2001cz]. The mass accuracy, $\delta m$ can be parametrised as: $$\delta m \simeq \Delta E \frac{1+0.36/\sqrt{N}}{\sqrt{18 N L \sigma}}$$ for S-wave processes, where the cross section rises as $\beta$ and as $$\delta m \simeq \Delta E \frac{1}{N^{1/4}} \frac{1+0.38/\sqrt{N}}{\sqrt{2.6 N L \sigma}}$$ for P-wave processes, where the cross section rises as $\beta^3$. The combination of these measurements allows the ILC to determine the $\chi^0_1$ mass to $\pm$0.05 GeV, which is two orders of magnitude better than the anticipated LHC accuracy, while the mass difference between the $\tilde \tau_1$ and the $\chi^0_1$ can be measured to $\pm$0.3 GeV, which is more than a factor ten better. Extension of ILC operation to 1 TeV gives access to the $e^+e^- \to H^0A^0$ process. As a result of the precision of these measurements, the ILC data at 0.5 TeV will allow to predict the neutralino relic density to $\pm$2 % and the addition of 1.0 TeV data will improve it to $\pm$0.25 %. It is suggestive that this accuracy is comparable, or better, than that expected by the improved CMB survey by the PLANCK mission. For comparison, the LHC data should provide a $\pm$7 % accuracy. This already a remarkable result, due the fact that, a large number of measurements will be available at the LHC and SUSY decay chains can be reconstructed. Still, the overall mass scale remains uncertain at the LHC. The direct mass measurements on the ILC data remove this uncertainty. The LCC1 point is characterised by the relatively low SUSY mass scale, most of the particles can be observed at the LHC and their masses accurately measured at the ILC. However, in more general scenarios, the information available from both collider will be more limited. This is the case at the LCC2 point, located in the focus point region, where masses of scalar quarks, sleptons and heavy Higgs bosons are very large, typically beyond the ILC but also the LHC reach, while gauginos masses are of the order of few hundreds GeV, thus within the kinematical domain of the ILC. In this specific scenario, the LHC will observe the SUSY process $\tilde g \to q \bar q \chi$ and the subsequent neutralino and chargino decays. Still the neutralino relic density can only be constrained within $\pm$40% and the hypothesis $\Omega_{\chi} h^2$ = 0, namely that the neutralino does not contribute to the observed dark matter density in the universe, cannot be ruled out, based only on LHC data. At a 0.5 TeV collider, the main SUSY reactions are $e^+e^- \to \chi^+_1 \chi^-_1$ and $e^+e^- \to \chi^0_2 \chi^0_3$. Operation at 1 TeV gives access also to $e^+e^- \to \chi^+_2 \chi^-_2$ and $e^+e^- \to \chi^0_3 \chi^0_4$. Not only the gaugino mass splittings but also the polarised neutralino and chargino production cross section can be accurately determined at the ILC [@Gray:2005ci]. These measurements fix the gaugino-Higgsino mixing angles, which play a major role in determining the neutralino relic density. The decoupling of the heavier, inaccessible part of the SUSY spectrum, can be insured with the data at the highest energy. The combined ILC data at 0.5 TeV and 1 TeV provide an estimate of the neutralino relic density to $\pm$8 % accuracy, which matches the current WMAP precision. The characteristics featured by the LCC2 point persist, while the SUSY masses increase, provided the gaugino-Higgsino mixing angle remains large enough. This DM-motivated region extends to SUSY masses which eventually exceed the LHC reach, highlighting an intriguing region of parameters where the ILC can still observe sizable production of supersymmetric particle, compatible with dark matter data, while the LHC may report no signals of New Physics [@Baer:2003ru]. Instead, the last two points considered, LCC3 and LCC4, are representative of those regions where the neutralino relic density is determined by accidental relationships between particle masses. Other such regions may also be motivated by baryogenesis constraints [@Balazs:2004bu]. The determination of the neutralino relic density, in such scenarios, depends crucially on the precision of spectroscopic measurements, due to the large sensitivity on masses and couplings. The conclusions of the current studies are that the LHC data do not provide quantitative constraints. On the contrary, the ILC can obtain interesting precision, especially when high energy data is available. \[tab:dmsummary\] The LCC3 point is in the so-called $\tilde \tau$ co-annihilation region. Here, the mass difference between the lightest neutralino, $\chi^0_1$, and the lightest scalar tau, $\tilde \tau_1$, is small enough that $\tilde \tau_1 \chi^0_1 \to \tau \gamma$ can effectively remove neutralinos in the early universe. The relative density of $\tilde \tau$ particles to neutralinos scales as $e^{-\frac{m_{\tilde \tau} - m_{\chi}}{m_{\chi}}}$, so this scenario tightly constrain the $m_{\tilde \tau} - m_{\chi}$ mass difference. Here, the precise mass determinations characteristic of LCC1 will not be available: at 0.5 TeV, the ILC will observe a single final state, $\tau^+ \tau^- + E_{missing}$, from the two accessible SUSY processes [@Khotilovich:2005gb], $e^+e^- \to \tilde \tau_1 \tilde \tau_1$, $\tilde \tau \to \tau \chi^0_1$ and $e^+e^- \to \chi^0_1 \chi^0_2$, $\chi^0_2 \to \chi^0_1 \tilde \tau \to \chi^0_1 \chi^0_1 \tau \tau$. The signal topology consists of two $\tau$-jets and missing energy. Background processes, such as $e^+e^- \to ZZ$ can be suppressed using cuts on event shape variables. The mass splitting can be determined by a study of the distribution of the invariant mass of the system made by the two $\tau$-jets and the missing energy vector, $M_{j_1 j_2 E_{missing}}$. In this variable, the remaining SM background is confined to low values and the shape and upper endpoint of the $\tilde \tau_1 \tilde \tau_1$ contribution depends on the stau-neutralino mass difference, $\Delta M = M_{\tilde \tau_1} - M_{\chi^0_1}$. Templates functions can be generated for different values of $\Delta M$ and the mass difference is extracted by a $\chi^2$ fit of these templates to the “data”. As the $\Delta M$ value decreases, the energy available to the $\tau$ leptons decreases. Since $\tau$ decays involve neutrinos, additional energy is lost from detection. When the $\tau \tau$ system becomes soft, the four fermion background process $ee \to ee \tau \tau$, the so-called $\gamma \gamma$ background which has cross sections at the nb level, makes its detection increasingly difficult. What makes possible to reject these $\gamma \gamma$ events is the presence of the two energetic primary electrons at small angle w.r.t. the beamline [@Bambade:2004tq]. This is a significant challenge for low angle calorimetry, since the electron has to be detected in an hostile environment populated by a large number of other electrons, of lower energy, arising from pairs created during the bunch collision [@Chen:1989ss; @Tauchi:1993tm]. A detailed study [@Khotilovich:2005gb], performed for a statistics of 500 fb$^{-1}$, shows that values of $\Delta M$ as small as 5 GeV can be measured at the ILC, provided the primary electrons can be vetoed down to 17 mrad. In the specific case of the LCC3 point, where the mass splitting, $\Delta M$, is 10.8 GeV, an accuracy of 1 GeV can be achieved. Heavier gauginos, as well as the $A^0$ boson, become accessible operating the ILC at 1 TeV. These data constrain both the mixing angles and $\tan \beta$. As a result the neutralino relic density can be estimated with an 18 % accuracy. Finally, the LCC4 point, chosen in the $A$ funnel, has the DM density controlled by the $\chi \chi \to A$ process. This point is rather instructive in terms of the discovery-driven evolution of a possible experimental program at the ILC. The ILC can obtain the neutralino and $\tilde \tau$ masses at 0.5 TeV, following the same technique as for LCC3. We would also expect LHC experiments to have observed the $A^0$ boson, but it is unlikely $M_A$ could be determined accurately in $pp$ collisions, since the available observation mode is the decay in $\tau$ lepton pairs. At this stage, it would be apparent that the mass relation between the neutralino mass, accurately measured by the ILC at 0.5 TeV, and the $A$ boson mass, from the LHC data, is compatible with $M_A \simeq 2 M_{\chi}$, as required for the s-channel annihilation process to be effective. Three more measurements have to be performed at the ILC: the $A^0$ mass, $M_A$, and width $\Gamma_A$ and the $\mu$ parameter, which is accessible through the mass splitting between heavier neutralinos, $\chi^0_3$, $\chi^0_4$ and the lighter $\chi^0_1$, $\chi^0_2$. All these measurements are available by operating the ILC at 1 TeV. $M_A$ and $\Gamma_A$ can be determined by studying the $A^0$ production in association with a $H^0$ boson, in the reaction $e^+e^- \to A^0 H^0 \to b \bar b b \bar b$. This process results in spectacular events with four $b$ jets, emitted almost symmetrically, due to low energy carried by the heavy Higgs bosons (see Figure \[fig12\]a). The cross section, for the parameters of LCC4 corresponding to $M_A$ = 419 GeV, is just 0.9 fb highlighting the need of large luminosity at the highest energy. Jet flavour tagging and event shape analysis significantly reduces the major multi-jet backgrounds, such as $WW$, $ZZ$ and $t \bar t$. The SM $b \bar b b \bar b$ electro-weak background has a cross section of $\sim$3 fb, but since it includes $Z^0$ or $h^0$ as intermediate states it can be efficiently removed by event shape and mass cuts. After event selection, the $A^0$ mass and width must be reconstructed from the measured di-jet invariant masses. This is achieved by pairing jets in the way that minimises the resulting di-jet mass difference, since the masses of the $A$ and $H$ bosons are expected to be degenerate within a few GeV, and the di-jet masses are computed by imposing constraints on energy and momentum conservation to improve the achievable resolution and gain sensitivity to the boson natural width (see Figure \[fig12\]b). The result is a determination of the $A$ mass to 0.2 % and of its width to $\simeq$15 % if a sample of 2 ab$^{-1}$ of data can be collected. The full set of ILC data provides a neutralino relic density evaluation with 19 % relative accuracy. The full details of how these numbers were obtained can be found in Ref. [@Baltz:2006fm] . SUSY offers a compelling example for investigating the complementarity in the search and discovery of new particles and in the study of their properties at the LHC and ILC. The connection to cosmology, through the study of dark matter brings precise requirements in terms of accuracy and completeness of the anticipated measurements and puts emphasis on scenarios at the edges of the parameter phase space. The interplay of satellite, ground-based and collider experiments in cosmology and particle physics will be unique and it will lead us to learn more about the structure of our Galaxy and of the Universe as well as of the underlying fundamental laws of the elementary particles. This quest will represent an major effort for science in the next several decades. The scenarios discussed above highlight the essential role of the ILC in this context. It will testing whether the particles observed at accelerators are responsible for making up a sizeable fraction of the mass of the Universe, through precision spectroscopic measurements. The data obtained at the ILC will effectively remove most particle physics uncertainties and become a solid ground for studying dark matter in our galaxy through direct and indirect detection experiments [@Feng:2005nz]. ### Indirect Sensitivity to New Physics at the ILC {#sec3.3} Beyond Supersymmetry there is a wide range of physics scenarios invoking new phenomena at, and beyond, the TeV scale. These may explain the origin of electro-weak symmetry breaking, if there is no light elementary Higgs boson, stabilise the SM, if SUSY is not realised in nature, or embed the SM in a theory of grand unification. The ILC, operating at high energy, represents an ideal laboratory for studying this New Physics in ways that are complementary to the LHC [@zp1; @dominici]. Not only it may directly produce some of the new particles predicted by these theories, the ILC also retains an indirect sensitivity, through precision measurements of virtual corrections to electro-weak observables, when the new particle masses exceed the available centre-of-mass energy. One of the simplest of such SM extensions consists of the introduction of an additional $U(1)$ gauge symmetry, as predicted in some grand unified theories [@Hewett:1993st; @Rizzo:2006nw]. The extra $Z'$ boson, associated to the symmetry, naturally mixes with the SM $Z^0$. The mixing angle is already strongly constrained, by precision electroweak data, and can be of the order of few mrad at most, while direct searches at Tevatron for a new $Z'$ boson set a lower limit on its mass around 800 GeV, which may reach 1 TeV by the time the LHC will start searching for such a state. The search for an extended gauge sector offers an interesting framework for studying the ILC sensitivity to scales beyond those directly accessible. It also raises the issue of the discrimination between different models, once a signal would be detected. The main classes of models with additional $Z'$ bosons include $E_6$ inspired models and left-right models (LR). In the $E_6$ models, the $Z'$ fermion couplings depend on the angle, $\theta_6$, defining the embedding of the extra $U(1)$ in the $E_6$ group. At the ILC, the indirect sensitivity to the mass of the new boson, $M_{Z'}$, can be parametrised in terms of the available integrated luminosity, ${\cal{L}}$, and centre-of-mass energy, $\sqrt{s}$. A scaling law for large values of $M_{Z'}$ can be obtained by considering the effect of the $Z'-\gamma$ interference in the two fermion production cross section $\sigma {\mathrm{(}}e^+e^- \to f \bar f {\mathrm{)}}$ ($\sigma_{f \bar f}$ in the following). For $s<< M_{Z'}^2$ and assuming the uncertainties $\delta \sigma$ to be statistically dominated, we obtain the following scaling for the difference between the SM cross section and that in presence of the $Z'$, in units of the statistical accuracy: $$\frac{\|\sigma^{SM}_{f \bar f} - \sigma^{SM+Z'}_{f \bar f}\|}{\delta \sigma} \propto \frac{1}{M^2_{Z'}}\sqrt{s {\cal{L}}}$$ from which we can derive that the indirect sensitivity to the $Z'$ mass scales with the square of the centre-of-mass energy and the luminosity as: $$M_{Z'} \propto (s {\cal{L}})^{1/4}. \label{resc}$$ In a full analysis, the observables sensitive to new physics contribution in two-fermion production are the cross section $\sigma_{f \bar f}$, the forward-backward asymmetries $A_{FB}^{f \bar f}$ and the left-right asymmetries $A_{LR}^{f \bar f}$. The ILC gives us the possibility to study a large number of reactions, $e^+_R e^-_L$, $e^+_R e^-_R \to (u \bar u~+~d \bar d),~s \bar s,~c \bar c,~b \bar b, ~t \bar t,~e^+e^-,~\mu^+\mu^-,~\tau^+\tau^-$ with final states of well defined flavour and, in several cases, helicity. In order to achieve this, jet flavour tagging is essential to separate $b$ quarks from lighter quarks and $c$ quarks from both $b$ and light quarks. Jet-charge and vertex-charge reconstruction allows then to tell the quark from the antiquark produced in the same event [@Ackerstaff:1997ke; @Abe:2004hx]. Similarly to LEP and SLC analyses, the forward-backward asymmetry can be obtained from a fit to the flow of the jet charge $Q^{jet}$, defined as $Q^{jet} = \frac{\sum_i q_i \|p_i T\|^k}{\sum_i \|p_i T\|^k}$, where $q_i$ is the particle charge, $p_i$ its momentum, $T$ the jet thrust axis and the sum is extended to all the particles in a given jet. Another possible technique uses the charge of secondary particles to determine the vertex charge and thus the quark charge. The application of this technique to the ILC has been studied in some details in relation to the optimisation of the Vertex Tracker [@Hillert:2005rp]. At ILC energies, the $e^+e^- \rightarrow f \bar f$ cross sections are significantly reduced, compared to those at LEP and SLC: at 1 TeV the cross section $\sigma(e^+e^- \to b \bar b)$ is only 100 fb, so high luminosity is essential and new experimental issues emerge. At 1 TeV, the ILC beamstrahlung parameter doubles compared to 0.5 TeV, beam-beam effects becoming important, and the primary $e^+e^-$ collision is accompanied by $\gamma \gamma \rightarrow {\mathrm{hadrons}}$ interactions [@Chen:1993db]. Being mostly confined in the forward regions, this background may reduce the polar angle acceptance for quark flavour tagging and dilute the jet charge separation using jet charge techniques. The statistical accuracy for the determination of $\sigma_{f \bar f}$, $A_{FB}^{f \bar f}$ and $A_{LR}^{f \bar f}$ has been studied, for $\mu^+\mu^-$ and $b \bar b$, taking the ILC parameters at $\sqrt{s}$ = 1 TeV. The additional particles from the $\gamma \gamma$ background cause a broadening of the $Q^{jet}$ distribution and thus a dilution of the quark charge separation. Detailed full simulation and reconstruction is needed to fully understand these effects. Despite these backgrounds, the anticipated experimental accuracy in the determination of the electro-weak observables in two-fermion processes at 1 TeV is of the order of a few percent, confirming the ILC role as the precision machine. Several scenarios of new physics have been investigated [@Riemann:1997py; @Battaglia:2001fr]. The analysis of the cross section and asymmetries at 1 TeV would reveal the existence of an additional $Z'$ boson up to $\simeq$ 6-15 TeV, depending on its couplings. As a comparison the LHC direct sensitivity extends up to approximately 4-5 TeV. The ILC indirect sensitivity also extends to different models on new physics, such as 5-dimensional extension of the SM with fermions on the boundary for a compactification where scales up to about 30 TeV can be explored. Finally, fermion compositeness or the exchange of very heavy new particles can be described in terms of effective four-fermion contact interactions [@Eichten:1983hw]. The interaction depends on a scale $\Lambda = M_X/g$, where $M_X$ is the mass of the new particle and $g$ the coupling. Limits to this scale $\Lambda$ can be set up to $\simeq$ 100 TeV, which shows that the ILC sensitivity to new phenomena can exceed its centre-of-mass energy by a significant factor. In order to maximise this indirect sensitivity to new physics, the precision of the SM predictions should match the experimental accuracy. Now, at TeV energies, well above the electroweak scale,the ILC will face the effects of large non-perturbative corrections. Large logarithms $\propto \alpha^n~log^{2n} (M^2/s)$ arise from the exchange of collinear, soft gauge bosons and are known as Sudakov logarithms [@Melles:2001ye]. At 1 TeV the logarithmically enhanced $W$ corrections to $\sigma_{b \bar b}$, of the form $\alpha~log^2 (M_W^2/s)$ and $\alpha~log (M_W^2/s)$ amount to 19% and -4% respectively. The effect of these large logarithmic corrections has been studied in some details [@Ciafaloni:1999ub; @Battaglia:2004mw]. It will be essential to promote a program of studies to reduce these theoretical uncertainties, to fully exploit the ILC potential in these studies. ### Run Plan Scenario {#sec3.4} One of the points of strength of the ILC is in its remarkable flexibility of running conditions. Not only the centre-of-mass energy can be changed over approximately an order of magnitude, but the beam particle and their polarization state can be varied to suit the need of the physics processes under study. At the same time, the ILC program is most diversified and data taken at the same centre-of-mass energy may be used for very different analyses, such as precise top mass determination, Higgs boson studies and reconstruction of SUSY decays. This has raised concerns whether the claimed ILC accuracies can be all achieved with a finite amount of data. A dedicated study was performed in 2001, under the guidance of Paul Grannis, taking two physics scenarios with Supersymmetry realised at relatively low mass, one being the LCC1 benchmark point, rich in pair-produced particles and requiring detailed threshold scans [@Battaglia:2002ey]. The study assumes a realistic profile for the delivered luminosity, which increases from 10 fb$^{-1}$ in the first year to 200 fb$^{-1}$ in the fifth year and 250 fb$^{-1}$ afterward, for a total integrated equivalent luminosity $\int {\cal{L}}$ = 1 ab$^{-1}$. The proposed run plan starts at the assumed maximum energy of 0.5 TeV for a first determination of the sparticle masses through the end-point study and then scans the relevant thresholds, including $t \bar t$ in short runs with tuned polarization states. A summary is given in Table \[tab:runplan\]. \[tab:runplan\] This plan devotes approximately two third of the total luminosity at, or near, the maximum energy, so the program will be sensitive to unexpected new phenomena at high energy, while providing accurate measurements of masses through dedicated scans. Sensors and Detectors for the ILC {#sec4} --------------------------------- The development of the ILC accelerator components and the definition of its physics case has been paralleled by a continuing effort in detector design and sensor R&D. This effort is motivated by the need to design and construct detectors which match the ILC promise to provide extremely accurate measurements over a broad range of collision energies and event topologies. It is important to stress that, despite more than a decade of detector R&D for the LHC experiments, much still needs to be done to obtain sensors matching the ILC requirements. While the focus of the LHC-motivated R&D has been on sensor radiation hardness and high trigger rate, the ILC, with its more benign background conditions and lower interaction cross sections, admits sensors of new technology which, in turn, have better granularity, smaller thickness and much improved resolution. Sensor R&D and detector design are being carried out world-wide and are starting deploying prototype detector modules on test beamlines. ### Detector Concepts {#sec4.1} The conceptual design effort for an optimal detector for the ILC interaction region has probed a wide spectrum of options which span from a spherical detector structure to improved versions of more orthodox barrel-shaped detectors. These studies have been influenced by the experience with SLD at the SLC, ALEPH, DELPHI and OPAL at LEP, but also with ATLAS and CMS at the LHC. The emphasis on accurate reconstruction of the particle flow in hadronic events and thus of the energy of partons is common to all designs. The main tracker technology drives the detector designs presently being studied. Four detector concepts have emerged, named GLD, LDC, SiD and 4$^{th}$ Concept [@concepts]. A large volume, 3D continuous tracking volume in a Time Projection Chamber is the centerpiece of the GLD, the LDC and the so-called 4$^{th}$ Concept designs. The TPC is followed by an highly segmented electro-magnetic calorimeter for which these three concepts are contemplating different technologies A discrete tracker made of layers of high precision Silicon microstrip detectors, and a larger solenoidal field, which allows to reduce the radius, and thus the size, of the calorimeter is being studied in the context of the SiD design. Dedicated detector design studies are being carried out internationally [@Behnke:2005re; @Abe:2006bz] to optimise, through physics benchmarks [@Battaglia:2006bv], the integrated detector concepts. Such design activities provide a bridge from physics studies to the assessment of priorities in detector R&D and are evolving towards the completion of engineered design reports at the end of this decade, synchronously with that foreseen for the ILC accelerator. ### Vertexing and Tracking {#sec4.2} The vertex and main tracker detectors must provide jet flavour identification and track momentum determination with the accuracy which makes the ILC such a unique facility for particle physics. The resolution in extrapolating charged particle trajectories to their production point, the so-called impact parameter, is dictated by the need to distinguish Higgs boson decays to $c \bar c$ from those to $b \bar b$ pairs, but also $\tau^+ \tau^-$ and gluon pairs, as discussed in section \[sec3.1\]. In addition, vertex charge measurements put emphasis on precise extrapolation of particle tracks down to very low momenta. Tagging of events with multiple $b$ jets, such as $e^+e^- \to H^0A^0 \to b \bar b b \bar b$, discussed in section \[sec3.2\], underscores the need of high tagging efficiency, $\epsilon_b$, since the overall efficiency scales as $\epsilon_b^N$, where $N$ is the number of jets to be tagged. This is best achieved by analysing the secondary vertex structures in hadronic jets. A $B$ meson, from a Higgs boson produced at 0.5 TeV, has an average energy of $x_B \sqrt{s}/4 \simeq$ 100 GeV, where $x_B \simeq 0.7$ represents the average $b$ fragmentation function, or a $\gamma$ value of $\simeq$ 70. Since $c \tau \simeq$ 500 $\mu$m, the average decay distance $\beta \gamma c \tau$ is 3.5 mm and the average impact parameter, $\beta \gamma c \tau \sin \theta$, is 0.5 mm. In comparison, a $D$ meson from a $H \to c \bar c$ decay has a decay length of 1.3 mm. More importantly, the average charged decay multiplicity for a $B$ meson is 5.1, while for a $D$ meson is 2.7. Turning these numbers into performance requirements sets the target accuracy for the asymptotic term $a$ and the multiple scattering term $b$ defining the track extrapolation resolution in the formula $$\sigma_{\mathrm{extrapolation}} = {\mathrm{a}}~\oplus \frac{{\mathrm{b}}}{p_t}$$ The ILC target values are compared to those achieved by the DELPHI experiment at LEP, those expected for ATLAS at the LHC and the best performance ever achieved at a collider experiment, that of SLD, in Table \[tab:ipres\]. \[tab:ipres\] This comparison shows that the improvements required for ILC on state-of-the-art technology is a factor 2-5 on asymptotic resolution and another factor 3-7 on the multiple scattering term. At the ILC, particle tracks in highly collimated jets contribute a local track density on the innermost layer of 0.2-1.0 hits mm$^{-2}$ at 0.5 TeV, to reach 0.4-1.5 hits mm$^{-2}$ at 1.0 TeV. Machine-induced backgrounds, mostly pairs, add about 3-4 hits mm$^{-2}$, assuming that the detector integrates 80 consecutive bunch crossings in a train. These values are comparable to, or even exceed, those expected on the innermost layer of the LHC detectors: 0.03 hits mm$^{-2}$ for proton collisions in ATLAS and 0.9 hits mm$^{-2}$ for heavy ion collisions in ALICE. Occupancy and point resolution set the pixel size to 20x20 $\mu$m$^2$ or less. The impact parameter accuracy sets the layer material budget to $\le 0.15\%~X_0$/layer. This motivates the development of thin monolithic pixel sensors. Charge coupled devices (CCD) have been a prototype architecture after the success of the SLD VXD3 [@Abe:1999ky]. However, to match the ILC requirements in terms of radiation hardness and read-out speed significant R&D is needed. New technologies, such as CMOS active pixels [@Turchetta:2001dy], SOI [@Marczewski:2005vy] and DEPFET [@Richter:2003dn] sensors, are emerging as promising, competitive alternatives, supported by an intensive sensor R&D effort promoted for the ILC [@Battaglia:2003kn]. The process $e^+e^- \to H^0Z^0$, $H^0 \to X$, $Z^0 \to \ell^+ \ell^-$ gives access to Higgs production, irrespective of the Higgs decay properties. Lepton momenta must be measured very accurately for the recoil mass resolution to be limited by the irreducible smearing due to beamstrahlung. Since the centre-of-mass energy $\sqrt{s} = E_{H} + E_{Z}$ is known and the total momentum $p_{H} + p_{Z} = 0$, the Higgs mass, $M_H$ can be written as: $$M_H^2 = E_H^2 - p_H^2 = (\sqrt{s} - E_Z)^2 - p_Z^2 = s + E_Z^2 - 2\sqrt{s} E_Z - p_Z^2 = s-2\sqrt{s} E_Z + M_Z^2$$ In the decay $Z^0 \to \mu^+ \mu^-$, $E_Z = E_{\mu^+} + E_{\mu^-}$ so that the resolution on $M_H$ depends on that on the muon momentum. In quantitative terms the resolution required is $$\delta p / p^2 < 2 \times 10^{-5}$$ A comparison with the performance of trackers at LEP and LHC is given in Table \[tab:pres\]. \[tab:pres\] The ability to tag Higgs bosons, independent on their decay mode is central to the ILC program in Higgs physics. A degraded momentum resolution would correspond to larger background, mostly from $e^+e^- \to ZZ^$, being accepted in the Higgs signal sample. This degrades the accuracy on the determination of the Higgs couplings both in terms of statistical and systematic uncertainties. The particle momentum is measured through its bending radius $R$ in the solenoidal magnetic field, $B$. The error on the curvature, $k=1/R$, for a particle track of high momentum, measured at $N$ equidistant points with an accuracy, $\sigma$, over a length L, applying the constraint that it does originate at the primary vertex (as for the leptons from the $Z^0$ in the Higgstrahlung reaction) is given by [@pdg]: $$\delta k = \frac{\sigma}{L^2}\sqrt{\frac{320}{N+4}}$$ This shows that the same momentum resolution can be achieved either by a large number of measurements, each of moderate accuracy, as in the case of a continuous gaseous tracker, or by a small number of points measured with high accuracy, as in the case of a discrete Si tracker. Continuous tracking capability over a large area, with timing information and specific ionization measurement, and its robust performance make the Time Projection Chamber an attractive option for precision tracking at the ILC. The introduction of Micro Pattern Gaseous Detectors [@Giomataris:1995fq; @Sauli:1997qp] (MPGD) offers significant improvements in terms of reduced $E \times B$, larger gains, ion suppression and faster, narrower signals providing better space resolution. Improving on the space resolution requires an optimal sampling of the collected charge, while the high solenoidal magnetic field reduces the diffusion effects. Several paths are presently being explored with small size prototypes operated on beamlines and in large magnetic fields [@Kappler:2004cg; @Colas:2004ks]. A multi-layered Si strip detector tracker in an high $B$ field may offer a competitive $\delta p/p^2$ resolution with reduced material budget and afford a smaller radius ECAL, thus reducing the overall detector cost. This is the main rationale promoting the development of an all-Si concept for the main tracker, which follows the spirit of the design of the CMS detector at LHC. Dedicated conceptual design and module R&D is being carried out as a world-wide program [@Kroseberg:2005ue]. There is also considerable R&D required for the engineering of detector ladders, addressing such issues as mechanical stability and integration of cooling and electrical services. These modules may also be considered as supplemental tracking devices in a TPC-based design to provide extra space points, with high resolution, and in end-cap tracking planes. Assessing the required detector performance involves realistic simulation and reconstruction code accounting for inefficiencies, noise, overlaps and backgrounds. ### Calorimetry {#sec4.3} The ILC physics program requires precise measurements of multi-jet hadronic events, in particular di-jet invariant masses to identify $W$, $Z$ and Higgs bosons, through their hadronic decays. An especially demanding reaction is $e^+e^- \to Z^0 H^0 H^0$, which provides access to the triple Higgs coupling as discussed in section \[sec3.1\]. The large background from $e^+e^- \to Z^0 Z^0 Z^0$ can be reduced only by an efficient $H^0$/$Z^0$ separation, based on their masses. This impacts the parton energy resolution through the measurement of hadronic jets. Detailed simulation [@Castanier:2001sf] shows that a jet energy resolution $\frac{\sigma_{E_{jet}}}{E_{jet}} \simeq \frac{0.30}{\sqrt{E}}$ is required, in order to achieve an interesting resolution on the $g_{HHH}$ coupling. The analysis of other processes, such as $e^+e^- \to W^+W^- \nu \bar \nu$ and Higgs hadronic decays, leads to similar conclusions [@Brient:2002gh]. In the case of the determination of $H^0 \to W^+W^-$ branching fractions, the statistical accuracy degrades by 22 % when changing the jet energy resolution from $\frac{0.30}{\sqrt{E}}$ to $\frac{0.60}{\sqrt{E}}$. Such performance is unprecedented and requires the development of an advanced calorimeter design as well as new reconstruction strategies. The most promising approach is based on the [particle flow algorithm]{} (PFA). The energy of each particle in an hadronic jet is determined based on the information of the detector which can measure it to the best accuracy. In the case of charged particles, this is achieved by measuring the particle bending in the solenoidal field with the main tracker. Electromagnetic neutrals ($\gamma$ and $\pi^0$) are measured in the electromagnetic calorimeter and hadronic neutrals ($K^0_L$, $n$) in the hadronic calorimeter. The jet energy is then obtained by summing these energies: $$E_{jet} = E_{charged} + E_{em~neutral} + E_{had~neutral}$$ each being measured in a specialised detector. The resolution is given by: $$\sigma^2_{E_{jet}} = \sigma^2_{charged} + \sigma^2_{em~neutral} + \sigma^2_{had~neutral} + \sigma^2_{confusion}.$$ Assuming the anticipated momentum resolution, $\sigma_{E} \simeq 0.11/\sqrt{E}$ for the e.m. calorimeter, $\sigma_E \simeq 0.40/\sqrt{E}$ for the hadronic calorimeter and the fractions of charged, e.m. neutral and hadronic neutral energy in an hadronic jet we get: $$\sigma^2_{charged} \simeq (0.02 {\mathrm{GeV}})^2 \frac{1}{10} \sum \frac{E^4_{charged}}{(10 {\mathrm{GeV}})^4}$$ $$\sigma^2_{em~neutral} \simeq (0.6 {\mathrm{GeV}})^2 \frac{E_{jet}}{100 {\mathrm{GeV}}}$$ $$\sigma^2_{had~neutral} \simeq (1.3 {\mathrm{GeV}})^2 \frac{E_{jet}}{100 {\mathrm{GeV}}}$$ In case of perfect energy-particle association this would correspond to a jet resolution $\simeq 0.14/\sqrt{E}$. But a major source of resolution loss turns out to be the confusion term, $\sigma_{confusion}$, which originates from inefficiencies, double-counting and fakes, which need to be minimised by an efficient pattern recognition. This strategy was pioneered by the ALEPH experiment at LEP, where a resolution $\simeq 0.60/\sqrt{E}$ was obtained, starting from the stochastic resolutions of $\sigma_{E} \simeq 0.18/\sqrt{E}$ for the e.m. calorimeter, and $\sigma_E \simeq 0.85/\sqrt{E}$ for the hadronic calorimeter [@Buskulic:1994wz]. At hadron colliders, the possible improvement from using tracking information together with calorimetric measurements is limited, due to underlying events and the shower core size. On the contrary, at the ILC these limitations can be overcome, by developing an imaging calorimeter, where spatial resolution becomes as important as energy resolution. The minimisation of the confusion rate can then be obtained by choosing a large solenoidal field, $B$, and calorimeter radius, $R$, to increase the separation between charged and neutral particles in dense jets, a small Moliere radius, $R_M$, for the e.m. calorimeter, to reduce the transverse shower spread and small cells, $R_{pixel}$, with large longitudinal segmentation. The distance between a neutral and a charged particle, of transverse momentum $p_t$, at the entrance of the e.m. calorimeter located at a radius $R$ is given by $0.15 B R^2 / p_t$, where $B$ is the solenoidal magnetic field. A useful figure of merit of the detector in terms of the particle flow reconstruction capability is then offered by: $$\frac{B R^2}{R_M^2 R_{pixel}^2}$$ which is a measure of the particle separation capability. The value of $BR^2$ is limited to about 60 Tm$^2$ by the mechanical stability. An optimal material in terms of Moliere radius is Tungsten, with $R_M$ = 9 mm. In four-jet events at $\sqrt{s}$=0.8 TeV, there are on average 28 GeV per di-jet carried by photons, which are deposited within 2.5 cm from a charged particle at the e.m. calorimeter radius. With pixel cells of order of $1 \times 1$ cm$^2$ to ensure sufficient transverse segmentation and 30 to 40 layers in depth, the e.m. calorimeter would consists of up to 30 M channels and 3000 m$^2$ of active Si. Due to the large amount of channels and the wish to use an absorber with the smallest possible Moliere radius, the e.m. calorimeter is the main cost-driver of the ILC detector and its optimisation in terms of performance and cost requires a significant R&D effort. A Silicon-Tungsten calorimeter (SiW) was first proposed in the framework of the TESLA study [@Videau:2000es; @Behnke:2001qq] and it is currently being pursued by large R&D collaborations in both Europe and the US. Alternative technologies are also being studied by the GLD and the 4$^{th}$ Concept. This R&D program involves design, prototyping and tests with high energy particle beams and it is being carried out world-wide [@Strom:2005id; @Mavromanolakis:2005yh; @Strom:2005xt], supported by efforts on detailed simulation and reconstruction. Epilogue -------- The ILC promises to complement and expand the probe into the TeV scale beyond the LHC capabilities, matching and improving its energy reach while adding precision. Its physics program will address many of the fundamental questions of today’s physics from the origin of mass, to the nature of Dark Matter. After more than two decades of intense R&D carried out world-wide, the $e^+e^-$ linear collider, with centre-of-mass energies up to 1 TeV, has become technically feasible and a costed reference design is now available. Detectors matching the precision requirements of its anticipated physics program are being developed in an intense R&D effort carried out world-wide. Now, theoretical predictions matching the anticipated experimental accuracies are crucially needed, as well as further clues on what physics scenarios could be unveiled by signals that the LHC may soon be observing. These will contribute to further define the physics landscape for the ILC. A TeV-scale electron-positron linear collider is an essential component of the research program that will provide in the next decades new insights into the structure of space, time, matter and energy. Thanks to the efforts of many groups from laboratories and universities around the world, the technology for achieving this goal is now in hand, and the prospects for the ILC success are extraordinarily bright. Acknowledgments {#acknowledgments .unnumbered} --------------- I am grateful to the TASI organisers, in particular to Sally Dawson and Rabindra N. Mohapatra, for their invitation and the excellent organization. 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--- abstract: 'Trapped ultracold neutrons (UCN) have for many years been the mainstay of experiments to search for the electric dipole moment (EDM) of the neutron, a critical parameter in constraining scenarios of new physics beyond the Standard Model. Because their energies are so low, UCN preferentially populate the lower region of their physical enclosure, and do not sample uniformly the ambient magnetic field throughout the storage volume. This leads to a substantial increase in the rate of depolarization, as well as to shifts in the measured frequency of the stored neutrons. Consequences for EDM measurements are discussed.' author: - 'P.G. Harris' - 'J.M. Pendlebury' - 'N.E. Devenish' bibliography: - 'neutron\_edm.bib' title: 'Gravitationally enhanced depolarization of ultracold neutrons in magnetic field gradients, and implications for neutron electric dipole moment measurements ' --- Introduction ============ Ultracold neutrons (UCN) are neutrons of extremely low energy, typically less than or of the order of 200 neV, which therefore have wavelengths that are long compared with the spacing between atomic nuclei in solids. The surfaces of many materials then appear as a positive potential barrier (the so-called Fermi potential) from which these neutrons reflect. This allows the storage of such neutrons in material bottles, typically for several minutes at a time, which in turn permits the study of their fundamental properties. One such study is the ongoing search for the electric dipole moment (EDM) of the neutron, of which the most recent measurement was carried out at the Institut Laue-Langevin, Grenoble, by a collaboration led by the University of Sussex and the Rutherford Appleton Laboratory,[@baker06] using apparatus at room temperature (in contrast to its cryogenic successor, now under development). The internal volume of the neutron trap used in the room-temperature EDM experiment (RT-nEDM) was an upright cylinder 12 cm high, with quartz walls 37 cm in diameter and a roof and floor of aluminium coated with diamond-like carbon. Crucial to the analysis of the experimental data was the fact that the UCN, being of very low energy, tended to populate preferentially the lower part of the storage volume, whereas the cohabiting mercury ($^{199}$Hg) magnetometer[@green98] filled the volume uniformly. Any vertical magnetic-field gradient $\dBzdz$ applied to the volume would affect the two species differently, such that the gyromagnetic-ratio-corrected ratio of the neutron and mercury Larmor precession frequencies $$\label{eqn:R} R = \left\| \frac{\nu_n}{\nu_{\rm Hg}}\cdot \frac{\gamma_{\rm Hg}}{\gamma_{n}} \right\|$$ would, to first order, be shifted by $$\label{eqn:DeltaR} \Delta R = \pm \Delta h \cdot \frac{\dBzdz}{B_{0_z}},$$ where $\Delta h$ is the (always positive) difference in height between the centre of mass of the mercury and that of the UCN, and the $\pm$ sign depends upon the relative directions of $B_{0_z}$ and $\dBzdz$: $R$ increases (i.e. $\Delta R$ becomes more positive) as the absolute magnitude of the field at the bottom of the storage cell (sampled preferentially by the neutrons) increases relative to that at the top of the cell. The Larmor precession frequency of the UCN was measured by means of the Ramsey method of separated oscilliatory fields, for which a time $T$ = 130 s between the two r.f. pulses was used consistently. During this period, the UCN would suffer some loss of their (transverse) polarization. This study looks at some of the mechanisms and consequences of this so-called $T_2$ relaxation. For the sake of example, all values of the various parameters used in modelling the phenomenon (storage cell size, Fermi potential, $B_{0_z}$ magnitude etc.) are those appropriate to the RT-nEDM experiment. UCN density distributions ========================= It is convenient to refer to the energy of UCN in terms of the maximum height attainable within Earth’s gravitational field. Phase space arguments can be used to demonstrate[@golub_UCN_book; @pendlebury94] that a population of trapped UCN each of energy $\epsilon$ has a density distribution with height $h$ of the form $$\label{eqn:height_dist} n(h) = \left(1-\frac{h}{\epsilon}\right)^{1/2}n(0).$$ Integration and inversion of this function shows that the height distribution of such UCN within a storage cell of height $H$ may be generated from numbers $X$ distributed uniformly between 0 and 1 via the equation $$\label{eqn:h_generator} h = \epsilon\left[1-\left(1-kX\right)^{2/3}\right].$$ The constant $k=1$ when $\epsilon<H$, and $k=1-\left(1-H/\epsilon\right)^{3/2}$ otherwise. We note that the average height of the UCN within this population is $$\label{eqn:average_height} \left<h\right> = -\frac{\epsilon}{k}\left[0.6-k-0.6\left(1-k\right)^{5/3}\right].$$ UCN may be generated by capturing the very low-energy tail of the Maxwell-Boltzmann distribution within a thermal source, or else by downscattering from e.g. liquid helium in a superthermal source. In either case, the energy distribution tends to be close to $$\label{eqn:energy_distn} n(\epsilon)d\epsilon \propto \epsilon^{1/2}d\epsilon.$$ By the time the UCN are stored, this distribution can change: for example, allowing the UCN to fall under gravity will shift the entire energy distribution upwards; or passage through a polarising foil can remove those of low energy. The top of the energy distribution tends to have a fairly sharp cut-off, corresponding to the Fermi potential of the storage vessel. In the case of RT-nEDM, the UCN rose under gravity after passage through a polarising foil, and the bottom of the storage cell was positioned such that those with just enough energy to pass through the foil would also have just enough energy to reach the cell. Here, therefore, the energy distribution is modelled with the simple function of \[eqn:energy\_distn\], using the 93 cm equivalent height Fermi potential of the quartz walls of the vessel as the cutoff energy. As above, integration and inversion yields a generating function $$\label{eqn:E_generator} \epsilon = \epsilon_{F}Y^{2/3},$$ where $\epsilon_{F}$ is the (Fermi potential) cut-off energy, and the numbers $Y$ are distributed uniformly between 0 and 1. The distribution of average heights of a population of UCN with such an energy distribution is shown in \[fig:height\_dist\]. The centre of mass of this modelled population of UCN is 3.0 mm below the centre of the storage trap, in good agreement with the 2.81 mm reported in Baker et al.[@baker06]. Some 4.6% of the UCN are not sufficiently energetic to reach the top of the trap. There is a small but extended tail, amounting to some 4.6% of the total population, of neutrons that do not have sufficient energy to reach the lid of the bottle. Over time, in a vertical magnetic-field gradient, two processes contributing to $T_2$ depolarisation come into play: (a) There is an energy dependence to the natural depolarisation rate in a magnetic-field gradient, because of the different rates at which the neutrons sample the measurement volume. This will be referred to as the [intrinsic]{} component, and is modelled in this study by means of a simulation. It is applicable even in the absence of a gravitational field. (b) Under gravity, UCN at different average heights effectively sample different magnetic fields, and therefore on average precess at different rates. This will be referred to as the [enhanced]{} component, and is here modelled by means of the analytic distributions described above. \[ht\] Simulation of UCN depolarisation in magnetic-field gradients ============================================================ Other studies have considered $T_1$ (longitudinal) and $T_2$ (transverse) relaxation rates of atoms in various configurations of electromagnetic fields and storage trap geometries.[@gamblin65; @schearer65; @cates88; @cates88b; @mcgregor90; @schmid08] An approach often adopted is that of the autocorrelation function, as outlined by McGregor.[@mcgregor90] In this instance, however, the situation is complicated by the parabolic nature of the orbits of the UCN moving under gravity. For this study, therefore, a Monte Carlo simulation has been developed, in which the UCN move in ballistic trajectories within the RT-nEDM cylindrical trap described above, and their spins evolve classically according to the solution $$\begin{aligned} \vec{\sigma}(t) &= \left(\vec{\sigma}_0-\frac{\left(\vec{\sigma}_0\cdot \vec{B}\right)\vec{B}}{B^2}\right)\cos\left(\omega t\right) \\ &+ \frac{\vec{\sigma}_0\times\vec{B}}{B}\sin\left(\omega t\right) \\ &+ \frac{\left(\vec{\sigma}_0\cdot\vec{B}\right) \vec{B}}{B^2}\end{aligned}$$ of the equation of motion $$\dot{\vec{\sigma}} = \gamma\vec{\sigma}\times\vec{B}$$ of the spin $\vec{\sigma}$ in a magnetic field $\vec{B}$. A vertical holding field $B_{0_z}$ of 1 $\mu$T was applied. Intuitively, since the polarisation is the ensemble average of projections $\cos(\delta \theta)\sim1-\delta\theta^2/2$, one can argue that the depolarisation rate should depend upon the variance of this quantity, and hence on the variance of $B$. Thus, one expects that the (intrinsic) $T_2$ should depend inversely upon $\left(\partial B_z/\partial z\right)^2$. \[fig:T2\_energy\] shows the values of $T_2\cdot(\partial B_z/\partial z)^2$, for a variety of different gradients, as a function of the UCN energy (represented by the maximum achievable height). For convenience, the gradients are in nT/m, which is appropriate for the magnitudes of gradients to be expected in such experiments. The scatter of the data points is representative of the uncertainty. The dependence upon $(\partial B_z/\partial z)^2$ provides an extremely good match across several orders of magnitude. The minimum $T_2$ corresponds to the point at which the UCN just have sufficient energy to reach the roof of the trap. For reference, the case in which gravity provides no influence is also shown. \[ht\] In the simulations underlying \[fig:T2\_energy\], completely specular reflections were assumed to occur 80% of the time. In fact, $T_2$ also has a significant dependence upon the specularity because of the inclination of specular reflections to lead to individual UCN lingering in particular orbits within the trap rather than sampling the volume uniformly. Representative sample points are shown (based upon a field gradient of 100 nT/m) from the equivalent curves with specularities ranging from 0% (perfectly diffuse reflections at each wall collision) to 60%. Effect upon frequency and polarization ====================================== In EDM measurements, the Larmor spin precession frequency is normally determined by the Ramsey method of separated oscillating fields. This allows a precise measurement of the ensemble average difference in accumulated phase (per unit time) between the spins of the UCN population and the reference oscillator providing the spin-manipulating r.f. fields. Over time, as different UCN sample different regions of the trap, the distribution of phases spreads out, and polarization is lost. In the absence of a gravitational field, all of the neutrons would normally sample all regions of the trap with equal probability. The distribution of accumulated phases would therefore be expected to be Gaussian, with the frequency determined by the phase at the peak of the distribution. In the case of UCN, however, the distribution is skewed by the low-energy tail. \[fig:phase\_dist\] shows this distribution of phases for a measurement of 130 s duration (as used in RT-nEDM) in a magnetic-field gradient of 5 nT/m. The numerical values of these phases are relative to that appropriate to the volume-averaged magnetic field, i.e. the field at the geometric centre of the trap. The solid curve shows the distribution excluding the intrinsic contribution: the latter provides a relatively small additional spreading of the phases. The reference phase $\hat{\phi}$, from which the frequency is determined, is given by $$\hat{\phi} = \tan^{-1}\left( \frac{\left<\sin\phi\right>}{\left<\cos\phi\right>} \right),$$ where $\cos\phi$ and $\sin\phi$ are averaged over all of the individual phases $\phi$. $\hat\phi$ is represented on the figure by the central (solid) vertical line, and it is clearly not at the peak of the distribution. Also indicated (by dashed vertical lines) are the phases $\pm \pi$ away from the reference phase. \[ht\] It will be noted that, as the phase distribution of \[fig:phase\_dist\] spreads, the population within the low-energy tail passes beyond $\pi$ radians from the reference phase. Since the Ramsey technique is sensitive only to phase modulo $2\pi$, these neutrons effectively reappear on the other side of the distribution, which pulls the reference phase back up towards the peak again, thus effectively [reducing]{} the frequency shift from the value (c.f. \[eqn:DeltaR\]) $$\Delta \nu = \gamma \Delta h \cdot \frac{\partial B_z}{\partial z}$$ that would naively be expected from the 3 mm height reduction of the centre of mass in combination with the applied magnetic field gradient. This effect is shown in \[fig:freq\_shift\]. \[ht\] The frequency shift also changes with time, as shown in \[fig:freq\_vs\_time\]. This effect is likewise a direct consequence of the asymmetric nature of the distribution of phases in \[fig:phase\_dist\]. If the reference phase $\hat{\phi}$ were simply the average accumulated phase, the frequency would be constant for a given gradient. The frequency change is initially fairly rapid, but then slows down as the tail of the distribution “wraps around” and starts to pull the reference phase back towards the peak. \[ht\] There is an additional second-order frequency-shift effect. Since the intrinsic contribution to the depolarization causes different parts of the energy spectrum to relax at different rates, and since depolarized UCN cannot contribute to the frequency measurement, the energy distribution of contributing UCN changes over time. This in turn changes the effective centre of mass of the polarised UCN, and thus, via the applied field gradient, produces a second-order frequency shift. The depolarization times $T_2$ in this scenario are however sufficiently long that this effect is negligible – about two orders of magnitude smaller than the shifts that we have been considering thus far. The polarization $\alpha$ is the average projection onto the reference phase of all of the spin vectors. $\alpha$ as a function of the applied gradient, again for a measurement period of 130 s, is shown in \[fig:alpha\_peak\]. The intrinsic contribution to the depolarization is shown explicitly as a separate set of points. The structure that is apparent at an applied gradient of 4-5 nT/m is another effect of the tail of the distribution wrapping around and moving towards the peak from the other side: temporarily at least, it reduces the average spread of the distribution, and thus moderates the fall in polarization. $\alpha$ as a function of time, for an applied gradient of 10 nT/m, is presented in \[fig:alpha\_vs\_time\]; once again, the intrinsic contribution (which falls off exponentially) is shown explicitly. \[ht\] \[ht\] Effect upon EDM systematic error calculations ============================================= Whilst any depolarization will reduce the sensitivity of an EDM measurement, it does not of itself bias the result. Likewise, a frequency shift that is independent of the electric field is not necessarily a cause for concern. Effective changes in the velocity spectrum due to differential depolarization could in principle influence the geometric-phase (GP) contribution, although any such effect is liable to be completely negligible: in RT-nEDM, the GP contribution arising from the UCN was approximately fifty times smaller than that from the cohabiting mercury magnetometer. A more significant concern is in the interpretation of GP-induced false-EDM signals. These are proportional to the applied field gradient $\partial B_z/\partial z$. In a real experiment such as RT-nEDM, this gradient can be most easily inferred from the ratio $R$ (see \[eqn:R\]) of the neutron frequency to the frequency of the cohabiting mercury magnetometer, which samples the volume uniformly. The measured EDM signals as a function of this ratio are shown in 13 of [@pendlebury04], and are fitted to the straight lines anticipated from \[eqn:DeltaR\] above. However, the frequency shifts due to the enhanced depolarization mean that the appropriate frequency ratio is more complex than this, and a function similar to that shown in \[fig:freq\_shift\] is required instead. Fitting to these lines should therefore be carried out with due care and attention, and only after careful modelling. It is clearly far preferable to undertake EDM measurements in conditions of very low magnetic-field gradients. Conclusion ========== UCN are of very low energy, and preferentially populate the lower regions of any trap within which they are contained. It has been shown that these gravitational effects result in a significant enhancement of the $T_2$ relaxation of the UCN, and can also lead to shifts in the measured Larmor precession frequency. Although there are potential impacts upon systematic-error calculations for EDM measurements, these are at a very manageable level; nonetheless, they underline the importance both of careful and precise modelling of the system, and also of keeping to an absolute minimum any magnetic-field gradients within the measurement apparatus. Acknowledgments {#acknowledgments .unnumbered} =============== This work was supported in part by grant no. ST/K001329/1 from the UK Science and Technology Facilities Council.	ArXiv
epsf [AMS, a particle spectrometer in space [^1]]{} [M. Buénerd]{}\ \ [for the AMS collaboration]{} Introduction ============ Accurate measurements of particle fluxes close to earth have been performed recently by the AMS experiment, bringing a body of excellent new data on the particle populations in the low altitude terrestrial environment. These results should rejuvenate the long standing interest of a broad community of scientists for the interactions between the cosmic ray (CR) flux and the atmosphere and for the dynamics of particles in the earth neighborhood. They certainly open new prospects for accurate studies of these phenomena to investigate the interaction mechanisms generating the observed populations. The AMS experiment took its first data during a precursor flight on june 2-12, 1998, on the Space Shuttle DISCOVERY. The flight was primitively intended as a qualification test for the spectrometer instrumentation. The orbit altitude was close to 370 km. During 100 Hours of counting, about 10$^8$ events were recorded providing new results of high quality on the particle distributions at the altitude of the detector. Some of these results were rather unexpected. They illustrate the discovery potential of the experiment in its future steps. This contribution is devoted to a general presentation of the project, of the results obtained during this first experimental test and of their interpretation, and of the goals and plans of the forthcoming phase II of the experimental program. The first part will deal with a description of the measurements performed and questions raised on the dynamics of the detected particles in the earth environment, by the results obtained. The second part will describe a phenomenological approach based on a simulation to account for the observed distributions. The third and last part will consist of a description of the phase II AMS spectrometer which will begin on the International Space Station as of next october 2003, which will be very different from the version flown on the shuttle, and of its physics program. The AMS01 precursor flight ========================== The spectrometer operation during the flight has been very successful with only a few instrumental defects, having no significant consequence on the quality of the measurements achieved. The spectrometer ---------------- Figure \[AMS01\] shows a cut view in perspective of the spectrometer which was flown on the shuttle. The apparatus included a cylindrical permanent magnet generating a 0.15 Tesla dipole field perpendicular to the axis of the cylinder inside its volume [@AIMANT]. The inner volume was mapped with a tracker consisting of 6 planes of silicon microstrips partially equipped at this stage, allowing the reconstruction of particle trajectories [@TRACK] . The tracker planes also provided dE/dX measurements of the particles. Above and below the magnet, two double planes of scintillator hodoscopes with perpendicular orientations of their paddles, provided both a measurement of the particle time of flight (TOF) and of their specific energy loss (dE/dX). The paddle location and the position sensitivity inside the paddles also provided a complementary determination of the particle hit coordinates, useful for background rejection. A skirt of scintillators around the tracker was used to veto on particles outside the fiducial angular acceptance of the counter. At the bottom of the device a threshold Cherenkov counter equipped with n=1.035 aerogel material allowed $p/e^+$ and $\bar{p}$/e$^-$ discrimination below the $p(\bar{p})$ threshold around 4 GeV/c particles [@ATC]. Results ------- Some of the results have already been published [@HEBAR; @PROT1; @PROT2; @LEPT; @HE]. The measured data are still under analysis however, and the physics issues addressed by the experiment are being actively investigated. Some of the latters are discussed on in the following. ### Search for antimatter The first claimed objective of the experiment is the search for primordial antimatter in space. It was then very important to investigate the capability of the spectrometer to identify antiparticles with Z$\geq$2, and to identify and reject background events.\ $\bullet$ [Antihelium [[@HEBAR]]{} - ]{} Figure \[AHE\] shows the spectral distribution of Z=2 particles as a function of their rigidity, i.e., momentum/charge, the sign of the charge being measured by the sign of the trajectory curvature in the tracker. Positive rigidities correspond to He particles, whereas antiHeliums are expected on the negative side. A few fake antiheliums due to soft interactions in the detector, were rejected by means of appropriate cuts on the energy deposit in the tracker planes. Finally the experiment has allowed to set a new lower limit on the $\overline{He}$/$He$ fraction in cosmic rays, of 1.1 10$^{-6}$. See [@BESS] for recent results from the BESS experiment.\ $\bullet$ [Antimatter nuclei Z$>$2 - ]{} The particle identification capabilities of the spectrometer have been used to search also for antimatter nuclei with Z$>$2. This search has been negative so far. The limit obtained will be reported in a future publication. ### Protons [[@PROT1; @PROT2]]{} The CR proton distribution was already very well known from previous experiments before the AMS flight. The measurements were intended to be used for checking and calibrating the experiments, no new result being expected. Figure \[PROTONS\] show the kinetic energy distributions of incoming particles (towards earth) measured by AMS in bins of latitude. The spectra show some expected features like the power law decrease with energy. The geomagnetic cutoff (GC) due to the sweeping away of particles by the earth magnetic field below a critical momentum, is clearly observed in the spectra, decreasing from about 15 GeV around the equator down to zero in the polar region. The spectrum at high latitudes is in good agreement with previous measurements. Although no significant flux was expected below GC, a strong rise of the spectra at low energy is observed at all low latitudes with a strong enhancement in the equatorial region. The Albedo (outgoing particles) spectra at the same latitude do not show as expected the high energy features due to the incoming CR flux, but they display one single component peaked at low energy and overlapping almost perfectly (to within 1%) with the low E component of the incoming flux. These features indicate that we are dealing with a population of trapped particles circling around earth magnetic field lines, exactly as in the Van Allen belts but at much higher energy and much closer to earth. This will be confirmed by the analysis reported below. ### Leptons [[@LEPT]]{} The flux of leptons has been measured up to about 100 GeV for electrons. It was limited to about 3 GeV for positrons by the $p/e^+$ discrimination range set by the Cherenkov counter threshold for protons. $\bullet$ [Electrons]{} The electron spectra show quite similar features as the proton spectra, with the low energy component of the downgoing flux and the upgoing flux almost perfectly overlapping in the equatorial region. In addition these components of the lepton flux have exactly the same shape to within statistical errors, as for protons, indicating that the particles are likely involved in the same dynamical process. $\bullet$ [Positrons]{} The positron spectra are similar to the electron’s over the range investigated. The surprising feature is that the positron to electron flux ratio is about 4 in the equator region, while in the cosmic flux it is about 0.1, and about one in the atmosphere. The origin of this feature is an open question which is being addressed by the groups of the collaboration. Figure \[LEPTONS\] shows the distributions of electrons and positrons over the positron ID range in the equatorial region (left) and the distribution of the e$^+$/e$^-$ ratio in latitude. ### Ions $\bullet$ [Deuterium [[@DEUT]]{} - ]{} The flux of deuterium has been measured and some preliminary results are available. $\bullet$ [Helium [[@HE]]{} - ]{} The measured flux of Helium is in agreement with previous measurements and doesn’t show a strong rise of flux below GC as the proton flux does. However a small flux of $^3$He is found below GC, which originates probably at least partly from the fragmentation of cosmic $^4$He (figure \[HELIUMS\]. A consistent picture of these population based on known nuclear reaction mechanisms is being invetsigated to account for these populations of light nuclei [@DERHE]. $\bullet$ [Z$>$2 Nuclei - ]{} Some significant samples of light ions with 2$<$Z$\leq\approx$10 have been measured during this run. They are still being analyzed. Origin of the measured proton flux [[@DER00]]{} =============================================== Simulation program ------------------ The inclusive spectrum of protons at the altitude of AMS (390-400km) has been calculated by means of a computer simulation program built to this purpose. CR particles are generated with their natural abundance and momentum distributions. They are propagated inside the earth magnetic field. Particles are allowed to interact with atmospheric nuclei and produce secondary protons with cross sections and multiplicities as discussed below. Each secondary proton is then propagated and allowed to collide as in the previous step. A reaction cascade can thus develop through the atmosphere. The reaction products are counted when they cross the virtual sphere at the altitude of the AMS spectrometer, upward and downward. Particles undergo energy loss by ionisation before and after the interaction. Multiple scattering effects have not been included at this stage. Each event is propagated until the particle disappears by either colliding with a nucleus, or being stopped in the atmosphere, or escaping to outer space beyond twice the production altitude. Note that particles are counted each time they cross the sphere of detection altitude. The contributions of trapped particles are thus weighted statistically with their numbers of crossings, which increases their contribution to the final spectrum. The secondary nucleon spectrum generated has to cover two orders of magnitudes in kinetic energy, between about 100 MeV and 10 GeV. The main component of proton production cross section was obtained by means of analytical relations fitted to 14.6 GeV $p+Be$ data. The scaling properties of the cross section have been checked with the FRITIOF/PYTHIA (Lünd) event generator. Since this generator is not expected to account for the very low energy and backward proton emission (target-like to negative rapidities), this latter component was incorporated using a parametrization. The respective contributions to the total multiplicity-weighted proton production cross-section, were 352 mb for the QE component and 88 mb for the DI components. Cross sections on atmospheric nuclei were renormalized from the original data or parametrizations obtained on different nuclei, using ratios of geometrical cross sections. Results ------- Many features of the dynamics of particles in the earth magnetic field appear in the simulation results. Some of them are discussed in the published paper. Others will be reported later. Figure \[DISTRIB\] shows the experimental kinetic energy distributions of downward (left) and upward (right) protons measured for some bins of latitude, compared to the results of the simulation. No free parameter was used for normalization to the data: The calculated results are entirely determined by the physics input to the calculation. It is seen that the agreement between the data and the simulation result is remarkably good, at all latitudes and for both the inward and outward flux. In particular, the cutoff region is particularly well reproduced, which indicates that the processing of the particle dynamics and kinematics is good. The shaded histograms in the figure correspond to secondary particles in the simulation. The fraction of events originating from the DI component of the proton production cross section described previously, vary from about 10% in the equatorial region up to 25% in the polar region, with a momentum distribution peaking at low kinetic energy and distributed below 500 MeV. It can be concluded from this result that the proton flux measured by AMS can be accounted for to a good accuracy by the single interaction of the incoming CR flux with the atmosphere. Future prospects ---------------- This successful result opens a world of new prospects on the phenomenology of particles in the earth environment. Beside the ongoing investigations of the other AMS results described above, some other issues of general or particular interest are being addressed or will be addressed soon, like the study of the atmospheric neutrino flux and the secondary antiproton populations close to earth. The same type of approach can be used also for particle propagation in the galactic interstellar medium and the study of the various astrophysical issues associated to this propagation. AMS02, a particle observatory in space ====================================== The second phase of the AMS experiment will begin on october 2003 with the launch of the spectrometer and its installation on the International Space station (ISS) for a 3 to 5 years campaign of measurements. Spectrometer structure ---------------------- The new spectrometer shown on figure \[AMS2\], will improve on the AMS01 (figure \[AMS01\]) version by many respects. Its main change consists of the permanent magnet (B$_{max}$=0.15 T) of AMS01 being replaced by a superconducting magnet in AMS02 (B$_{max}$=1 T) which will result in a 6 times better resolution and 6 times larger momentum range, because of this larger magnetic field. In addition, several new detectors will be implemented in AMS02: A transition radiation detector (TRD) will provide lepton identification up to above 300 GeV and an improved tracking accuracy. A Cherenkov imager (RICH) will allow nuclear isotope identification up to about 13 GeV/c for mass around carbon [@RICH]. An electromagnetic calorimeter (ECAL) will provide the energy measurement of electromagnetic particles $\gamma$, leptons, and their discrimination from hadrons up to the TeV range. The synchrotron radiation detector (SRD) would provide e$^+$/e$^-$ identification/discrimination at very high energies [@SRD]. Physics program --------------- The physics program to be covered with the new instrument is wide, with a high discovery potential and a significant probability of unexpected results and new findings. Basically the spectrometer will be able to accumulate statistics larger by 3 to 4 orders of magnitudes than those measured so far by other embarked experiments, for all the species studied. The range in rigidity will extend from around 300 MV up to 3 TV, depending on the particle species, with a good identification capability for leptons, hadrons, and ions, provided by the spectrometer instrumentation. The TRD counter will provide lepton-hadron (mainly positron-proton) discrimination up to the TRD proton threshold around 300 GeV. The RICH will allow both charge and mass measurements: The charge can be obtained from the photon yield, proportionnal to Z$^2$, for momenta going from threshold up to the upper limit of the counter momentum resolution. The mass measurement is obtained from the ring image processing, over a range limited by the space resolution of the counter as shown in the table (see [@RICH] for details). The final performances and range will depend on the choice of Cherenkov radiators. The electromagnetic calorimeter will allow the energy measurement of electromagnetic particles, leptons and photons, up to the TeV range, and lepton-hadron discrimination from the measured P/E ratio over the same range. These features will allow a high statistics study of many cosmic ray species including $e^+$, $e^-$, $p$, $\bar{p}$, and the lightest ions $d,t,^{3,4}$He. Heavier light ions will also be studied with mass identification up to A$\approx 20$, and elements up to Z$\approx 20$ depending on the final performances of the instrumentation of the spectrometer (RICH in particular). Unstable ions like $^{10}$Be, and $^{26}$Al are of particular interest since they provide a measurement of the time of confinement of charged particles in the galaxy (galactic chronometers) [@BO00]. The corresponding antimatter nuclei will be searched with equivalent instrumental performances in identification and kinematic range. The metrological perspectives are summarized in table \[PROSP\]. \[CHARAC\] [llll]{} Particles & P$_{min}$ & P$_{max}$ & Comments\ e$^-$ & $\approx$0.3 & $\approx$3000 & Upper limit set by rigidity resolution\ e$^+$ & $\approx$0.3 & $\approx$300 & Upper limit set by TRD\ proton & $\approx$0.3 & $\approx$3000 & Upper limit set by rigidity resolution\ \ Ions Z<$\approx$20 & $\approx$0.3 & $\approx$1500 ? & Depending on RICH performances\ \ Ions A<4 & 1 to 4 & $\approx$20 & Depending on RICH performances\ Ions 4<A<$\approx$20 & 1 to 4 & $\approx$12 &\ \ $\bar{p}$ & $\approx$0.3 & $\approx$3000 & Depending on $\bar{p}/e^-$ discrimination\ $\overline{ions}$ & $\approx$0.3 & $\approx$1500 & $\overline{He}$, $\overline{C}$\ \[PROSP\] These capabilities will allow to address with an unmatched sensitivity the main scientific objectives of the program:\ 1) The search for antimatter in space [@MBAR]: The experimental signature for the detection of an antinucleus basically requires a determination of its charge in module and sign, the key point being the sign of the charge provided by the radius of curvature of the trajectory in the tracker, its accuracy and contamination by various backgrounds. The results discussed previously have shown that the background level is under control (see details in [@HEBAR]). It will be improved with the updated the tracker equipment in number of planes and readout electronics.\ 2) The search for dark matter in space through the signature of neutralino annihilations. The latter are expected to generate kinematic structures in the spectra of their annihilation products [@DARKM]. Such structures will be searched for in $\bar{p}$ and e$^+$ spectra. 3) In addition, the study of the various ions within the spectrometer (RICH) range of identification, will be achieved, in particular for the $^{10}$Be isotope. This is illustrated on figure \[BE10\] with the result of a simulation incorporating the ID resolution of the spectrometer provide by the RICH and a theoretical $^{10}$Be distribution in momentum. It is seen that the measured sample for 6 weeks of counting time on the ISS would provide highly accurate data over a range totally unexplored so far by previous experiments. The study of this sample will provide an estimate of the accuracy on the propagation parameters to be expected [@BO00]. See ref [@GAMMAS] for a study of a possible high energy gamma ray astronomy program with AMS. SUMMARY and CONCLUSION ====================== In summary, it has been shown that the first engineering flight of the AMS experiment has been very successful, both instrumentally and scientifically. This first step has provided a significant number of new unexpected physics results on the particle populations close to Earth. Although no new physics has emerged so far from these data, they are new and important and expected to provide some significant improvements in our understanding of the particle flux in the environment close to earth at the completion of the study. These result obtained in a region extensively explored by previous balloon or satellite experiments, illustrate the discovery potential of the spectrometer due to its instrumental characteristics: large geometrical acceptance, large momentum range and dynamics and particle identification capability, and to the long duration of the measurement campaign scheduled for the main phase of the experiment on the ISS, where close to 3 orders magnitude more statistics than in the first flight will be collected in much better instrumental conditions. [99]{} S. Ahlen et al., Nucl. Inst. and Meth. in Phys. A350(1994)351 D. Alvisi et al., Nucl. Inst. and Meth. in Phys. A437(1999)212 B. Alpat et al., Nucl. Inst. and Meth. in Phys. A446(2000)522; ibid, Nucl. Inst. and Meth. in Phys. A439(2000)53 D. Barancourt et al., Nucl. Inst. and Meth. in Phys., in press ; preprint astro-ph/0010242 The AMS collaboration, J.Alcaraz et al., Phys. Lett. B461(1999)387 M. Nozaki et al., Proc of 26th Int. Cosm. Ray Conf., Salt-Lake city, Aug, 17-25, 1999, vol 3, p 85. The AMS collaboration, J.Alcaraz et al., Phys. Lett., B472(2000)215 The AMS collaboration, J.Alcaraz et al., Phys. Lett. B490(2000)27 The AMS collaboration, J.Alcaraz et al., Phys. Lett. B484(2000)10 G. Lamanna, AMS note 2000-07-02 The AMS collaboration, J.Alcaraz et al., Phys. Lett. B494(2000)193 L. Derome and M. Buénerd, in preparation. L. Derome et al., Phys. Lett. 489(2000)1; L. Derome and M. Buénerd, Nucl. Phys. A, in Press. see http://ams.cern.ch/AMS/ams\_homepage.html M. Buénerd and Z. Ren, Nucl. Inst. and Meth. in Phys. A454(2000)476; T. Thuillier et al., Nucl. Inst. and Meth. in Phys. A442(2000)74; T. Thuillier, PhD thesis, Université J.Fourier, Grenoble (France), 1999; M. Pohl and H. Hofer Nucl. Inst. and Meth. in Phys. A416(1998)59 See the references in H. Kurki-Suhonio and E. Sihlova Phys. Rev. D62:103508,2000 E.Diehl et al., Phys.Rev.D52(1995)4223; S. Rudaz and F.W. Stecker, ApJ 325(1988)16; see E.A. Baltz et al, Phys. Rev. D61:023514,2000; and Phys. Rev. D59(1999)023511 for a recent status of the field A. Bouchet et al., Nucl. Phys. A, in press. R. Battiston et al., Astropart. Phys. 13(2000)51 [^1]: Talk given at the XXIV Symposium on Nuclear Physics, January 3-6, 2001, Taxco, Mexico.	ArXiv
--- abstract: 'We classify all possible limits of families of translates of a fixed, arbitrary complex plane curve. We do this by giving a set-theoretic description of the projective normal cone (PNC) of the base scheme of a natural rational map, determined by the curve, from the ${{\mathbb{P}}}^8$ of $3\times 3$ matrices to the ${{\mathbb{P}}}^N$ of plane curves of degree $d$. In a sequel to this paper we determine the multiplicities of the components of the PNC. The knowledge of the PNC as a cycle is essential in our computation of the degree of the ${\text{\rm PGL}}(3)$-orbit closure of an arbitrary plane curve, performed in [@MR2001h:14068].' address: - 'Dept. of Mathematics, Florida State University, Tallahassee FL 32306, U.S.A.' - 'Inst. för Matematik, Kungliga Tekniska Högskolan, S-100 44 Stockholm, Sweden' author: - 'Paolo Aluffi, Carel Faber' bibliography: - 'ghizzIbib.bib' title: 'Limits of PGL(3)-translates of plane curves, I' --- Introduction {#intro} ============ In this paper we determine the possible [limits]{} of a fixed, arbitrary complex plane curve ${{\mathscr C}}$, obtained by applying to it a family of translations $\alpha(t)$ centered at a singular transformation of the plane. In other words, we describe the curves in the boundary of the ${\text{\rm PGL}}(3)$-orbit closure of a given curve ${{\mathscr C}}$. Our main motivation for this work comes from enumerative geometry. In [@MR2001h:14068] we have determined the [degree]{} of the ${\text{\rm PGL}}(3)$-orbit closure of an arbitrary (possibly singular, reducible, non-reduced) plane curve; this includes as special cases the determination of several characteristic numbers of families of plane curves, the degrees of certain maps to moduli spaces of plane curves, and isotrivial versions of the Gromov-Witten invariants of the plane. A description of the limits of a curve, and in fact a more refined type of information is an essential ingredient of our approach. This information is obtained in this paper and in its sequel [@ghizzII]; the results were announced and used in [@MR2001h:14068]. The set-up is as follows. Consider the natural action of ${\text{\rm PGL}}(3)$ on the projective space of plane curves of a fixed degree. The orbit closure of a curve ${{\mathscr C}}$ is dominated by the closure ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8$ of the graph of the rational map $c$ from the ${{\mathbb{P}}}^8$ of $3\times3$ matrices to the ${{\mathbb{P}}}^N$ of plane curves of degree $d$, associating to $\varphi\in {\text{\rm PGL}}(3)$ the translate of ${{\mathscr C}}$ by $\varphi$. The boundary of the orbit consists of limits of ${{\mathscr C}}$ and plays an important role in the study of the orbit closure. Our computation of the degree of the orbit closure of ${{\mathscr C}}$ hinges on the study of ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8$, and especially of the scheme-theoretic inverse image in ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8$ of the base scheme ${{\mathscr S}}$ of $c$. Viewing ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8$ as the blow-up of ${{\mathbb{P}}}^8$ along ${{\mathscr S}}$, this inverse image is the exceptional divisor, and may be identified with the projective normal cone (PNC) of ${{\mathscr S}}$ in ${{\mathbb{P}}}^8$. A description of the PNC leads to a description of the limits of ${{\mathscr C}}$: the image of the PNC in ${{\mathbb{P}}}^N$ is contained in the set of limits, and the complement, if nonempty, consists of easily identified ‘stars’ (that is, unions of concurrent lines). This paper is devoted to a set-theoretic description of the PNC for an arbitrary curve. This suffices for the determination of the limits, but does not suffice for the enumerative applications in [@MR2001h:14068]; these applications require the full knowledge of the PNC [as a cycle,]{} that is, the determination of the multiplicities of its different components. We obtain this additional information in [@ghizzII]. The final result of our analysis (including multiplicities) was announced in §2 of [@MR2001h:14068]. The proofs of the facts stated there are given in the present article and its sequel. The main theorem of this paper (Theorem \[mainmain\], in §\[proof\]) gives a precise set-theoretic description of the PNC, relying upon five types of families and limits identified in §\[germlist\]. In this introduction we confine ourselves to formulating a weaker version, focusing on the determination of limits. In [@ghizzII] (Theorem 2.1), we compute the multiplicities of the corresponding five types of components of the PNC. The limits of a curve ${{\mathscr C}}$ are necessarily curves with [small linear orbit,]{} that is, curves with infinite stabilizer. Such curves are classified in §1 of [@MR2002d:14084]; we reproduce the list of curves obtained in [@MR2002d:14084] in an appendix at the end of this paper (§\[appendix\]). For another classification, from a somewhat different viewpoint, we refer to [@MR1698902]. For these curves, the limits can be determined using the results in [@MR2002d:14083] (see also §\[boundary\]). The following statement reduces the computation of the limits of an arbitrary curve ${{\mathscr C}}$ to the case of curves with small orbit. \[main\] Let ${{\mathscr X}}$ be a limit of a plane curve ${{\mathscr C}}$ of degree $d$, obtained by applying to it a ${{\mathbb{C}}}((t))$-valued point of ${\text{\rm PGL}}(3)$ with singular center. Then ${{\mathscr X}}$ is in the orbit closure of a star (reproducing projectively the $d$-tuple cut out on ${{\mathscr C}}$ by a line meeting it properly), or of curves with small orbit determined by the following features of ${{\mathscr C}}$: - The linear components of the support ${{{{\mathscr C}}'}}$ of ${{\mathscr C}}$; - The nonlinear components of ${{{{\mathscr C}}'}}$; - The points at which the tangent cone of ${{\mathscr C}}$ is supported on at least $3$ lines; - The Newton polygons of ${{\mathscr C}}$ at the singularities and inflection points of ${{{{\mathscr C}}'}}$; - The Puiseux expansions of formal branches of ${{\mathscr C}}$ at the singularities of ${{{{\mathscr C}}'}}$. The limits corresponding to these features may be described as follows. In cases I and III they are unions of a star and a general line, that we call ‘fans’; in case II, they are supported on the union of a nonsingular conic and a tangent line; in case IV, they are supported on the union of the coordinate triangle and several curves from a pencil $y^c=\rho\, x^{c-b} z^b$, with $b<c$ coprime positive integers; and in case V they are supported on unions of quadritangent conics and the distinguished tangent line. The following picture illustrates the limits in cases IV and V: ![image](pictures/forintro) A more precise description of the limits is given in §\[germlist\], referring to the classification of these curves obtained in §1 of [@MR2002d:14084] and reproduced in §\[appendix\] of this paper. The proof of Theorem \[main\] (or rather of its more precise form given in Theorem \[mainmain\]) is by an explicit reduction process, and goes along the following lines. The stars mentioned in the statement are obtained by families of translations $\alpha(t)$ (‘germs’) centered at an element $\alpha(0)\not\in{{\mathscr S}}$. To analyze germs centered at points of ${{\mathscr S}}$, we introduce a notion of equivalence of germs (Definition \[equivgermsnew\]), such that equivalent germs lead to the same limit. We then prove that every germ centered at a point of ${{\mathscr S}}$ is essentially equivalent to one with matrix representation $$\begin{pmatrix} 1 & 0 & 0\\ q(t) & t^b & 0\\ r(t) & s(t)t^b & t^c \end{pmatrix}$$ with $0\le b\le c$ and $q$, $r$, and $s$ polynomials. Here, coordinates are chosen so that the point $p=(1:0:0)$ belongs to ${{\mathscr C}}$. Studying the limits obtained by applying such germs to ${{\mathscr C}}$, we identify five specific types of families (the [marker germs]{} listed in §\[germlist\]), reflecting the features of ${{\mathscr C}}$ at $p$ listed in Theorem \[main\], and with the stated kind of limit. We prove that unless the germ is of one of these types, the corresponding limit is already accounted for (for example, it is in the orbit closure of a star of the type mentioned in the statement). In terms of the graph of the rational map $c$ mentioned above, we prove that every component of the PNC is hit at a general point by the lift in ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8$ of one of the five distinguished types of germs. This yields our set-theoretic description of the PNC. In fact, the lifts intersect the corresponding components transversally, and this will be important in our determination of the multiplicities of the components in [@ghizzII]. The procedure underlying the proof of Theorem \[mainmain\] may be applied to any given plane curve, producing a list of its limits. In practice, one needs to find the marker germs for the curve; these determine the components of the PNC. The two examples in §\[twoexamples\] illustrate this process, and show that components of all types may already occur on curves of degree $4$. Here is a simpler example, for a curve of degree $3$. \[exone\] Consider the irreducible cubic ${{\mathscr C}}$ given by the equation $$xyz+y^3+z^3=0\,.$$ It has a node at $(1:0:0)$ and three inflection points. According to Theorem \[mainmain\] and the list in §\[germlist\], the PNC for ${{\mathscr C}}$ has one component of type II and several of type IV. The latter correspond to the three inflection points and the node. A list of representative marker germs for the component of type II and for the component of type IV due to the node may be obtained by following the procedure explained in §\[setth\]: $${\rm II}: \begin{pmatrix} -2 & -t & 0\\ 1 & t & 0\\ 1 & 0 & t^2 \end{pmatrix};\quad {\rm IV}: \begin{pmatrix} 1 & 0 & 0\\ 0 & t & 0\\ 0 & 0 & t^2 \end{pmatrix} \,,\, \begin{pmatrix} 1 & 0 & 0\\ 0 & t^2 & 0\\ 0 & 0 & t \end{pmatrix}\,.$$ The latter two marker germs, corresponding to the two lines in the tangent cone at the node, have the same center and lead to projectively equivalent limits, hence they contribute the same component of the PNC. Equations for the limits of ${{\mathscr C}}$ determined by the germs listed above are $$x(xz+2y^2)=0,\quad y(y^2+xz)=0,\quad \text{and} \quad z(z^2+xy)=0\,,$$ respectively: a conic with a tangent line, and a conic with a transversal line (two limits). The inflection points also contribute components of type IV; the limits in that case are cuspidal cubics. According to Theorem \[main\], all limits of ${{\mathscr C}}$ (other than stars of lines) are projectively equivalent to one of these curves, or to limits of them (cf. §\[boundary\]). Necessary preliminary considerations, and the full statement of the main theorem, are found in §\[prelim\]. The determination of the limits by successive reductions of a given family of curves, proving the result, is worked out in §\[setth\] and §\[typeVcomps\]. In §\[boundary\] we summarize the more straightforward situation for curves with small orbits. Harris and Morrison ([@MR99g:14031], p. 138) pose the [flat completion problem]{} for families of embedded curves, asking for the determination of all curves in ${{\mathbb{P}}}^n$ that can arise as flat limits of a family of embedded stable curves over the punctured disc. The present article solves the isotrivial form of this problem, for plane curves. In principle, a solution of the isotrivial flat completion problem for plane curves can already be found in the marvelous article [@ghizz] by Aldo Ghizzetti, dating back to the 1930s. However, Ghizzetti’s results do not lead to a description of the PNC, which is necessary for our application in [@MR2001h:14068], and which is the main result of this paper and of its sequel. Caporaso and Sernesi use our determination of the limits in [@MR2003k:14035] (Theorem 5.2.1). Hacking [@MR2078368] and Hassett [@MR2000j:14045] study the limits of families of nonsingular plane curves of a given degree, by methods different from ours: they allow the plane to degenerate together with the curve. It would be interesting to compare their results to ours. However, there are fundamental differences between the phenomena we study and those addressed in [@MR2078368] and [@MR2000j:14045]; for example, our families are constant in moduli, and our results apply to [arbitrary]{} plane curves. By the same token, neither Hacking-stability nor GIT-stability play an important role in our study. Consider the case of a plane curve with an analytically irreducible singularity. The determination of the contribution of the singularity to the PNC of the curve requires both its [linear]{} type and [all]{} its Puiseux pairs, see §5 of [@MR2001h:14068]. In general, the stability conditions mentioned above require strictly less (cf. Kim-Lee [@MR2090618]). For example, a singularity analytically isomorphic to $y^2=x^5$ on a [quartic]{} leads necessarily to a component of type V (cf. Example \[extwo\]), whereas on a quintic, it leads to either a component of type IV or a component of type V, according to the order of contact with the tangent line. For GIT-stability, see also Remark \[GITrem\]. The enumerative problem considered in [@MR2001h:14068], as well as the question of limits of PGL-translates, makes sense for hypersurfaces of projective space of any dimension. The case of configurations of points in ${{\mathbb{P}}}^1$ is treated in [@MR1244973]. The degree of the orbit closure of a configuration of planes in ${{\mathbb{P}}}^3$ is computed in [@MR2455792]. In general, these problems appear to be very difficult. The techniques used in this paper could in principle be used in arbitrary dimension, but the case-by-case analysis (which is already challenging for curves in ${{\mathbb{P}}}^2$) would likely be unmanageable in higher dimension. By contrast, the techniques developed in [@ghizzII] should be directly applicable: once ‘marker germs’ have been determined, computing the multiplicities of the corresponding components of the PNC should be straightforward, using the techniques of [@ghizzII]. [^1] We also thank the referee of this paper and [@ghizzII], for the careful reading of both papers and for comments that led to their improvement. Set-theoretic description of the PNC {#prelim} ==================================== Limits of translates {#rough} -------------------- We work over ${{\mathbb{C}}}$. We choose homogeneous coordinates $(x:y:z)$ in ${{\mathbb{P}}}^2$, and identify ${\text{\rm PGL}}(3)$ with the open set of nonsingular matrices in the space ${{\mathbb{P}}}^8$ parametrizing $3\times 3$ matrices. We consider the right action of ${\text{\rm PGL}}(3)$ on the space ${{\mathbb{P}}}^N={{\mathbb{P}}}H^0({{\mathbb{P}}}^2, \mathcal O(d))$ of degree-$d$ plane curves; if $F(x,y,z)=0$ is an equation for a plane curve ${{\mathscr C}}$, and $\alpha\in {\text{\rm PGL}}(3)$, we denote by ${{\mathscr C}}\circ\alpha$ the curve with equation $F(\alpha(x,y,z))=0$. We will consider families of plane curves over the punctured disk, of the form ${{\mathscr C}}\circ\alpha(t)$, where $\alpha(t)$ is a $3\times 3$ matrix with entries in ${{\mathbb{C}}}[t]$, such that $\alpha(0)\ne 0$, $\det\alpha(t) \not\equiv 0$, and $\det\alpha(0)=0$. Simple reductions show that studying these families is equivalent to studying all families ${{\mathscr C}}\circ\alpha(t)$, where $\alpha(t)$ is a ${{\mathbb{C}}}((t))$-valued point of ${{\mathbb{P}}}^8$ such that $\det\alpha(0)=0$. We also note that if ${{\mathscr C}}$ is a smooth curve of degree $d\ge 4$, then any family of curves of degree $d$ parametrized by the punctured disk and whose members are abstractly isomorphic to ${{\mathscr C}}$, i.e., an isotrivial family, is essentially of this type (cf. [@MR770932], p. 56). The arcs of matrices $\alpha(t)$ will be called [germs,]{} and viewed as germs of curves in ${{\mathbb{P}}}^8$. The flat limit $\lim_{t\to 0}\,{{\mathscr C}}\circ \alpha(t)$ of a family ${{\mathscr C}}\circ\alpha(t)$ as $t \to 0$ may be computed concretely by clearing common powers of $t$ in the expanded expression $F(\alpha(t))$, and then setting $t=0$. Our goal is the determination of all possible limits of families as above, for a given arbitrary plane curve ${{\mathscr C}}$. The Projective Normal Cone {#ident} -------------------------- The set of all translates ${{\mathscr C}}\circ\alpha$ is the [linear orbit]{} of ${{\mathscr C}}$, which we denote by ${{\mathscr O_{{\mathscr C}}}}$; the complement of ${{\mathscr O_{{\mathscr C}}}}$ in its closure ${\overline{{{\mathscr O_{{\mathscr C}}}}}}$ is the [boundary]{} of the orbit of ${{\mathscr C}}$. By the [limits of ${{\mathscr C}}$]{} we will mean the limits of families ${{\mathscr C}}\circ\alpha(t)$ with $\alpha(0)\not\in {\text{\rm PGL}}(3)$. For every curve ${{\mathscr C}}$, the boundary is a subset of the set of limits; if $\dim{{\mathscr O_{{\mathscr C}}}}=8$ (the stabilizer of ${{\mathscr C}}$ is finite), then these two sets coincide. If $\dim{{\mathscr O_{{\mathscr C}}}}<8$ (the stabilizer is infinite, and the orbit is [small,]{} in the terminology of [@MR2002d:14083] and [@MR2002d:14084]) then there are families with limit equal to ${{\mathscr C}}$; in this case, the whole orbit closure ${\overline{{{\mathscr O_{{\mathscr C}}}}}}$ consists of limits of ${{\mathscr C}}$. The set of limit curves is itself a union of orbits of plane curves; our goal is a description of representative elements of these orbits; in particular, this will yield a description of the boundary of ${{\mathscr O_{{\mathscr C}}}}$. In this section we relate the set of limits of ${{\mathscr C}}$ to the [projective normal cone]{} mentioned in the introduction. Points of ${{\mathbb{P}}}^8$, that is, $3\times 3$ matrices, may be viewed as rational maps ${{\mathbb{P}}}^2 \dashrightarrow {{\mathbb{P}}}^2$. The kernel of a singular matrix $\alpha\in {{\mathbb{P}}}^8$ determines a line of ${{\mathbb{P}}}^2$ (if $\operatorname{rk}\alpha=1$) or a point (if $\operatorname{rk}\alpha=2$); $\ker\alpha$ will denote this locus. Likewise, the image of $\alpha$ is a point of ${{\mathbb{P}}}^2$ if $\operatorname{rk}\alpha=1$, or a line if $\operatorname{rk}\alpha=2$. The action map $\alpha \mapsto {{\mathscr C}}\circ\alpha$ for $\alpha\in {\text{\rm PGL}}(3)$ defines a rational map $$c: {{\mathbb{P}}}^8 \dashrightarrow {{\mathbb{P}}}^N\quad.$$ We denote by ${{\mathscr S}}$ the base scheme of this rational map. The closure of the graph of $c$ may be identified with the blow-up ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8$ of ${{\mathbb{P}}}^8$ along ${{\mathscr S}}$. The support of ${{\mathscr S}}$ consists of the matrices $\alpha$ such that (with notation as above) $F(\alpha(x,y,z))\equiv 0$; that is, matrices whose image is contained in ${{\mathscr C}}$. The [projective normal cone]{} (PNC) of ${{\mathscr S}}$ in ${{\mathbb{P}}}^8$ is the exceptional divisor $E$ of this blow-up. We have the following commutative diagram: $$\xymatrix@M=10pt{ E \ar@{->>}[d] \ar@<-.5ex>@{^(->}[r] & {{{{\widetilde{{{\mathbb{P}}}}}}}}^8 \ar@{->>}[d]^\pi \ar@<-.5ex>@{^(->}[r] & {{\mathbb{P}}}^8\times {{\mathbb{P}}}^N \ar@{->>}[d] \\ {{\mathscr S}}\ar@<-.5ex>@{^(->}[r] & {{\mathbb{P}}}^8 \ar@{-->}[r]^c & {{\mathbb{P}}}^N }$$ Therefore, as a subset of ${{\mathbb{P}}}^8\times{{\mathbb{P}}}^N$, the support of the PNC is $$\begin{gathered} \|E\|=\{(\alpha,{{\mathscr X}})\in {{\mathbb{P}}}^8\times{{\mathbb{P}}}^N : \text{${{\mathscr X}}$ is a limit of ${{\mathscr C}}\circ \alpha(t)$}\\ \text{for some germ $\alpha(t)$ centered at $\alpha\in {{\mathscr S}}$ and not contained in ${{\mathscr S}}$}\}\quad.\end{gathered}$$ \[PNCtolimits\] The set of limits of ${{\mathscr C}}$ consists of the image of the PNC in ${{\mathbb{P}}}^N$, and of limits of families ${{\mathscr C}}\circ \alpha(t)$ with $\alpha=\alpha(0)$ a singular matrix whose image is not contained in ${{\mathscr C}}$. In the latter case: if $\alpha$ has rank 1, the limit consists of a multiple line supported on $\ker\alpha$; if $\alpha$ has rank 2, the limit consists of a star of lines through $\ker\alpha$, reproducing projectively the tuple of points cut out by ${{\mathscr C}}$ on the image of $\alpha$. The PNC dominates the set of limits of families ${{\mathscr C}}\circ \alpha(t)$ for which $\alpha(t)$ is centered at a point of indeterminacy of $c$. This gives the first statement. To verify the second assertion, assume that $\alpha(t)$ is centered at a singular matrix $\alpha$ at which $c$ [is]{} defined; $\alpha$ is then a rank-1 or rank-2 matrix such that $F(\alpha(x,y,z))\not\equiv 0$. After a coordinate change we may assume without loss of generality that $$\alpha=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \quad\text{or}\quad \alpha=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix}$$ and $F(x,0,0)$, resp. $F(x,y,0)$ are not identically zero. These are then the forms defining the limits of the corresponding families, and the descriptions given in the statement are immediately verified in these cases. The second part of Lemma \[PNCtolimits\] may be viewed as the analogue in our context of an observation of Pinkham (‘sweeping out the cone with hyperplane sections’, [@MR0376672], p. 46). \[eluding\] Denote by $R$ the proper transform in ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8$ of the set of singular matrices in ${{\mathbb{P}}}^8$. Lemma \[PNCtolimits\] asserts that the set of limits of ${{\mathscr C}}$ is the image of the union of the PNC and $R$. A more explicit description of the image of $R$ has eluded us; for a smooth curve ${{\mathscr C}}$ of degree $\ge 5$ these ‘star limits’ have two moduli. It would be interesting to obtain a classification of curves ${{\mathscr C}}$ with smaller ‘star-moduli’. The image of the [intersection]{} of $R$ and the PNC will play an important role in this paper. Curves in the image of this locus will be called ‘rank-$2$ limits’; we note that the set of rank-$2$ limits has dimension $\le 6$. Lemma \[PNCtolimits\] translates the problem of finding the limits for families of plane curves ${{\mathscr C}}\circ\alpha(t)$ into the problem of describing the PNC for the curve ${{\mathscr C}}$. Each component of the PNC is a $7$-dimensional irreducible subvariety of ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8\subset {{\mathbb{P}}}^8\times {{\mathbb{P}}}^N$. We will describe it by listing representative points of the component. More precisely, note that ${\text{\rm PGL}}(3)$ acts on ${{\mathbb{P}}}^8$ by right multiplication, and that this action lifts to a right action of ${\text{\rm PGL}}(3)$ on ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8$. Each component of the PNC is a union of orbits of this action. For each component, we will list germs $\alpha(t)$ lifting on ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8$ to germs $\tilde\alpha(t)$ so that the union of the orbits of the centers $\tilde\alpha(0)$ is dense in that component. Marker germs {#germlist} ------------ In a coarse sense, the classification of limits into ‘types’ as in Theorem \[main\] depends on the image of the center $\alpha(0)$ of the family: this will be a subset of ${{\mathscr C}}$ (cf. Lemma \[PNCtolimits\]), hence it will either be a (linear) component of ${{\mathscr C}}$ (type I), or a point of ${{\mathscr C}}$ (general for type II, singular or inflectional for types III, IV, and V). We will now list germs determining the components of the PNC in the sense explained above. We will call such a germ a [marker germ,]{} as the center of its lift to ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8$ (the corresponding [marker center]{}) ‘marks’ a component of the PNC. The first two types depend on global features of ${{\mathscr C}}$: its linear and nonlinear components. The latter three depend on local features of ${{\mathscr C}}$: inflection points and singularities of (the support of) ${{\mathscr C}}$. That there are only two global types is due to the fact that the order of contact of a nonlinear component and the tangent line at a general point equals two (in characteristic zero). The three local types are due to linear features at singularities of ${{\mathscr C}}$ (type III), single nonlinear branches at special points of ${{\mathscr C}}$ (type IV), and collections of several matching nonlinear branches at singularities of ${{\mathscr C}}$ (type V). Only type V leads to limits with additive stabilizers, and the absence of further types is due to the fact, shown in [@MR2002d:14084], that in characteristic zero only one kind of curves with small orbit has additive stabilizers (also cf. §\[appendix\]). \[GITrem\] A plane curve with small orbit is not GIT-stable. Whether it is strictly semistable or unstable is not directly related to the questions we are considering here. For example, the curves $xyz$ and $x^2yz$ have similar behavior from the point of view of this paper; yet the former is strictly semistable, the latter is unstable. Similarly, consider the union of a general quartic and a multiple line in general position. This has 8-dimensional orbit; it is stable in degree 5, strictly semistable in degree 6, and unstable in higher degrees. But the multiplicity of the line does not affect the behavior from our point of view in any substantial way. The lesson we draw from these examples is that there is no direct relation between the considerations in this paper and GIT. We should point out that the referee of this paper suggests otherwise, noting that closures of orbits are of interest in both contexts, curves with small orbits play a key role, and the mechanics of finding the limits is somewhat similar in the two situations. The referee asks: [which marker germs would be relevant in a GIT analysis?]{} We pass this question on to the interested reader. The terminology employed in the following matches the one in §2 of [@MR2001h:14068]; for example, a [fan]{} is the union of a star and a general line. In four of the five types, $\alpha=\alpha(0)$ is a rank-1 matrix and the line $\ker\alpha$ plays an important role; we will call this ‘the kernel line’. [Type I.]{} Assume ${{\mathscr C}}$ contains a line, defined by a linear polynomial $L$. Write a generator of the ideal of ${{\mathscr C}}$ as $$F(x,y,z)=L(x,y,z)^m G(x,y,z)$$ with $L$ not a factor of $G$. Type I limits are obtained by germs $$\alpha(t)=\alpha(0)+t\beta(t)\quad,$$ where $\alpha(0)$ has rank 2 and image the line defined by $L$. As we are assuming (cf. §\[rough\]) that $\det\alpha(t)\not\equiv 0$, the image of $\beta(t)$ is not contained in $\operatorname{im}\alpha(0)$, so that the limit $\lim_{t\to 0}L\circ\beta(t)$ is a well-defined line $\ell$. The limit $\lim_{t\to 0} {{\mathscr C}}\circ\alpha(t)$ consists of the $m$-fold line $\ell$, and a star of lines through the point $\ker\alpha(0)$. This star reproduces projectively the tuple cut out on $L$ by the curve defined by $G$. ![image](pictures/typeIlim) The limit is in general a fan, and degenerates to a star if the $m$-fold line $\ell$ contains the point $\ker\alpha(0)$. Fans and stars are studied in [@MR2002d:14084], and are the only kinds of curves with small orbit that consist of lines; they are items (1) through (5) in our classification of curves with small orbit, see §\[appendix\]. For types II—V we choose coordinates so that $p=(1:0:0)$ is a point of ${{\mathscr C}}$; for types II, IV, and V we further require that $z=0$ is a chosen component $\ell$ of the tangent cone to ${{\mathscr C}}$ at $p$. [Type II.]{} Assume that $p$ is a nonsingular, non-inflectional point of the support ${{{{\mathscr C}}'}}$ of ${{\mathscr C}}$, contained in a nonlinear component, with tangent line $z=0$. Let $$\alpha(t)=\begin{pmatrix} 1 & 0 & 0 \\ 0 & t & 0 \\ 0 & 0 & t^2 \end{pmatrix}\quad.$$ Then the ideal of $\lim_{t\to 0}{{\mathscr C}}\circ \alpha(t)$ is generated by $$x^{d-2S}(y^2+\rho x z)^S\quad,$$ where $S$ is the multiplicity of the component in ${{\mathscr C}}$, and $\rho\ne 0$; that is, the limit consists of a (possibly multiple) nonsingular conic tangent to the kernel line, union (possibly) a multiple of the kernel line. ![image](pictures/typeIIlim) Such curves are items (6) and (7) in the classification reproduced in §\[appendix\]. The extra kernel line is present precisely when ${{\mathscr C}}$ is not itself a multiple nonsingular conic. [Type III.]{} Assume that $p$ is a singular point of ${{{{\mathscr C}}'}}$ of multiplicity $m$ in ${{\mathscr C}}$, with tangent cone supported on at least three lines. Let $$\alpha(t)=\begin{pmatrix} 1 & 0 & 0 \\ 0 & t & 0 \\ 0 & 0 & t \end{pmatrix}\quad.$$ Then $\lim_{t\to 0} {{\mathscr C}}\circ\alpha(t)$ is a fan consisting of a star centered at $(1:0:0)$ and projectively equivalent to the tangent cone to ${{\mathscr C}}$ at $p$, and of a residual $(d-m)$-fold line supported on the kernel line $x=0$. ![image](pictures/typeIIIlim) [Type IV.]{} Assume that $p$ is a singular or inflection point of the support of ${{\mathscr C}}$. Germs of type IV are determined by the choice of the line $\ell$ in the tangent cone to ${{\mathscr C}}$ at $p$, and by the choice of a side of a corresponding Newton polygon, with slope strictly between $-1$ and $0$. This procedure is explained in more detail in §\[details\]. Let $b<c$ be relatively prime positive integers such that $-b/c$ is the slope of the chosen side. Let $$\alpha(t)=\begin{pmatrix} 1 & 0 & 0 \\ 0 & t^b & 0 \\ 0 & 0 & t^c \end{pmatrix}\quad.$$ Then the ideal of $\lim_{t\to 0} {{\mathscr C}}\circ\alpha(t)$ is generated by a polynomial of the form $$x^{\overline e}y^fz^e \prod_{j=1}^S(y^c+\rho_j x^{c-b}z^b)\quad,$$ with $\rho_j \ne 0$. The number $S$ of ‘cuspidal’ factors in the limit curve is the number of segments cut out by the integer lattice on the selected side of the Newton polygon. ![image](pictures/typeIVlim2) The germ listed above contributes a component of the PNC unless $b/c=1/2$ and the limit curve is supported on a conic union (possibly) the kernel line. The limit curves arising in this way are items (7) through (11) listed in §\[appendix\]. (In particular, the picture drawn above does not capture the possible complexity of the situation: several cuspidal curves may appear in the limit, as well as all lines of the basic triangle.) These limit curves are studied enumeratively in [@MR2002d:14083]. The limit curves contributing components to the PNC in this fashion are precisely the curves that contain nonlinear components and for which the maximal connected subgroup of the stabilizer of the union of the curve and the kernel line is the multiplicative group ${{\mathbb{G}}}_m$. [Type V.]{} Assume $p$ is a singular point of the support of ${{\mathscr C}}$. Germs of type V are determined by the choice of the line $\ell$ in the tangent cone to ${{\mathscr C}}$ at $p$, the choice of a formal branch $z=f(y)=\gamma_{\lambda_0}y^{\lambda_0}+\dots$ for ${{\mathscr C}}$ at $p$ tangent to $\ell$, and the choice of a certain ‘characteristic’ rational number $C>\lambda_0$ (assuming these choices can be made). This procedure is also explained in more detail in §\[details\]. For $a<b<c$ positive integers such that $\frac ca=C$ and $\frac ba= \frac{C-\lambda_0}2+1$, let $$\alpha(t)=\begin{pmatrix} 1 & 0 & 0 \\ t^a & t^b & 0 \\ \underline{f(t^a)} & \underline{f'(t^a)t^b} & t^c \end{pmatrix}$$ where $\underline{\cdots}$ denotes the truncation modulo $t^c$. The integer $a$ is chosen to be the minimal one for which all entries in this germ are polynomials. Then $\lim_{t\to 0}{{\mathscr C}}\circ \alpha(t)$ is given by $$x^{d-2S}\prod_{i=1}^S\left(zx-\frac {\lambda_0(\lambda_0-1)}2 \gamma_{\lambda_0}y^2 -\frac{\lambda_0+C}2 \gamma_{\frac{\lambda_0+C}2}yx-\gamma_C^{(i)}x^2\right)\quad,$$ where $S$ and $\gamma_C^{(i)}$ are defined in §\[details\]. ![image](pictures/typeVlim) These curves consist of [at least two]{} ‘quadritangent’ conics—that is, nonsingular conics meeting at exactly one point—and (possibly) a multiple kernel line. (Again, the picture drawn here does not capture the subtlety of the situation: these limits may occur already for irreducible singularities.) These curves are item (12) in the list in §\[appendix\], and are studied enumeratively in [@MR2002d:14083], §4.1. They are precisely the curves for which the maximal connected subgroup of the stabilizer is the additive group ${{\mathbb{G}}}_a$. Details for types IV and V {#details} -------------------------- [Type IV:]{} Let $p=\operatorname{im}\alpha(0)$ be a singular or inflection point of the support of ${{\mathscr C}}$; choose a line in the tangent cone to ${{\mathscr C}}$ at $p$, and choose coordinates $(x:y:z)$ as before, so that $p=(1:0:0)$ and the selected line in the tangent cone has equation $z=0$. The [Newton polygon]{} for ${{\mathscr C}}$ in the chosen coordinates is the boundary of the convex hull of the union of the positive quadrants with origin at the points $(j,k)$ for which the coefficient of $x^iy^jz^k$ in the generator $F$ for the ideal of ${{\mathscr C}}$ is nonzero (see [@MR88a:14001], p. 380). The part of the Newton polygon consisting of line segments with slope strictly between $-1$ and $0$ does not depend on the choice of coordinates fixing the flag $z=0$, $p=(1:0:0)$. The limit curves are then obtained by choosing a side of the polygon with slope strictly between $-1$ and $0$, and setting to $0$ the coefficients of the monomials in $F$ [not]{} on that side. These curves are studied in [@MR2002d:14083]; typically, they consist of a union of cuspidal curves. The kernel line is part of the distinguished triangle of such a curve, and in fact it must be one of the distinguished tangents. Here is the Newton polygon for the curve of Example \[exone\], with respect to the point $(1:0:0)$ and the line $z=0$: ![image](pictures/newton) Setting to zero the coefficient of $z^3$ produces the limit $y(y^2+xz)$. [Type V:]{} Let $p=\operatorname{im}\alpha(0)$ be a singular point of the support of ${{\mathscr C}}$, and let $m$ be the multiplicity of ${{\mathscr C}}$ at $p$. Again choose a line in the tangent cone to ${{\mathscr C}}$ at $p$, and choose coordinates $(x:y:z)$ so that $p=(1:0:0)$ and $z=0$ is the selected line. We may describe ${{\mathscr C}}$ near $p$ as the union of $m$ ‘formal branches’, cf. §\[formalbranches\]; those that are tangent to the line $z=0$ (but not equal to it) may be written $$z=f(y)=\sum_{i\ge 0} \gamma_{\lambda_i} y^{\lambda_i}$$ with $\lambda_i\in {{\mathbb{Q}}}$, $1<\lambda_0<\lambda_1<\dots$, and $\gamma_{\lambda_0}\ne 0$. The choices made above determine a finite set of rational numbers, which we call the ‘characteristics’ for ${{\mathscr C}}$ (w.r.t. $p$ and the line $z=0$): these are the numbers $C$ for which there exist two branches ${{\mathscr B}}$, ${{\mathscr B}}'$ tangent to $z=0$ that agree modulo $y^C$, differ at $y^C$, and have $\lambda_0<C$. (Formal branches are called ‘pro-branches’ in [@MR2107253], Chapter 4; the numbers $C$ are ‘exponents of contact’.) Let $S$ be the number of branches that agree with ${{\mathscr B}}$ (and ${{\mathscr B}}'$) modulo $y^C$. The initial exponents $\lambda_0$ and the coefficients $\gamma_{\lambda_0}$, $\gamma_{\frac {\lambda_0+C}2}$ for these $S$ branches agree. Let $\gamma_C^{(1)},\dots,\gamma_C^{(S)}$ be the coefficients of $y^C$ in these branches (so that at least two of these numbers are distinct, by the choice of $C$). Then the limit is defined by $$x^{d-2S}\prod_{i=1}^S\left(zx-\frac {\lambda_0(\lambda_0-1)}2 \gamma_{\lambda_0}y^2 -\frac{\lambda_0+C}2 \gamma_{\frac{\lambda_0+C}2}yx-\gamma_C^{(i)}x^2\right)\quad.$$ This is a union of quadritangent conics with (possibly) a multiple of the distinguished tangent, which must be supported on the kernel line. The main theorem, and the structure of its proof {#proof} ------------------------------------------------ Simple dimension counts show that, for each type as listed in §\[germlist\], the union of the orbits of the marker centers is a set of dimension $7$ in ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8\subset {{\mathbb{P}}}^8\times{{\mathbb{P}}}^N$; hence it is a dense set in a component of the PNC. In fact, marker centers of type I, III, IV, and V have 7-dimensional orbit, so the corresponding components of the PNC are the orbit closures of these points. Type II marker centers are points $(\alpha, {{\mathscr X}})\in {{\mathbb{P}}}^8\times{{\mathbb{P}}}^N$, where $\alpha$ is a rank-1 matrix whose image is a general point of a nonlinear component of ${{\mathscr C}}$. The support of ${{\mathscr X}}$ contains a conic tangent to the kernel line; this gives a 1-parameter family of 6-dimensional orbits in ${{\mathbb{P}}}^8\times{{\mathbb{P}}}^N$, accounting for a component of the PNC. We can now formulate a more precise version of Theorem \[main\]: \[mainmain\] Let ${{\mathscr C}}\subset {{\mathbb{P}}}^2_{{\mathbb{C}}}$ be an arbitrary plane curve. The marker germs listed in §\[germlist\] determine components of the PNC for ${{\mathscr C}}$, as explained above. Conversely, all components of the PNC are determined by the marker germs of type I–V listed in §\[germlist\]. By the considerations in §\[ident\], this statement implies Theorem \[main\]. The first part of Theorem \[mainmain\] has been established above. In order to prove the second part, we will define a simple notion of ‘equivalence’ of germs (Definition \[equivgermsnew\]), such that, in particular, equivalent germs $\alpha(t)$ lead to the same component of the PNC. We will show that any given germ $\alpha(t)$ centered at a point of ${{\mathscr S}}$ either is equivalent (after a parameter change, if necessary) to one of the marker germs, or its lift in ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8$ meets the PNC at a point of $R$ (cf. Remark \[eluding\]) or of the boundary of the orbit of a marker center. In the latter cases, the center of the lift varies in a locus of dimension $<7$, hence such germs do not contribute components to the PNC. The following lemma allows us to identify easily limits in the intersection of $R$ and the PNC. \[rank2lemma\] Assume that $\alpha(0)$ has rank $1$. If $\lim_{t\to 0}{{\mathscr C}}\circ\alpha(t)$ is a star with center on $\ker\alpha(0)$, then it is a rank-2 limit. Assume ${{\mathscr X}}=\lim_{t\to 0}{{\mathscr C}}\circ\alpha(t)$ is a star with center on $\ker\alpha(0)$. We may choose coordinates so that $x=0$ is the kernel line, and the generator for the ideal of ${{\mathscr X}}$ is a polynomial in $x,y$ only. If $$\alpha(t)=\begin{pmatrix} a_{11}(t) & a_{12}(t) & a_{13}(t) \\ a_{21}(t) & a_{22}(t) & a_{23}(t) \\ a_{31}(t) & a_{32}(t) & a_{33}(t) \end{pmatrix}\quad,$$ then ${{\mathscr X}}=\lim_{t\to 0}{{\mathscr C}}\circ\beta(t)$ for $$\beta(t)=\begin{pmatrix} a_{11}(t) & a_{12}(t) & 0 \\ a_{21}(t) & a_{22}(t) & 0 \\ a_{31}(t) & a_{32}(t) & 0 \end{pmatrix}\quad.$$ Since $\alpha(0)$ has rank 1 and kernel line $x=0$, $$\alpha(0)=\begin{pmatrix} a_{11}(0) & 0 & 0 \\ a_{21}(0) & 0 & 0 \\ a_{31}(0) & 0 & 0 \end{pmatrix}=\beta(0)\quad.$$ Now $\beta(t)$ is contained in the rank-2 locus, verifying the assertion. A limit $\lim_{t\to 0}{{\mathscr C}}\circ\alpha(t)$ as in this lemma will be called a ‘kernel star’. Sections \[setth\] and \[typeVcomps\] contain the successive reductions bringing a given germ $\alpha(t)$ centered at a point of ${{\mathscr S}}$ into one of the forms given in §\[germlist\], or establishing that it does not contribute a component of the PNC. This analysis will conclude the proof of Theorem \[mainmain\]. Two examples {#twoexamples} ------------ The two examples that follow illustrate the main result, and show that components of all types may already occur on curves of degree 4. Simple translations are used to bring the marker germs provided by §\[germlist\] into the form given here. Consider the reducible quartic ${{\mathscr C}}_1$ given by the equation $$(y+z)(xy^2+xyz+xz^2+y^2z+yz^2)=0\,.$$ It consists of an irreducible cubic with a node at $(1:0:0)$ and a line through the node and the inflection point $(0:1:-1)$. The other inflection points are $(0:1:0)$ and $(0:0:1)$. According to Theorem \[mainmain\] and the list in §\[germlist\], the PNC for ${{\mathscr C}}_1$ has one component of type I, one component of type II, one component of type III, corresponding to the triple point $(1:0:0)$, and four components of type IV: one for each of the inflection points $(0:1:0)$ and $(0:0:1)$, one for the node $(0:1:-1)$ and the tangent line $x=y+z$ to the cubic at that point, and one for the triple point $(1:0:0)$ and the two lines in the tangent cone $y^2+yz+z^2=0$ to the cubic at that point. Here is a schematic drawing of the curve, with features marked by the corresponding types (four points are marked as $\text{IV}_i$, since four different points are responsible for the presence of type IV components): ![image](pictures/quaE1) A list of representative marker germs is as follows: $${\rm I}: \begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & -1 & t \end{pmatrix};\quad {\rm II}: \begin{pmatrix} 2 & 0 & 0\\ -3 & t & 0\\ 6 & 0 & t^2 \end{pmatrix};\quad {\rm III}: \begin{pmatrix} 1 & 0 & 0\\ 0 & t & 0\\ 0 & 0 & t \end{pmatrix};$$ and, for type IV: $$\begin{pmatrix} t & 0 & 0\\ 0 & 1 & 0\\ -t & 0 & t^3 \end{pmatrix},\quad \begin{pmatrix} t & 0 & 0\\ -t & t^3 & 0\\ 0 & 0 & 1 \end{pmatrix},\quad \begin{pmatrix} t & 0 & 0\\ 0 & 1 & 0\\ t & -1 & t^3 \end{pmatrix},\quad \begin{pmatrix} 1 & 0 & 0\\ 0 & \rho t & 0\\ 0 & t & t^2 \end{pmatrix},\quad \begin{pmatrix} 1 & 0 & 0\\ 0 & \rho^2 t & 0\\ 0 & t & t^2 \end{pmatrix}$$ (where $\rho$ is a primitive third root of unity). The latter two marker germs have the same center and lead to projectively equivalent limits, hence they contribute the same component of the PNC. The corresponding limits of ${{\mathscr C}}_1$ are given by $$xy^2z,\quad x^2(8y^2-9xz),\quad x(y+z)(y^2+yz+z^2),\quad y(y^2z+x^3),\quad z(yz^2+x^3),$$$$x(y^2z-x^3),\quad y^2(y^2-(\rho+2)xz),\quad \text{and} \quad y^2(y^2-(\rho^2+2)xz),$$ respectively: a triangle with one line doubled, a conic with a double tangent line, a fan with star centered at $(1:0:0)$, a cuspidal cubic with its cuspidal tangent (two limits), a cuspidal cubic with the line through the cusp and the inflection point, and finally a conic with a double transversal line (two limits). Schematically, the limits may be represented as follows: ![image](pictures/listE1s) According to Theorem \[main\], all limits of ${{\mathscr C}}_1$ (other than stars of lines) are projectively equivalent to one of these curves, or to limits of them (cf. §\[boundary\]). \[extwo\] Consider the irreducible quartic ${{\mathscr C}}_2$ given by the equation $$(y^2-xz)^2=y^3z.$$ It has a ramphoid cusp at $(1:0:0)$, an ordinary cusp at $(0:0:1)$, and an ordinary inflection point at $(3^3 5{:}{-}2^6 3^2{:}{-}2^{12})$; there are no other singular or inflection points. The PNC for ${{\mathscr C}}_2$ has one component of type II, two components of type IV, corresponding to the inflection point and the ordinary cusp, and one component of type V, corresponding to the ramphoid cusp. (Note that there is no component of type IV corresponding to the ramphoid cusp.) Representative marker germs for the latter two components are $${\rm IV}: \begin{pmatrix} 0 & t^3 & 0\\ t^2 & 0 & 0\\ 0 & 0 & 1 \end{pmatrix}\quad{\rm and}\quad {\rm V}: \begin{pmatrix} 1 & 0 & 0\\ t^4 & t^5 & 0\\ t^8 & 2t^9 & t^{10} \end{pmatrix}$$ and the corresponding limits of ${{\mathscr C}}_2$ are given by $$z(y^2z-x^3)\quad{\rm and}\quad(y^2-xz+x^2)(y^2-xz-x^2),$$ respectively: a cuspidal cubic with its inflectional tangent and a pair of quadritangent conics. The connected component of the stabilizer of the latter limit is the additive group. The germ with entries $1$, $t$, and $t^2$ on the diagonal and zeroes elsewhere leads to the limit $(y^2-xz)^2$, a double conic; its orbit is too small to produce an additional component of type IV. Proof of the main theorem: key reductions and components of type I–IV {#setth} ===================================================================== Outline {#preamble} ------- In this section we show that, for a given curve ${{\mathscr C}}$, any germ $\alpha(t)$ contributing to the PNC is ‘equivalent’ (up to a coordinate and parameter change, if necessary) to a marker germ as listed in §\[germlist\]. As follows from §\[rough\] and Lemma \[PNCtolimits\], we may assume that $\det\alpha(t)\not\equiv 0$ and that the image of $\alpha(0)$ is contained in ${{\mathscr C}}$. Observe that if the center $\alpha(0)$ has rank 2 and is a point of ${{\mathscr S}}$, then $\alpha(t)$ is already of the form given in §\[germlist\], Type I; it is easy to verify that the limit is then as stated there. This determines completely the components of type I. Thus, we will assume in most of what follows that $\alpha(0)$ has rank 1, and its image is a point of ${{\mathscr C}}$. ### Equivalence of germs \[equivgermsnew\] Two germs $\alpha(t)$, $\beta(t)$ are [equivalent]{} if $\beta(t\nu(t))\equiv \alpha(t)\circ m(t)$, with $\nu(t)$ a unit in ${{\mathbb{C}}}[[t]]$, and $m(t)$ a germ such that $m(0)=I$ (the identity). For example: if $n(t)$ is a ${{\mathbb{C}}}[[t]]$-valued point of ${\text{\rm PGL}}(3)$, then $\alpha(t)\circ n(t)$ is equivalent to $\alpha(t)\circ n(0)$. We will frequently encounter this situation. \[stequivnew\] Let ${{\mathscr C}}$ be any plane curve, with defining homogeneous ideal $(F(x,y,z))$. If $\alpha(t)$, $\beta(t)$ are equivalent germs, then the initial terms in $F\circ\alpha(t)$, $F\circ\beta(t)$ coincide up to a nonzero multiplicative constant; in particular, the limits $\lim_{t\to 0}{{\mathscr C}}\circ \alpha(t)$, $\lim_{t\to 0}{{\mathscr C}}\circ \beta(t)$ are equal. If $\alpha$ and $\beta$ are equivalent germs, note that $\alpha(0)= \beta(0)$; by Lemma \[stequivnew\] it follows that, for every curve ${{\mathscr C}}$, $\alpha$ and $\beta$ lift to germs in ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8$ centered at the same point. ### Summary of the argument The general plan for the rest of this section is as follows: we will show that every contributing $\alpha(t)$ centered at a rank-1 matrix is equivalent (in suitable coordinates, and possibly up to a parameter change) to one of the form $$\alpha(t)=\begin{pmatrix} 1 & 0 & 0\\ 0 & t^b & 0\\ 0 & 0 & t^c \end{pmatrix}\quad\text{or}\quad \begin{pmatrix} 1 & 0 & 0 \\ t^a & t^b & 0 \\ \underline{f(t^a)} & \underline{f'(t^a) t^b} & t^c\end{pmatrix} \quad,$$ where $b\le c$ resp. $a<b\le c$ are positive integers, $z=f(y)$ is a formal branch for ${{\mathscr C}}$ at $(1:0:0)$, and $\underline{\cdots}$ denotes the truncation modulo $t^c$ (cf. §\[germlist\] and §\[details\]). The main theorem will follow from further analyses of these forms, identifying which do [not]{} contribute components to the PNC, and leading to the restrictions explained in §\[germlist\] and §\[details\]. Specifically, the germs on the left lead to components of type II, III, and IV (§\[1PS\]); those on the right lead to components of type V. The latter germs require a subtle study, performed in §\[typeVcomps\], leading to the definition of ‘characteristics’ and to the description given in §\[details\] (cf. Proposition \[typeV\]). Linear algebra -------------- ### This subsection is devoted to the proof of the following result. \[keyreduction\] Every germ as specified in §\[preamble\] is equivalent to one which, up to a parameter change, has matrix representation $$\begin{pmatrix} 1 & 0 & 0\\ q(t) & t^b & 0\\ r(t) & s(t)t^b & t^c \end{pmatrix}$$ in suitable coordinates, with $1\le b\le c$ and $q,r,s$ polynomials such that $\deg(q)<b$, $\deg(r)<c$, $\deg(s)<c-b$, and $q(0)=r(0)=s(0)=0$. A refined version of this statement is given in Lemma \[faber\]. We will deal with $3\times 3$ matrices with entries in ${{\mathbb{C}}}[[t]]$, that is, ${{\mathbb{C}}}[[t]]$-valued points of $\operatorname{Hom}(V,W)$, for $V$, $W$ 3-dimensional complex vector spaces with chosen bases. Every such matrix $\alpha(t)$ determines a germ in ${{\mathbb{P}}}^8$. A generator $F$ of the ideal of ${{\mathscr C}}$ will be viewed as an element of ${\text{\rm Sym}}^d W^$, for $d=\deg{{\mathscr C}}$; the composition $F\circ\alpha(t)$, a ${{\mathbb{C}}}[[t]]$-valued point of ${\text{\rm Sym}}^d V^$, generates the ideal of ${{\mathscr C}}\circ\alpha(t)$. We will call matrices of the form $$\lambda(t)=\begin{pmatrix} t^a & 0 & 0\\ 0 & t^b & 0\\ 0 & 0 & t^c \end{pmatrix}$$ ‘1-PS’, as they correspond to 1-parameter subgroups of ${\text{\rm PGL}}(3)$. We will say that two matrices $\alpha(t)$, $\beta(t)$ are equivalent if the corresponding germs are equivalent in the sense of Definition \[equivgermsnew\]. The following lemma will allow us to simplify matrix expressions of germs up to equivalence. Define the degree of the zero polynomial to be $-\infty$. \[MPI\] Let $$h_1(t) =\begin{pmatrix} u_1 & b_1 & c_1 \\ a_2 & u_2 & c_2 \\ a_3 & b_3 & u_3 \end{pmatrix}$$ be a matrix with entries in ${{\mathbb{C}}}[[t]]$, such that $h_1(0)=I$, and let $a\le b\le c$ be integers. Then $h_1(t)$ can be written as a product $h_1(t)=h(t)\cdot j(t)$, with $$h(t)=\begin{pmatrix} 1 & 0 & 0 \\ q & 1 & 0 \\ r & s & 1 \end{pmatrix} \quad,\quad j(t)=\begin{pmatrix} v_1 & e_1 & f_1 \\ d_2 & v_2 & f_2 \\ d_3 & e_3 & v_3 \end{pmatrix}$$ where $q$, $r$, $s$ are [polynomials,]{} satisfying 1. $h(0)=j(0)=I$; 2. $\deg(q)<b-a$, $\deg(r)<c-a$, $\deg(s)<c-b$; 3. \[refMPI\] $d_2\equiv0\pmod{t^{b-a}}$, $d_3\equiv0\pmod{t^{c-a}}$, $e_3\equiv0\pmod{t^{c-b}}$. Necessarily $v_1=u_1, e_1=b_1$ and $f_1=c_1$. Use division with remainder to write $ v_1^{-1}a_2=D_2t^{b-a}+q $ with $\deg(q)<b-a$, and let $d_2=v_1D_2t^{b-a}$ (so that $qv_1+d_2=a_2$). This defines $q$ and $d_2$, and uniquely determines $v_2$ and $f_2$. (Note that $q(0)=d_2(0)=f_2(0)=0$ and that $v_2(0)=1$.) Similarly, we let $r$ be the remainder of $(v_1v_2-e_1d_2)^{-1}(v_2a_3-d_2b_3)$ after division by $t^{c-a}$; and $s$ be the remainder of $(v_1v_2-e_1d_2)^{-1}(v_1b_3-e_1a_3)$ after division by $t^{c-b}$. Then $\deg(r)<c-a$, $\deg(s)<c-b$ and $r(0)=s(0)=0$; moreover, we have $$v_1r+d_2s\equiv a_3\pmod{t^{c-a}},\qquad e_1r+v_2s\equiv b_3 \pmod{t^{c-b}},$$ so we take $d_3=a_3-v_1r-d_2s$ and $e_3=b_3-e_1r-v_2s$. This defines $r$, $s$, $d_3$ and $e_3$, and uniquely determines $v_3$. \[MPIcorol\] Let $h_1(t)$ be a matrix with entries in ${{\mathbb{C}}}[[t]]$, such that $h_1(0)=I$, and let $a\le b\le c$ be integers. Then there exists a constant invertible matrix $L$ such that the product $$h_1(t)\cdot \begin{pmatrix} t^a & 0 & 0 \\ 0 & t^b & 0 \\ 0 & 0 & t^c \end{pmatrix}$$ is equivalent to $$\begin{pmatrix} 1 & 0 & 0 \\ q & 1 & 0 \\ r & s & 1 \end{pmatrix}\cdot \begin{pmatrix} t^a & 0 & 0 \\ 0 & t^b & 0 \\ 0 & 0 & t^c \end{pmatrix}\cdot L$$ where $q$, $r$, $s$ are polynomials such that $\deg(q)<b-a$, $\deg(r)<c-a$, $\deg(s)<c-b$, and $q(0)=r(0)=s(0)=0$. With notation as in Lemma \[MPI\] we have $$j(t)\cdot \begin{pmatrix} t^a & 0 & 0 \\ 0 & t^b & 0 \\ 0 & 0 & t^c \end{pmatrix} = \begin{pmatrix} v_1 t^a & e_1 t^b & f_1 t^c \\ d_2 t^a & v_2 t^b & f_2 t^c \\ d_3 t^a & e_3 t^b & v_3 t^c \end{pmatrix} = \begin{pmatrix} t^a & 0 & 0 \\ 0 & t^b & 0 \\ 0 & 0 & t^c \end{pmatrix} \cdot \ell(t)\quad,$$ with $$\ell(t)=\begin{pmatrix} v_1 & e_1 t^{b-a} & f_1 t^{c-a} \\ d_2 t^{a-b} & v_2 & f_2 t^{c-b} \\ d_3 t^{a-c} & e_3 t^{b-c} & v_3 \end{pmatrix}\quad.$$ By (\[refMPI\]) in Lemma \[MPI\], $\ell(t)$ has entries in ${{\mathbb{C}}}[[t]]$ and is invertible; in fact, $L=\ell(0)$ is lower triangular, with 1’s on the diagonal. Therefore Lemma \[MPI\] gives $$h_1(t)\cdot \begin{pmatrix} t^a & 0 & 0 \\ 0 & t^b & 0 \\ 0 & 0 & t^c \end{pmatrix}= \begin{pmatrix} 1 & 0 & 0 \\ q & 1 & 0 \\ r & s & 1 \end{pmatrix}\cdot \begin{pmatrix} t^a & 0 & 0 \\ 0 & t^b & 0 \\ 0 & 0 & t^c \end{pmatrix}\cdot \ell(t)\quad,$$ from which the statement follows. The gist of this result is that, up to equivalence, matrices ‘to the left of a 1-PS’ and centered at the identity may be assumed to be lower triangular, and to have polynomial entries, with controlled degrees. ### We denote by $v$ the order of vanishing at $0$ of a polynomial or power series; we define $v(0)$ to be $+\infty$. The following statement is a refined version of Proposition \[keyreduction\]. \[faber\] Let $\alpha(t)$ be a $\/3\times 3$ matrix with entries in ${{\mathbb{C}}}[[t]]$, such that $\alpha(0)\ne 0$ and $\det \alpha(t)\not\equiv 0$. Then there exist constant invertible matrices $H$, $M$ such that $\alpha(t)$ is equivalent to $$\beta(t)=H\cdot \begin{pmatrix} 1 & 0 & 0 \\ q & 1 & 0 \\ r & s & 1 \end{pmatrix} \cdot \begin{pmatrix} 1 & 0 & 0 \\ 0 & t^b & 0 \\ 0 & 0 & t^c \end{pmatrix} \cdot M\quad,$$ with - $b\le c$ nonnegative integers, $q,r,s$ polynomials; - $\deg(q)<b$, $\deg(r)<c$, $\deg(s)<c-b$; - $q(0)=r(0)=s(0)=0$. If, further, $b=c$ and $q$, $r$ are not both zero, then we may assume that $v(q)<v(r)$. Finally, if $q(t)\not\equiv 0$ then we may choose $q(t)=t^a$, with $a=v(q)<b$ (and thus $a<v(r)$ if $b=c$). By standard diagonalization of matrices over Euclidean domains, every $\alpha(t)$ as in the statement can be written as a product $$h_0(t)\cdot \begin{pmatrix} 1 & 0 & 0 \\ 0 & t^b & 0 \\ 0 & 0 & t^c \end{pmatrix} \cdot k(t)\quad,$$ where $b\le c$ are nonnegative integers, and $h_0(t)$, $k(t)$ are invertible (over ${{\mathbb{C}}}[[t]]$). Letting $H=h_0(0)$, $h_1(t)=H^{-1}\cdot h_0(t)$, and $K=k(0)$, this shows that $\alpha(t)$ is equivalent to $$H\cdot h_1(t) \cdot \begin{pmatrix} 1 & 0 & 0 \\ 0 & t^b & 0 \\ 0 & 0 & t^c \end{pmatrix} \cdot K$$ with $h_1(0)=I$, and $K$ constant and invertible. By Corollary \[MPIcorol\], this matrix is equivalent to $$\beta(t)=H\cdot \begin{pmatrix} 1 & 0 & 0 \\ q & 1 & 0 \\ r & s & 1 \end{pmatrix}\cdot \begin{pmatrix} 1 & 0 & 0 \\ 0 & t^b & 0 \\ 0 & 0 & t^c \end{pmatrix} \cdot L\cdot K$$ with $L$ invertible, and $q$, $r$, $s$ polynomials satisfying the needed conditions. Letting $M=L\cdot K$ gives the statement in the case $b<c$. If $b=c$, then the condition that $\deg s<c-b=0$ forces $s\equiv 0$. When $q$ and $r$ are not both $0$, the inequality $v(q)<v(r)$ may be obtained by conjugating with a constant matrix. If $q(t)\not\equiv 0$ and $v(q)=a$, then we can extract its $a$-th root as a power series. It follows that there exists a unit $\nu(t)\in{{\mathbb{C}}}[[t]]$ such that $q(t\nu(t))=t^a$. Therefore, $$\beta(t\nu(t))=H\cdot \begin{pmatrix} 1 & 0 & 0 \\ t^a & 1 & 0 \\ r(t\nu(t)) & s(t\nu(t)) & 1 \end{pmatrix}\cdot \begin{pmatrix} 1 & 0 & 0 \\ 0 & t^b & 0 \\ 0 & 0 & t^c \end{pmatrix} \cdot \begin{pmatrix} 1 & 0 & 0 \\ 0 & \nu(t)^b & 0 \\ 0 & 0 & \nu(t)^c \end{pmatrix} \cdot M\quad.$$ Another application of Corollary \[MPIcorol\] allows us to truncate the power series $r(t\nu(t))$ and $s(t\nu(t))$ to obtain polynomials $\underline r$, $\underline s$ satisfying the same conditions as $r$, $s$, at the price of multiplying to the right of the 1-PS by a constant invertible matrix $\underline K$: that is, $\beta(t\nu(t))$ (and hence $\alpha(t)$) is equivalent to $$H\cdot \begin{pmatrix} 1 & 0 & 0 \\ t^a & 1 & 0 \\ \underline r & \underline s & 1 \end{pmatrix}\cdot \begin{pmatrix} 1 & 0 & 0 \\ 0 & t^b & 0 \\ 0 & 0 & t^c \end{pmatrix} \cdot \left[ \underline K \cdot \begin{pmatrix} 1 & 0 & 0 \\ 0 & \nu(0)^b & 0 \\ 0 & 0 & \nu(0)^c \end{pmatrix} \cdot M \right]\quad.$$ Renaming $r=\underline r$, $s=\underline s$, and absorbing the factors on the right into $M$ completes the proof of Lemma \[faber\]. The matrices $H$, $M$ appearing in Lemma \[faber\] may be omitted by changing the bases of $W$ and $V$ accordingly. Further, we may assume that $b>0$, since we are already reduced to the case in which $\alpha(0)$ is a rank-1 matrix. This concludes the proof of Proposition \[keyreduction\]. In what follows, we will assume that $\alpha$ is a germ in the standard form given above. Components of type II, III, and IV {#1PS} ---------------------------------- It will now be convenient to switch to affine coordinates centered at the point $(1:0:0)$. We write $$F(1:y:z)=F_m(y,z)+F_{m+1}(y,z)+\cdots +F_d(y,z)\quad,$$ with $d=\deg {{\mathscr C}}$, $F_i$ homogeneous of degree $i$, and $F_m\ne 0$. Thus, $F_m(y,z)$ generates the ideal of the [tangent cone]{} of ${{\mathscr C}}$ at $p$. We first consider the case in which $q=r=s=0$, that is, in which $\alpha(t)$ is itself a 1-PS: $$\alpha(t)=\begin{pmatrix} 1 & 0 & 0\\ 0 & t^b & 0\\ 0 & 0 & t^c \end{pmatrix}$$ with $1\le b \le c$. Also, we may assume that $b$ and $c$ are coprime: this only amounts to a reparametrization of the germ by $t \mapsto t^{1/gcd(b,c)}$; the new germ is not equivalent to the old one in terms of Definition \[equivgermsnew\], but clearly achieves the same limit. Germs with $b=c$ $(=1)$ lead to components of type III, cf. §\[germlist\] (also cf. [@MR2001h:14068], §2, Fact 4(i)): \[tgcone\] If $q=r=s=0$ and $b=c$, then $\lim_{t\to 0} {{\mathscr C}}\circ\alpha(t)$ is a fan consisting of a star projectively equivalent to the tangent cone to ${{\mathscr C}}$ at $p$, and of a residual $(d-m)$-fold line supported on $\ker\alpha$. The composition $F\circ\alpha(t)$ is $$F(x:t^by:t^bz)=t^{bm}x^{d-m}F_m(y,z)+t^{b(m+1)}x^{d-(m+1)} F_{m+1}(y,z)+\cdots+ t^{dm}F_d(y,z)\quad.$$ By definition of limit, $\lim_{t\to 0}{{\mathscr C}}\circ\alpha(t)$ has ideal $(x^{d-m}F_m(y,z))$, proving the assertion. The case $b<c$ corresponds to the germs of type II and type IV in §\[germlist\]. We have to prove that contributing germs of this type are precisely those satisfying the further restrictions specified there: specifically, $-b/c$ must be a slope of one of the Newton polygons for ${{\mathscr C}}$ at the point. We first show that $z=0$ must be a component of the tangent cone: If $q=r=s=0$ and $b<c$, and $z=0$ is not contained in the tangent cone to ${{\mathscr C}}$ at $p$, then $\lim_{t\to 0} {{\mathscr C}}\circ\alpha(t)$ is a rank-2 limit. The condition regarding $z=0$ translates into $F_m(1,0)\ne 0$. Applying $\alpha(t)$ to $F$, we find: $$F(x:t^by:t^cz)=t^{bm}x^{d-m} F_m(y,t^{c-b}z)+t^{b(m+1)} x^{d-(m+1)} F_{m+1}(y,t^{c-b}z)+\cdots$$ Since $F_m(1,0)\ne 0$, the dominant term on the right-hand-side is $x^{d-m}y^m$. This proves the assertion, by Lemma \[rank2lemma\]. Components of the PNC that arise due to 1-PS with $b<c$ may be described in terms of the [Newton polygon]{} for ${{\mathscr C}}$ at $(0,0)$ relative to the line $z=0$, which we may now assume to be part of the tangent cone to ${{\mathscr C}}$ at $p$. The Newton polygon for ${{\mathscr C}}$ in the chosen coordinates is the boundary of the convex hull of the union of the positive quadrants with origin at the points $(j,k)$ for which the coefficient of $x^iy^jz^k$ in the equation for ${{\mathscr C}}$ is nonzero (see [@MR88a:14001], p. 380). The part of the Newton polygon consisting of line segments with slope strictly between $-1$ and $0$ does not depend on the choice of coordinates fixing the flag $z=0$, $p=(0,0)$. \[Newtonsides\] Assume $q=r=s=0$ and $b<c$. - If $-b/c$ is not a slope of the Newton polygon for ${{\mathscr C}}$, then the limit $\lim_{t\to 0} {{\mathscr C}}\circ\alpha(t)$ is supported on (at most) three lines; these curves do not contribute components to the PNC. - If $-b/c$ is a slope of a side of the Newton polygon for ${{\mathscr C}}$, then the ideal of the limit $\lim_{t\to 0}{{\mathscr C}}\circ\alpha(t)$ is generated by the polynomial obtained by setting to $0$ the coefficients of the monomials in $F$ [not]{} on that side. Such polynomials are of the form $$G=x^{\overline e}y^fz^e \prod_{j=1}^S(y^c+\rho_j x^{c-b}z^b) \quad.$$ For the first assertion, simply note that under the stated hypotheses only one monomial in $F$ is dominant in $F\circ\alpha(t)$; hence, the limit is supported on the union of the coordinate axes. A simple dimension count shows that such limits may span at most a 6-dimensional locus in ${{\mathbb{P}}}^8\times{{\mathbb{P}}}^N$, and it follows that such germs do not contribute a component to the PNC. For the second assertion, note that the dominant terms in $F\circ\alpha(t)$ are precisely those on the side of the Newton polygon with slope equal to $-b/c$. It is immediate that the resulting polynomial can be factored as stated. If the point $p=(1:0:0)$ is a singular or an inflection point of the support of ${{\mathscr C}}$, and $b/c\ne 1/2$, we find the type IV germs of §\[germlist\]; also cf. [@MR2001h:14068], §2, Fact 4(ii). The number $S$ of ‘cuspidal’ factors in $G$ is the number of segments cut out by the integer lattice on the selected side of the Newton polygon. If $b/c=1/2$, then a dimension count shows that the corresponding limit will contribute a component to the PNC (of type IV) unless it is supported on a conic union (possibly) the kernel line. If $p$ is a [nonsingular, non-inflectional]{} point of the support of ${{\mathscr C}}$, then the Newton polygon consists of a single side with slope $-1/2$; these are the type II germs of §\[germlist\]. Also cf. [@MR2001h:14068], Fact 2(ii). Components of type V {#typeVcomps} ==================== Having dealt with the 1-PS case in the previous section, we may now assume that $$\tag{$\dagger$} \alpha(t)=\begin{pmatrix} 1 & 0 & 0\\ q(t) & t^b & 0\\ r(t) & s(t)t^b & t^c \end{pmatrix}$$ with the conditions listed in Lemma \[faber\], and further [such that $q,r$, and $s$ do not all vanish identically.]{} Our task is to show that contributing germs of this kind must in fact be of the form specified in §\[germlist\] and §\[details\]. We will show that a germ $\alpha(t)$ as above leads to a rank-2 limit (and hence does not contribute a component to the PNC) unless $\alpha(t)$ and certain formal branches (cf. [@MR88a:14001] and [@MR1836037], Chapter 6 and 7) of the curve are closely related. More precisely, we will prove the following result. \[standardform\] Let $\alpha(t)$ be as specified above, and assume that $\lim_{t\to 0}\mathcal C\circ\alpha(t)$ is not a rank-2 limit. Then ${{\mathscr C}}$ has a formal branch $z=f(y)$, tangent to $z=0$, such that $\alpha$ is equivalent to a germ $$\begin{pmatrix} 1 & 0 & 0 \\ t^a & t^b & 0 \\ \underline{f(t^a)} & \underline{f'(t^a) t^b} & t^c\end{pmatrix} \quad,$$ with $a<b<c$ positive integers. Further, it is necessary that $\frac ca\le \lambda_0+2(\frac ba-1)$, where $\lambda_0>1$ is the (fractional) order of the branch. For a power series $g(t)$ with fractional exponents, we write here $\underline{g(t)}$ for its truncation modulo $t^c$. (The truncations appearing in the statement are in fact polynomials.) The proof of the proposition requires the analysis of several cases. We will first show that under the hypothesis that $\lim_{t\to 0} {{\mathscr C}}\circ\alpha(t)$ is not a rank-2 limit we may assume that $q(t)\not\equiv 0$, and this will allow us to replace it with a power of $t$; next, we will deal with the $b=c$ case; and finally we will see that if $b<c$ and $\alpha(t)$ is not in the stated form, then the limit of every irreducible branch of ${{\mathscr C}}$ is a star with center $(0:0:1)$. This will imply that the limit of ${{\mathscr C}}$ is a kernel star in this case, proving the assertion by Lemma \[rank2lemma\]. This analysis is carried out in §\[formalbranches\]–\[Eop\]. In §\[charaV\] we determine germs of the form given in Proposition \[standardform\] that can lead to components of type V, obtaining the description given in §\[germlist\]. In §\[quadritangent\] we complete the proof of Theorem \[mainmain\], recovering the description given in §\[details\] of the limits obtained along these germs. Limits of formal branches {#formalbranches} ------------------------- In this subsection we recall the notion of formal branches and define the ‘limit’ of a formal branch. The limit of a curve ${{\mathscr C}}$ will be expressed in terms of the limits of its formal branches. Choose affine coordinates $(y,z)=(1:y:z)$ so that $p=(0,0)$, and let $\Phi(y,z)=F(1:y:z)$ be the generator for the ideal of ${{\mathscr C}}$ in these coordinates. Decompose $\Phi(y,z)$ in ${{\mathbb{C}}}[[y,z]]$: $$\Phi(y,z)=\Phi_1(y,z)\cdot\cdots\cdot \Phi_r(y,z)$$ with $\Phi_i(y,z)$ irreducible power series. These define the [irreducible branches]{} of ${{\mathscr C}}$ at $p$. Each $\Phi_i$ has a unique tangent line at $p$; if this tangent line is [not]{} $y=0$, by the Weierstrass preparation theorem we may write (up to a unit in ${{\mathbb{C}}}[[y,z]]$) $\Phi_i$ as a monic polynomial in $z$ with coefficients in ${{\mathbb{C}}}[[y]]$, of degree equal to the multiplicity $m_i$ of the branch at $p$ (cf. for example [@MR1836037], §6.7). If $\Phi_i$ [is]{} tangent to $y=0$, we may likewise write it as a polynomial in $y$ with coefficients in ${{\mathbb{C}}}[[z]]$; [mutatis mutandis,]{} the discussion which follows applies to this case as well. Concentrating on the first case, let $$\Phi_i(y,z)\in {{\mathbb{C}}}[[y]][z]$$ be a monic polynomial of degree $m_i$, defining an irreducible branch of ${{\mathscr C}}$ at $p$, not tangent to $y=0$. Then $\Phi_i$ splits (uniquely) as a product of linear factors over the ring ${{\mathbb{C}}}[[y^]]$ of power series with [rational nonnegative]{} exponents: $$\Phi_i(y,z)=\prod_{j=1}^{m_i} \left(z-f_{ij}(y)\right)\quad,$$ with each $f_{ij}(y)$ of the form $$f(y)=\sum_{k\ge 0} \gamma_{\lambda_k} y^{\lambda_k}$$ with $\lambda_k\in {{\mathbb{Q}}}$, $1\le \lambda_0<\lambda_1<\dots$, and $\gamma_{\lambda_k}\ne 0$. We call each such $z=f(y)$ a [formal branch]{} of ${{\mathscr C}}$ at $p$. The branch is [tangent to $z=0$]{} if the dominating exponent $\lambda_0$ is $>1$. The terms $z-f_{ij}(y)$ in this decomposition are the Puiseux series for ${{\mathscr C}}$ at $p$. We will need to determine $\lim_{t\to 0}{{\mathscr C}}\circ\alpha(t)$ as a union of ‘limits’ of the individual formal branches at $p$. The difficulty here resides in the fact that we cannot perform an arbitrary change of variable in a power series with fractional exponents. In the case in which we will need to do this, however, $\alpha(t)$ will have the following special form: $$\alpha(t)=\begin{pmatrix} 1 & 0 & 0 \\ t^a & t^b & 0 \\ r(t) & s(t)t^b & t^c \end{pmatrix}$$ with $a<b\le c$ positive integers and $r(t)$, $s(t)$ polynomials (satisfying certain restrictions, which are immaterial here). The difficulty we mentioned may be circumvented by the following [ad hoc]{} definition. \[branchlimit\] The [limit]{} of a formal branch $z=f(y)$, along a germ $\alpha(t)$ as above, is defined by the dominant term in $$(r(t)+s(t)t^by+t^cz)-f(t^a)-f'(t^a)t^by-f''(t^a)t^{2b}\frac{y^2}2 -\cdots$$ where $f'(y)=\sum \gamma_{\lambda_k}\lambda_k y^{\lambda_k-1}$ etc. By ‘dominant term’ we mean the coefficient of the lowest power of $t$ after cancellations. This coefficient is a polynomial in $y$ and $z$, giving the limit of the branch according to our definition. This definition behaves as expected: that is, the limit of ${{\mathscr C}}$ is the union of the limits of its individual formal branches. This fact will be used several times in the rest of the paper, and may be formalized as follows. \[limitlemma\] Let $\Phi(y,z)\in {{\mathbb{C}}}[[y]][z]$ be a monic polynomial, $$\Phi(y,z)=\prod_i \left(z-f_i(y)\right)$$ a decomposition over ${{\mathbb{C}}}[[y^]]$, and let $\alpha(t)$ have the special form above. Then the dominant term in $\Phi\circ\alpha(t)$ is the product of the limits of the formal branches $z=f_i(y)$ along $\alpha$, as in Definition \[branchlimit\]. Let $m$ be the multiplicity of ${{\mathscr C}}$ at $p=(0,0)$. For simplicity, we assume that no branches of ${{\mathscr C}}$ are tangent to the line $y=0$, leaving to the reader the necessary adjustments in the presence of such branches. We write the generator $F$ for the ideal of ${{\mathscr C}}$ as a product of formal branches $F =\prod_{i=1}^m (z-f_i(y))$. We will focus on the formal branches that are tangent to the line $z=0$, which may be written explicitly as $$z=f(y)=\sum_{k\ge 0} \gamma_{\lambda_k} y^{\lambda_k}$$ with $\lambda_k\in {{\mathbb{Q}}}$, $1<\lambda_0<\lambda_1<\dots$, and $\gamma_{\lambda_k}\ne 0$. Now we begin the proof of Proposition \[standardform\]. Reduction to $q\ne 0$ --------------------- The first remark is that, under the assumptions that $q$, $r$, and $s$ do not all vanish, we may in fact assume that $q(t)$ is not identically zero. \[qnot0\] If $\alpha(t)$ is as in $(\dagger)$, and $q= 0$, then $\lim_{t\to 0}\mathcal C\circ\alpha(t)$ is a rank-2 limit. (Sketch.) Assume $q=0$, and study the action of $\alpha(t)$ on individual monomials $x^Ay^Bz^C$ in an equation for ${{\mathscr C}}$: $$m_{ABC}:=x^A y^B (r(t) x+s(t) t^b y+t^c z)^C t^{bB}\quad.$$ There are various possibilities for the vanishing of $r$ and $s$, but the dominant terms in $m_{ABC}$ are always kernel stars, which are rank-2 limits by Lemma \[rank2lemma\]. Reduction to $b<c$ ------------------ By Lemma \[qnot0\] and the last part of Lemma \[faber\] we may replace $\alpha(t)$ with an equivalent germ $$\begin{pmatrix} 1 & 0 & 0 \\ t^a & t^b & 0 \\ r(t) & s(t)t^b & t^c\end{pmatrix}$$ with $a<b\le c$, and $r(t)$, $s(t)$ polynomials of degree $<c$, $<(c-b)$ respectively and vanishing at $t=0$. Next, we have to show that if $\lim_{t\to 0}{{\mathscr C}}\circ\alpha(t)$ is not a rank-2 limit then $b<c$ and $r(t)$, $s(t)$ are as stated in Proposition \[standardform\]. \[b=c\] Let $\alpha(t)$ be as above. If $b=c$, then $\lim_{t\to 0} {{\mathscr C}}\circ\alpha(t)$ is a rank-2 limit. Decompose $F(1:y:z)$ in ${{\mathbb{C}}}[[y,z]]$: write $F(1:y:z)=G(y,z)\cdot H(y,z)$, where $G(y,z)$ collects the branches that are [not]{} tangent to $z=0$. If $b=c$, then necessarily $s=0$: $$\alpha(t)=\begin{pmatrix} 1 & 0 & 0 \\ t^a & t^b & 0 \\ r(t) & 0 & t^b \end{pmatrix}\quad,$$ and further $a<v(r)$ (cf. Lemma \[faber\]). The reader can verify that the limits of the branches collected in $G$ are supported on the kernel line $x=0$. The limit of each (formal) branch collected in $H(y,z)$ may be computed as in Definition \[branchlimit\], and is found to be given by a homogeneous equation in $x$ and $z$ only: that is, a $(0:1:0)$-star. It follows that the limit of ${{\mathscr C}}$ is again a kernel star, hence a rank-2 limit by Lemma \[rank2lemma\]. End of the proof of Proposition \[standardform\] {#Eop} ------------------------------------------------ By Lemma \[b=c\], we may now assume that $\alpha$ is given by $$\alpha(t)=\begin{pmatrix} 1 & 0 & 0\\ t^a & t^b & 0\\ r(t) & s(t)t^b & t^c \end{pmatrix}$$ with the usual conditions on $r(t)$ and $s(t)$, and further $a<b<c$. The limit of ${{\mathscr C}}$ under $\alpha$ is analyzed by studying limits of formal branches. \[otherbranches\] The limits of formal branches that are not tangent to the line $z=0$ are necessarily $(0:0:1)$-stars. Further, if $a<v(r)$, then the limit of such a branch is the kernel line $x=0$. \[tangentbranches\] The limit of a formal branch $z=f(y)$ tangent to the line $z=0$ is a $(0:0:1)$-star unless - $r(t)\equiv f(t^a)\pmod{t^c}$; - $s(t)\equiv f'(t^a)\pmod{t^{c-b}}$. The limit of the branch is given by the dominant terms in $$r(t)+s(t)t^by+t^cz=f(t^a)+f'(t^a)t^by+\dots$$ If $r(t)\not\equiv f(t^a)\pmod{t^c}$, then the weight of the branch is necessarily $<c$, so the ideal of the limit is generated by a polynomial in $x$ and $y$, as needed. The same reasoning applies if $s(t)\not\equiv f'(t^a)\pmod{t^{c-b}}$. To verify the condition on $\frac ca$ stated in Proposition \[standardform\], note that the limit of the formal branch $z=f(y)$ is now given by the dominant term in $$r(t)+s(t) t^b y+t^c z=f(t^a)+f'(t^a)t^b y+\frac{f''(t^a)t^{2b}y^2}2 +\dots:$$ the dominant weight will be less than $c$ (causing the limit to be a $(0:0:1)$-star) if $c>2b+v(f''(t^a))=2b+a(\lambda_0-2)$. The stated condition follows at once, completing the proof of Proposition \[standardform\]. Characterization of type V germs {#charaV} -------------------------------- In the following, we will replace $t$ by a root of $t$ in the germ obtained in Proposition \[standardform\], if necessary, in order to ensure that the exponents appearing in its expression are relatively prime integers; the resulting germ determines the same component of the PNC. In order to complete the characterization of type V germs given in §\[germlist\], we need to determine the possible triples $a<b<c$ yielding germs contributing components of the PNC. This determination is best performed in terms of $B=\frac ba$ and $C=\frac ca$. Let $$z=f(y)=\sum_{k\ge 0} \gamma_{\lambda_k} y^{\lambda_k}$$ with $\lambda_k\in {{\mathbb{Q}}}$, $1<\lambda_0<\lambda_1<\dots$, and $\gamma_{\lambda_k}\ne 0$, be a formal branch tangent to $z=0$. Every choice of such a branch and of a rational number $C=\frac ca>1$ determines a truncation $$f_{(C)}(y):=\sum_{\lambda_k<C} \gamma_{\lambda_k} y^{\lambda_k}\quad.$$ The choice of a rational number $B=\frac ba$ satisfying $1<B<C$ and $B\ge \frac{C-\lambda_0}2+1$ determines now a germ as prescribed by Proposition \[standardform\]: $$\alpha(t)=\begin{pmatrix} 1 & 0 & 0 \\ t^a & t^b & 0 \\ \underline{f(t^a)} & \underline{f'(t^a) t^b} & t^c\end{pmatrix}$$ (choosing the smallest positive integer $a$ for which the entries of this matrix have integer exponents). Observe that the truncation $\underline{f(t^a)}=f_{(C)}(t^a)$ is identically 0 if and only if $C\le\lambda_0$. Also observe that $\underline{f'(t^a)t^b}$ is determined by $f_{(C)}(t^a)$, as it equals the truncation to $t^c$ of ${(f_{(C)})}'(t^a)t^b$. \[abc\] If $C\le\lambda_0$ or $B\ne \frac{C-\lambda_0}2+1$, then $\lim_{t\to 0}{{\mathscr C}}\circ\alpha(t)$ is a rank-2 limit. We deal with the different cases separately. \[Clelambda0\] If $C\le\lambda_0$, then $\lim_{t\to 0}{{\mathscr C}}\circ\alpha(t)$ is a $(0:1:0)$-star. If $C=\frac ca\le\lambda_0$, then $f_{(C)}(y)=0$, so $$\alpha(t)=\begin{pmatrix} 1 & 0 & 0 \\ t^a & t^b & 0 \\ 0 & 0 & t^c\end{pmatrix}\quad.$$ The statement follows by computing the limit of individual formal branches, using Definition \[branchlimit\]. By Lemma \[rank2lemma\], the limits obtained in Lemma \[Clelambda0\] are rank-2 limits, so the first part of Proposition \[abc\] is proved. As for the second part, the limit of a branch tangent to $z=0$ depends on whether the branch truncates to $f_{(C)}(y)$ or not. These cases are studied in the next two lemmas. Recall that, by our choice, $B\ge \frac{C-\lambda_0}2+1$. \[nottrunc\] Assume $C>\lambda_0$, and let $z=g(y)$ be a formal branch tangent to $z=0$, such that $g_{(C)}(y)\ne f_{(C)}(y)$. Then the limit of the branch is supported on a kernel line. The limit of the branch is determined by the dominant terms in $$\underline{f(t^a)}+\underline{f'(t^a)t^b}y+t^cz=g(t^a)+g'(t^a) t^by+\dots .$$ As the truncations $g_{(C)}$ and $f_{(C)}$ do not agree, the dominant term is independent of $z$. Under our hypotheses on $B$ and $C$, it is found to be independent of $y$ as well, as needed. \[dominant\] Assume $C>\lambda_0$, and let $z=g(y)$ be a formal branch tangent to $z=0$, such that $g_{(C)}(y)= f_{(C)}(y)$. Denote by $\gamma_C^{(g)}$ the coefficient of $y^C$ in $g(y)$. - If $B> \frac{C-\lambda_0}2+1$, then the limit of the branch $z=g(y)$ by $\alpha(t)$ is the line $$z=(C-B+1)\gamma_{C-B+1} y+\gamma_C^{(g)}\quad.$$ - If $B= \frac{C-\lambda_0}2+1$, then the limit of the branch $z=g(y)$ by $\alpha(t)$ is the conic $$z=\frac {\lambda_0(\lambda_0-1)}2\gamma_{\lambda_0}y^2+\frac {\lambda_0+C}2\gamma_{\frac{\lambda_0+C}2}y+\gamma_C^{(g)}\quad.$$ Rewrite the expansion whose dominant terms give the limit of the branch as: $$t^c z=(g(t^a)-\underline{f(t^a)})+(g'(t^a)t^b-\underline{f'(t^a)t^b})y +\frac{g''(t^a)}2 t^{2b} y^2+\dots$$ The dominant term has weight $c=Ca$ by our choices; if $B> \frac{C-\lambda_0}2+1$ then the weight of the coefficient of $y^2$ exceeds $c$, so it does not survive the limiting process, and the limit is a line. If $B= \frac{C-\lambda_0}2+1$, the term in $y^2$ is dominant, and the limit is a conic. The explicit expressions given in the statement are obtained by reading the coefficients of the dominant terms. We can now complete the proof of Proposition \[abc\]: \[completion\] Assume $C>\lambda_0$. If $B>\frac{C-\lambda_0}2+1$, then the limit $\lim_{t\to 0} {{\mathscr C}}\circ\alpha(t)$ is a rank-2 limit. We will show that the limit is necessarily a kernel star, which gives the statement by Lemma \[rank2lemma\]. As $B>1$, the coefficient $\gamma_{C-B+1}$ is determined by the truncation $f_{(C)}$, and in particular it is the same for all formal branches with that truncation. Since $B>\frac{C-\lambda_0}2+1$, by Lemma \[dominant\] the branches considered there contribute lines through the fixed point $(0:1:(C-B+1)\gamma_{C-B+1})$. We are done if we check that all other branches contribute a kernel line $x=0$: and this is implied by Lemma \[otherbranches\] for branches that are not tangent to $z=0$ (note $a<v(r)$ for the germs we are considering), and by Lemma \[nottrunc\] for formal branches $z=g(y)$ tangent to $z=0$ but whose truncation $g_{(C)}$ does not agree with $f_{(C)}$. Quadritangent conics {#quadritangent} -------------------- We are ready to complete the proof of Theorem \[mainmain\], by determining the limits of the last contributing germs. These have been reduced to the form listed as type V in §\[germlist\] (up to a coordinate change, and replacing $t$ by a root of $t$): $$\alpha(t)=\begin{pmatrix} 1 & 0 & 0 \\ t^a & t^b & 0 \\ \underline{f(t^a)} & \underline{f'(t^a)t^b} & t^c \end{pmatrix}$$ for some branch $z=f(y)=\gamma_{\lambda_0}y^{\lambda_0}+\dots$ of ${{\mathscr C}}$ tangent to $z=0$ at $p=(0,0)$, and further satisfying $C>\lambda_0$ and $B=\frac{C-\lambda_0}2+1$ for $B=\frac ba$, $C=\frac ca$. Type V components of the PNC will arise depending on the limit $\lim_{t\to 0}{{\mathscr C}}\circ\alpha(t)$, which we now determine. \[quadconics\] If $C>\lambda_0$ and $B=\frac{C-\lambda_0}2+1$, then the limit $\lim_{t\to 0}{{\mathscr C}}\circ\alpha(t)$ consists of a union of quadritangent conics, with distinguished tangent equal to the kernel line $x=0$, and of a multiple of the distinguished tangent line. Both $\gamma_{\lambda_0}$ and $\gamma_{\frac{\lambda_0+C}2}$ are determined by the truncation $f_{(C)}$ (since $C>\lambda_0$); hence the equations of the conics $$z=\frac {\lambda_0(\lambda_0-1)}2\gamma_{\lambda_0}y^2+\frac {\lambda_0+C}2\gamma_{\frac{\lambda_0+C}2}y+\gamma_C$$ contributed (according to Lemma \[dominant\]) by different branches with truncation $f_{(C)}$ may only differ in the coefficient $\gamma_C$. It is immediately verified that all such conics are tangent to the kernel line $x=0$, at the point $(0:0:1)$, and that any two distinct such conics meet only at the point $(0:0:1)$; thus they are necessarily quadritangent. Finally, the branches that do not truncate to $f_{(C)}(y)$ must contribute kernel lines, by Lemmas \[otherbranches\] and \[nottrunc\]. The degenerate case in which only one conic arises corresponds to germs not contributing components of the projective normal cone, by dimension considerations. A component is present as soon as there are two or more conics, that is, as soon as two branches contribute distinct conics to the limit. This leads to the description given in §\[germlist\]. We say that a rational number $C$ is ‘characteristic’ for ${{\mathscr C}}$ (with respect to $z=0$) if at least two formal branches of ${{\mathscr C}}$ (tangent to $z=0$) have the same nonzero truncation, but different coefficients for $y^C$. \[typeV\] The set of characteristic rationals is finite. The limit $\lim_{t\to 0}{{\mathscr C}}\circ\alpha(t)$ obtained in Lemma \[quadconics\] determines a component of the projective normal cone precisely when $C$ is characteristic. If $C\gg 0$, then branches with the same truncation must in fact be identical, hence they cannot differ at $y^C$, hence $C$ is not characteristic. Since the set of exponents of any branch is discrete, the first assertion follows. The second assertion follows from Lemma \[quadconics\]: if $C>\lambda_0$ and $B=\frac{C-\lambda_0}2+1$, then the limit is a union of a multiple kernel line and conics with equation $$z=\frac {\lambda_0(\lambda_0-1)}2\gamma_{\lambda_0}y^2+\frac {\lambda_0+C}2\gamma_{\frac{\lambda_0+C}2}y+\gamma_C\quad:$$ these conics are different precisely when the coefficients $\gamma_C$ are different, and the statement follows. Proposition \[typeV\] leads to the procedure giving components of type V explained in §\[details\] (also cf. [@MR2001h:14068], §2, Fact 5), concluding the proof of Theorem \[mainmain\]. Boundaries of orbits {#boundary} ==================== We have now completed the set-theoretic description of the PNC determined by an arbitrary plane curve ${{\mathscr C}}$. As we have argued in §\[prelim\], this yields in particular a description of the boundary of ${{\mathscr O_{{\mathscr C}}}}$. In this section we include a few remarks aimed at making this description more explicit. If $\dim{{\mathscr O_{{\mathscr C}}}}=8$, then the boundary of ${{\mathscr O_{{\mathscr C}}}}$ consists of the image of the union of the PNC and of the proper transform $R$ of the complement of ${\text{\rm PGL}}(3)$ in ${{\mathbb{P}}}^8$ (cf. Remark \[eluding\]). Curves in the image of $R$ are stars (Lemma \[PNCtolimits\]). Curves in the image of the components of the PNC belong to the orbit closures of the limits of the marker germs listed in §\[germlist\]. We have proved that this list is exhaustive; therefore, the boundary of a given curve ${{\mathscr C}}$ may be determined (up to stars) by identifying the marker germs for ${{\mathscr C}}$, and taking the union of the orbit closures of the (finitely many) corresponding limits. This reduces the determination of the curves in the boundary of the orbit of a given curve to the determination for curves with [small]{} orbit (i.e., of dimension $\le 7$). We note that, for a curve ${{\mathscr C}}$ with small orbit, some components of the PNC will in fact dominate ${\overline{{{\mathscr O_{{\mathscr C}}}}}}$: indeed, in this case ${{\mathscr C}}$ has positive dimensional stabilizer in ${\text{\rm PGL}}(3)$; the limit of a germ centered at a singular matrix and otherwise contained in the stabilizer is ${{\mathscr C}}$ itself. This germ can be chosen to be equivalent to a marker germ, identifying a component of the PNC which dominates the orbit closure. As mentioned in the introduction, the boundary for a curve with small orbit may be determined by a direct method. Indeed, for such a curve we have constructed in [@MR2002d:14083] explicit sequences of blow-ups at nonsingular centers which resolve the indeterminacies of the basic rational map, and hence dominate the corresponding graph ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8$. The boundary of the curve may be determined by studying the image in ${{\mathbb{P}}}^N$ of the various exceptional divisors of these blow-ups. The result may be summarized by indicating which types of curves with small orbits are in the boundary of a given curve with small orbit. Figure \[figure1\] expresses part of this relation in terms of the representative pictures for curves with small orbit shown in §\[appendix\]. The five columns represent curves with orbits of dimension 7, 6, 5, 4, 2 respectively. Arrows indicate specialization: for example, the figure indicates that the boundary of the orbit of the union of a conic and a tangent line contains stars, but not single conics. Stars with more than three lines are not displayed, to avoid cluttering the picture; the three kinds of curves displayed in the leftmost column all degenerate to such stars, the only exceptions being the special cases of the second picture given by the union of a conic and a transversal line, and by a single cuspidal cubic. The situation illustrated here is precisely what one would expect from naive considerations; it is confirmed by the study of the blow-ups mentioned above. Slightly more refined phenomena (for example, involving multiplicities of the components) are not represented in this figure; in general, they can be easily established by applying the results of this paper or by analyzing the blow-ups of [@MR2002d:14083]. ![[]{data-label="figure1"}](pictures/smallpichor3) We close by pointing out one such phenomenon. In general, the union of a set of quadritangent conics and a tangent line can specialize to the union of a conic and a tangent line in two ways: (i) by type II germs aimed at a general point of one of the conic components, and (ii) by a suitable type IV germ aimed at the tangency point. The multiplicity of the conic in the limit is then the multiplicity of the selected component in case (i), and the sum of the multiplicities of all conic components in case (ii). If the curve consists solely of quadritangent conics, it degenerates to a multiple conic in case (ii). This possibility occurs in the boundary of the orbit of the quartic curve from Example \[extwo\], represented in Figure \[figure2\]. We have omitted the set of stars of four distinct lines also in this figure; in this case, it is a $6$-dimensional union of $5$-dimensional orbits. ![[]{data-label="figure2"}](pictures/exapic) Appendix: curves with small linear orbits {#appendix} ========================================= For the convenience of the reader, we reproduce here the description of plane curves with small linear orbits given in [@MR2002d:14084]. That reference contains a proof that this list is exhaustive, and details on the stabilizer of each type of curve (as well as enumerative results for orbits of curves consisting of unions of lines, items (1)–(5) in the following list). Let ${{\mathscr C}}$ be a curve with small linear orbit. We list all possibilities for ${{\mathscr C}}$, together with the dimension of the orbit ${{\mathscr O_{{\mathscr C}}}}$ of ${{\mathscr C}}$. The irreducible components of the curves described here may appear with arbitrary multiplicities. 1. ${{\mathscr C}}$ consists of a single line; $\dim{{\mathscr O_{{\mathscr C}}}}=2$. 2. ${{\mathscr C}}$ consists of 2 (distinct) lines; $\dim{{\mathscr O_{{\mathscr C}}}}=4$. 3. ${{\mathscr C}}$ consists of 3 or more concurrent lines; $\dim{{\mathscr O_{{\mathscr C}}}}=5$. (We call this configuration a [star]{}.) 4. ${{\mathscr C}}$ is a triangle (consisting of 3 lines in general position); $\dim{{\mathscr O_{{\mathscr C}}}}=6$. 5. ${{\mathscr C}}$ consists of 3 or more concurrent lines, together with 1 other (non-concurrent) line; $\dim{{\mathscr O_{{\mathscr C}}}}=7$. (We call this configuration a [fan]{}.) 6. ${{\mathscr C}}$ consists of a single conic; $\dim{{\mathscr O_{{\mathscr C}}}}=5$. 7. ${{\mathscr C}}$ consists of a conic and a tangent line; $\dim{{\mathscr O_{{\mathscr C}}}}=6$. 8. ${{\mathscr C}}$ consists of a conic and 2 (distinct) tangent lines; $\dim{{\mathscr O_{{\mathscr C}}}}=7$. 9. ${{\mathscr C}}$ consists of a conic and a transversal line and may contain either one of the tangent lines at the 2 points of intersection or both of them; $\dim{{\mathscr O_{{\mathscr C}}}}=7$. 10. ${{\mathscr C}}$ consists of 2 or more bitangent conics (conics in the pencil $y^2+\lambda x z$) and may contain the line $y$ through the two points of intersection as well as the lines $x$ and/or $z$, tangent lines to the conics at the points of intersection; again, $\dim{{\mathscr O_{{\mathscr C}}}}=7$. 11. ${{\mathscr C}}$ consists of 1 or more (irreducible) curves from the pencil $y^b+\lambda z^a x^{b-a}$, with $b\ge 3$, and may contain the lines $x$ and/or $y$ and/or $z$; $\dim{{\mathscr O_{{\mathscr C}}}}=7$. 12. ${{\mathscr C}}$ contains 2 or more conics from a pencil through a conic and a double tangent line; it may also contain that tangent line. In this case, $\dim{{\mathscr O_{{\mathscr C}}}}=7$. The last case is the only one in which the maximal connected subgroup of the stabilizer is the additive group ${{\mathbb{G}}}_a$; this fact was mentioned in §\[germlist\]. The following picture represents schematically the curves described above. ![image](pictures/twelve1new) ![image](pictures/twelve2new) [^1]: [Acknowledgments.]{} Work on this paper was made possible by support from Mathematisches Forschungsinstitut Oberwolfach, the Volkswagen Stiftung, the Max-Planck-Institut für Mathematik (Bonn), Princeton University, the Göran Gustafsson foundation, the Swedish Research Council, the Mittag-Leffler Institute, MSRI, NSA, NSF, and our home institutions. We thank an anonymous referee of our first article on the topic of linear orbits of plane curves, [@MR94e:14032], for bringing the paper of Aldo Ghizzetti to our attention.	ArXiv
--- abstract: 'We study explosive percolation (EP) on Erdös-Rényi network for product rule (PR) and sum rule (SR). Initially, it was claimed that EP describes discontinuous phase transition, now it is well-accepted as a probabilistic model for thermal continuous phase transition (CPT). However, no model for CPT is complete unless we know how to relate its observable quantities with those of thermal CPT. To this end, we define entropy, specific heat, re-define susceptibility and show that they behave exactly like their thermal counterparts. We obtain the critical exponents $\nu, \alpha, \beta$ and $\gamma$ numerically and find that both PR and SR belong to the same universality class and they obey Rushbrooke inequality.' author: - 'M. K. Hassan and M. M. H. Sabbir' title: 'Product-Sum universality and Rushbrooke inequality in explosive percolation ' --- The notion of percolation is omnipresent in many seemingly disparate natural and man-made systems [@ref.Stauffer]. Examples include spread of forest fire, flow of fluid through porous media, spread of biological and computer viruses etc. [@ref.saberi; @ref.Newman_virus; @ref.Moore_virus]. Besides such direct applications, percolation is best known as a paradigmatic model for phase transition. One of the simplest models for percolation is the classical random percolation (RP) on Erdös-Rényi (ER) network in which one starts with $N$ labeled nodes that are initially all isolated [@ref.erdos]. Then at each step a link, say $e_{ij}$, is picked at random from all the possible pair of links and occupy it to connect nodes $i$ and $j$. As the number of occupied links $n=tN$ increases from zero we find that clusters, i.e. contiguous nodes connected by occupied links, are formed and on the average grown. In the process, the largest cluster $s_{{\rm max}}$ undergoes a transition across $t_c=0.5$ from minuscule size ($s_{{\rm max}}\sim \log N$) to giant size ($s_{{\rm max}} \sim N$). The emergence of such threshold value $t_c$ is found to be accompanied by a sudden change in the order parameter $P$, the ratio of the largest cluster to the network size, such that $P=0$ at $t\leq t_c$ and $P>0$ at $t>t_c$ in the limit $N\rightarrow \infty$. This is reminiscent of the second order or continuous phase transition (CPT). In 2009, Achlioptas [et al.]{} proposed a class of percolation model in which two links are picked randomly instead of one at each step [@ref.Achlioptas] . However, ultimately only one of the links, that results in the smaller clustering, is occupied and the other is discarded for future picking. One of the key features of this rule, which is now known as the Achlioptas process (AP), is that it discourages the growth of the larger clusters and encourages the smaller ones which inevitably delays the transition. Eventually, when it reaches near the critical point it is so unstable that occupation of one or two links triggers an explosion of growth. It leads to the emergence of a giant cluster with a bang and hence it is called “explosive percolation" (EP). Indeed, the corresponding $P$, in contrast to its classical counterpart, undergoes such an abrupt transition that it was at first mistaken as a discontinuity and suggested to exhibit the first order or discontinuous transition. Their results jolted the scientific community through a series of claims, unclaims and counter-claims [@ref.Friedman; @ref.ziff_1; @ref.radicchi_1; @ref.Costa_2; @ref.souza; @ref.cho_1; @ref.ara; @ref.da_Costa; @ref.Grassberger; @ref.Bastas]. It is now well settled that the explosive percolation transition is actually continuous but with first order like finite-size effects [@ref.Grassberger; @ref.Bastas; @ref.Riordan; @ref.bastas_review; @ref.Choi]. In general, scientists use theoretical model, just like architects use geometric model before building large expensive structure, because it provides useful insights into the real-world systems. The real systems that percolation represent is complex as it often involves quantum and many particle interaction effects. However, modeling is only useful if we know how to relate its various observable quantities to those of the real-world systems. To this end, Kasteleyn and Fortuin used the mapping of the percolation problem onto the $q$-state Potts model in order to relate its observable to the thermal quantities of the Potts model [@ref.Kasteleyn]. Owing to that mapping we know that $P$ is the order parameter, mean cluster size $\langle s\rangle$ is the susceptibility etc. but not equivalent counterpart of entropy. In thermal CPT, the entropy $S$ and the order parameter (OP) complement each other as $S$, that measures the degree of disorder, is maximum where OP is zero and OP, that measures the extent of order, is maximum where $S$ is zero. A similar behaviour in percolation is also expected in order to elucidate whether it is also an order-disorder transition or not. Universality is another aspect that we find common in the thermal CPT and in the random percolation. In the case of EP, we are yet to find universality of any type or any kind. Another interesting aspect of thermal CPT is that its critical exponents $\alpha, \beta$ and $\gamma$ obey the Rushbrooke inequality $\alpha+2\beta+\gamma\geq 2$ which reduces to equality under static scaling hypothesis [@ref.Stanley]. Whether it holds in explosive percolation or not, is also an interesting issue. In this article, we investigate EP on the ER networks for product rule (PR) and sum rule (SR) and find their critical exponents numerically. First, we define susceptibility $\chi$ as the ratio of the successive jump $\Delta P$ of $P$ and the magnitude of successive intervals $\Delta t$ instead of using the mean cluster size $\langle s\rangle$ as susceptibility. Then we obtain the critical exponents $\nu$ of the correlation length, $\gamma$ of $\chi$, and $\beta$ of $P$. Note that $\langle s\rangle$ exhibits the expected divergence only if the largest cluster size is excluded from it and even then it gives too large a value of $\gamma$. Realizing these drawbacks, many researchers are already considering alternative definitions [@ref.radicchi_1; @ref.ziff_3; @ref.qian]. Second, we define entropy $H$ for EP and find that it is continuous across the whole spectrum of the control parameter $t$ which clearly reveals that EP transition is indeed continuous in nature. We then define the specific heat as $C=q{{dH}\over{dq}}$ where $q=(1-t)$ and find that it diverges with positive critical exponent $\alpha$. The most intriguing and unexpected findings of this work is that PR and SR belong to the same universality class. Besides, we find that the elusive Rushbrooke inequality holds in EP. Recently, using the the same definitions for entropy, specific heat and susceptibility we have shown that the Rushbrooke inequality holds in the random percolation too [@ref.hassan_didar]. Finding that RI also holds in EP on random network provides a clear testament of how robust our results are. Percolation is all about clusters as every observable quantity of it is related, this way or another, to the clusters by virtue of definition. Initially, all the labeled nodes are considered isolated so that every node is a cluster of its own size. The process starts by picking two distinct links, say $e_{ij}$ and $e_{kl}$, randomly at each step. To apply the PR, we then calculate the products, $\Pi_{ij}=s_i\times s_j$ and $\Pi_{kl}= s_k \times s_l$, of the size of the clusters that the two nodes on either side of each link contain. The link with the smaller value of the products $\Pi_{ij}$ and $\Pi_{kl}$ is occupied. On the other hand, if we find $\Pi_{ij}=\Pi_{kl}$ then we occupy one of the two links at random with equal probability. In the case of SR, we take the sum $\Sigma_{ij}=s_i+s_j$ and $\Sigma_{kl}=s_k+s_l$ instead of the product and do the rest exactly in the same way as we did for PR. Each time we occupy a link, either the size of an existing cluster grows due to occupation of an inter-cluster link or the cluster size remains the same due to addition of an intra-cluster link. In either case, the growth of large clusters are always disfavoured which is in sharp contrast to its RP counterpart. Thus, the emergence of a giant cluster is considerably slowed down but eventually when it happens, it happens abruptly but without discontinuity. We first investigate not the $P$ itself but its successive jump $\Delta P$ within successive interval $\Delta t=1/N$. The idea of successive jump size $\Delta P$ was first introduced by Manna [@ref.manna]. We use it to define susceptibility as $$\chi(t)={{\Delta P}\over{\Delta t}},$$ which essentially becomes the derivative of $P$ in the limit $N\rightarrow \infty$. In Figs. (\[fig:1a\]) and (\[fig:1b\]) we show plots of $\chi$ versus $t$ for both PR and SR model. According to the finite-size scaling (FSS) hypothesis, the susceptibility $\chi_{{\rm max}}$ at $t=t_c$ increases following a power-law $\chi_{{\rm max}}\sim N^{\gamma/\nu}$. In an attempt to verify this we plot $\log(\chi_{{\rm max}})$ vs $\log(N)$, see insets of Figs. (\[fig:1a\]) and (\[fig:1b\]), and find straight lines with slopes $\gamma/\nu=0.480(3)$ for PR and $\gamma/\nu=0.475(4)$ for SR. Following the procedures in Ref. [@ref.Hassan_Rahman_1] we also get a rough estimate of the exponent $1/\nu = 0.535(5)$ for PR and $1/\nu = 0.537(1)$ for SR. The FSS theory further suggests that if we now plot $\chi N^{-\gamma/\nu}$ vs $(t_c-t)N^{1/\nu}$, all the distinct plots of Figs. (\[fig:1a\]) and (\[fig:1b\]) should collapse into their respective universal curves. Indeed, by tuning ${\gamma/\nu}$ and $1/\nu$ we find excellent data collapse, see Figs. (\[fig:1c\]) and (\[fig:1d\]), if we use $\gamma/\nu=0.478$ and $1/\nu=0.535$ for PR and SR respectively. Note that $t_c$ value also affects the data collapse and hence tuning the initial estimates for $t_c$ we get the best data-collapse if we use $t_c=0.88850$ for PR and $t_c= 0.86018$ for SR. The quality of data collapse itself provides a clear testament to the extent of accuracy of these values. What is most noteworthy, however, is that both PR and SR share the same value for the exponents $\gamma$ and $\nu$. Using now the relation $N\sim (t-t_c)^{-\nu}$ in $\chi\sim N^{\gamma/\nu}$ we find that $$\chi\sim (t-t_c)^{-\gamma},$$ where $\gamma=0.893$ for both PR and SR rules within the acceptable limit of error. We find that the susceptibility now diverges even without the exclusion of the largest cluster and that too with the same $\gamma$. Now, we consider the order parameter $P$ itself and plot it as a function of $t$ in Figs. (\[fig:2a\]) and (\[fig:2b\]) for PR and SR respectively. We follow the same standard procedure as in Ref. [@ref.Hassan_Rahman_1; @ref.Hassan_Rahman_explosive] and find $\beta/\nu=0.045$ for both the variants. It is well-known that $P(t,N)$ exhibits finite-size scaling. One way of testing it is to plot $PN^{\beta/\nu}$ vs $(t-t_c)N^{1/\nu}$ and check if all the distinct curves of $P$ vs $t$ curves collapse or not. Indeed, Figs. (\[fig:2c\]) and (\[fig:2d\]) suggest that they all collapse superbly with $\beta/\nu=0.045$ and $1/\nu=0.535$ values regardless of whether it is PR or SR. Substituting the relation $N\sim (t-t_c)^{-\nu}$ in $P\sim N^{-\beta/\nu}$ we get $$P(t)\sim (t-t_c)^\beta.$$ This is exactly how the order parameter behaves near critical point in the thermal CPT as well. We once again find that both PR and SR rules share the same exponent $\beta =0.084$ within the acceptable limits of error. Such unusually low value of $\beta$ compared to that of the RP on ER where $\beta=1$ is the hallmark of EP transition [@ref.mori]. Note also that Grassberger [et al.]{} obtained $1/\nu=0.5$ and $\beta=0.0861(5)$ for PR on ER [@ref.Grassberger]. Our values are quite close to their values; however little differences are there which marks a significant improvement in the quality of data-collapse. Phase transitions always entail a change in entropy and hence no model for phase transition is complete without a proper definition for it. To this end, we find that the most suitable choice for entropy in percolation is the Shannon entropy which is defined as $$\label{eq:shannon_entropy} H(t)=-K\sum_i^m \mu_i\log \mu_i,$$ where we choose $K=1$ since it merely amounts to a choice of a unit of measure of entropy [@ref.shannon]. Although there is no explicit restriction per se on the choice of $\mu_i$ there exist some implicit restrictions. The text-book definitions of thermal entropy $S$ and the specific heat $c$ suggest that the $S$ vs $T$ plot must always have a sigmoidal shape with positive slope [@ref.Stanley]. Recently, Vieira [et al.]{} used the probability $w_s$, that a node picked at random belongs to a cluster exactly of size $s$, in Eq. (\[eq:shannon\_entropy\]) to measure Shannon entropy for explosive percolation and found that the entropy increases from zero at $t=0$ to reach its maximum value at $t_c$ followed by sharp decrease above $t_c$ [@ref.Vieira]. We also know that the order parameter is also zero at $t=0$. It means that the system is ordered and disordered at the same time which is not possible. Besides, the bell-shaped like entropy curve also violates the second law of thermodynamics. The problem lies in the fact that the sum in Eq. (\[eq:shannon\_entropy\]) is over each individual cluster not over a class of cluster of size $s$ and hence one cannot use $w_s$ to measure entropy. Note that the Shannon entropy measures how much information is contained in each cluster like in each message in the information theory. To find the appropriate probability $\mu_i$ for Eq. (\[eq:shannon\_entropy\]), we assume that for a given $t$ there are $m$ distinct and disjoint labeled clusters $i=1,2,...,m$ of size $s_1,s_2,....,s_m$ respectively. We then propose a labeled cluster picking probability (CPP) $\mu_i$, that a node picked at random belongs to cluster $i$, and assume that it depends on the size $s_i$ of the cluster $i$ itself, so that $\mu_i=s_i/\sum_i s_i$ where $\sum_i s_i=N$. Incorporating $\mu_i=s_i/N$ in Eq. (\[eq:shannon\_entropy\]) we obtain entropy for explosive percolation. To visualize we plot it in Figs. (\[fig:3a\]) and (\[fig:3b\]) as a function of $q=1-t$ for PR and SR respectively. We observe that the maximum entropy occurs at $q=1$ where $\mu_i=1/N$ $\ \forall \ i$ which means that every node has the same probability to be picked if we hit one at random. This is exactly like the state of the isolated ideal gas since here too all accessible microstates are equally probable. The $q=1$ state is thus the most confused or disordered state. Now as we lower the $q$ value, we see that entropy decreases slowly but as we approach towards $q_c=1-t_c$ we observe a dramatic decrease in entropy. This is because as we approach $q_c$ from higher $q$ value we find that many moderately large sized clusters get accumulated as the AP discourages growth of large clusters and encourages the smaller ones. Eventually the crowding of the moderately large clusters reaches a critical state at $q=q_c$ where addition of a few links triggers the growth of the largest cluster in an explosive fashion. We find that at $q=0$ the entropy $H$ is minimally low but the order parameter $P$ is maximally high and hence it is clearly the ordered phase. We thus see that at $q=1$ the entropy is maximally high but the order parameter $P=0$ and hence it correspond to the disordered phase. On the other hand at $q=0$, the order parameter is maximally high and entropy is minimally low. The term percolation therefore refers to the transition from ordered phase characterized by vanishingly small entropy at $q<q_c$ to disordered phase characterized by $P=0$ at $q>q_c$ as one tunes the control parameter $q$. We thus find that in percolation too, like in the thermal CPT, entropy $H$ and order parameter $P$ compliments each other. Once we know the entropy, we can find specific heat as we can define it as $$C(t)=(1-t){{dH}\over{d(1-t)}}$$ in analogy with the definition of its thermal counterpart. Taking differentiation of $H$ from first principles and multiplying that value with the corresponding value of $(1-t)$, we can immediately obtain $C(t)$. We then plot $C(t)$ in Figs. (\[fig:4a\]) and (\[fig:4b\]) as a function of $t$ for PR and SR respectively. To compute the corresponding critical exponent $\alpha$ once again we use the FSS hypothesis and find $\alpha/\nu=0.535$ for both PR and SR. Finally, we plot $CL^{-\alpha/\nu}$ vs $(t-t_c)L^{1/\nu}$ and obtain a perfect data-collapse with $\alpha/\nu=0.535$ and $1/\nu=0.535$ for both PR and SR as shown in Figs. (\[fig:4c\]) and (\[fig:4d\]). We then use the relation $L\sim (t-t_c)^{-\nu}$ in $C(t)\sim L^{\alpha/\nu}$ and immediately find that the specific heat diverges like $$C(t)\sim (t-t_c)^{-\alpha},$$ where $\alpha=1$ for both PR and SR. The quality of data-collapse is a clear testament of the accuracy of $\alpha$ value. Classifications of any system into universality classes is always an interesting proposition. To this end, finding that PR and SR of explosive percolation belongs to the same universality class is a significant development especially when we know that the most expected site-bond universality breaks down even in the lattice. To check whether the Rushbrooke inequality holds in EP or not, we substitute our values of $\alpha=1$, $\gamma=0.893$ and $\beta=0.084$ in the Rushbrooke relation and find $\alpha+2\beta+\gamma=2.061$. Thus, we find that the Rushbrooke inequality not only holds but also its value is close to equality, within the acceptable range of errors. Recently, we applied the same approach to the square and weighted planar stochastic (WPS) lattices where we found that RI holds in RP for both the lattices albeit they belong to different universality classes [@ref.hassan_didar]. Moreover, in both the cases, we find that RI holds almost as an equality like in the thermal CPT. Thus, finding that RI holds for three different universality classes that include a class as exotic as EP provides sufficient confidence in our results. It implies that explosive percolation is indeed a paradigmatic model for continuous phase transition with some unusual finite-size behaviours since we find hysteresis loops in its forward and reverse processes, doublehump in the distributions of the order parameter $P$, which, however, disappears in the thermodynamic limit [@ref.Grassberger; @ref.Bastas; @ref.Riordan; @ref.bastas_review; @ref.Choi]. Besides, we also know that the time difference $\Delta=t_2-t_2$ between the last step $t_2$ for which the largest cluster $C <N^{1/2}$ and the first step $t_1$ for which $C>0.5N$ is not extensive while it is extensive for RP on ER . For all these reasons explosive percolation is indeed a non-trivial paradigmatic model for CPT. To summarize, we have used our recently defined entropy, specific heat and re-defined susceptibility in explosive percolation on random network. Until now we could only quantify the extent of order in percolation by measuring the order parameter $P$. Thanks to the definition of entropy, we can now quantify the other phase too. It is so high in the phase where $P=0$ that we can regard it as disordered phase. It implies that the high-$q$ phase is more disordered, i.e., has a higher symmetry than the low-$q$ phase thus revealing that percolation is an order-disorder transition like ferromagnetic transition. We have also shown that the specific heat and susceptibility diverge at the critical point without having to exclude the largest cluster which is in sharp contrast to the mean cluster size which also diverges at the critical point but only if we exclude the largest cluster from it. We obtained the critical exponents $\alpha$, $\beta$, $\gamma$ and $\nu$ numerically and found that their values for PR and SR are the same, revealing that they belong to the same universality class. Such PR-SR universality is highly intriguing and unexpected especially against the background of the breakdown of the usual site-bond universality even in the lattice. We have also shown that the value of the critical exponents $\alpha$, $\beta$, $\gamma$ obey the Rushbrooke inequality. 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--- abstract: 'We calculate the ratio $R_{\ell\ell}$ of same sign (SS) to opposite sign (OS) dileptons in type I and generalized inverse seesaw models and show that it can be anywhere between 0 and 1 depending on the detailed texture of the right-handed neutrino mass matrix. Measurement of $R_{\ell\ell}$ in hadron colliders can therefore provide a way to probe the nature of seesaw mechanism and also to distinguish between the two types of seesaw mechanisms. We work within the framework of left-right symmetric model as an example. We emphasize that coherence of the final states in the $W_R$ decay is crucial for this discussion and it requires the right-handed neutrinos to be highly degenerate. We isolate the range of parameters in the model where this effect is observable at the LHC and future colliders.' author: - Arindam Das - 'P. S. Bhupal Dev' - 'Rabindra N. Mohapatra' title: Same Sign versus Opposite Sign Dileptons as a Probe of Low Scale Seesaw Mechanisms --- Introduction ============ Different kinds of seesaw mechanism have been proposed as ultraviolet (UV)-complete theories that lead to the dimension-5 Weinberg operator [@weinberg] for understanding small neutrino masses. Two of them are the so called type-I [@seesaw1; @seesaw2; @seesaw3; @seesaw4; @seesaw5] and inverse seesaw [@ISS1; @ISS2], which have been widely discussed in the literature. The type-I seesaw involves adding SM-singlet heavy fermions $N$ with Majorana masses that violate lepton number maximally, whereas in the inverse seesaw, one adds two SM-singlet heavy neutrinos $N$ and $S$ and a small $L$-violating mass for one set of the new singlet fermions. A simple UV-complete extension of the Standard Model (SM) that incorporates all the key ingredients of both type I seesaw and inverse seesaw and leads to them naturally is the left-right symmetric model [@LR1; @LR2; @LR3]. No extra symmetries need to be added to generate the right texture for getting tiny neutrino masses. The right-handed neutrino (RHN), predicted by anomaly considerations in this theory, couples to the right-handed (RH) gauge boson $W_R$ and is the source of the lepton number violating (LNV) signal [@KS] we will discuss. In this paper we will work within the framework of the minimal left-right model and assume that $W_R$ is kinematically accessible to the colliders. In other words, for $\sqrt s=14$ TeV LHC, we assume the mass of the $W_R$ boson to be less than 5 TeV or so [@Ferrari:2000sp; @Deppisch:2015qwa]. A key predictions for the TeV-scale left-right type-I seesaw model is that it leads to a spectacular LNV signal in hadron colliders in the form of two same-sign leptons and two jets with no missing energy [@KS]. This arises from the production and decay of heavy RHNs, both mediated by the $W_R$ gauge boson in the $s$-channel. The Majorana nature of the RHN dictates that the final states with same-sign (SS) dileptons ($\ell^\pm\ell^\pm$) appear in equal number with opposite-sign (OS) dilepton states ($\ell^\pm\ell^\mp$). In other words, the minimal left-right type-I seesaw prediction is that the ratio of the number of events in the two final states, $R_{\ell\ell}\equiv N_{\rm SS}/N_{\rm OS}=1$. This in fact is considered a ‘smoking gun’ signal for TeV-scale type-I seesaw in general[^1] and, more specifically, for the left-right seesaw model and has been extensively studied in the literature, both for the LHC [@KS; @Ferrari:2000sp; @Deppisch:2015qwa; @Nemevsek:2011hz; @Das:2012ii; @AguilarSaavedra:2012gf; @Han:2012vk; @Chen:2013fna; @Khachatryan:2014dka; @Ng:2015hba; @Dev:2015kca; @Gluza:2016qqv; @Mitra:2016kov; @Ruiz:2017nip; @Aad:2015xaa; @Dev:2016dja; @Roitgrund:2017byx], as well as other future colliders [@Lindner:2016lxq; @Mondal:2015zba; @Biswal:2017nfl; @Golling:2016gvc]. On the other hand, in the inverse seesaw mechanism, lepton number breaking is very small, because the heavy singlet neutrino ($N$) is paired with another singlet fermion ($S$) to form a pseudo-Dirac pair and the Majorana nature of the neutrino emerges from a keV-scale Majorana mass $\mu_S$ of $S$ fermion (for TeV-scale seesaw). This model when embedded into the TeV-scale left-right framework exhibits some interesting features. There are two versions of this model: the minimal version where there is no majorana mass for the $N$ [@Dev:2009aw; @An:2011uq; @Chen:2011hc] and a second more general one where there is a Majorana mass $\mu_R$ for $N$ [@Dev:2012sg; @Awasthi:2013ff; @Dev:2015pga]. In the minimal version, the leading order prediction for collider signal is that final states will approximately conserve lepton number, implying that $R_{\ell\ell}\simeq 0$ [@Chen:2011hc]. In the more general inverse seesaw, which can also arise from left-right seesaw models [@Dev:2015pga], the neutrino mass formula remains unaffected at the tree-level, although there is an unavoidable one-loop contribution from electroweak radiative corrections [@Dev:2012sg]; however the $N$ fermion has a potentially large Majorana mass that breaks lepton number by two units. The question remains as to how do the dilepton final states look like in this general case i.e. is $R_{\ell\ell}=1$ or different? This question has been recently studied in some special cases [@Dev:2015pga; @Anamiati:2016uxp; @Antusch:2017ebe] and was shown that due to interference between two heavy Majorana neutrino mass eigenstates, one could in principle realize a scenario with $R_{\ell\ell}$ anywhere between 0 and 1. The goal of this study is to do a more general analysis and discuss whether analyzing dilepton states in a hadron collider via production of a $W_R$ boson, one can probe the details of the RHN mass matrix and distinguish between the type-I and general inverse seesaw mechanisms. The rest of the paper is organized as follows. In Section \[sec:coh\] we discuss the coherence condition for interference between two heavy Majorana neutrino mass eigenstates, which plays a crucial role in our discussion. In Section \[sec:typeI\], we apply the coherence conditions to discuss the nature of dilepton final states in type-I seesaw. In Section \[sec:inv\] we explain the general inverse seesaw model. In Section \[sec:inv2\] we apply the coherence conditions for the inverse seesaw case to get the $R_{\ell\ell}$ as a function of parameters of inverse seesaw model. We give our conclusions in Section \[sec:con\]. Some useful three-body decay widths for the RHN are listed in Appendix \[sec:app\]. Coherence Conditions for Interference {#sec:coh} ===================================== When a $W_R$ gauge boson is produced in proton-proton collisions, it decays into flavor eigenstates of the RHNs $N_{\ell}$ along with the corresponding charged lepton $\ell_R$ (where $\ell=e,\mu,\tau$). For simplicity, let us consider two RHNs, say $N_e$ and $N_\mu$. When these flavor eigenstates evolve, they do so as linear combination of mass eigenstates $N_{1,2}$. The $N_{1,2}$ are linear combinations of $N_e$ and $N_\mu$ in the type-I seesaw case and of $N$ and $S$ in the inverse seesaw case. The $N_{1,2}$ are Majorana fermions and they will evolve and interfere as they produce the charged leptons (along with two jets) in their final state. Only if the coherence condition (discussed below) is satisfied, they will interfere; otherwise they will simply give equal number of SS and OS dilepton final states. The coherence conditions for light neutrinos have been discussed inRefs. [@Kayser:1981ye; @Akhmedov:2007fk]. There are two conditions that must be satisfied for interference between the two states to take place: (i) coherence in emission and (ii) the coherence must be maintained till the RHNs decay i.e. for their full decay length. The results imply that the first condition is satisfied when the uncertainty in their mass square exceeds their actual mass difference. We now transplant their argument to the case of two RHNs at hand. Denoting by $\sigma_{m^2}$ the mass uncertainty, we get for the coherence condition $\sigma_{m^2}$ $\geq \Delta M^2\equiv \|M^2_1-M^2_2\|$. The $\sigma_{m^2}$ in this case is estimated to be $2\sqrt{2} E\Gamma_{W_R}$ where $E$ is the average energy of the RHN eigenstates and $\Gamma_{W_R}$ is the width of the $W_R$ which causes the uncertainly in the energy of the produced heavy neutrino state. Thus, in our case, coherence in emission occurs when $$\begin{aligned} \Delta M^2 \ \leq \ 2\sqrt{2}E\Gamma_{W_R} \, . \label{eq:coh1}\end{aligned}$$ For TeV-scale $W_R$ and RHNs, this is satisfied when the mass difference between the states is less than few hundred GeV, where we have estimated $\Gamma_{W_R}\simeq (g^2/12\pi) M_{W_R}$, setting the $SU(2)_L$ and $SU(2)_R$ couplings to be equal, i.e. $g_L=g_R\equiv g$. Turning to the second condition, we take the decay length $L$ as $L=1/\Gamma_N$ and using the results of Ref. [@Akhmedov:2007fk], require that $L\leq \sigma_x/\delta v_g$, where $\sigma_x$ is the size of the RHN wave packets and $\delta v_g$ is the difference between the group velocities of the individual RHNs. We have $\sigma_x \sim ( \sigma_E )^{-1}\sim (\Delta M^2/2\sqrt{2}E)^{-1}$ and $\delta v_g\sim (\Delta M^2/2E^2)$. Putting theses together, we get $$\begin{aligned} L \ \equiv \ \frac{1}{\Gamma_N} \ < \ \frac{4\sqrt{2}E^3}{(\Delta M^2)^2} \, . \label{eq:coh2}\end{aligned}$$ This implies a stringent condition on the mass difference between the two interfering RHNs. For instance, for $M_{W_R}=5$ TeV, $M_{N}\simeq 1$ TeV, and $E\sim 2$ TeV, we get the coherence condition $\Delta M\equiv \|M_1-M_2\|\leq $ GeV. Note that this condition is more stringent than what condition (i) alone would have implied \[cf. Eq. \] and requires a degeneracy of one part in $10^3$ between the two RHN masses for interference to take place. In deriving this, we have used the decay width formula for the RHNs given in Appendix \[sec:app\]. From this discussion, we conclude that if interference effect is observed, it will imply constraints on the mass matrix of both the type I and inverse seesaw, helping to further elucidate the nature of the seesaw. It will for example imply that there are at least two nearly degenerate RHN states, consistent with the general expectation from many TeV-scale seesaw models [@Pilaftsis:1991ug; @Gluza:2002vs; @Kersten:2007vk; @Xing:2009in; @He:2009ua; @Adhikari:2010yt; @Ibarra:2010xw; @Mitra:2011qr; @Dev:2013oxa; @Chattopadhyay:2017zvs], which require the quasi-degeneracy to satisfy the neutrino oscillation data. This is also the requirement for successful resonant leptogenesis via the out-of-equilibrium decay of TeV scale RHNs [@Pilaftsis:2003gt; @Dev:2017wwc]. Same Sign vs Opposite Sign Dilepton Events in Type-I seesaw {#sec:typeI} =========================================================== Let us first briefly recapitulate the well known field theoretic argument of why for Majorana RHNs the final states in its decay have equal number of both sign leptons. For concreteness, we illustrate this in the context of left-right model but the argument is general. In the left-right model, the decay of $N$ can be assumed to occur via the emission of a virtual $W_R$ boson and it comes from the RH gauge interaction $$\begin{aligned} {\cal L}_I \ = \ \frac{g}{\sqrt{2}}\bar{\ell}_R\gamma_\mu N W^{-,\mu}_R+\frac{g}{\sqrt{2}} N^TC^{-1}\gamma_\mu\ell_RW^{+,\mu}_R \label{eq:1}\end{aligned}$$ The second term in the above equation is nothing but the hermitian conjugate of the first one after we use the Majorana condition for $N$ i.e. $N=C\bar{N}^T$ (where $C$ is the charge conjugation operator). Now note that in both terms the $N$ field is annihilated but the final state from the first term is an $\ell^-$ whereas that from the second term is an $\ell^+$, while both the amplitudes are the same i.e. $g/\sqrt{2}$. This is the basic reason for equal number of SS and OS dileptons in the final states which for a $pp$ collision leads to their ratio $R_{\ell\ell}=1$. To see how interference between two RHN states affects the ratio $R_{\ell\ell}$, let us consider the simple case of type-I seesaw with only two heavy neutrino flavors $(N_e,N_\mu)$. This case has been discussed in some details in Refs. [@Bray:2007ru; @Gluza:2015goa; @Carmona:2016oxx; @Gluza:2016qqv]. Here we emphasize the importance of the coherence condition and present new analytic results on the effect on different flavor combinations of the final states. One can easily generalize this to more flavors, but the main conclusion of this section remains unchanged. Including the effect of $CP$ violation, we can write the flavor eigenstates as the following combinations of the mass eigenstates: $$\begin{aligned} N_e \ & = & \ c_\theta N_1+s_\theta e^{i\delta}N_2\, , \nonumber \\ N_\mu\ & = & \ -s_\theta N_1+c_\theta e^{i \delta}N_2 \, , \label{eq:3.2}\end{aligned}$$ where $\delta$ is the $CP$ phase in the RHN mixing, $\theta$ is the mixing angle in this sector, and $c_\theta\equiv \cos\theta,\, s_\theta \equiv \sin\theta$. For the general $2\times 2$ RHN mass matrix $$\begin{aligned} {\cal M}_N \ = \ \begin{pmatrix} M_1 & Me^{i\phi} \\ Me^{i\phi} & M_2 \end{pmatrix}\, , \label{mass}\end{aligned}$$ the mixing angle is given by $$\begin{aligned} \theta \ = \ \frac{1}{2}\tan^{-1}\left\|\frac{2M}{M_1-M_2}\right\| \, . \label{eq:theta}\end{aligned}$$ Substituting Eqs. in the interaction Lagrangian for RHNs in Eq. , we get $$\begin{aligned} \mathcal{L}_I &\ = \ &\frac{g}{\sqrt{2}}\Big[\bar{e}_R\gamma_\mu (c_\theta N_1+s_\theta e^{i\delta}N_2)W^{-,\mu}_R+(c_\theta N_1+s_\theta e^{-i\delta}N_2)^TC^{-1}\gamma_\mu e_RW^{+,\mu}_R \nonumber \\ &+&\bar{\mu}_R\gamma_\mu (-s_\theta N_1+c_\theta e^{i\delta}N_2)W^{-,\mu}_R+ (-s_\theta N_1+c_\theta e^{-i\delta}N_2)^TC^{-1}\gamma_\mu\mu_R W^{+,\mu}_R \Big] \end{aligned}$$ where we have assumed that RH charged leptons are the mass eigenstate. Using the coherence conditions, we can write the time evolution of the amplitudes for SS and OS final states as follows: $$\begin{aligned} A_{{\rm OS},ee}(t)&\ = \ &c^2_\theta e^{-iE_1 t -\frac{\Gamma_1t}{2}}+s^2_\theta e^{-iE_2 t -\frac{\Gamma_2t}{2}} \, , \label{eq:OSee} \\ A_{{\rm SS},ee}(t)&\ = \ &c^2_\theta e^{-iE_1 t -\frac{\Gamma_1t}{2}}+s^2_\theta e^{-2i\delta} e^{-iE_2 t -\frac{\Gamma_2t}{2}} \, ,\label{eq:SSee} \\ A_{{\rm OS},\mu\mu}(t)&\ = \ &s^2_\theta e^{-iE_1 t -\frac{\Gamma_1t}{2}}+c^2_\theta e^{-iE_2 t -\frac{\Gamma_2t}{2}} \, ,\label{eq:OSmumu} \\ A_{{\rm SS},\mu\mu}(t)&\ = \ &s^2_\theta e^{-iE_1 t -\frac{\Gamma_1t}{2}}+c^2_\theta e^{-2i\delta}e^{-iE_2 t -\frac{\Gamma_2t}{2}} \, , \label{eq:SSmumu}\\ A_{{\rm OS},e\mu}(t)&=&-c_\theta s_\theta \Big[ e^{-iE_1 t -\frac{\Gamma_1t}{2}}- e^{-iE_2 t -\frac{\Gamma_2t}{2}}\Big] \ =\ A_{{\rm OS},\mu e}(t)\, , \label{eq:OSemu} \\ A_{{\rm SS},e\mu}(t)&\ = \ &-c_\theta s_\theta \Big[ e^{-iE_1 t -\frac{\Gamma_1t}{2}}- e^{-2i\delta}e^{-iE_2 t -\frac{\Gamma_2t}{2}}\Big]\ = \ A_{{\rm SS},\mu e} (t) \, ,\label{eq:SSemu}\end{aligned}$$ where $\Gamma_{1,2}$ are the total decay widths of the two mass eigenstates $N_{1,2}$. We adopt the following procedure to get the ratio of SS and OS final states [@Anamiati:2016uxp]: $$\begin{aligned} R_{\ell\ell} \ = \ \frac{\int^\infty_0 dt\left \|A_{{\rm SS},\ell\ell}(t)\right\|^2}{\int^\infty_0 dt\left \|A_{{\rm OS},\ell\ell}(t)\right\|^2} \ \equiv \ \frac{N_{{\rm SS},\ell\ell}}{N_{{\rm OS},\ell\ell}} \, . \label{Rll}\end{aligned}$$ In order to illustrate the effect of the interference between the two states, we make the simplifying assumption that the two RHNs are non-relativistic (which is a good approximation when the $W_R$ mass is slightly larger than two times the RHN mass) and approximate $E_{1,2}\simeq M_{N_{1,2}}\simeq M_N\pm \Delta M/2$, where $\Delta M\equiv M_{N_1}-M_{N_2}$ is the mass splitting between the two RHN mass eigenstates. The eigenvalues $M_{N_{1,2}}$ can be obtained by calculating the eigenvalues of Eq. . Then from Eqs. and , the number of SS and OS dielectron events are respectively given by $$\begin{aligned} N_{{\rm OS}, ee} & \ = \ & \Gamma_{\rm avg}\left[\frac{c^4_\theta}{\Gamma_1}+\frac{s^4_\theta}{\Gamma_2}+c^2_\theta s^2_\theta\frac{\Gamma_1+\Gamma_2}{\left(\frac{\Gamma_1+\Gamma_2}{2}\right)^2+(\Delta M)^2}\right] \, ,\\ N_{{\rm SS},ee} & \ = \ & \Gamma_{\rm avg}\left[\frac{c^4_\theta}{\Gamma_1}+\frac{s^4_\theta}{\Gamma_2}+c^2_\theta s^2_\theta\left\{\frac{(\Gamma_1+\Gamma_2)\cos{2\delta}}{\left(\frac{\Gamma_1+\Gamma_2}{2}\right)^2+(\Delta M)^2}-\frac{2\Delta M \sin{2\delta} }{\left(\frac{\Gamma_1+\Gamma_2}{2}\right)^2+(\Delta M)^2}\right\}\right] \, ,\end{aligned}$$ where $\Gamma_{\rm avg}\equiv (\Gamma_1+\Gamma_2)/2$. Similarly, for dimuon events, we have from Eqs. and respectively, $$\begin{aligned} N_{{\rm OS},\mu\mu} & \ = \ & \Gamma_{\rm avg}\left[\frac{s^4_\theta}{\Gamma_1}+\frac{c^4_\theta}{\Gamma_2}+c^2_\theta s^2_\theta\frac{\Gamma_1+\Gamma_2}{\left(\frac{\Gamma_1+\Gamma_2}{2}\right)^2+(\Delta M)^2}\right] \, , \\ N_{{\rm SS},\mu\mu} & \ = \ & \Gamma_{\rm avg}\left[\frac{s^4_\theta}{\Gamma_1}+\frac{c^4_\theta}{\Gamma_2}+c^2_\theta s^2_\theta\left\{\frac{(\Gamma_1+\Gamma_2)\cos{2\delta}}{\left(\frac{\Gamma_1+\Gamma_2}{2}\right)^2+(\Delta M)^2}-\frac{2\Delta M\sin{2\delta} }{\left(\frac{\Gamma_1+\Gamma_2}{2}\right)^2+(\Delta M)^2}\right\}\right] \, .\end{aligned}$$ Finally, for the $e\mu$ events, we have from Eqs. and respectively $$\begin{aligned} N_{{\rm OS},e\mu} & \ = \ & N_{{\rm OS}, \mu e} \ = \ \Gamma_{\rm avg} \: c^2_\theta s^2_{\theta}\left[\frac{1}{\Gamma_1}+\frac{1}{\Gamma_2}-\frac{\Gamma_1+\Gamma_2}{\left(\frac{\Gamma_1+\Gamma_2}{2}\right)^2+(\Delta M)^2}\right] \, ,\\ \ N_{{\rm SS},e\mu}& \ = \ & N_{{\rm SS}, \mu e} \ = \ \Gamma_{\rm avg} \: c^2_\theta s^2_{\theta}\left[\frac{1}{\Gamma_1}+\frac{1}{\Gamma_2}-\left\{\frac{(\Gamma_1+\Gamma_2)\cos{2\delta}}{\left(\frac{\Gamma_1+\Gamma_2}{2}\right)^2+(\Delta M)^2}-\frac{2\Delta M\sin{2\delta} }{\left(\frac{\Gamma_1+\Gamma_2}{2}\right)^2+(\Delta M)^2}\right\}\right] \, .\nonumber\\\end{aligned}$$ Expanding these equations out, we find that if there is no $CP$ phase i.e. $\delta=0$, we get $$\begin{aligned} N_{{\rm OS},ee} \ = \ N_{{\rm SS},ee}\, ; \quad N_{{\rm OS},\mu\mu} \ = \ N_{{\rm SS},\mu\mu} \, ; \quad N_{{\rm OS}, e\mu} \ = \ N_{{\rm SS},e\mu} \, , \end{aligned}$$ as expected for purely Majorana RHNs. However, in the presence of a non-zero $CP$ phase, we find $$\begin{aligned} N_{{\rm OS},\ell\ell} \ \neq \ N_{{\rm SS},\ell\ell} \, , \quad {\rm or} \quad R_{\ell\ell}\neq 1 \, , \end{aligned}$$ as illustrated in Figure \[fig:Rll-typeI\]. We emphasize again that these arguments are true only if the two RHN states satisfy the coherence conditions and . Let us apply our findings to the special case where the RHN mass matrix is of the form $M_N=M\tau_1$ where $\tau_1$ is the first Pauli matrix. In this case $\theta=\pi/4$ and $\delta=\pi/2$. Also in this case, $\Gamma_1=\Gamma_2$ and $\Delta M=0$. Substituting this in Eq. , we get $N_{{\rm SS},ee}=N_{{\rm SS},\mu\mu}=0$ and only $N_{{\rm SS},e\mu}\neq 0$ as we would expect from the structure of the RHN mass matrix.[^2] In Figure \[fig:Rll-typeI\], we show the variation of $R_{\ell\ell}$ (for $\ell\ell=ee,\mu\mu$) as a function of the $CP$ phase $\delta$ for different values of $\Delta M/\Gamma_{\rm avg}$. As for the RHN decay widths, we have used the three-body decay widths of $N_\ell\to W_R^\ell \to q\bar{q}'\ell$ (see Appendix \[sec:app\]). For numerical purposes, we have chosen a fixed value of $M_{W_R}=5$ TeV and $M_{N_1}=500$ GeV, but our main results are independent of the choice of the exact mass values, as long as $M_{N_{1,2}}< M_{W_R}$, which is anyway required from vacuum stability arguments [@Mohapatra:1986pj; @Maiezza:2016bzp]. We find that for $\delta=0$ and $\pi$ (i.e. no $CP$ violation), $R_{\ell\ell}=1$, as discussed above. But for $\delta=\pi/2$ and $\Delta M=0$, $R_{\ell\ell}=0$, i.e. there is a completely destructive interference between the two RHN mass eigenstates in the SS channel. The degree of interference decreases rapidly as we increase $\Delta M$ and as $\Delta M$ becomes much larger than $\Gamma_{\rm avg}$, there is virtually no interference, leading to the limit $R_{\ell\ell}\to 1$, as expected for purely Majorana RHNs. In the intermediate range of $\Delta M/\Gamma_{\rm avg}$, we have $R_{\ell\ell}>1$, i.e. enhanced SS signal even compared to the purely Majorana case, for certain choices of the $CP$ phase $\delta$, when the constructive interference is maximum in Eqs. and . Note here that both $ee$ and $\mu\mu$ channels lead to almost identical predictions for the ratio $R_{\ell\ell}$ because for $\Delta M\ll M$ in Eq. , $\theta\approx \pi/4$, so $c_\theta\simeq s_\theta\simeq 1/\sqrt{2}$. General Inverse Seesaw Case {#sec:inv} =========================== We start this section by briefly reviewing the inverse seesaw extension of the left-right symmetric model. The model is based on the gauge group $SU(3)_c\times SU(2)_L\times SU(2)_R \times U(1)_{B-L}$ gauge group [@LR1; @LR2; @LR3] with scalar sector consisting of two $SU(2)$ doublets $\chi_R^{}(1,1,2,+1)$, together with a bidoublet $\phi(2,2,0)$ and a $B-L=2$ triplet $\Delta_R(1,3,+2)$, while the fermion sector contains not only the usual $SU(2)$ doublets of the left-right model i.e. $Q_L^{}(3,2,1,+\frac{1}{3})$, $Q_R^{}(3,1,2,+\frac{1}{3})$, $L^{}(1,2,1,-1)$ and $R^{}(1,1,2,-1)$, but also additional $SU(2)$ singlets $S_a$ (with $a=1,2,3$). Note that we are working with a model where parity symmetry breaking scale $M_P$ and the $SU(2)_R$ symmetry breaking scale $v_R$ are different with $M_P\gg v_R$ [@Chang:1983fu]. To discuss the inverse seesaw in this model, we need the leptonic Yukawa couplings: $$\begin{aligned} {\cal L}_Y \ = \ h_l\bar{L}\phi R+h_\nu \bar{R}\chi_R S +f\bar{R}^C \Delta_R R+\mu_s \bar{S}^CS+{\rm H.c.} \end{aligned}$$ After symmetry breaking by the vacumm expectation values of the Higgs fields i.e. ${\rm Diag} \langle\phi \rangle=(\kappa, \kappa')$; $\langle \chi^0_R\rangle = \sigma_R$ and $\langle\Delta^0_R\rangle = v_R$, we get the neutral fermion mass matrix of the form: $$\begin{aligned} {\cal M} \ &=& \ \left(\begin{array}{ccc} 0 & M_D & 0\\ M^T_D & \mu_R & M_N\\ 0 & M^T_N & \mu_S \end{array}\right) \label{eq:4.2}\end{aligned}$$ where $M_D=h_l\sqrt{\kappa^2+\kappa'^2}$, $\mu_R=fv_R$ and $M_N=h_\nu \sigma_R$.[^3] It leads to the formula for light neutrino mass matrix at tree-level:[^4] $$\begin{aligned} M_\nu \ = \ (M_DM_N^{-1}) \mu_S(M_DM_N^{-1})^T \, . \label{eq:4.3}\end{aligned}$$ This is the inverse seesaw mechanism at work for the most general case where each entry in Eq. is a $3\times 3$ matrix corresponding to three flavors. For simplicity below we consider a single family version of this matrix to illustrate our discussion of SS and OS dilepton plus two jets in $pp$ collision. We note that this analysis can be applicable to realistic situation with flavor in the following way: Consider the case when $M_D, M_N, \mu_R$ are all diagonal $3\times 3$ matrices and let all neutrino flavor mixings reside in the $\mu_S$ matrix. This is the so-called flavor-diagonal scenario. Proper choice of the $\mu_S$ matrix can explain the observed neutrino oscillation results but since each element of the $\mu_S$ matrix is very small compared to other matrices in the problem i.e. $\mu_R, M_N$, they will not affect our conclusions about the ratio $R_{\ell\ell}$ for each flavor. Of course one could also consider flavor structures in $\mu_R$ and/or $M_N$. The analysis is then more complicated and we do not consider it here.[^5] $R_{\ell\ell}$ in the Inverse Seesaw Case {#sec:inv2} ========================================= In order to study the final states with SS or OS dileptons, we consider a simplified yet realistic case where $\mu_R$ and $M_N$ are $2\times 2$ diagonal matrices so that all neutrino mixings arise from the matrix $\mu_S$, which does not have any effect on $R_{\ell\ell}$. We consider the eigenstates of the mass matrix . We do this in stages and for the parameter domain where $M_N\gg \mu_R \gg M_D \gg \mu_S$, we can first diagonalize the lower $2\times 2$ matrix and get the following eigenstates with real eigenvalues $$\begin{aligned} {\cal N}_{1} \ &=& \ c_\alpha N+s_\alpha S \, ; \\ {\cal N}_{2} \ &=& \ i(-s_\alpha N+c_\alpha S) \, .\end{aligned}$$ Using these we can rewrite the $W_R$-induced charged-current interactions as $$\begin{aligned} {\cal L}_I \ = \ \frac{g}{\sqrt{2}}\bar{\ell}_R\gamma_\mu (c_\alpha {\cal N}_1+is_\alpha {\cal N}_2)W^{-,\mu}_R+\frac{g}{\sqrt{2}} (c_\alpha {\cal N}_1-is_\alpha {\cal N}_2)^TC\gamma_\mu\ell_RW^{+,\mu}_R \label{eq:laginv}\end{aligned}$$ To calculate the OS and SS event numbers, we need to discuss whether there is coherence between the decays of $\mathcal{N}_{1,2}$ when produced in $W_R$ decay as well as the requirement for maintaining coherence over the decay length of the RHNs. For the inverse seesaw case, the discussions in Ref. [@Akhmedov:2007fk] lead us to the same coherence condition in the emission as stated in Sec. \[sec:coh\] except in this case, $\Delta M$ denotes the mass difference between the $N$ and $S$ fermions. The condition on parameters from coherence length considerations are different here since the Dirac Yukawa couplings which dominate the decay of $N,S$ states are expected to be much larger for inverse seesaw than the type I case. The decay length in this case is therefore much shorter than the type I case i.e. $L\sim \frac{12\pi}{h^2 M_N}$. Choosing $h\sim 0.1$ and $M_N\sim $ TeV, we get for $\Delta M\sim 100$ GeV, as compared to 1 GeV or so in the type I case. When the coherence condition is satisfied, recalling that the first term in the Lagrangian is responsible for the production of OS events and second one for SS events, we can write the amplitudes for OS and SS events as follows: $$\begin{aligned} A_{\rm OS}(t) \ &=& \ c^2_\alpha e^{-iE_1t-\frac{\Gamma_1}{2}t}+s^2_\alpha e^{-iE_2t-\frac{\Gamma_2}{2}t} \, , \nonumber \\ A_{\rm SS}(t) \ &=& \ c^2_\alpha e^{-iE_1t-\frac{\Gamma_1}{2}t}-s^2_\alpha e^{-iE_2t-\frac{\Gamma_2}{2}t} \, .\end{aligned}$$ We approximate $E_{1,2}\simeq M_{1,2}\pm \Delta M/2$ as before and use the expression in Eq. to obtain for the OS and SS events respectively $$\begin{aligned} N_{\rm OS} \ & = \ \Gamma_{\rm avg} \left[ \frac{c_\alpha^4}{\Gamma_1}+\frac{s_\alpha^4}{\Gamma_2}-\frac{c_\alpha^2 s_\alpha^2(\Gamma_1+\Gamma_2)}{\frac{(\Gamma_1+\Gamma_2)^2}{4}+(\Delta M)^2}\right], \label{eq:OS} \\ N_{\rm SS} \ & = \ \Gamma_{\rm avg} \left[ \frac{c_\alpha^4}{\Gamma_1}+\frac{s_\alpha^4}{\Gamma_2}+\frac{c_\alpha^2s_\alpha^2(\Gamma_1+\Gamma_2)}{\frac{(\Gamma_1+\Gamma_2)^2}{4}+(\Delta M)^2} \right] \, . \label{eq:SS}\end{aligned}$$ Using the RHN decay widths given in Appendix \[sec:app\], we have plotted in Fig. \[fig:Rll-ISS\] the ratio $R_{\ell\ell}=N_{\rm SS}/N_{\rm OS}$ as a function of $\mu_R$ for different RHN masses. Here we have chosen a fixed value of $M_{W_R}=5$ TeV for illustration. We find that smaller values of $\mu_R$ favors the OS signal whereas higher values of $\mu_R$ favor the SS signal. For lower values of $M_N$, the range of $\mu_R$ increases where $R_{\ell\ell}\to 1$. Now we can look at three special cases: [Case (i): $\mu_R=0$]{}: This is the case which has been considered in Refs. [@Anamiati:2016uxp; @Antusch:2017ebe]. In this case for a TeV $M_N$, fitting neutrino mass scale requires that $\mu_S\leq $ keV. This means that $\Delta M \sim \mu_S\sim $ keV and the coherence condition is very well satisfied. Furthermore, in this case $c_\alpha=s_\alpha = \frac{1}{\sqrt{2}}$. Using the fact that we have also $\Gamma_1=\Gamma_2$, we get from Eqs. and , that $$\begin{aligned} R_{\ell\ell} \ = \ \frac{(\Delta M)^2}{2\Gamma^2+(\Delta M)^2} \end{aligned}$$ in agreement with the result in Ref. [@Anamiati:2016uxp]. Note that for TeV-scale $M_{N}$, typically $\Gamma\sim 10-100$ keV and $\Delta M\sim 1$ keV, leading to $R_{\ell\ell}\lesssim 1\%$. Thus to get large $R_{\ell\ell}$ in inverse seesaw models, one must include the effect of $\mu_R$. [Case (ii) $\mu_R\ll M_N$]{}: In this case, in general $\alpha$ is different from $\pi/4$ and we do not expect $\Gamma_1$ and $\Gamma_2$ to be equal. If we assume that $\Gamma_1\sim \Gamma_0 c^2_\alpha$ and $\Gamma_2\sim \Gamma_0 s^2_\alpha$, we get $$\begin{aligned} R_{\ell\ell} \ = \ \frac{\cos^2 2\alpha+\frac{4(\Delta M)^2}{\Gamma^2_0}}{1+\sin^22\alpha+\frac{4(\Delta M)^2}{\Gamma^2_0}} \, . \end{aligned}$$ For the case when $\frac{4(\Delta M)^2}{\Gamma^2_0}\ll 1$, it reduces to the formula in Ref. [@Dev:2015kca]. In this case, $R_{\ell\ell}$ can be significant; see Figure \[fig:Rll-ISS\]. [Case (iii): Hierarchical masses i.e. $\mu_R \gg M_N$:]{} In this case, the two eigenstates ${\cal N}_{1,2}$ have a large mass difference i.e. $(\Delta M)^2\gg \Gamma^2_{1,2}$. In this case, there is no coherence and we have therefore $R_{\ell\ell}=1$ as in the type-I seesaw case since the two Majorana eigenstates both lead to equal number of SS and OS dilepton states. Conclusion {#sec:con} ========== We show that in generic TeV scale $W_R$ models for type I and general inverse seesaw models, the ratio $R_{\ell\ell}$, of the number of same sign ($N_{\rm SS}$) and opposite sign ($N_{\rm OS}$) dilepton states need not be the same when summed over different flavors, contrary to general expectations. This can happen when there is a high degree of degeneracy between the RHNs produced in $W_R$ decay. The degree of degeneracy depends on whether it is type I or inverse seesaw case, and is determined by the coherence condition which in turn depends on the magnitude of the Dirac Yukawa couplings in the theory. For generic choice of parameters, in the first case, the degeneracy has to be at the level of one part in a thousand for TeV scale RHNs whereas in the case of inverse seesaw, it can be a factor of ten or less. Thus observation of the ratio $R_{\ell\ell}$ can in principle, allow us to probe the deeper structure of the RHN mass matrix in the type I seesaw case and the $(N,S)$ sector mass matrix in the inverse seesaw case. We find that in the case of type I seesaw, one needs $CP$ violation to get $R_{\ell\ell}$ different from one, whereas for the inverse seesaw, it is the parameter $\mu_R$ which governs $R_{\ell\ell}$. We believe that the connection between $R_{\ell\ell}\neq 1$ and near degeneracy of RHN states is already an important conclusion, since it is known that low scale leptogenesis in TeV scale seesaw models already requires near degeneracy of RHN states. Our main goal in this work was to derive the analytic results for $R_{\ell\ell}$ in the singlet seesaw scenario, and to show as a proof of principle that it can be different from 1 in the parameter space relevant for the LHC. This result is valid irrespective of the details of the collider simulation of the OS and SS events, with their respective signal and background efficiencies, which can be done in a straightforward manner for any given benchmark point following the existing experimental analyses; see e.g. Ref. [@Khachatryan:2014dka]. Also in the case of inverse seesaw, we have ignored detailed flavor effects, since our goal has been merely to illustrate an interesting phenomenon involving lepton [number]{} violation. A detailed collider analysis (including flavor effects) is a bit premature at this stage and might be more appropriate in scrutinizing the different seesaw models, only if there is a statistically significant observation of dilepton plus two jet signal (either SS or OS) in the future.[^6] Acknowledgement {#acknowledgement .unnumbered} =============== We gratefully acknowledge the local hospitality provided at the ACFI workshop on ‘Neutrinos at the High-Energy Frontier’ at UMass, Amherst, where part of this work was done. R.N.M. was supported by the US National Science Foundation under Grant No. PHY1620074. Partial Decay Widths of $N$ {#sec:app} =========================== In the left-right model, the RHN has three-body decays through an off-shell $W_R$ (for $M_N<M_{W_R}$): $N_\ell\to W_R^\ell \to \ell q\bar{q}^{\prime}$. This is in addition to the usual two-body decay modes of the RHN: $N\to W\ell, \, Z\nu, \, h\nu$, induced by its mixing with the light neutrinos. In this analysis, we choose the region of parameter space where the light-heavy neutrino mixing is small enough to ensure that the three-body decay is dominant over the two-body one [@Chen:2013fna]. For light-quark final states, the corresponding three-body decay width is given by [@Gluza:2015goa; @Das:2016akd] $$\Gamma(N \to q\bar{q}^\prime \ell ) \ = \ \frac{g_R^4}{2048 \pi^3} M_{N} \frac{12}{x}\left[1-\frac{x}{2}-\frac{x^2}{6}+\frac{1-x}{x}\ln (1-x)\right] ~,$$ with $x = M_{N}^2/M_{W_R}^2$. Here we neglect the SM quark and lepton masses. 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CMS Collaboration, CMS-PAS-EXO-16-045. CMS Collaboration, CMS-PAS-EXO-17-011. B. A. Dobrescu and P. J. Fox, JHEP [1605]{}, 047 (2016) \[arXiv:1511.02148 \[hep-ph\]\]. A. Das, N. Nagata and N. Okada, JHEP [1603]{}, 049 (2016) \[arXiv:1601.05079 \[hep-ph\]\]. [^1]: The minimal TeV-scale type-I seesaw (without any additional gauge or Higgs interactions) requires large light-heavy neutrino mixing in order to have an observable signal at colliders [@Deppisch:2015qwa; @Atre:2009rg; @Antusch:2016ejd]. [^2]: However, in this special case, $R_{e\mu}$ is ill-defined, because $N_{{\rm OS},e\mu}=0$. [^3]: Note that the left-right symmetry does not allow a (1,3) entry in Eq. , which would otherwise lead to the linear seesaw [@Barr:2003nn; @Malinsky:2005bi]. [^4]: The $\mu_R$ term leads to unavoidable one-loop corrections to the light neutrino mass matrix, but for a given $\mu_R$, we can carefully choose $\mu_S$ so that the light neutrino oscillation data is always satisfied [@Dev:2012sg]. 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--- abstract: \| We study the finite sample behavior of Lasso-based inference methods such as post double Lasso and debiased Lasso. Empirically and theoretically, we show that these methods can exhibit substantial omitted variable biases (OVBs) due to Lasso not selecting relevant controls. This phenomenon can be systematic in finite samples and occur even when the coefficients are very sparse and the sample size is large and larger than the number of controls. Therefore, relying on the existing asymptotic inference theory can be problematic in empirical applications. We compare the Lasso-based inference methods to modern high-dimensional OLS-based methods and provide practical guidance. Keywords: Lasso, post double Lasso, debiased Lasso, OLS, omitted variable bias, limited variability, finite sample analysis author: - 'Kaspar Wüthrich[^1] Ying Zhu[^2]\' bibliography: - 'bibliography.bib' date: 'First draft on arXiv: March 20, 2019. This draft: .' title: 'Omitted variable bias of Lasso-based inference methods: A finite sample analysis[^3] ' --- Introduction ============ Researchers are often interested in making statistical inferences on a single parameter (for example, the effect of a treatment or a policy), while controlling for confounding factors. In more and more economic applications, the number of potential control variables ($p$) is becoming large relative to the sample size ($n$), either due to the inherent richness of the data, the desire of researchers to specify flexible functional forms, or both. In such problems, a natural approach is to use the least absolute shrinkage and selection operator (Lasso), introduced by @tibsharini1996regression, to select the relevant controls (i.e., those with nonzero coefficients) and then run OLS with the selected controls. However, this approach has been criticized because, unless the magnitude of the coefficients associated with the relevant controls is very small, it requires these coefficients to be well separated from zero to ensure that Lasso selects them. This critique has motivated the development of post double Lasso [@belloni2014inference] and debiased Lasso [@javanmard2014confidence; @vandergeer2014asymptotically; @Zhang_Zhang]. The breakthrough in this literature is that it does not require the aforementioned separation condition, and the Lasso not selecting relevant controls yields negligible asymptotic biases under certain conditions on $n$, $p$, and the degree of sparsity. Since their introduction, post double Lasso and debiased Lasso have quickly become the most popular inference methods for problems with many control variables. Given the rapidly growing (asymptotic) theoretical and applied literature on these methods, it is crucial to take a step back and examine the performance of these new procedures in empirically relevant settings as well as to better understand their merits and limitations relative to other alternatives. In particular, there is a misconception that the post double Lasso and debiased Lasso are immune to under-selection of the Lasso because they do not require the above-mentioned separation condition. Empirically and theoretically, this paper shows that, in finite samples, under-selection can result in substantial OVBs of these methods and yield invalid inferences. We also compare the post double Lasso and debiased Lasso to modern high-dimensional OLS-based inference procedures. Let us consider the linear model $$\begin{aligned} Y_{i} & = & D_{i}\alpha^{}+X_{i}\beta^{}+\eta_{i},\label{eq:main-y}\\ D_{i} & = & X_{i}\gamma^{}+v_{i}.\label{eq:main-d}\end{aligned}$$ Here $Y_{i}$ is the outcome, $D_{i}$ is the scalar treatment variable of interest, and $X_{i}$ is a $(1\times p)$-dimensional vector of additional control variables. To focus on the impact of under-selection and facilitate the exposition, we assume that (\[eq:main-y\]) and (\[eq:main-d\]) share the same set of $k$ non-zero coefficients. In this paper, we study the performance of post double Lasso and the debiased Lasso for estimating and making inferences (e.g., constructing confidence intervals) on the treatment effect $\alpha^{\ast}$. We present extensive simulation evidence demonstrating that post double Lasso and debiased Lasso can exhibit substantial OVBs relative to the standard deviations due to the Lasso not selecting all the relevant controls. Our simulation results can be summarized as follows. (i) Large OVBs are persistent across a range of empirically relevant settings and can occur even when $n$ is large and larger than $p$, and $k$ is small (e.g., when $n=10000$, $p=4000$, $k=5$). (ii) For the same $(n,p,k)$, noise variances, and magnitude of coefficients, there can be no OVBs at all, small OVBs, or substantial OVBs, depending on the variance of the relevant controls. (iii) When the controls exhibit limited variability, the performance of Lasso-based inference methods can be very sensitive to the choice of the regularization parameters; under sufficient variability, post double Lasso is less sensitive. (iv) There is no simple recommendation for how to choose the regularization parameters.[^4] (v) The OVBs can lead to invalid inferences and under-coverage of confidence intervals. In addition to the simulations, we conduct Monte Carlo studies based on two empirical applications: The analysis of the effect of 401(k) plans on savings by @belloni2017program and the study of the racial test score gap by @fryerlevitt2013. We draw samples of different size from the large original datasets and compare the subsample estimates to the estimates based on the original data. This exercise mimics random sampling from a large super-population. In both applications, we find substantial biases even when $n$ is considerably larger than $p$, and document that the magnitude of the biases varies substantially depending on the regularization choice. The existing (asymptotic) theory provides little insight about the OVBs of the Lasso-based inference methods documented in our simulation studies. In terms of formal results, it only implies an upper bound of $\texttt{constant}\cdot\frac{k\log p}{n}$ for the bias. Here the (positive) $\texttt{constant}$ does not depend on $(n,p,k)$ and bears little meaning in the existing theory which simply assumes $\frac{k\log p}{\sqrt{n}}\rightarrow0$ among other sufficient conditions.[^5] The asymptotic upper bound $\texttt{constant}\cdot\frac{k\log p}{n}$ is only informative about the least favorable* case and does not explain the following practically relevant questions: (i) When do OVBs arise? (ii) Why can the OVBs be drastically different despite $(n,p,k)$, noise variances, and absolute values of coefficients being the same? (iii) What is the magnitude of OVBs in the most favorable cases? (iv) How severe can the OVBs be in finite samples where $\frac{k\log p}{n}$ is not small enough? To explain (i) and (ii), we provide theoretical conditions under which the OVBs occur systematically and establish a novel result on the under-selection of the Lasso. To explain (iii) and (iv), we derive new informative lower and upper bounds on the OVBs of post double Lasso and the debiased Lasso proposed by @vandergeer2014asymptotically. Our analyses are non-asymptotic and allow us to study the OVBs for fixed $(n,p,k)$, but are also informative when $\frac{k\log p}{n}\rightarrow0$ or $\frac{k\log p}{n}\rightarrow\infty$. Our theoretical results reveal that, in finite samples, the OVBs are not just simple linear functions of $\frac{k\log p}{n}$ but depend on $n$, $p$, and $k$ in a more complex way. In one of our results, we derive explicit universal constants, allowing us to compute precise lower bounds and perform “comparative statics” given features of the underlying empirical problems. This lower bound analysis of the OVBs is the first of its kind in the literature. In contrast to upper bound analyses, it is informative about the most favorable cases and thus the finite sample limitations of Lasso-based inference methods. Our results suggest that the OVBs can be substantial relative to the standard deviation derived from the existing theory. As a consequence, the confidence intervals proposed in the literature can exhibit under-coverage. In the main part of the paper, we focus on post double Lasso and present results for the debiased Lasso in the appendix. Post double Lasso consists of two Lasso selection steps: A Lasso regression of $Y_{i}$ on $X_{i}$ and a Lasso regression of $D_{i}$ on $X_{i}$. In the third step, the estimator of $\alpha^{\ast}$, $\tilde{\alpha}$, is the OLS regression of $Y_{i}$ on $D_{i}$ and the union of controls selected in the two Lasso steps. For the setup of (\[eq:main-y\])–(\[eq:main-d\]), post double has a clear advantage over a post (single) Lasso OLS. As @belloni2014inference [p.614] put it: “Intuitively, this procedure \[post double Lasso\] works well since we are more likely to recover key controls by considering selection of controls from both equations instead of just considering selection of controls from the single equation”. However, OVBs can still arise whenever the relevant controls are selected in neither Lasso steps. To formally study the OVBs, one has to first understand when such “double under-selection” is likely to occur in finite samples. This task is difficult because it requires necessary results on the Lasso’s inclusion to show that double under-selection can occur with high probability. To our knowledge, no formal results exist that can explain the phenomenon “double under-selection with high probability”, and this is likely why the literature provides no characterization of the finite sample OVBs. In this paper, under some classical assumptions, we prove that if the ratios of the absolute values of the non-zero coefficients and the variances of the controls are no greater than half the regularization parameters[^6] (or, in other words, if the products of the absolute values of the non-zero coefficients and the standard deviations of the corresponding controls are small relative to the noise’s standard deviation), Lasso fails to select these controls in both steps with high probability. Our new result on the under-selection of the Lasso is the key ingredient for deriving a lower bound formula for the OVB of the post double Lasso estimator $\tilde{\alpha}$. The OVB lower bound is characterized by the complicated interplay between the probability of double under-selection and the omitted coefficients, and is not a simple linear function of $\frac{k\log p}{n}$ in general. It can be substantial compared to the standard deviation obtained from the asymptotic distribution in @belloni2014inference[^7], even in settings where $k$ is very small, $n$ is large and larger than $p$. As $\frac{k\log p}{n}$ gets large, depending on the configuration of $\left(\beta_{K}^{},\,\gamma_{K}^{},\,\alpha^{}\right)$, the OVB lower bounds scale as $\left\|\alpha^{}\right\|$ or as $\frac{\sigma_{\eta}}{\sigma_{v}}$, where $\sigma_{\eta}^{2}$ is the variance of the $\eta_{i}$s in (\[eq:main-y\]) and $\sigma_{v}^{2}$ is the variance of the $v_{i}$s in (\[eq:main-d\]). This raises the question of how severe the OVBs can be when $\frac{k\log p}{n}\rightarrow\infty$ and both Lasso steps are inconsistent in terms of the prediction errors (and $\ell_{2}$-errors). To answer this question, we would like a meaningful upper bound (other than just $\infty$) on the OVBs, but there is no such result in the existing literature. Interestingly enough, under this undesirable regime, we show that the upper bounds on OVBs scale as $\left\|\alpha^{}\right\|$ or $\frac{\sigma_{\eta}}{\sigma_{v}}$ with high probability. The occurrence of double under-selection and the magnitude of OVBs both depend on the absolute values of the non-zero coefficients and the standard deviations of the corresponding controls in both Lasso steps through their products. Therefore, under-selection cannot be avoided or mitigated by rescaling the covariates and the OVB lower bound remains the same after rescaling. Any normalization of $X_{i}$ simply leads to rescaled coefficients and vice versa, while their products stay the same. This result suggests an equivalence between “small” (nonzero) coefficient problems and problems with “limited” variability in the relevant controls. By rescaling the controls, the former can always be recast as the latter and vice versa. Given our theoretical results, all else equal, limited variability in the relevant controls makes it more likely for the Lasso to omit them. Limited variability is ubiquitous in applied economic research and there are many instances where it occurs by design. First, it naturally arises from small cells (i.e., when there are only a few observations in some of the cells defined by covariate values). Small cells are prevalent in specifications that include many two-way interactions and are saturated in at least a subset of controls.[^8] When the controls are discrete, limited overlap a major concern in research designs relying on unconfoundedness-type identification assumptions can be viewed as a small cell problem [e.g., @rothe2017robust]. Moreover, categorical controls, when incorporated through a set of indicator variables, give rise to small cells if some of the categories are sparsely populated. Second, when researchers perform subsample analyses, there are often controls that exhibit limited variability within subsamples. Third, in times series and “large $T$” panel data applications, persistence in the controls over time can lead to limited variability. Finally, many empirical settings feature high-dimensional fixed effects that suffer from limited variability. Some authors propose to penalize the fixed effects [e.g., @kock2019uniform] with the Lasso regularization, while others do not [e.g., @belloni2016cluster]. The results in this paper suggest that penalizing fixed effects with the Lasso regularization can be problematic. Our theoretical analyses of the OVBs of the post double Lasso have important implications for inference based on the procedure proposed in @belloni2014inference. On the one hand, @belloni2014inference show that the assumptions $\frac{k\log p}{\sqrt{n}}\rightarrow0$, $\left\|\alpha^{}\right\|$ being bounded from above, and $\left(\sigma_{\eta},\,\sigma_{v}\right)$ being bounded away from zero and above (among other regularity conditions) are sufficient for establishing the asymptotic normality and unbiasedness of their post double Lasso procedure, regardless of whether under-selection is present or not. On the other hand, we show empirically relevant settings where double under-selection occurs with high probability, and also show that in such settings, small $\frac{k\log p}{\sqrt{n}}$ and $\frac{\left\|\alpha^{}\right\|\sigma_{v}}{\sigma_{\eta}}$ are almost necessary* for a good performance of the post double Lasso. Unfortunately in practice, because $k$ and $\alpha^{}$ are fundamentally unknown, it is impossible to know whether or not $\frac{k\log p}{\sqrt{n}}$ and $\frac{\left\|\alpha^{}\right\|\sigma_{v}}{\sigma_{\eta}}$ are small enough. Moreover, the requirement of small $\frac{k\log p}{\sqrt{n}}$ is actually quite demanding in high dimensional settings. For example, even with $n=10000>p=4000$ and $k=5$, we have $\frac{k\log p}{\sqrt{n}}\approx0.41$. In view of our theoretical results, simulations, and empirical evidence, a natural question is how to make statistical inference in a reliable manner, especially when limited variability is present. The case where $p$ is comparable to but still smaller than $n$ deserves special attention because of its prevalence in applied economic research.[^9] In this case, recently developed high-dimensional OLS-based inference methods [e.g., @cattaneo2018inference; @jochmans2018heteroscedasticity; @kline2018leaveout; @dadamo2018cluster] constitute a natural alternative to Lasso-based procedures. These modern OLS-based methods are based on (unbiased) OLS estimators and variance estimators that are robust to the inclusion of many controls.[^10] Based on extensive simulations, we find that OLS with standard errors proposed by @cattaneo2018inference is unbiased (as expected) and demonstrates excellent size accuracy, irrespective of the degree of variability in the controls. Another advantage of OLS-based methods is that, unlike the Lasso-based inference methods, they do not rely on any sparsity assumptions. This is important because sparsity assumptions may not be satisfied in applications and, as we show in this paper, the OVBs of the Lasso-based inference procedures can be substantial even when $k$ is very small, $n$ is very large and larger than $p$. However, OLS yields somewhat wider confidence intervals than the Lasso-based inference methods, which suggests that there is a trade-off between coverage accuracy and the width of the confidence intervals. Our analyses suggest several recommendations concerning the use of Lasso-based inference methods in empirical studies. First, one should always perform robustness checks with respect to the choice of the regularization parameters. Our simulation and theoretical results suggest the following heuristic: If the estimates of $\alpha^{}$ are robust to increasing the theoretically recommended regularization parameters in the two Lasso steps, post double Lasso could be a reliable and efficient method. Second, our findings highlight the usefulness of augmenting the final OLS regression in post double Lasso with control variables motivated by economic theory and prior knowledge, as suggested by @belloni2014inference. Third, when $p$ is smaller than $n$, modern high-dimensional OLS-based inference methods constitute a viable alternative to Lasso-based inference methods. Our paper highlights the intrinsic limitations of basing inference procedures on variable selection in empirically relevant settings, and opens up interesting avenues for future research. For instance, we expect under-selection to have important implications for the finite sample behavior of many other linear and nonlinear inference procedures based on the Lasso [e.g., @belloni2012sparse; @belloni2014inference; @farrell2015robust; @belloni2017program; @chernozhukov2018double]. Furthermore, our paper motivates systematic analyses of the practical usefulness of Lasso-based inference procedures and other modern high-dimensional methods. For example, @angrist2019machine investigate the usefulness of post double Lasso for regression-based sensitivity analyses, and present simulation evidence on the finite sample behavior of Lasso-based instrumental variables (IV) methods. Finally, we emphasize that our results do not contradict the existing asymptotic theory on post double Lasso and the debiased Lasso, but rather offer a better understanding of when these methods perform well in finite samples and when they do not. This paper should not be viewed as a criticism of these methods and there is no doubt that they provide significant theoretical improvements over earlier procedures (e.g., post single Lasso OLS). Instead, our findings should be viewed as a warning that relying on the existing asymptotic inference theory for post double Lasso and debiased Lasso can be problematic in empirical applications and these methods can still suffer from the consequence of under-selection in finite samples. Lasso and post double Lasso {#sec:lasso_post_double_lassp} =========================== The Lasso {#sec:lasso} --------- Consider the following linear regression model $$Y_{i}=X_{i}\theta^{}+\varepsilon_{i},\qquad i=1,\dots,n,\label{eq:1}$$ where $\left\{ Y_{i}\right\} _{i=1}^{n}=Y$ is an $n$-dimensional response vector, $\left\{ X_{i}\right\} _{i=1}^{n}=X$ is an $n\times p$ matrix of covariates with $X_{i}$ denoting the $i$th row of $X$, $\left\{ \varepsilon_{i}\right\} _{i=1}^{n}=\varepsilon$ is a zero-mean error vector, and $\theta^{}$ is a $p$-dimensional vector of unknown coefficients. The Lasso estimator of $\theta^{}$ is given by $$\hat{\theta}\in\arg\min_{\theta\in\mathbb{R}^{p}}\frac{1}{2n}\sum_{i=1}^{n}\left(Y_{i}-X_{i}\theta\right)^{2}+\lambda\sum_{j=1}^{p}\left\|\theta_{j}\right\|,\label{eq:las}$$ where $\lambda$ is the regularization parameter. For example, if $\varepsilon\sim\mathcal{N}\left(0_{n},\,\sigma^{2}I_{n}\right)$ and $X$ is a fixed design matrix with normalized columns (i.e., $\frac{1}{n}\sum_{i=1}^{n}X_{ij}^{2}=b$ for all $j=1,\dots,p$), @bickel2009simultaneous set $\lambda=2\sigma\sqrt{\frac{2b\left(1+\tau\right)\log p}{n}}$ (where $\tau>0$) to establish upper bounds on $\sqrt{\sum_{j=1}^{p}(\hat{\theta}_{j}-\theta_{j}^{})^{2}}$ with high probability guarantee. @wainwright2009sharp sets $\lambda$ proportional to $\frac{\sigma}{\phi}\sqrt{\frac{b\log p}{n}}$, where $\phi\in(0,\,1]$ is a measure of correlation between the covariates with nonzero coefficients and those with zero coefficients, to establish perfect selection. Besides the classical choices in @bickel2009simultaneous and @wainwright2009sharp mentioned above, other choices of $\lambda$ are available in the literature. For instance, @belloni2013least develop a data-dependent approach and @belloni2012sparse and @belloni2016cluster propose choices that accommodate heteroscedastic and clustered errors. In the case of nearly orthogonal $X$ (which is typically required to ensure a good performance of the Lasso), these choices of $\lambda$ have a similar magnitude as those in @bickel2009simultaneous and @wainwright2009sharp. Finally, a very popular practical approach for choosing $\lambda$ is cross-validation. However, only a few theoretical results exist on the properties of Lasso when $\lambda$ is chosen using cross-validation; see, for example, @homrighausen2013thelasso [@homrighausen2014leaveoneout] and @chetverikov2017cv. Post double Lasso ----------------- The model implies the following reduced form model for $Y_{i}$: $$\begin{aligned} Y_{i} & = & X_{i}\pi^{}+u_{i},\label{eq:reduce-1}\end{aligned}$$ where $\pi^{}=\gamma^{}\alpha^{}+\beta^{}$ and $u_{i}=\eta_{i}+\alpha^{}v_{i}$. The post double Lasso, introduced by @belloni2014inference, essentially exploits the Frisch-Waugh theorem, where the regressions of $Y$ on $X$ and $D$ on $X$ are implemented with the Lasso: $$\begin{aligned} \hat{\pi} & \in & \textrm{arg}\min_{\pi\in\mathbb{R}^{p}}\frac{1}{2n}\sum_{i=1}^{n}\left(Y_{i}-X_{i}\pi\right)^{2}+\lambda_{1}\sum_{j=1}^{p}\left\|\pi_{j}\right\|,\label{eq:las-1}\\ \hat{\gamma} & \in & \textrm{arg}\min_{\gamma\in\mathbb{R}^{p}}\frac{1}{2n}\sum_{i=1}^{n}\left(D_{i}-X_{i}\gamma\right)^{2}+\lambda_{2}\sum_{j=1}^{p}\left\|\gamma_{j}\right\|.\label{eq:las-2}\end{aligned}$$ The final estimator $\tilde{\alpha}$ of $\alpha^{}$ is then obtained from an OLS regression of $Y$ on $D$ and the union of selected controls $$\left(\tilde{\alpha},\,\tilde{\beta}\right)\in\textrm{arg}\min_{\alpha\in\mathbb{R},\beta\in\mathbb{R}^{p}}\frac{1}{2n}\sum_{i=1}^{n}\left(Y_{i}-D_{i}\alpha-X_{i}\beta\right)^{2}\quad\textrm{s.t. }\beta_{j}=0\;\forall j\notin\left\{ \hat{I}_{1}\cup\hat{I}_{2}\right\} ,\label{eq:double}$$ where $\hat{I}_{1}=\textrm{supp}\left(\hat{\pi}\right)=\left\{ j:\,\hat{\pi}_{j}\neq0\right\} $ and $\hat{I}_{2}=\textrm{supp}\left(\hat{\gamma}\right)=\left\{ j:\,\hat{\gamma}_{j}\neq0\right\} $. Evidence on the OVB of post double Lasso {#sec:evidence} ======================================== This section presents evidence on the under-selection of the Lasso and the OVB of post double Lasso. Section \[sec:numerical\_evidence\] analyzes a simple numerical example, Section \[sec:simulation\_evidence\] presents simulation evidence for different choices of the regularization parameters, and Section \[sec:empirical\_evidence\] revisits two empirical applications. In our simulations, we vary the variance of the control variables while fixing their coefficients. In particular, we consider settings where the control variables exhibit limited variability. It is important to note that all the results remain unchanged when we rescale the controls and transform the limited variability problem into a small coefficients problem; see Section \[sec:inference\] for a further discussion. Numerical example {#sec:numerical_evidence} ----------------- To study the under-selection of the Lasso, we simulate data according to the linear model , where $X_{i}\sim \mathcal{N}\left(0,\sigma_x^2I_{p}\right)$ is independent of $\varepsilon_{i}\sim\mathcal{N}(0,1)$ and $\left\{ X_{i},\varepsilon_{i}\right\} _{i=1}^{n}$ consists of i.i.d. entries. We set $n=500$, $p=200$, and consider a sparse setting where $\theta^{}=(\underbrace{1,\dots,1}_{k},0,\dots,0)^{T}$ and $k=5$. We employ the classical recommendation for the regularization parameter by @bickel2009simultaneous.[^11] Figure \[fig:performance\_illu\] displays the average number of selected (relevant and irrelevant) covariates as a function of the degree of variability, $\sigma_x$. The variability of the covariates significantly affects the selection performance of the Lasso. The average number of selected covariates is monotonically increasing in $\sigma_x$, ranging from approximately zero when $\sigma_x=0.05$ up to five when $\sigma_x=0.5$. \[fig:performance\_illu\] Next, we investigate the implications of the under-selection of Lasso for post double Lasso. We simulate data according to the structural model –, where $X_{i}\sim \mathcal{N}\left(0,\sigma_x^2I_{p}\right)$, $\eta_{i}\sim\mathcal{N}(0,1)$, and $v_{i}\sim \mathcal{N}(0,1)$ are independent of each other and $\left\{ X_{i},\eta_{i},v_{i}\right\} _{i=1}^{n}$ consists of i.i.d. entries. Our object of interest is $\alpha^{}$. We set $n=500$, $p=200$, $\alpha^{}=0$, and consider a sparse setting where $\beta^{}=\gamma^{}=(\underbrace{1,\dots,1}_{k},0,\dots,0)^{T}$ and $k=5$. In both Lasso steps, we employ the recommendation for the regularization parameter by @bickel2009simultaneous. Figure \[fig:fsd\_dml\] displays the finite sample distribution of post double Lasso for different values of $\sigma_x$. For comparison, we plot the distribution of the “oracle estimator” of $\alpha^\ast$, a regression of $Y_i-X_i\pi^\ast$ on $D_i-X_i\gamma^\ast$. \[fig:fsd\_dml\] For $\sigma_x=0.05$, the post double Lasso estimator exhibits a small bias. Increasing $\sigma_x$ to $0.15$ shifts the distribution to the right, resulting in a large bias relative to the standard deviation. For $\sigma_x=0.3$, post double Lasso is biased and exhibits a larger standard deviation. Finally, for $\sigma_x=0.5$, the post double Lasso estimator is approximately unbiased and its distribution is centered at the true value $\alpha^{}=0$. We emphasize that these striking differences arise despite $(n,p,k,\sigma_v,\sigma_\eta)$ and the coefficients being the same in these four designs. The bias of post double Lasso is caused by the OVB arising from the two Lasso steps not selecting all the relevant covariates. Figure \[fig:hist\_sel\_dml\] displays the number of selected relevant covariates (i.e., the cardinality of $\hat{I}_{1}\cup\hat{I}_{2}$ in ). \[fig:hist\_sel\_dml\] With very high probability, none of the relevant controls get selected when $\sigma_x\le 0.15$. The selection performance improves as $\sigma_x$ increases, until, with high probability, all relevant regressors get selected when $\sigma_x=0.5$.[^12] We further note that the shape of the finite sample distribution depends on $\sigma_x$. This distribution is a mixture of the distributions of OLS conditional on the two Lasso steps selecting different combinations of covariates.[^13] For $\sigma_x=0.05$, $\sigma_x=0.15$, and $\sigma_x=0.5$, the finite sample distributions are well-approximated by normal distributions (albeit centered at different values). The reason is that, when $\sigma_x=0.05$ or $\sigma_x=0.15$, none of the relevant controls get selected with high probability, and, when $\sigma_x=0.5$, the two Lasso steps almost always select all the relevant controls. In between these two cases, when $\sigma_x=0.3$, the finite sample distribution is a mixture of distributions with different means (depending on how many controls get selected) and has a larger standard deviation. The event where none of the relevant covariates are selected is of particular interest. Figure \[fig:cond\_bias\] displays the probability of this event as a function of $\sigma_x$ as well as the OVB conditional on this event, which is computed as $$\frac{1}{\sum_{r=1}^{R}S_{r}}\sum_{r=1}^{R}\left(\tilde{\alpha}_{r}-\alpha^\ast\right)\cdot S_{r}.$$ In the above formula, $R$ is the total number of simulation repetitions, $S_{r}$ is an indicator which is equal to one if nothing gets selected and zero otherwise, and $\tilde{\alpha}_{r}$ is the estimate of $\alpha^\ast$ in the $r$[th]{} repetition. As the variability in the controls increases from $\sigma_x=0.05$ to $\sigma_x=0.4$, the probability that nothing gets selected is decreasing from one to zero. The OVB increases until $\sigma_x=0.35$ and is not defined for $\sigma_x>0.35$ because the empirical probability that nothing gets selected is zero in this case. \[fig:cond\_bias\] Importantly, the issue documented here is not a “small sample” phenomenon, but persists even in large sample settings. To illustrate, Figures \[fig:fsd\_large\] and \[fig:hist\_sel\_large\] display the finite sample distributions and the number of selected relevant covariates when $(n,p,k)=(10000,4000,5)$. Even in large samples, the finite sample distribution may not be centered at the true value and the bias can be large relative to the standard deviation because, with high probability, none of the relevant controls are selected.[^14] Compared to the results for $(n,p,k)=(500,200,5)$, under-selection and large biases occur at lower values of $\sigma_x$, and all the relevant controls get selected when $\sigma_x=0.15$. \[fig:fsd\_large\] \[fig:hist\_sel\_large\] Simulation evidence {#sec:simulation_evidence} ------------------- Section \[sec:numerical\_evidence\] considers a simple numerical example and the classical regularization choice by @bickel2009simultaneous based on the true $\sigma^2$. Here we present simulation evidence for three popular feasible choices of the regularization parameter: The heteroscedasticity-robust proposal in @belloni2012sparse ($\lambda_{\text{BCCH}}$)[^15], the regularization parameter with the minimum cross-validated error ($\lambda_{\text{min}}$), and the regularization parameter with the minimum cross-validation error plus one standard deviation ($\lambda_{\text{1se}}$). We start by investigating the selection performance. We simulate data based on the DGP of Section \[sec:numerical\_evidence\], where $(n,p,k)=(500,200,5)$. To illustrate the impact of changing the sample size, we also show results for $n=1000$ and $n=200$. The results are based on 1,000 simulation repetitions. Figure \[fig:sel\] displays the average number of selected covariates as a function of $\sigma_x$. Lasso with $\lambda_{\text{BCCH}}$ selects the lowest number of covariates. Choosing $\lambda_{\text{1se}}$ leads to a somewhat higher number of selected covariates and results in moderate over-selection for larger values of $\sigma_x$. Lasso with $\lambda_{\text{min}}$ selects the highest number of covariates and exhibits substantial over-selection. Figure \[fig:sel\_rel\] shows the corresponding numbers of selected relevant covariates. We note that, when $\sigma_x=0.1$, the Lasso does not select all the relevant covariates even with $\lambda_{\text{min}}$ which can result in substantial over-selection. For all regularization choices, the selection performance improves as the sample size increases. \[fig:sel\] \[fig:sel\_rel\] To investigate the impact of under-selection on post double Lasso, we simulate data according to the DGP of Section \[sec:numerical\_evidence\], where $(n,p,k)=(500,200,5)$, and also consider $n=1000$ and $n=200$. The results are based on 1,000 simulation repetitions. Appendix \[app:additional\_simulations\] presents additional simulation evidence, where we vary $k$, the distribution of $X_i$, the error terms $(\eta_i,v_i)$, the true value $\alpha^\ast$, and also consider a heteroscedastic DGP. We employ the same regularization choice in both Lasso steps. Figure \[fig:dml\_bias\_std\] presents evidence on the bias of post double Lasso. To make the results easier to interpret, we report the ratio of the bias to the empirical standard deviation. \[fig:dml\_bias\_std\] Under limited variability, post double Lasso with $\lambda_{\text{BCCH}}$ can exhibit biases that are up to two times larger than the standard deviation when $n=200$ and still comparable to the standard deviation when $n=1000$. The relationship between $\sigma_x$ and the ratio of bias to standard deviation is non-monotonic: It is increasing for small $\sigma_x$ and decreasing for larger $\sigma_x$. The bias is somewhat smaller than the standard deviation when $\lambda=\lambda_{\text{1se}}$. Setting $\lambda=\lambda_{\text{min}}$ yields the smallest ratio of bias to standard deviation. For all choices of the regularization parameters, the ratio of bias to standard deviation is decreasing in the sample size. Finally, we note that when $\sigma_x$ is large enough such that there is no under-selection, post double Lasso performs well and is approximately unbiased for all regularization parameters. The additional simulation evidence reported in Appendix \[app:additional\_simulations\] confirms these results and further shows that $\alpha^\ast$ is an important determinant of the performance of post double Lasso because of its direct effect on the magnitude of the coefficients and the error variance in the reduced form equation . Moreover, we show that, while choosing $\lambda=\lambda_{\text{min}}$ works well when $\alpha^\ast=0$, this choice can yield poor performances when $\alpha^\ast\ne 0$ (c.f. Figure \[fig:dml\_bias\_std\_app\], DGP A5).[^16] Thus, there is no simple recommendation for how to choose the regularization parameters in practice. The substantive performance differences between the three regularization choices suggest that post double Lasso is sensitive to the regularization parameters when $\sigma_x$ is small (but not quite zero). To further investigate this issue, we compare the results for $\lambda_{\text{BCCH}}$, $0.5 \lambda_{\text{BCCH}}$, and $1.5 \lambda_{\text{BCCH}}$. Figures \[fig:sel\_sensitivity\] and \[fig:sel\_rel\_sensitivity\] display the average numbers of selected and selected relevant covariates. The differences in the selection performance are substantial. Lasso with $0.5 \lambda_{\text{BCCH}}$ over-selects for all $n$ and $\sigma_x$, while Lasso with $1.5 \lambda_{\text{BCCH}}$ under-selects unless $\sigma_x$ and $n$ are large. The differences get smaller as the sample size increases. Figure \[fig:dml\_bias\_std\_sens\] displays the ratio of bias and standard deviation for post double Lasso. Choosing $0.5 \lambda_{\text{BCCH}}$ yields small biases relative to the standard deviations for all $\sigma_x$. By contrast, choosing $1.5 \lambda_{\text{BCCH}}$ yields biases that can be more than four times larger than the standard deviations when $n=200$, and still be substantial when $n=1000$. For larger values of $\sigma_x$, post double Lasso is less sensitive to the choice of the regularization parameters. In Section \[sec:inference\], based on our theoretical results, we discuss how to interpret and use robustness checks with respect to the regularization parameters in empirical applications. \[fig:sel\_sensitivity\] \[fig:sel\_rel\_sensitivity\] \[fig:dml\_bias\_std\_sens\] In sum, our simulation evidence shows (i) that under-selection can lead to large biases relative to the standard deviation, (ii) that, under limited (but not quite zero) variability, the performance of post double Lasso is very sensitive to the choice of the regularization parameters, and (iii) that there is no simple recommendation for how to choose the regularization parameters in practice. Empirical evidence {#sec:empirical_evidence} ------------------ ### The effect of 401k plans on savings {#sec:401k} Here we revisit the analysis of the causal effect of 401(k) plans on individual savings. We use the same dataset as in @CH2004 and @belloni2017program, which we refer to for more details and descriptive statistics. The data contain information about $n=9915$ observations from a sample of households from the 1991 Survey of Income and Program Participation (SIPP). We focus on the effect of eligibility for 401(k) plans ($D$) on total wealth ($Y$).[^17] We consider two different specifications of the control variables ($X$). 1. Two-way interactions specification. We use the same set of controls as in @Benjamin2003 and @CH2004: Seven dummies for income categories, five dummies for age categories, family size, four dummies for education categories, as well as indicators of marital status, two-earner status, defined benefit pension status, individual retirement account (IRA) participation status, and homeownership. Following common empirical practice, we augment this baseline specification with all two-way interactions. After removing collinear columns there are $p=167$ control variables. 2. Quadratic spline & interactions specification. This specification is due to @belloni2017program. It contains indicators of marital status, two-earner status, defined benefit pension status, IRA participation status, and homeownership status, second-order polynomials in family size and education, a third-order polynomial in age, a quadratic spline in income with six breakpoints, as well as interactions of all the non-income variables with each term in the income spline. After removing collinear columns there are $p=272$ control variables. Table \[tab:results\_401k\] presents post double Lasso estimates based on the whole sample with $\lambda_{\text{BCCH}}$, $0.5\lambda_{\text{BCCH}}$ and $1.5\lambda_{\text{BCCH}}$. For comparison, we also report OLS estimates with and without covariates. For both specifications, the results are qualitatively similar across the different regularization choices and similar to OLS with all controls. This is not surprising given our simulation evidence, since $n$ is much larger than $p$. Nevertheless, there are some non-negligible quantitative differences between the point estimates. A comparison to OLS without control variables shows that omitting controls can yield substantial OVBs in this application. [1.25]{}[1]{} [lcc]{}\[tab:results\_401k\] & & [\ ]{} [\ ]{} Method & Point estimate & Robust std. error [\ ]{} Post double Lasso ($\lambda_{\text{BCCH}}$) & 6624.47 & 2069.73\ Post double Lasso ($0.5\lambda_{\text{BCCH}}$) & 6432.36 & 2073.35\ Post double Lasso (1.5$\lambda_{\text{BCCH}}$) & 7474.51 & 2053.00\ OLS with all covariates & 6751.91 & 2067.86\ OLS without covariates & 35669.52 & 2412.02\ [\ ]{} Method & Point estimate & Robust std. error [\ ]{} Post double Lasso ($\lambda_{\text{BCCH}}$) & 4646.58 & 2014.63\ Post double Lasso ($0.5\lambda_{\text{BCCH}}$) & 5648.14 & 1988.81\ Post double Lasso (1.5$\lambda_{\text{BCCH}}$) & 4472.32 & 2027.72\ OLS with all covariates & 5988.41 & 2033.02\ OLS without covariates & 35669.52 & 2412.02\ [1.25]{}[1.4]{} To investigate the impact of under-selection, we perform the following exercise. We draw random subsamples of size $n_s\in \{400,800,1600\}$ with replacement from the original dataset. This exercise mimics random sampling from a large super-population. Based on each subsample, we estimate $\alpha^\ast$ using post double Lasso with $\lambda_{\text{BCCH}}$, $0.5\lambda_{\text{BCCH}}$ and $1.5\lambda_{\text{BCCH}}$ and compute the bias as the difference between the average subsample estimate and the point estimate based on the original data with the same regularization parameters (cf. Table \[tab:results\_401k\]). The results are based on 2,000 simulation repetitions. Figures \[fig:app\_401k\_1\] and \[fig:app\_401k\_2\] display the bias and the ratio of bias to standard deviation for both specifications. We find that post double Lasso can exhibit large finite sample biases. The biases under the quadratic spline & interactions specification tend to be smaller (in absolute value) than the biases under the two-way interactions specification. Interestingly, the ratio of bias to standard deviation may not be monotonically decreasing in $n_s$ (in absolute value) due to the standard deviation decaying faster than the bias. Finally, we find that post double Lasso can be very sensitive to the choice of the regularization parameters. \[fig:app\_401k\_1\] \[fig:app\_401k\_2\] ### Racial differences in the mental ability of children In this section, we revisit @fryerlevitt2013’s analysis of the racial differences in the mental ability of young children. @fryerlevitt2013 use data from two sources: The Early Childhood Longitudinal Study Birth Cohort (ECLS-B) and the US Collaborative Perinatal Project (CPP). Here we focus on the CCP data, which contains information on women who gave birth in 12 medical centers from 1959 to 1965. We restrict the sample to black and white children and focus on the black-white test score gap. Our final sample includes $n=30002$ observations. We choose the standardized test score in the Stanford-Binet and Wechsler Intelligence Test at the age of seven as our outcome variable ($Y$).[^18] The variable of interest ($D$) is an indicator for black. Control variables ($X$) include rich information on socio-demographic characteristics, the home environment, and the prenatal environment. We refer to Table 1B in @fryerlevitt2013 for descriptive statistics. We use the same specification as in @fryerlevitt2013, excluding interviewer fixed effect.[^19] After removing collinear terms there are $p=78$ controls. Table \[tab:results\_testscores\] shows the results for post double Lasso with $\lambda_{\text{BCCH}}$, $0.5\lambda_{\text{BCCH}}$ and $1.5\lambda_{\text{BCCH}}$, as well as OLS with and without covariates based on the whole sample. As expected, since $n=30002$ is much larger than $p=78$, all methods except for OLS without covariates yield similar results. [1.25]{}[1]{} \[tab:results\_testscores\] ------------------------------------------------ ---------------- ------------------- Method Point estimate Robust std. error Post double Lasso ($\lambda_{\text{BCCH}}$) -0.6770 0.0114 Post double Lasso ($0.5\lambda_{\text{BCCH}}$) -0.6762 0.0114 Post double Lasso (1.5$\lambda_{\text{BCCH}}$) -0.6778 0.0114 OLS with all covariates -0.6694 0.0115 OLS without covariates -0.8538 0.0105 : Results based on original data [1.25]{}[1.4]{} To investigate the impact of under-selection, we draw random subsamples of size $n_s\in \{400,800,1600\}$ with replacement from the original dataset. In each sample, we estimate $\alpha^\ast$ using post double Lasso with $\lambda_{\text{BCCH}}$, $0.5\lambda_{\text{BCCH}}$ and $1.5\lambda_{\text{BCCH}}$ and compute the bias as the difference between the average estimate based on the subsamples and the estimate based on the original data with the same regularization parameters. The results are based on 2,000 simulation repetitions. Figure \[fig:app\_test\] displays the bias and the ratio of bias to standard deviation. While the bias is decreasing in $n_s$, it can be substantial and larger than the standard deviation when $n_s$ is small. Moreover, the performance of post double Lasso is very sensitive to the choice of the regularization parameters. For $\lambda=0.5\lambda_{\text{BCCH}}$, post double Lasso is approximately unbiased for all $n_s$, whereas, for $\lambda=1.5\lambda_{\text{BCCH}}$, the bias is comparable to the standard deviation even when $n_s=1600$. \[fig:app\_test\] Theoretical analysis {#sec:inference} ==================== This section provides a theoretical explanation for the findings in Section \[sec:evidence\]. We first establish a new necessary result for the Lasso’s inclusion and then derive lower and upper bounds on the OVBs of post double Lasso. These results hold for fixed $\left(n,\,p,\,k\right)$, and are also informative when $\frac{k\log p}{n}\rightarrow0$ or $\frac{k\log p}{n}\rightarrow\infty$. Throughout this section, we assume the regime where $p$ is comparable to or even much larger than $n$; that is, $p\asymp n$ or $p\gg n$. For the convenience of the reader, here we collect the notation to be used in the theoretical analyses. Let $1_{m}$ denote the $m-$dimensional (column) vector of “1”s and $0_{m}$ is defined similarly. The $\ell_{1}-$norm of a vector $v\in\mathbb{R}^{m}$ is denoted by $\left\|v\right\|_{1}:=\sum_{i=1}^{m}\left\|v_{i}\right\|$. The $\ell_{\infty}$ matrix norm (maximum absolute row sum) of a matrix $A$ is denoted by $\left\Vert A\right\Vert _{\infty}:=\max_{i}\sum_{j}\left\|a_{ij}\right\|$. For a vector $v\in\mathbb{R}^{m}$ and a set of indices $T\subseteq\left\{ 1,\dots,m\right\} $, let $v_{T}$ denote the sub-vector (with indices in $T$) of $v$. For a matrix $A\in\mathbb{R}^{n\times m}$, let $A_{T}$ denote the submatrix consisting of the columns with indices in $T$. For a vector $v\in\mathbb{R}^{m}$, let $\textrm{sgn}(v):=\left\{ \textrm{sgn}(v_{j})\right\} _{j=1,...,m}$ denote the sign vector such that $\textrm{sgn}(v_{j})=1$ if $v_{j}>0$, $\textrm{sgn}(v_{j})=-1$ if $v_{j}<0$, and $\textrm{sgn}(v_{j})=0$ if $v_{j}=0$. We denote $\max\left\{ a,\,b\right\} $ by $a\vee b$ and $\min\left\{ a,\,b\right\} $ by $a\wedge b$. Model setup ----------- We consider the structural model , which can be written in matrix notation as $$\begin{aligned} Y & = & D\alpha^{}+X\beta^{}+\eta,\label{eq:20}\\ D & = & X\gamma^{}+v.\label{eq:21-1}\end{aligned}$$ Following standard practice, we work with centered data, i.e., $\bar{D}=\frac{1}{n}\sum_{i=1}^{n}D_{i}=0$, $\bar{X}=\left\{ \frac{1}{n}\sum_{i=1}^{n}X_{ij}\right\} _{j=1}^{p}=0_{p}$, and $\bar{Y}=\frac{1}{n}\sum_{i=1}^{n}Y_{i}=0$. In matrix notation, the reduced form becomes $$\begin{aligned} Y & = & X\pi^{}+u,\label{eq:reduce}\end{aligned}$$ where $\pi^{}=\gamma^{}\alpha^{}+\beta^{}$ and $u=\eta+\alpha^{}v$. We make the following assumptions about model (\[eq:20\])(\[eq:21-1\]). \[ass:inference\_normality\] The error terms $\eta$ and $v$ consist of independent entries drawn from $\mathcal{N}\left(0,\,\sigma_{\eta}^{2}\right)$ and $\mathcal{N}\left(0,\,\sigma_{v}^{2}\right)$, respectively, where $\eta$ and $v$ are independent of each other. \[ass:inference\_sparsity\_incoherence\] The following are satisfied: (i) $\beta^{}$ and $\gamma^{}$ are exactly sparse with $k\left(\le\min\left\{ n,\,p\right\} \right)$ non-zero coefficients and $K=\left\{ j:\,\beta_{j}^{}\neq0\right\} =\left\{ j:\,\gamma_{j}^{}\neq0\right\} \neq\emptyset$; (ii) $$\left\Vert \left(X_{K^{c}}^{T}X_{K}\right)\left(X_{K}^{T}X_{K}\right)^{-1}\right\Vert _{\infty}=1-\phi\label{eq:14}$$ for some $\phi\in(0,\,1]$, where $K^{c}$ is the complement of $K$; (iii) $X_{K}^{T}X_{K}$ is a diagonal matrix with the diagonal entries $X_{j}^{T}X_{j}=s\neq0$ for all $j\in K$, and $X_{j}^{T}X_{j}\leq s$ for all $j\in K^{c}$. Known as the incoherence condition due to @wainwright2009sharp, part (ii) in Assumption \[ass:inference\_sparsity\_incoherence\] is needed for the exclusion of the irrelevant controls. Note that if the columns in $X_{K^{c}}$ are orthogonal to the columns in $X_{K}$ (but within $X_{K^{c}}$, the columns need not be orthogonal to each other), then $\phi=1$. Obviously a special case of this is when the entire $X$ consists of mutually orthogonal columns (which is possible if $n\geq p$). To provide some intuition for (\[eq:14\]), let us consider the simple case where $k=1$ and $K=\left\{ 1\right\} $, $X$ is centered (such that $\left\{ \frac{1}{n}\sum_{i=1}^{n}X_{ij}\right\} _{j=1}^{p}=0_{p}$), and the columns in $X_{-1}=\left[X_{2},\,X_{3},\dots,\,X_{p}\right]$ are normalized such that the standard deviations of $X_{1}$ and $X_{j}$ (for any $j\in\left\{ 2,3,\dots,p\right\} $) are identical. Then, $1-\phi$ is simply the maximum of the absolute (sample) correlations between $X_{1}$ and each of the $X_{j}$s with $j\in\left\{ 2,3,\dots,p\right\} $. Assumptions \[ass:inference\_normality\] and \[ass:inference\_sparsity\_incoherence\] are classical; in particular, parts (ii) and (iii) of Assumption \[ass:inference\_sparsity\_incoherence\] are often viewed favorable to the performance of Lasso. Our goal here is to demonstrate that, even in these simple yet classical settings, the finite sample OVBs of post double Lasso can be substantial relative to the standard deviation provided in the existing literature. Stronger necessary results on the Lasso’s inclusion {#sec:nonasymptotic} --------------------------------------------------- \[sec:main\_results\] Post double Lasso exhibits OVBs whenever the relevant controls are selected in neither nor . To the best of our knowledge, there are no formal results strong enough to show that, with high probability, Lasso can fail to select the relevant controls in both steps. Therefore, we first establish a new necessary result for the (single) Lasso’s inclusion in Lemma \[prop:fixed\_design\]. Throughout Section \[sec:inference\], we focus on fixed designs (of $X$) to highlight the essence of the problem; see Appendix \[sec:Random-design\] for an extension to random designs. \[prop:fixed\_design\] In model , suppose the $\varepsilon_{i}$s are independent over $i=1,\dots,n$ and $\varepsilon_{i}\sim\mathcal{N}\left(0,\sigma^{2}\right)$, where $\sigma\in\left(0,\,\infty\right)$;[^20] $\theta^{}$ is exactly sparse with at most $k\left(\le\min\left\{ n,\,p\right\} \right)$ non-zero coefficients and $K=\left\{ j:\,\theta_{j}^{}\neq0\right\} \neq\emptyset$. Let Assumption \[ass:inference\_sparsity\_incoherence\](ii)-(iii) hold. We solve the Lasso (\[eq:las\]) with $\lambda\geq\frac{2\sigma}{\phi}\sqrt{\frac{s}{n}}\sqrt{\frac{2\left(1+\tau\right)\log p}{n}}$ (where $\tau>0$). Let $E_{1}$ denote the event that $\textrm{sgn}\left(\hat{\theta}_{j}\right)=-\textrm{sgn}\left(\theta_{j}^{}\right)$ for at least one $j\in K$, and $E_{2}$ denote the event that $\textrm{sgn}\left(\hat{\theta}_{l}\right)=\textrm{sgn}\left(\theta_{l}^{}\right)$ for at least one $l\in K$ with $$\left\|\theta_{l}^{}\right\|\leq\frac{\lambda n}{2s}.\label{eq:min}$$ Then, we have $$\mathbb{P}\left(E_{1}\cap\mathcal{\mathcal{E}}\right)=\mathbb{P}\left(E_{2}\cap\mathcal{\mathcal{E}}\right)=0\label{eq:zero}$$ where $\mathcal{E}$ is defined in (\[eq:event\]) of Appendix \[appendix a1\] and $\mathbb{P}\left(\mathcal{E}\right)\geq1-\frac{1}{p^{\tau}}$. If (\[eq:min\]) holds for all $l\in K$, we have $$\mathbb{P}\left(\hat{\theta}=0_{p}\right)\geq1-\frac{1}{p^{\tau}}.\label{eq:nec}$$ Lemma \[prop:fixed\_design\] shows that for large enough $p$, Lasso fails to select any of the relevant covariates with high probability if (\[eq:min\]) holds for all $l\in K$. If such conditions hold with respect to both and , then Lemma \[prop:fixed\_design\] implies that the relevant controls are selected in neither nor with high probability, i.e., at least $1-\frac{2}{p^{\tau}}$ (cf. Panels (a) and (b) of Figure \[fig:hist\_sel\_dml\] and Panel (a) of Figure \[fig:hist\_sel\_large\]). Suppose that $$\lambda=\frac{2\sigma}{\phi}\sqrt{\frac{s}{n}}\sqrt{\frac{2\left(1+\tau\right)\log p}{n}}.\label{eq:choice}$$ Then becomes $$\left\|\theta_{l}^{}\right\|\sqrt{\frac{s}{n}}\leq\phi^{-1}\sigma\sqrt{\frac{2\left(1+\tau\right)\log p}{n}},\,\,\,\,l\in K.$$ The product “$\left\|\theta_{l}^{}\right\|\sqrt{\frac{s}{n}}$” suggests that normalizing $X_{j}$ to make $\frac{1}{n}\sum_{i=1}^{n}X_{ij}^{2}=1$ for all $j=1,\dots,p$ does not change the conclusions in Lemma \[prop:fixed\_design\]. Such normalization simply leads to rescaled coefficients and estimates (by a factor of $\sqrt{\frac{s}{n}}$). In particular, the choice of $\lambda\geq\frac{2\sigma}{\phi}\sqrt{\frac{s}{n}}\sqrt{\frac{2\left(1+\tau\right)\log p}{n}}$ in Lemma \[prop:fixed\_design\] becomes $\lambda=\lambda_{norm}\geq\frac{2\sigma}{\phi}\sqrt{\frac{2\left(1+\tau\right)\log p}{n}}$; also, $\left\|\theta_{j}^{}\right\|\leq\frac{\lambda n}{2s}$ (where $\lambda\geq\frac{2\sigma}{\phi}\sqrt{\frac{s}{n}}\sqrt{\frac{2\left(1+\tau\right)\log p}{n}}$ without normalization) is replaced by $\sqrt{\frac{s}{n}}\left\|\theta_{j}^{}\right\|\leq\frac{\lambda_{norm}}{2}$ (where $\lambda_{norm}\geq\frac{2\sigma}{\phi}\sqrt{\frac{2\left(1+\tau\right)\log p}{n}}$ with normalization). For $n=10000$, $p=4000$, $\sigma=1$, $\phi=0.95$, and $\tau=0.5$ in (\[eq:choice\]), Lemma \[prop:fixed\_design\] says that, with probability at least $0.97$, none of the relevant covariates are selected if $\max_{l\in K}\left\|\theta_{l}^{}\right\|\leq0.05$ for $\sqrt{\frac{s}{n}}=1$ and $\max_{l\in K}\left\|\theta_{l}^{}\right\|\leq0.5$ for $\sqrt{\frac{s}{n}}=0.1$. Therefore, everything else equal, limited variability in the relevant controls makes it more likely for the Lasso to omit them. Limited variability in the irrelevant controls $X_{K^{c}}$ does not matter for the selection performance of the Lasso because, for $j\in K^{c}$, $\left\|\theta_{j}^{}\right\|\sqrt{\frac{1}{n}X_{j}^{T}X_{j}}=0$ regardless. Note that (\[eq:zero\]) implies $\mathbb{P}\left(\hat{\theta}_{l}\neq0\right)\leq\frac{1}{p^{\tau}}$ for any $l\in K$ subject to (\[eq:min\]). In comparison, @wainwright2009sharp shows that whenever $\theta_{l}^{}\in\left(\lambda\frac{n}{s}\textrm{sgn}\left(\theta_{l}^{}\right),\,0\right)$ or $\theta_{l}^{}\in\left(0,\,\lambda\frac{n}{s}\textrm{sgn}\left(\theta_{l}^{}\right)\right)$ for some $l\in K$, $$\mathbb{P}\left[\textrm{sgn}\left(\hat{\theta}_{K}\right)=\textrm{sgn}\left(\theta_{K}^{}\right)\right]\leq\frac{1}{2}.\label{eq:wainwright}$$ Constant bounds in the form of (\[eq:wainwright\]) cannot explain that, with high probability, Lasso fails to select the relevant covariates in both and when $p$ is sufficiently large. \[more choices\]Under the assumptions in Lemma \[prop:fixed\_design\], the choices of regularization parameters $\lambda_{1}$ and $\lambda_{2}$ coincide with those in @bickel2009simultaneous when $\phi=1$; e.g., the columns in $X_{K^{c}}$ are orthogonal to the columns in $X_{K}$ (but within $X_{K^{c}}$, the columns need not be orthogonal to each other). If $\phi\approx 1$, similar results as those in Lemma \[prop:fixed\_design\] as well as in Sections \[sec:bias\_post\_double\_lasso\], \[sec:post\_double\_upper\_bounds\], and \[sec:bias\_debiased\_lasso\] hold for any choices of regularization parameters derived from the principle that $\lambda$ should be no smaller than $2\max_{j=1,\dots,p}\left\|\frac{X_{j}^{T}\varepsilon}{n}\right\|$ with high probability. These choices constitute what has been used in the vast majority of literature [e.g., @bickel2009simultaneous; @wainwright2009sharp; @belloni2012sparse; @belloni2013least; @belloni2014inference]. Choosing regularization parameters according to the principle discussed in Remark \[more choices\] prevents the inclusion of overly many irrelevant controls. This property is needed to ensure a good performance of post double Lasso, as discussed in Section \[sec:simulation\_evidence\] and Footnote \[footnote:over-selection\]. Lower bounds on the OVBs {#sec:bias_post_double_lasso} ------------------------ Proposition \[prop:bias\_post\_double\_formula\] derives a lower bound formula for the OVB of post double Lasso. We focus on the case where $\alpha^{}=0$ because the conditions required to derive the explicit formula are difficult to interpret when $\alpha^{}\ne0$. The reason is that the error in the reduced form equation involves $\alpha^{}$, such that the choice of $\lambda_{1}$ in depends on the unknown $\alpha^{}$. On the other hand, it is possible to provide easy-to-interpret scaling results (without explicit constants) for cases where $\alpha^{}\ne0$, as we will show in Propositions \[prop:bias\_post\_double\_selection\] and \[prop:bias\_post\_double\_selection-1\]. \[prop:bias\_post\_double\_formula\] Let Assumptions \[ass:inference\_normality\] and \[ass:inference\_sparsity\_incoherence\] hold. Suppose $\lambda_{1}=2\phi^{-1}\sigma_{\eta}\sqrt{\frac{s}{n}}\sqrt{\frac{2\left(1+\tau\right)\log p}{n}}$, $\lambda_{2}=2\phi^{-1}\sigma_{v}\sqrt{\frac{s}{n}}\sqrt{\frac{2\left(1+\tau\right)\log p}{n}}$; for all $j\in K$ and $\left\|a\right\|,\left\|b\right\|\in(0,\,1]$, $$\text{both}\quad\beta_{j}^{}\sqrt{\frac{s}{n}}=a\phi^{-1}\sigma_{\eta}\sqrt{\frac{2\left(1+\tau\right)\log p}{n}}\quad\text{ and}\quad\gamma_{j}^{}\sqrt{\frac{s}{n}}=b\phi^{-1}\sigma_{v}\sqrt{\frac{2\left(1+\tau\right)\log p}{n}}.\label{eq:16}$$ In terms of $\tilde{\alpha}$ obtained from (\[eq:double\]), we have $$\left\|\mathbb{E}\left(\tilde{\alpha}-\alpha^{}\vert\mathcal{M}\right)\right\|\geq\underset{:=\underline{\text{OVB}}}{\underbrace{\max_{r\in(0,1]}T_{1}\left(r\right)T_{2}\left(r\right)}}$$ where $$\begin{aligned} T_{1}\left(r\right) & = & \frac{\left(1+\tau\right)\left\|ab\right\|\phi^{-2}\sigma_{\eta}\frac{k\log p}{n}}{4\left(1+\tau\right)\phi^{-2}b^{2}\sigma_{v}\frac{k\log p}{n}+\left(1+r\right)\sigma_{v}},\\ T_{2}\left(r\right) & = & 1-k\exp\left(\frac{-b^{2}\left(1+\tau\right)\log p}{4\phi^{2}}\right)-\frac{1}{p^{\tau}}-\exp\left(\frac{-nr^{2}}{8}\right), \end{aligned}$$ for any $r\in(0,\,1]$, and $\mathcal{M}$ is an event with $\mathbb{P}\left(\mathcal{M}\right)\geq1-k\exp\left(\frac{-b^{2}\left(1+\tau\right)\log p}{4\phi^{2}}\right)-\frac{2}{p^{\tau}}$.[^21] \[rem:absenceOVB\] Let Assumptions \[ass:inference\_normality\] and \[ass:inference\_sparsity\_incoherence\] hold. As in Proposition \[prop:bias\_post\_double\_formula\], let $\lambda_{1}=2\phi^{-1}\sigma_{\eta}\sqrt{\frac{s}{n}}\sqrt{\frac{2\left(1+\tau\right)\log p}{n}}$ and $\lambda_{2}=2\phi^{-1}\sigma_{v}\sqrt{\frac{s}{n}}\sqrt{\frac{2\left(1+\tau\right)\log p}{n}}$. For all $j\in K,\;a>3,\,b>3$, if $$\text{either}\quad\left\|\beta_{j}^{}\right\|\sqrt{\frac{s}{n}}=a\phi^{-1}\sigma_{\eta}\sqrt{\frac{2\left(1+\tau\right)\log p}{n}}\quad\text{or}\quad\left\|\gamma_{j}^{}\right\|\sqrt{\frac{s}{n}}=b\phi^{-1}\sigma_{v}\sqrt{\frac{2\left(1+\tau\right)\log p}{n}},\label{eq:15}$$ then $\mathbb{P}\left[\textrm{supp}\left(\hat{\pi}\right)=\textrm{supp}\left(\pi^{}\right)\right]\geq1-\frac{1}{p^{\tau}}$ or $\mathbb{P}\left[\textrm{supp}\left(\hat{\gamma}\right)=\textrm{supp}\left(\gamma^{}\right)\right]\geq1-\frac{1}{p^{\tau}}$ by standard arguments. As a result, $\mathbb{P}\left(\left\{ \hat{I}_{1}\cup\hat{I}_{2}\right\} =K\right)\geq1-\frac{1}{p^{\tau}}$, where $\hat{I}_{1}$ and $\hat{I}_{2}$ are defined in (\[eq:double\]); i.e., the final OLS step (\[eq:double\]) includes all the relevant controls with high probability. By similar argument as in Section \[sec:appendix\_post\_double\_lasso\], on the high probability event $\left\{ \left\{ \hat{I}_{1}\cup\hat{I}_{2}\right\} =K\right\} $, the OVB of $\tilde{\alpha}$ is zero. Panels (d) of Figures \[fig:fsd\_dml\]\[fig:hist\_sel\_dml\] and Panels (b) of Figures \[fig:fsd\_large\]\[fig:hist\_sel\_large\] illustrate this phenomenon. Proposition \[prop:bias\_post\_double\_formula\] and Remark \[rem:absenceOVB\] show that the OVBs can be drastically different when we fix $\left(n,\,p,\,k,\,\sigma_{v},\,\sigma_{\eta},\,\alpha^{}\right)$ in (\[eq:20\])(\[eq:21-1\]), but vary the products of the absolute values of the non-zero coefficients and the standard deviations of the corresponding controls: - Under (\[eq:16\]), $\mathbb{P}\left(\hat{I}_{1}=\hat{I}_{2}=\emptyset\right)\geq1-\frac{2}{p^{\tau}}$ and the OVBs are bounded from below by $\underline{OVB}$, which decreases as $\left\|a\right\|$ decreases. By fixing $\left(n,p,\phi,\sigma_{\eta},\sigma_{v},\tau\right)$, one can easily see from (\[eq:16\]) that the lower bound $\underline{OVB}$ depends on $\left\|\beta_{j}^{}\right\|\sqrt{\frac{s}{n}}$ through $\left\|a\right\|$ and $\left\|\gamma_{j}^{}\right\|\sqrt{\frac{s}{n}}$ through $\left\|b\right\|$, for $j\in K$. Therefore, under-selection cannot be avoided or mitigated by rescaling the covariates and $\underline{OVB}$ remains the same after rescaling. This fact suggests that the problem of “limited‘" variability can be recast as a “small‘" coefficient problem and vice versa. - Under (\[eq:15\]), $\mathbb{P}\left(\left\{ \hat{I}_{1}\cup\hat{I}_{2}\right\} =K\right)\geq1-\frac{1}{p^{\tau}}$ and the OVBs are zero. Under Assumptions \[ass:inference\_normality\]–\[ass:inference\_sparsity\_incoherence\] and if $\sigma_{v}$ is bounded away from zero and $\sigma_{\eta}$ is bounded from above, by contrast, the existing theory would imply that the biases of post double Lasso are bounded from above by $\texttt{constant}\cdot\frac{k\log p}{n}$, irrespective of whether Lasso fails to select the relevant controls or not, and how small $\left\|a\right\|$ and $\left\|b\right\|$ are. The (positive) $\texttt{constant}$ here does not depend on $\left(n,\,p,\,k,\,\frac{s}{n},\,\beta_{K}^{},\,\gamma_{K}^{},\,\alpha^{}\right)$, and bears little meaning in the asymptotic framework which simply assumes $\frac{k\log p}{\sqrt{n}}\rightarrow0$ among other sufficient conditions. \[The existing theoretical framework makes it difficult to derive an informative $\texttt{constant}$, and to our knowledge, the literature provides no such derivation.\] The asymptotic upper bound $\texttt{constant}\cdot\frac{k\log p}{n}$ corresponds to the least favorable case and, thus, is uninformative about the most favorable cases that could vary in $\left(\frac{s}{n},\,\beta_{K}^{},\,\gamma_{K}^{},\,\alpha^{}\right)$. By contrast, our lower bound analyses are informative about the most favorable cases, which are crucial for understanding the finite sample limitations of post double Lasso. For sufficiently large $p$, the configuration of $\left\|\beta_{K}^{}\right\|$, $\left\|\gamma_{K}^{}\right\|$, and $\frac{s}{n}$ in Proposition \[prop:bias\_post\_double\_formula\] leads to large $\mathbb{P}\left(\hat{I}_{1}=\hat{I}_{2}=\emptyset\right)$; meanwhile, the lower bound $\underline{OVB}$ is characterized by the interplay between $\mathbb{P}\left(\hat{I}_{1}=\hat{I}_{2}=\emptyset\right)$ (related to $T_{2}$) and the omitted coefficients (related to $T_{1}$). To gauge the magnitude of the OVB and explain why the confidence intervals proposed in the literature can exhibit under-coverage, it is instructive to compare $\underline{OVB}$ with $\ensuremath{\sigma_{\tilde{\alpha}}=\frac{1}{\sqrt{n}}\frac{\sigma_{\eta}}{\sigma_{v}}}$, the standard deviation (of $\tilde{\alpha}$) obtained from the asymptotic distribution in @belloni2014inference.[^22] Let us consider the following examples: $a=b=1$, $\sigma_{\eta}=\sigma_{v}=1$, $n=10000$, $p=4000$, $\tau=0.5$, and $\phi=0.95$; if $\left\|\beta_{j}^{}\right\|\sqrt{\frac{s}{n}}\approx0.05$ and $\left\|\gamma_{j}^{}\right\|\sqrt{\frac{s}{n}}\approx0.05$ for all $j\in K$, then $\frac{\underline{OVB}}{\sigma_{\tilde{\alpha}}}\approx0.52$ for $k=5$ and $\frac{\underline{OVB}}{\sigma_{\tilde{\alpha}}}\approx0.12$ for $k=1$. It is important to bear in mind that, the calculations of $\frac{\underline{OVB}}{\sigma_{\tilde{\alpha}}}$ are based on our theoretical lower* bounds (for the OVBs) corresponding to the most favorable cases. The result $\frac{\underline{OVB}}{\sigma_{\tilde{\alpha}}}\approx0.52$ equals to the ratio of $\texttt{constant}\cdot\frac{k\log p}{n}$ (the typical asymptotic upper bound for the bias) to $\frac{1}{\sqrt{n}}$ (the typical scaling of the standard deviation $\sigma_{\tilde{\alpha}}$), with $\texttt{constant}=1.25$; for the second example with $k=1$, $\frac{\underline{OVB}}{\sigma_{\tilde{\alpha}}}\approx1.47\frac{k\log p}{\sqrt{n}}$. This comparison suggests that these most favorable cases are essentially the least favorable case. Now, let us recall the first example where $k=5$; changing $a=1$ to $a=0.1$ there yields $\frac{\underline{OVB}}{\sigma_{\tilde{\alpha}}}\approx0.12\frac{k\log p}{\sqrt{n}}$ when $\left\|\beta_{j}^{}\right\|\sqrt{\frac{s}{n}}\approx0.005$, while changing $a=1$ to $a=3.1$ yields no OVBs when $\left\|\beta_{j}^{}\right\|\sqrt{\frac{s}{n}}\approx0.16$, according to Remark \[rem:absenceOVB\]. These examples suggest that the most favorable cases may differ substantially from the least favorable case. In view of Proposition \[prop:bias\_post\_double\_formula\], $\underline{OVB}$ is not a simple linear function of $\frac{k\log p}{n}$ in general, but rather depends on $\left(n,p,k\right)$ and other factors in a more complex way. As $\frac{k\log p}{n}\rightarrow\infty$, we will show in Propositions \[prop:bias\_post\_double\_selection\] and \[prop:bias\_post\_double\_selection-1\] that, the OVB lower bounds scale as $\left\|\alpha^{}\right\|$ or as $\frac{\sigma_{\eta}}{\sigma_{v}}$, depending on the configuration of $\left(\beta_{K}^{},\,\gamma_{K}^{},\,\alpha^{}\right)$; as a consequence, $\frac{\underline{OVB}}{\sigma_{\tilde{\alpha}}}$ scales as $\frac{\sqrt{n}\left\|\alpha^{}\right\|\sigma_{v}}{\sigma_{\eta}}$ or as $\sqrt{n}$. While the scaling $\sqrt{n}$ as $\frac{k\log p}{n}\rightarrow\infty$ can be easily seen from Proposition \[prop:bias\_post\_double\_formula\] where $\alpha^{}=0$, Propositions \[prop:bias\_post\_double\_selection\] and \[prop:bias\_post\_double\_selection-1\] also consider cases where $\alpha^{}\neq0$. When double under-selection occurs with high probability and $\frac{k\log p}{\sqrt{n}}$ is not small enough, according to the results above, $\tilde{\alpha}$ will perform poorly in general, except in one (albeit) extreme case. By Lemma \[prop:fixed\_design\] and (\[eq:16\]), $\left\|\hat{\beta}-\beta^{}\right\|_{1}=\left\|a\right\|\frac{k\lambda_{1}n}{2s}$ and $\left\|\hat{\gamma}-\gamma^{}\right\|_{1}=\left\|b\right\|\frac{k\lambda_{2}n}{2s}$ with probability at least $1-\frac{2}{p^{\tau}}$. If $\sigma_{v}$ is bounded away from zero, $\sigma_{\eta}$ is bounded from above, and $$\left\|a\right\|=\left\|b\right\|=o\left(1\right),\label{eq: small_OVB}$$ by similar argument in @belloni2014inference, we can show that $\sqrt{n}\left(\tilde{\alpha}-\alpha^{}\right)$ is approximately normal and unbiased, even if $\frac{k\log p}{\sqrt{n}}$ scales as a constant. If $\frac{k\log p}{n}$ scales as a constant, then replacing (\[eq: small\_OVB\]) with $$\left\|a\right\|=\left\|b\right\|=o\left(\frac{1}{\sqrt{n}}\right),\label{eq: smaller_OVB}$$ yields the same conclusion. Holding other factors constant, the magnitude of OVBs decreases as $\left\|\beta_{K}^{}\right\|$ and $\left\|\gamma_{K}^{}\right\|$ decrease (i.e., as $\left\|a\right\|$ and $\left\|b\right\|$ decrease). As $\left\|a\right\|$ and $\left\|b\right\|$ become very small, the relevant controls become essentially irrelevant. The next results, Propositions \[prop:bias\_post\_double\_selection\] and \[prop:bias\_post\_double\_selection-1\], provide the scaling of OVB lower bounds under two different setups. These results consider cases where $\alpha^{}\neq0$. Depending on the setups, some of the cases behave similarly to the case where $\alpha^{}=0$ and some behave differently. For functions $f(n)$ and $g(n)$, we write $f(n)\succsim g(n)$ to mean that $f(n)\geq cg(n)$ for a universal constant $c\in(0,\,\infty)$ and similarly, $f(n)\precsim g(n)$ to mean that $f(n)\leq c^{'}g(n)$ for a universal constant $c^{'}\in(0,\,\infty)$; $f(n)\asymp g(n)$ when $f(n)\succsim g(n)$ and $f(n)\precsim g(n)$ hold simultaneously. As a general rule, $c$ constants denote positive universal constants that are independent of $n$, $p$, $k$, $\sigma_{\eta}$, $\sigma_{v}$, $s$, $\alpha^{}$ and may change from place to place. These propositions are useful for understanding how the OVB lower bounds behave roughly as a function of $\left(n,\,p,\,k,\,\sigma_{\eta},\,\sigma_{v},\,\left\|\alpha^{}\right\|\right)$, by abstracting from Proposition \[prop:bias\_post\_double\_formula\] all the finite sample subtleties (such as various universal constants) and assuming $\left\|a\right\|,\,\left\|b\right\|\asymp1$ (while $\left\|a\right\|,\,\left\|b\right\|\le1$) and $\phi^{-1}\precsim1$ (hence, $\phi^{-1}\asymp1$). Propositions \[prop:bias\_post\_double\_selection\] and \[prop:bias\_post\_double\_selection-1\] suggest that the OVB lower bounds do not simply scale as $\frac{k\log p}{n}$ even if $\left\|a\right\|,\,\left\|b\right\|\asymp1$ and $\phi^{-1}\precsim1$: When $\frac{k\log p}{n}\rightarrow0$, they scale as $\frac{\sigma_{\eta}}{\sigma_{v}}\frac{k\log p}{n}$ (in Proposition \[prop:bias\_post\_double\_selection\]) and as $\left\|\alpha^{}\right\|\frac{k\log p}{n}$ (in Proposition \[prop:bias\_post\_double\_selection-1\]); when $\frac{k\log p}{n}\rightarrow\infty$, they scale as $\frac{\sigma_{\eta}}{\sigma_{v}}$ (in Proposition \[prop:bias\_post\_double\_selection\]) and as $\left\|\alpha^{}\right\|$ (in Proposition \[prop:bias\_post\_double\_selection-1\]), instead of $\infty$. \[prop:bias\_post\_double\_selection\] Let Assumptions \[ass:inference\_normality\] and \[ass:inference\_sparsity\_incoherence\] hold. Suppose $\phi^{-1}\precsim1$ in (\[eq:14\]); the regularization parameters in (\[eq:las-1\]) and (\[eq:las-2\]) are chosen in a similar fashion as in Lemma \[prop:fixed\_design\] such that $\lambda_{1}\asymp\phi^{-1}\sigma_{\eta}\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}}$[^23] and $\lambda_{2}\asymp\phi^{-1}\sigma_{v}\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}}$; for all $j\in K$, $\left\|\beta_{j}^{}\right\|\leq\frac{\lambda_{1}n}{2s}$ and $\left\|\gamma_{j}^{}\right\|\leq\frac{\lambda_{2}n}{2s}$, but $\left\|\beta_{j}^{}\right\|\asymp\sigma_{\eta}\sqrt{\frac{n}{s}}\sqrt{\frac{\log p}{n}}$ and $\left\|\gamma_{j}^{}\right\|\asymp\sigma_{v}\sqrt{\frac{n}{s}}\sqrt{\frac{\log p}{n}}$. Let us consider $\tilde{\alpha}$ obtained from (\[eq:double\]). \(i) If $\alpha^{}=0$, then there exist positive universal constants $c^{\dagger},c_{1},c_{2},c_{3},c^{},c_{0}^{}$ such that $$\left\|\mathbb{E}\left(\tilde{\alpha}-\alpha^{}\vert\mathcal{M}\right)\right\|\geq c^{\dagger}\frac{\sigma_{\eta}}{\sigma_{v}}\left(\frac{k\log p}{n}\wedge1\right)\left[1-c_{1}k\exp\left(-c_{2}\log p\right)-\exp\left(-c_{3}n\right)\right],\label{eq:22}$$ where $\mathcal{M}$ is an event with $\mathbb{P}\left(\mathcal{M}\right)\geq1-c^{}k\exp\left(-c_{0}^{}\log p\right)$. \(ii) If $\alpha^{}\gamma_{j}^{}\in(0,\,-\beta_{j}^{}]$, $\beta_{j}^{}<0$ for $j\in K$ (or, $\alpha^{}\gamma_{j}^{}\in[-\beta_{j}^{},\,0)$, $\beta_{j}^{}>0$ for $j\in K$), then for some positive universal constants $c^{\dagger},c_{1},c_{2},c_{3},c^{},c_{0}^{}$, (\[eq:22\]) holds with $\mathbb{P}\left(\mathcal{M}\right)\geq1-c^{}k\exp\left(-c_{0}^{}\log p\right)$. To motivate the next proposition, note that as long as $\hat{I}_{1}=K$ or $\hat{I}_{2}=K$, the final OLS step (\[eq:double\]) corresponds to the oracle estimator. Under sufficient variability in $X_{K}$, when $\left\|\pi_{K}^{}\right\|$ are small enough (so $\hat{I}_{1}=\emptyset$ with high probability), but $\left\|\gamma_{K}^{}\right\|$ in (\[eq:21-1\]) are large enough (so $\hat{I}_{2}=K$ with high probability), post double Lasso coincides with the oracle estimator while OLS post (the single) Lasso (\[eq:las-1\]) omits the relevant controls with high probability. As we will show below, if $\left\|\gamma_{K}^{}\right\|$ are also small enough, then like the OLS post (\[eq:las-1\]), post double Lasso can also yield substantial OVBs relative to the standard deviation. \[prop:bias\_post\_double\_selection-1\] Let Assumptions \[ass:inference\_normality\] and \[ass:inference\_sparsity\_incoherence\] hold. Suppose $\phi^{-1}\precsim1$ in (\[eq:14\]); the regularization parameters in (\[eq:las-1\]) and (\[eq:las-2\]) are chosen in a similar fashion as in Lemma \[prop:fixed\_design\] such that $\lambda_{1}\asymp\frac{\left(\sigma_{\eta}+\left\|\alpha^{}\right\|\sigma_{v}\right)}{\phi}\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}}$ and $\lambda_{2}\asymp\phi^{-1}\sigma_{v}\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}}$; for all $j\in K$, $\left\|\gamma_{j}^{}\right\|\leq\frac{\lambda_{2}n}{2s}$ but $\left\|\gamma_{j}^{}\right\|\asymp\sigma_{v}\sqrt{\frac{n}{s}}\sqrt{\frac{\log p}{n}}$. Let us consider $\tilde{\alpha}$ obtained from (\[eq:double\]). \(i) If $\pi_{j}^{}=0$ for all $j\in K$, then there exist positive universal constants $c^{\dagger},c_{1},c_{2},c_{3},c^{},c_{0}^{}$ such that $$\left\|\mathbb{E}\left(\tilde{\alpha}-\alpha^{}\vert\mathcal{M}\right)\right\|\geq c^{\dagger}\left\|\alpha^{}\right\|\left(\frac{k\log p}{n}\wedge1\right)\left[1-c_{1}k\exp\left(-c_{2}\log p\right)-\exp\left(-c_{3}n\right)\right],\label{eq:23}$$ where $\mathbb{P}\left(\mathcal{M}\right)\geq1-c^{}k\exp\left(-c_{0}^{}\log p\right)$. \(ii) For all $j\in K$, suppose $\left\|\pi_{j}^{}\right\|\asymp\left(\sigma_{\eta}+\left\|\alpha^{}\right\|\sigma_{v}\right)\sqrt{\frac{n}{s}}\sqrt{\frac{\log p}{n}}$, and we have either (1) $\alpha^{}<0$, $\beta_{j}^{}>0$, $\gamma_{j}^{}>0$, $0<\pi_{j}^{}\leq\frac{\lambda_{1}n}{2s}$, or (2) $\alpha^{}>0$, $\beta_{j}^{}<0$, $\gamma_{j}^{}>0$, $-\frac{\lambda_{1}n}{2s}<\pi_{j}^{}<0$. Then there exist positive universal constants $c^{\dagger},c_{1},c_{2},c_{3},c^{},c_{0}^{}$ such that (\[eq:23\]) holds with $\mathbb{P}\left(\mathcal{M}\right)\geq1-c^{}k\exp\left(-c_{0}^{}\log p\right)$. The different scaling in Propositions \[prop:bias\_post\_double\_selection\] and \[prop:bias\_post\_double\_selection-1\] comes from the fact that, the former constrains the magnitude of $\pi_{K}^{}$ through constraining the magnitude of $\left(\beta_{K}^{},\,\gamma_{K}^{},\,\alpha^{}\right)$, and the latter allows large magnitude of $\left(\beta_{K}^{},\,\alpha^{}\right)$ while constraining the magnitude of $\left(\pi_{K}^{},\,\gamma_{K}^{}\right)$. Proposition \[prop:bias\_post\_double\_selection-1\] suggests that, even if $\frac{k\log p}{\sqrt{n}}$ is small, the OVB lower bounds can still be substantial if $\left\|\alpha^{}\right\|$ is large enough. To see this, suppose $\left\|\alpha^{}\right\|\asymp\frac{n}{k\log p}$ and $\frac{k\log p}{n}=o\left(\frac{1}{\sqrt{n}}\right)$ (so that $\frac{k\log p}{\sqrt{n}}=o\left(1\right)$). Then the OVB lower bounds in Proposition \[prop:bias\_post\_double\_selection-1\] scale as a constant bounded away from zero. Even if $\frac{\sigma_{\eta}}{\sigma_{v}}\asymp1$, the ratios of these lower bounds to $\ensuremath{\sigma_{\tilde{\alpha}}=\frac{1}{\sqrt{n}}\frac{\sigma_{\eta}}{\sigma_{v}}}$ scale as $\frac{\left\|\alpha^{}\right\|\sigma_{v}}{\sigma_{\eta}}\frac{k\log p}{\sqrt{n}}\asymp\sqrt{n}$. By contrast, in the setup of Proposition \[prop:bias\_post\_double\_selection\], the ratios of its lower bounds to $\ensuremath{\sigma_{\tilde{\alpha}}}$ scale as $\frac{k\log p}{\sqrt{n}}=o\left(1\right)$ (irrespective of $\frac{\sigma_{\eta}}{\sigma_{v}}$), whenever $\frac{k\log p}{n}=o\left(\frac{1}{\sqrt{n}}\right)$. Propositions \[prop:bias\_post\_double\_formula\]\[prop:bias\_post\_double\_selection-1\] on the OVB lower bounds have important implications for inference based on the post double Lasso procedure proposed in @belloni2014inference. On the one hand, the existing theoretical results on post double Lasso such as @belloni2014inference prove that the assumptions $\frac{k\log p}{\sqrt{n}}=o\left(1\right)$, $\left\|\alpha^{}\right\|\precsim 1$, $\left\|\sigma_{v}\right\|\asymp 1$, and $\left\|\sigma_{\eta}\right\|\asymp 1$ (among other regularity conditions) are sufficient* for establishing asymptotic normality and unbiasedness, regardless of whether under-selection is present or not. On the other hand, our Propositions \[prop:bias\_post\_double\_formula\]\[prop:bias\_post\_double\_selection-1\] show settings where double under-selection occurs with high probability, and suggest that in such settings, small $\frac{k\log p}{\sqrt{n}}$ and $\frac{\left\|\alpha^{}\right\|\sigma_{v}}{\sigma_{\eta}}$ are almost necessary* for a reliable performance of the post double Lasso, under any regularization choice derived from the principle mentioned in Remark \[more choices\]. \[rem:when\_post\_double\_work\] In view of our theoretical results and simulation evidence, if increasing $\lambda_{1}$ and $\lambda_{2}$ from the theoretically recommended choices (such as the one in @belloni2014inference) yield similar $\tilde{\alpha}$s, then the underlying model could be in the regime where either under-selection in both Lasso steps is unlikely, or the OVBs are simply negligible. Outside this regime, the magnitude and performance of post double Lasso can be quite sensitive to the increase of $\lambda_{1}$ and $\lambda_{2}$. The rationale behind this heuristic lies in that the final step of post double Lasso, (\[eq:double\]), is simply an OLS regression of $Y$ on $D$ and the union of selected controls from (\[eq:las-1\])(\[eq:las-2\]). One might ask by how much $\lambda_{1}$ and $\lambda_{2}$ should be increased for the robustness checks. For the regularization choice proposed in @belloni2014inference, our simulations suggest that an increase by $50\%$ works well in practice. As an example, Figure \[fig:dml\_bias\_std\_sens\] shows that the ratio of bias to standard deviation for the post double Lasso is less sensitive to increasing the regularization parameters when $\sigma_{x}\ge0.5$ because the selection performance of Lasso is more robust when $\sigma_{x}\ge 0.5$ (cf. Figures \[fig:sel\_sensitivity\] and \[fig:sel\_rel\_sensitivity\]). [\[]{}In Figures \[fig:sel\_sensitivity\], \[fig:sel\_rel\_sensitivity\], and \[fig:dml\_bias\_std\_sens\], we fix $\beta_{K}^{}$ and $\gamma_{K}^{}$ while varying $\sigma_{x}$, which is equivalent to varying the magnitude of $\beta_{K}^{}$ and $\gamma_{K}^{}$ while fixing $\sigma_{x}$, as discussed previously. In other words, we can simply replace the label $\sigma_{x}$ for the horizontal axis by the magnitude of $\beta_{K}^{}$ and $\gamma_{K}^{}$.[\]]{} Upper bounds on the OVBs {#sec:post_double_upper_bounds} ------------------------ Proposition \[prop:bias\_post\_double\_selection\] suggests that, the lower bounds on the OVBs scale as $\frac{\sigma_{\eta}}{\sigma_{v}}$ as $\frac{k\log p}{n}\rightarrow\infty$. Proposition \[prop:bias\_post\_double\_selection-1\] suggests that, the lower bounds on the OVBs scale as $\left\|\alpha^{}\right\|$ as $\frac{k\log p}{n}\rightarrow\infty$. This prompts the question of how large the OVBs can be under the regime where $\frac{k\log p}{n}\rightarrow\infty$. To answer this question, we would like a meaningful upper bound (other than just $\infty$) on the OVBs, but there is no such result in the existing literature. Interestingly enough, we show that the upper bounds on the OVBs scale as $\frac{\sigma_{\eta}}{\sigma_{v}}$ or $\left\|\alpha^{}\right\|$ with high probability, even if $\frac{k\log p}{n}\rightarrow\infty$ and the Lasso is inconsistent in the sense $\sqrt{\frac{1}{n}\sum_{i=1}^{n}\left(X_{i}\hat{\pi}-X_{i}\pi^{}\right)^{2}}\rightarrow\infty$, $\sqrt{\frac{1}{n}\sum_{i=1}^{n}\left(X_{i}\hat{\gamma}-X_{i}\gamma^{}\right)^{2}}\rightarrow\infty$ with high probability. To illustrate how the OVB upper bounds behave roughly as a function of $\left(n,\,p,\,k,\,\sigma_{\eta},\,\sigma_{v},\,\left\|\alpha^{}\right\|\right)$, we again abstract from all the finite sample subtleties (such as various universal constants) and assume $\left\|a\right\|,\,\left\|b\right\|\asymp1$ (while $\left\|a\right\|,\,\left\|b\right\|\le1$) and $\phi^{-1}\precsim1$. \[prop:bias\_post\_double\_selection-upper\]Let Assumptions \[ass:inference\_normality\] and \[ass:inference\_sparsity\_incoherence\] hold. Suppose $\phi^{-1}\precsim1$ in (\[eq:14\]); the regularization parameters in (\[eq:las-1\]) and (\[eq:las-2\]) are chosen in a similar fashion as in Proposition \[prop:bias\_post\_double\_selection\] such that $\lambda_{1}\asymp\phi^{-1}\sigma_{\eta}\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}}$ and $\lambda_{2}\asymp\phi^{-1}\sigma_{v}\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}}$; for all $j\in K$, $\gamma_{j}^{}=\gamma^{}$, $\left\|\beta_{j}^{}\right\|\leq\frac{\lambda_{1}n}{2s}$ and $\left\|\gamma^{}\right\|\leq\frac{\lambda_{2}n}{2s}$, but $\left\|\beta_{j}^{}\right\|\asymp\sigma_{\eta}\sqrt{\frac{n}{s}}\sqrt{\frac{\log p}{n}}$ and $\left\|\gamma^{}\right\|\asymp\sigma_{v}\sqrt{\frac{n}{s}}\sqrt{\frac{\log p}{n}}$. Let us consider $\tilde{\alpha}$ obtained from (\[eq:double\]). Then for either $\alpha^{}=0$, or $\alpha^{}\neq0$ subject to the conditions in part (ii) of Proposition \[prop:bias\_post\_double\_selection\], there exist positive universal constants $c_{1},c_{2},c_{3},c_{4},c^{},c_{0}^{}$ such that $$\mathbb{P}\left(\left\|\tilde{\alpha}-\alpha^{}\right\|\leq\overline{OVB}\vert\hat{I}_{1}=\hat{I}_{2}=\emptyset\right)\geq1-c_{1}k\exp\left(-c_{2}\log p\right)-c_{3}\exp\left(-c_{4}nr^{2}\right)$$ where $\mathbb{P}\left(\hat{I}_{1}=\hat{I}_{2}=\emptyset\right)\geq1-c^{}\exp\left(-c_{0}^{}\log p\right)$ and $$\overline{OVB}\asymp\max\left\{ \frac{\sigma_{\eta}}{\sigma_{v}}\left(\frac{k\log p}{n}\wedge1\right),\,\frac{\sigma_{v}\sigma_{\eta}\left(r\vee\frac{k\log p}{n}\right)}{\left(\frac{k\log p}{n}\vee1\right)\sigma_{v}^{2}}\right\}$$ for any $r\in(0,\,1]$. \[prop:bias\_post\_double\_selection-upper-1\]Let Assumptions \[ass:inference\_normality\] and \[ass:inference\_sparsity\_incoherence\] hold. Suppose $\phi^{-1}\precsim1$ in (\[eq:14\]); the regularization parameters in (\[eq:las-1\]) and (\[eq:las-2\]) are chosen in a similar fashion as in Proposition \[prop:bias\_post\_double\_selection-1\] such that $\lambda_{1}\asymp\frac{\sigma_{\eta}+\left\|\alpha^{}\right\|\sigma_{v}}{\phi}\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}}$ and $\lambda_{2}\asymp\phi^{-1}\sigma_{v}\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}}$; for all $j\in K$, $\gamma_{j}^{}=\gamma^{}$, $\left\|\gamma^{}\right\|\leq\frac{\lambda_{2}n}{2s}$, but $\left\|\gamma^{}\right\|\asymp\sigma_{v}\sqrt{\frac{n}{s}}\sqrt{\frac{\log p}{n}}$. Let us consider $\tilde{\alpha}$ obtained from (\[eq:double\]). \(i) If $\pi_{j}^{}=0$ for all $j\in K$, then there exist positive universal constants $c_{1},c_{2},c_{3},c_{4},c^{},c_{0}^{}$ such that $$\mathbb{P}\left(\left\|\tilde{\alpha}-\alpha^{}\right\|\leq\overline{OVB}\vert\hat{I}_{1}=\hat{I}_{2}=\emptyset\right)\geq1-c_{1}k\exp\left(-c_{2}\log p\right)-c_{3}\exp\left(-c_{4}nr^{2}\right) \label{eq:OVB_upper}$$ where $\mathbb{P}\left(\hat{I}_{1}=\hat{I}_{2}=\emptyset\right)\geq1-c^{}\exp\left(-c_{0}^{}\log p\right)$ and $$\overline{OVB}\asymp\max\left\{ \left\|\alpha^{}\right\|\left(\frac{k\log p}{n}\wedge1\right),\,\frac{\sigma_{v}\sigma_{\eta}\left(r\vee\frac{k\log p}{n}\right)}{\left(\frac{k\log p}{n}\vee1\right)\sigma_{v}^{2}}\right\}$$ for any $r\in(0,\,1]$. \(ii) If $0<\left\|\pi_{j}^{}\right\|\leq\frac{\lambda_{1}n}{2s}$ but $\left\|\pi_{j}^{}\right\|\asymp\left(\sigma_{\eta}+\left\|\alpha^{}\right\|\sigma_{v}\right)\sqrt{\frac{n}{s}}\sqrt{\frac{\log p}{n}}$ for all $j\in K$, then there exist positive universal constants $c_{1},c_{2},c_{3},c_{4},c^{},c_{0}^{}$ such that (\[eq:OVB\_upper\]) holds with $\mathbb{P}\left(\hat{I}_{1}=\hat{I}_{2}=\emptyset\right)\geq1-c^{}\exp\left(-c_{0}^{}\log p\right)$ and $$\overline{OVB}\asymp\max\left\{ \left(\left\|\alpha^{}\right\|\vee\frac{\sigma_{\eta}}{\sigma_{v}}\right)\left(\frac{k\log p}{n}\wedge1\right),\,\frac{\sigma_{v}\sigma_{\eta}\left(r\vee\frac{k\log p}{n}\right)}{\left(\frac{k\log p}{n}\vee1\right)\sigma_{v}^{2}}\right\}$$ for any $r\in(0,\,1]$. Suppose $\sigma_{v}\asymp1$ and $\frac{c^{'}k}{p^{c^{''}}}$ is small for some positive universal constants $c^{'}$ and $c^{''}$. As $\frac{k\log p}{n}\rightarrow\infty$ (where both Lasso steps are inconsistent in the sense that $\sqrt{\frac{1}{n}\sum_{i=1}^{n}\left(X_{i}\hat{\pi}-X_{i}\pi^{}\right)^{2}}\rightarrow\infty$ and $\sqrt{\frac{1}{n}\sum_{i=1}^{n}\left(X_{i}\hat{\gamma}-X_{i}\gamma^{}\right)^{2}}\rightarrow\infty$ with high probability), Proposition \[prop:bias\_post\_double\_selection-upper\] implies that $\overline{OVB}\asymp\frac{\sigma_{\eta}}{\sigma_{v}}$ and Proposition \[prop:bias\_post\_double\_selection-upper-1\] implies that $\overline{OVB}\asymp\left(\left\|\alpha^{}\right\|\vee\frac{\sigma_{\eta}}{\sigma_{v}}\right)$, and $\mathbb{P}\left(\left\|\tilde{\alpha}-\alpha^{}\right\|\leq\overline{OVB}\vert\hat{I}_{1}=\hat{I}_{2}=\emptyset\right)$ is large. Implications for inference and comparison to high-dimensional OLS-based methods =============================================================================== As shown in Section \[sec:inference\], the OVBs of post double Lasso have important consequences for making inference. In terms of simulation evidence, Figure \[fig:dml\_cov\] displays the coverage rates of 90% confidence intervals based on the DGP in Section \[sec:simulation\_evidence\] and shows that the OVB of post double Lasso can cause substantial under-coverage even when $n=1000$. \[fig:dml\_cov\] These results and those in Section \[sec:inference\] prompt the question of how to make inference in a reliable manner when under-selection and OVBs is a concern, such as, when limited variability is present. In many economic applications, $p$ is comparable to but still smaller than $n$. In such moderately high dimensional settings, OLS-based inference procedures provide a natural alternative to Lasso-based methods. Under classical conditions, OLS is the best linear unbiased estimator. Moreover, under normality and homoscedasticity, OLS admits exact finite sample inference for any fixed $(n,\,p)$ as long as $\frac{p+1}{n}\leq1$ (recalling that the number of regression coefficients is $p+1$ in ).[^24] Furthermore, unlike the Lasso-based inference methods, OLS does not rely on any sparsity assumptions. This is important because sparsity assumptions may not be satisfied in applications and, as we show in this paper, the OVBs of Lasso-based inference procedures can be substantial even when $k$ is very small, $n$ is large and larger than $p$. While OLS is unbiased, constructing standard errors is challenging when $p$ is large. For instance, @cattaneo2018inference show that the usual versions of Eicker-White robust standard errors are inconsistent under asymptotics where the number of controls grows as fast as the sample size. This result motivates a recent literature to develop high-dimensional OLS-based inference procedures that are valid in settings with many controls [e.g., @cattaneo2018inference; @jochmans2018heteroscedasticity; @kline2018leaveout]. Figures \[fig:dml\_bias\_std\_ols\]–\[fig:dml\_cov\_ols\] compare the finite sample performance of post double Lasso and OLS with the heteroscedasticity robust HCK standard errors proposed by @cattaneo2018inference. We do not report results for $n=200$ since OLS cannot be applied in this case. OLS is unbiased (as expected) and exhibits close-to-exact empirical coverage rates irrespective of the degree of variability in the controls. The additional simulation evidence in Appendix \[app:additional\_simulations\] confirms the excellent performance of OLS with HCK standard errors. Figure \[fig:dml\_ci\_ols\] displays the average length of 90% confidence intervals and shows that OLS yields somewhat wider confidence intervals than post double Lasso. \[fig:dml\_bias\_std\_ols\] \[fig:dml\_cov\_ols\] \[fig:dml\_ci\_ols\] In sum, our simulation results suggest that modern OLS-based inference methods that accommodate many controls constitute a viable alternative to Lasso-based inference methods. These methods are unbiased and demonstrate an excellent size accuracy, irrespective of the degree of variability in the relevant controls. However, there is a trade-off because OLS yields somewhat wider confidence intervals than post double Lasso. Recommendations for empirical practice {#sec:recommendation} ====================================== Here we summarize the practical implications of our results and provide guidance for empirical researchers. First, the simulation evidence in Section \[sec:evidence\] and Appendix \[app:additional\_simulations\] and the theoretical results in Section \[sec:bias\_post\_double\_lasso\] (cf. Remark \[rem:when\_post\_double\_work\]) suggest the following heuristic: If the estimates of $\alpha^{}$ are robust to increasing the theoretically recommended regularization parameters in the two Lasso steps, post double Lasso could be a reliable and efficient method. Therefore, we recommend to always check whether empirical results are robust to increasing the regularization parameters. Based on our simulations, a simple rule of thumb is to increase by $50\%$ the regularization parameters proposed in @belloni2014inference. Robustness checks are standard in other contexts (e.g., bandwidth choices in regression discontinuity designs) and our results highlight the importance of such checks in the context of Lasso-based inference methods. Second, we show that even if the relevant controls are associated with large coefficients, the Lasso may not select such controls due to limited variability. Therefore, following @belloni2014inference, we recommend to always augment the union of selected controls with an “amelioration” set of controls motivated by economic theory and prior knowledge to mitigate the OVBs. Third, our simulations show that in moderately high-dimensional settings where $p$ is comparable to but smaller than $n$, recently developed OLS-based inference methods that are robust to the inclusion of many controls exhibit better size properties and not much wider confidence intervals than Lasso-based inference methods. This suggests that high-dimensional OLS-based procedures constitute a viable alternative to Lasso-based inference methods. Conclusion ========== Given the rapidly increasing popularity of Lasso and Lasso-based inference methods in empirical economic research, it is crucial to better understand the merits and limitations of these new tools, and how they compare to other alternatives such as the high-dimensional OLS-based procedures. This paper presents empirical and theoretical evidence on the finite sample behavior of post double Lasso and the debiased Lasso (in the appendix). Specifically, we analyze the finite sample OVBs arising from the Lasso not selecting all the relevant control variables. Our results have important practical implications, and we provide guidance for empirical researchers. We focus on the implications of under-selection for post double Lasso and the debiased Lasso in linear regression models. However, our results on the under-selection of the Lasso also have important implications for other inference methods that rely on Lasso as a first-step estimator. Towards this end, an interesting avenue for future research would be to investigate the impact of under-selection on the performance of the non-linear Lasso-based approaches proposed by @belloni2014inference, @farrell2015robust, @belloni2017program, and @chernozhukov2018double. In moderately high-dimensional settings where $p$ is smaller than but comparable to $n$, it would also be interesting to compare the non-linear treatment effects estimators in @belloni2014inference to the robust finite sample methods proposed by @rothe2017robust. Finally, this paper motivates further examinations of the practical usefulness of Lasso-based inference procedures and other modern high-dimensional methods. For example, in the context of the study of elite college effects by @dale2002estimating, @angrist2019machine find that post double Lasso can be useful for regression-based sensitivity analyses when there is an abundance of potential control variables. They further present interesting simulation evidence on the finite sample behavior of Lasso-based IV methods [e.g., @belloni2012sparse]. [Appendix to “Omitted variable bias of Lasso-based inference methods: A finite sample analysis” (for online publication)]{} Notation {#notation .unnumbered} ======== Here we collect additional notation that is not provided in the main text. The $\ell_{q}-$norm of a vector $v\in\mathbb{R}^{m}$ is denoted by $\left\|v\right\|_{q}$, $1\leq q\leq\infty$ where $\left\|v\right\|_{q}:=\left(\sum_{i=1}^{m}\left\|v_{i}\right\|^{q}\right)^{1/q}$ when $1\leq q<\infty$ and $\left\|v\right\|_{q}:=\max_{i=1,\dots,m}\left\|v_{i}\right\|$ when $q=\infty$. For a matrix $A\in\mathbb{R}^{n\times m}$, the $\ell_{2}-$operator norm of $A$ is defined as $\left\Vert A\right\Vert _{2}:=\sup_{v\in S^{m-1}}\left\|Av\right\|_{2}$, where $S^{m-1}=\left\{ v\in\mathbb{R}^{m}:\,\left\|v\right\|_{2}=1\right\} $. For a square matrix $A\in\mathbb{R}^{m\times m}$, let $\lambda_{\min}(A)$ and $\lambda_{\max}(A)$ denote its minimum eigenvalue and maximum eigenvalue, respectively. Proofs for the main results =========================== Lemma \[prop:fixed\_design\] {#appendix a1} ---------------------------- ### Preliminary {#preliminary .unnumbered} We will exploit the following Gaussian tail bound: $$\mathbb{P}\left(\mathcal{Z}\geq t\right)\leq\frac{1}{2}\exp\left(\frac{-t^{2}}{2\sigma^{2}}\right)$$ for all $t\geq0$, where $\mathcal{Z}\sim\mathcal{N}\left(0,\,\sigma^{2}\right)$. Note that the constant “$\frac{1}{2}$” cannot be improved uniformly. Given $\lambda\geq\frac{2\sigma}{\phi}\sqrt{\frac{s}{n}}\sqrt{\frac{2\left(1+\tau\right)\log p}{n}}$ where $\tau>0$ and the tail bound $$\mathbb{P}\left(\left\|\frac{X^{T}\varepsilon}{n}\right\|_{\infty}\geq t\right)\leq\exp\left(\frac{-nt^{2}}{2\sigma^{2}s/n}+\log p\right)\leq\frac{1}{p^{\tau}}$$ for $t=\frac{\sigma}{\phi}\sqrt{\frac{s}{n}}\sqrt{\frac{2\left(1+\tau\right)\log p}{n}}$, we have $$\lambda\geq2\left\|\frac{X^{T}\varepsilon}{n}\right\|_{\infty}\label{eq:lambda}$$ with probability at least $1-\frac{1}{p^{\tau}}$. Let the event $$\mathcal{E}=\left\{ \left\|\frac{X^{T}\varepsilon}{n}\right\|_{\infty}\leq\frac{\sigma}{\phi}\sqrt{\frac{s}{n}}\sqrt{\frac{2\left(1+\tau\right)\log p}{n}}\right\} .\label{eq:event}$$ Note that $\mathbb{P}\left(\mathcal{E}\right)\geq1-\frac{1}{p^{\tau}}$. Lemma \[prop:fixed\_design\] relies on the following intermediate results. (i) On the event $\mathcal{E}$, (\[eq:las\]) has a unique optimal solution $\hat{\theta}$ such that $\hat{\theta}_{j}=0$ for $j\notin K$.* (ii) If $\mathbb{P}\left(\left\{ \hat{\theta}_{j}\neq0,\,j\in K\right\} \cap\mathcal{E}\right)>0$, conditioning on $\left\{ \hat{\theta}_{j}\neq0,\,j\in K\right\} \cap\mathcal{E}$, we must have $$\left\|\hat{\theta}_{j}-\theta_{j}^{}\right\|\geq\frac{\lambda n}{2s}.\label{eq:37-1}$$* Claim (i) above follows from the argument in @wainwright_2019. To show claim (ii), we develop our own proof. The proof for claim (i) above is based on a construction called Primal-Dual Witness (PDW) method developed by @wainwright2009sharp. The procedure is described as follows. 1. Set $\hat{\theta}_{K^{c}}=0_{p-k}$. 2. Obtain $(\hat{\theta}_{K},\,\hat{\delta}_{K})$ by solving $$\hat{\theta}_{K}\in\arg\min_{\theta_{K}\in\mathbb{R}^{k}}\left\{ \underset{:=g(\theta_{K})}{\underbrace{\frac{1}{2n}\left\|Y-X_{K}\theta_{K}\right\|_{2}^{2}}}+\lambda\left\|\theta_{K}\right\|_{1}\right\} ,\label{eq:sub}$$ and choosing $\hat{\delta}_{K}\in\partial\left\|\theta_{K}\right\|_{1}$ such that $\nabla g(\theta_{K})\vert_{\theta_{K}=\hat{\theta}_{K}}+\lambda\hat{\delta}_{K}=0$.[^25] 3. Obtain $\hat{\delta}_{K^{c}}$ by solving $$\frac{1}{n}X^{T}(X\hat{\theta}-Y)+\lambda\hat{\delta}=0,\label{eq:42}$$ and check whether or not $\left\|\hat{\delta}_{K^{c}}\right\|_{\infty}<1$ (the strict dual feasibility condition) holds. Lemma 7.23 from Chapter 7 of @wainwright_2019 shows that, if the PDW construction succeeds, then $\hat{\theta}=(\hat{\theta}_{K},\,0_{p-k})$ is the unique optimal solution of program (\[eq:las\]). To show that the PDW construction succeeds on the event $\mathcal{E}$, it suffices to show that $\left\|\hat{\delta}_{K^{c}}\right\|_{\infty}<1$. The details can be found in Chapter 7.5 of @wainwright_2019. In particular, under the choice of $\lambda$ stated in Lemma \[prop:fixed\_design\], we obtain that $\left\|\hat{\delta}_{K^{c}}\right\|_{\infty}<1$ and hence the PDW construction succeeds conditioning on $\mathcal{E}$ where $\mathbb{P}\left(\mathcal{E}\right)\geq1-\frac{1}{p^{\tau}}$. In summary, conditioning on $\mathcal{E}$, under the choice of $\lambda$ stated in Lemma \[prop:fixed\_design\], program (\[eq:las\]) has a unique optimal solution $\hat{\theta}$ such that $\hat{\theta}_{j}=0$ for $j\notin K$. We now show (\[eq:37-1\]). By construction, $\hat{\theta}=(\hat{\theta}_{K},\,0_{p-k})$, $\hat{\delta}_{K}$, and $\hat{\delta}_{K^{c}}$ satisfy (\[eq:42\]) and therefore we obtain $$\begin{aligned} \frac{1}{n}X_{K}^{T}X_{K}\left(\hat{\theta}_{K}-\theta_{K}^{}\right)-\frac{1}{n}X_{K}^{T}\varepsilon+\lambda\hat{\delta}_{K} & = & 0_{k},\label{eq:7-1}\\ \frac{1}{n}X_{K^{c}}^{T}X_{K}\left(\hat{\theta}_{K}-\theta_{K}^{}\right)-\frac{1}{n}X_{K^{c}}^{T}\varepsilon+\lambda\hat{\delta}_{K^{c}} & = & 0_{p-k}.\label{eq:8}\end{aligned}$$ Solving the equations above yields $$\hat{\theta}_{K}-\theta_{K}^{}=\left(\frac{X_{K}^{T}X_{K}}{n}\right)^{-1}\frac{X_{K}^{T}\varepsilon}{n}-\lambda\left(\frac{X_{K}^{T}X_{K}}{n}\right)^{-1}\hat{\delta}_{K}.\label{eq:49}$$ In what follows, we will condition on $\left\{ \hat{\theta}_{j}\neq0,\,j\in K\right\} \cap\mathcal{E}$ and make use of (\[eq:lambda\])-(\[eq:event\]). Let $\Delta=\frac{X_{K}^{T}\varepsilon}{n}-\lambda\hat{\delta}_{K}$. Note that $$\left\|\hat{\theta}_{K}-\theta_{K}^{}\right\|\geq\left\|\left(\frac{X_{K}^{T}X_{K}}{n}\right)^{-1}\right\|\left\|\left\|\lambda\hat{\delta}_{K}\right\|-\left\|\frac{X_{K}^{T}\varepsilon}{n}\right\|\right\|,\label{eq:7}$$ where the inequality uses the fact that $\left(\frac{X_{K}^{T}X_{K}}{n}\right)^{-1}$ is diagonal. In Step 2 of the PDW procedure, $\hat{\delta}_{K}$ is chosen such that $\left\|\hat{\delta}_{j}\right\|=1$ for any $j\in K$ with $\hat{\theta}_{j}\neq0$; we therefore obtain $$\left\|\hat{\theta}_{j}-\theta_{j}^{}\right\|\geq\frac{n}{s}\left\|\left\|\lambda\right\|-\left\|\frac{X_{j}^{T}\varepsilon}{n}\right\|\right\|\geq\frac{\lambda n}{2s}$$ where the second inequality follows from (\[eq:lambda\]). ### Main proof {#main-proof .unnumbered} In what follows, we let $$\begin{aligned} E_{1} & = & \left\{ \textrm{sgn}\left(\hat{\theta}_{j}\right)=-\textrm{sgn}\left(\theta_{j}^{}\right),\,\textrm{for some }j\in K\right\} ,\\ E_{2} & = & \left\{ \textrm{sgn}\left(\hat{\theta}_{j}\right)=\textrm{sgn}\left(\theta_{j}^{}\right),\,\textrm{for some }j\in K\textrm{ such that (\ref{eq:min}) holds}\right\} ,\\ E_{3} & = & \left\{ \textrm{sgn}\left(\hat{\theta}_{j}\right)=\textrm{sgn}\left(\theta_{j}^{}\right),\,\textrm{for some }j\in K\right\} .\end{aligned}$$ To show (\[eq:nec\]) in (iv), recall we have established that conditioning on $\mathcal{E}$, (\[eq:las\]) has a unique optimal solution $\hat{\theta}$ such that $\hat{\theta}_{j}=0$ for $j\notin K$. Therefore, conditioning on $\mathcal{E}$, the KKT condition for (\[eq:las\]) implies $$\frac{s}{n}\left(\theta_{j}^{}-\hat{\theta}_{j}\right)=\lambda\textrm{sgn}\left(\hat{\theta}_{j}\right)-\frac{X_{j}^{T}\varepsilon}{n}\label{eq:kkt}$$ for $j\in K$ such that $\hat{\theta}_{j}\neq0$. We first show that $\mathbb{P}\left(E_{1}\cap\mathcal{E}\right)=0$. Suppose $\mathbb{P}\left(E_{1}\cap\mathcal{E}\right)>0$. We may then condition on the event $E_{1}\cap\mathcal{E}$. Case (i): $\theta_{j}^{}>0$ and $\hat{\theta}_{j}<0$. Then, the LHS of (\[eq:kkt\]), $\frac{s}{n}\left(\theta_{j}^{}-\hat{\theta}_{j}\right)>0$; consequently, the RHS, $\lambda\textrm{sgn}\left(\hat{\theta}_{j}\right)-\frac{X_{j}^{T}\varepsilon}{n}=-\lambda-\frac{X_{j}^{T}\varepsilon}{n}>0$. However, given the choice of $\lambda$, conditioning on $\mathcal{E}$, $\lambda\geq2\left\|\frac{X^{T}\varepsilon}{n}\right\|_{\infty}$ and consequently, $-\lambda-\frac{X_{j}^{T}\varepsilon}{n}\leq-\frac{\lambda}{2}<0$. This leads to a contradiction. Case (ii): $\theta_{j}^{}<0$ and $\hat{\theta}_{j}>0$. Then, the LHS of (\[eq:kkt\]), $\frac{s}{n}\left(\theta_{j}^{}-\hat{\theta}_{j}\right)<0$; consequently, the RHS, $\lambda\textrm{sgn}\left(\hat{\theta}_{j}\right)-\frac{X_{j}^{T}\varepsilon}{n}=\lambda-\frac{X_{j}^{T}\varepsilon}{n}<0$. However, given the choice of $\lambda$, conditioning on $\mathcal{E}$, $\lambda\geq2\left\|\frac{X^{T}\varepsilon}{n}\right\|_{\infty}$ and consequently, $\lambda-\frac{X_{j}^{T}\varepsilon}{n}\geq\frac{\lambda}{2}>0$. This leads to a contradiction. It remains to show that $\mathbb{P}\left(E_{2}\cap\mathcal{E}\right)=0$. We first establish a useful fact under the assumption that $\mathbb{P}\left(E_{3}\cap\mathcal{E}\right)>0$. Let us condition on the event $E_{3}\cap\mathcal{E}$. If $\theta_{j}^{}>0$, we have $\frac{s}{n}\left(\theta_{j}^{}-\hat{\theta}_{j}\right)=\lambda-\frac{X_{j}^{T}\varepsilon}{n}\geq\frac{\lambda}{2}>0$ (i.e., $\theta_{j}^{}\geq\hat{\theta}_{j}$); similarly, if $\theta_{j}^{}<0$, then we have $\frac{s}{n}\left(\theta_{j}^{}-\hat{\theta}_{j}\right)=-\lambda-\frac{X_{j}^{T}\varepsilon}{n}\leq-\frac{\lambda}{2}<0$ (i.e., $\theta_{j}^{}\leq\hat{\theta}_{j}$). Putting the pieces together implies that, for $j\in K$ such that $\textrm{sgn}\left(\hat{\theta}_{j}\right)=\textrm{sgn}\left(\theta_{j}^{}\right)$, $$\left\|\theta_{j}^{}-\hat{\theta}_{j}\right\|=\left\|\theta_{j}^{}\right\|-\left\|\hat{\theta}_{j}\right\|.\label{eq:equal}$$ We now show that $\mathbb{P}\left(E_{2}\cap\mathcal{E}\right)=0$. Suppose $\mathbb{P}\left(E_{2}\cap\mathcal{E}\right)>0$. We may then condition on the event that $E_{2}\cap\mathcal{E}$. Because of (\[eq:min\]) and (\[eq:equal\]), we have $\left\|\theta_{j}^{}-\hat{\theta}_{j}\right\|<\frac{\lambda n}{2s}$. On the other hand, (\[eq:37-1\]) implies that $\left\|\theta_{j}^{}-\hat{\theta}_{j}\right\|\geq\frac{\lambda n}{2s}$. We have arrived at a contradiction. Consequently, we must have $\mathbb{P}\left(E_{2}\cap\mathcal{E}\right)=0$. In summary, we have shown that $\mathbb{P}\left(E_{1}\cap\mathcal{E}\right)=0$ and $\mathbb{P}\left(E_{2}\cap\mathcal{E}\right)=0$. Claim (i) in “Preliminary” implies that $\mathbb{P}\left(E_{4}\vert\mathcal{E}\right)=0$ where $E_{4}$ denotes the event that $\hat{\theta}_{j}\neq0$ for some $j\notin K$. Therefore, on $\mathcal{E}$, none of the events $E_{1}$, $E_{2}$ and $E_{4}$ can happen. This fact implies that, if (\[eq:min\]) is satisfied for all $l\in K$, we must have $$\mathbb{P}\left(\hat{\theta}=0_{p}\right)\geq1-\mathbb{P}\left(\mathcal{E}^{c}\right)\geq1-\frac{1}{p^{\tau}}.$$ Proposition \[prop:bias\_post\_double\_formula\] {#sec:appendix_post_double_lasso} ------------------------------------------------ We first show the case where $ab>0$. Let the events $$\begin{aligned} \mathcal{E}_{t_{1}} & = & \left\{ \left\|\frac{X_{K}^{T}v}{n}\right\|_{\infty}\leq t_{1},\,t_{1}>0\right\} ,\label{eq:50}\\ \mathcal{E}_{t_{2}}^{'} & = & \left\{ \frac{1}{n}\sum_{i=1}^{n}v_{i}^{2}\leq\sigma_{v}^{2}+t_{2},\:t_{2}\in(0,\,\sigma_{v}^{2}]\right\} .\nonumber \end{aligned}$$ By tail bounds for Gaussian and Chi-Square variables, we have $$\begin{aligned} \mathbb{P}\left(\mathcal{E}_{t_{1}}\right) & \geq & 1-k\exp\left(\frac{-nt_{1}^{2}}{2\frac{s}{n}\sigma_{v}^{2}}\right),\label{eq:32}\\ \mathbb{P}\left(\mathcal{E}_{t_{2}}^{'}\right) & \geq & 1-\exp\left(\frac{-nt_{2}^{2}}{8\sigma_{v}^{4}}\right).\nonumber \end{aligned}$$ In the following proof, we exploit the bound $$\begin{aligned} \mathbb{P}\left(\mathcal{E}_{t_{2}}^{'}\vert\hat{I}_{2}=\emptyset,\,\mathcal{E}_{t_{1}}\right) & \geq & \mathbb{P}\left(\mathcal{E}_{t_{2}}^{'}\cap\mathcal{E}_{t_{1}}\cap\left\{ \hat{I}_{2}=\emptyset\right\} \right)\nonumber \\ & \geq & \mathbb{P}\left(\mathcal{E}_{t_{1}}\right)+\mathbb{P}\left(\mathcal{E}_{t_{2}}^{'}\right)+\mathbb{P}\left(\hat{I}_{2}=\emptyset\right)-2\nonumber \\ & \geq & 1-\frac{1}{p^{\tau}}-k\exp\left(\frac{-nt_{1}^{2}}{2\frac{s}{n}\sigma_{v}^{2}}\right)-\exp\left(\frac{-nt_{2}^{2}}{8\sigma_{v}^{4}}\right)\label{eq:58}\end{aligned}$$ where the third inequality follows from Lemma \[prop:fixed\_design\], which implies $\hat{I}_{2}=\emptyset$ with probability at least $1-\frac{1}{p^{\tau}}$. Note that $\mathbb{P}\left(\mathcal{E}_{t_{1}}\cap\left\{ \hat{I}_{2}=\emptyset\right\} \right)\geq\mathbb{P}\left(\mathcal{E}_{t_{1}}\right)+\mathbb{P}\left(\hat{I}_{2}=\emptyset\right)-1\geq1-\frac{1}{p^{\tau}}-k\exp\left(\frac{-nt_{1}^{2}}{2\frac{s}{n}\sigma_{v}^{2}}\right)$, which is a “high probability” guarantee for sufficiently large $p$ and $t_{1}$. Thus, working with $\mathbb{P}\left(\mathcal{E}_{t_{2}}^{'}\vert\hat{I}_{2}=\emptyset,\,\mathcal{E}_{t_{1}}\right)$ is sensible under an appropriate choice of $t_{1}$ (as we will see below). We first bound $\frac{\frac{1}{n}D^{T}X_{K}}{\frac{1}{n}D^{T}D}\beta_{K}^{}$. Note that $$\begin{aligned} \frac{\frac{1}{n}D^{T}X_{K}}{\frac{1}{n}D^{T}D}\beta_{K}^{} & = & \left(\frac{D^{T}D}{n}\right)^{-1}\left[\frac{1}{n}\left(X_{K}\gamma_{K}^{}+v\right)^{T}X_{K}\beta_{K}^{}\right]\\ & = & \left(\frac{D^{T}D}{n}\right)^{-1}\left[\frac{1}{n}\gamma_{K}^{T}X_{K}^{T}X_{K}\beta_{K}^{}+\frac{1}{n}v^{T}X_{K}\beta_{K}^{}\right]\\ & = & \frac{\frac{s}{n}\gamma_{K}^{T}\beta_{K}^{}+\frac{1}{n}v^{T}X_{K}\beta_{K}^{}}{\frac{1}{n}\left(X_{K}\gamma_{K}^{}+v\right)^{T}\left(X_{K}\gamma_{K}^{}+v\right)},\end{aligned}$$ and that $\frac{s}{n}\gamma_{K}^{T}\beta_{K}^{}=2\left(1+\tau\right)ab\phi^{-2}\sigma_{\eta}\sigma_{v}\frac{k\log p}{n}$. Moreover, applying (\[eq:58\]) with $t_{1}=\left\|b\right\|\frac{\lambda_{2}}{4}$ and $t_{2}=r\sigma_{v}^{2}$ (where $r\in(0,\,1]$) yields $$\begin{aligned} \frac{s}{n}\gamma_{K}^{T}\beta_{K}^{}+\frac{1}{n}v^{T}X_{K}\beta_{K}^{} & \geq & \frac{s}{n}\gamma_{K}^{T}\beta_{K}^{}-\left\|\frac{1}{n}v^{T}X_{K}\right\|_{\infty}\left\|\beta_{K}^{}\right\|_{1}\nonumber \\ & \geq & \left(1+\tau\right)ab\phi^{-2}\sigma_{\eta}\sigma_{v}\frac{k\log p}{n}\label{eq:pos}\end{aligned}$$ as well as $$\begin{aligned} \frac{1}{n}\left(X_{K}\gamma_{K}^{}+v\right)^{T}\left(X_{K}\gamma_{K}^{}+v\right) & \leq & \frac{s}{n}\gamma_{K}^{T}\gamma_{K}^{}+\left\|\frac{2}{n}v^{T}X_{K}\right\|_{\infty}\left\|\gamma_{K}^{}\right\|_{1}+\frac{1}{n}v^{T}v\nonumber \\ & \leq & 4\left(1+\tau\right)\phi^{-2}b^{2}\sigma_{v}^{2}\frac{k\log p}{n}+\sigma_{v}^{2}+r\sigma_{v}^{2}\label{eq:den}\end{aligned}$$ with probability at least $$1-k\exp\left(\frac{-b^{2}\left(1+\tau\right)\log p}{4\phi^{2}}\right)-\frac{1}{p^{\tau}}-\exp\left(\frac{-nr^{2}}{8}\right):=T_{2}\left(r\right).$$ Conditioning on $\mathcal{E}_{t_{1}}\cap\left\{ \hat{I}_{2}=\emptyset\right\} $ with $t_{1}=t^{}=\left\|b\right\|\frac{\lambda_{2}}{4}$, putting the pieces together yields $$\frac{D^{T}X_{K}}{D^{T}D}\beta_{K}^{}\geq\frac{\left(1+\tau\right)ab\phi^{-2}\sigma_{\eta}\frac{k\log p}{n}}{4\left(1+\tau\right)\phi^{-2}b^{2}\sigma_{v}\frac{k\log p}{n}+\sigma_{v}+r\sigma_{v}}:=T_{1}\left(r\right),\label{eq:47}$$ with probability at least $T_{2}\left(r\right)$. That is, $$\mathbb{P}\left(\frac{D^{T}X_{K}}{D^{T}D}\beta_{K}^{}\geq T_{1}\left(r\right)\vert\hat{I}_{2}=\emptyset,\,\mathcal{E}_{t^{}}\right)\geq T_{2}\left(r\right).$$ When $\alpha^{}=0$ in (\[eq:20\]), the reduced form coefficients $\pi^{}$ in (\[eq:reduce\]) coincide with $\beta^{}$ and $u$ coincides with $\eta$. Given the conditions on $X,$ $\eta$, $v$, $\beta_{K}^{}$ and $\gamma_{K}^{}$, we can then apply (\[eq:nec\]) in Lemma \[prop:fixed\_design\] and the fact $\mathbb{P}\left(\hat{I}_{1}=\hat{I}_{2}=\emptyset\right)\geq\mathbb{P}\left(\hat{I}_{1}=\emptyset\right)+\mathbb{P}\left(\hat{I}_{2}=\emptyset\right)-1$ to show that $E=\left\{ \hat{I}_{1}=\hat{I}_{2}=\emptyset\right\} $ occurs with probability at least $1-\frac{2}{p^{\tau}}$. Note that with the choice $t_{1}=t^{}=\left\|b\right\|\frac{\lambda_{2}}{4}$, $\mathbb{P}\left(E\cap\mathcal{E}_{t^{}}\right)\geq\mathbb{P}\left(E\right)+\mathbb{P}\left(\mathcal{E}_{t^{}}\right)-1\geq1-k\exp\left(\frac{-b^{2}\left(1+\tau\right)\log p}{4\phi^{2}}\right)-\frac{2}{p^{\tau}}$, which is a “high probability” guarantee given sufficiently large $p$.[^26] Therefore, it is sensible to work with $\mathbb{E}\left(\tilde{\alpha}-\alpha^{}\vert\mathcal{M}\right)$ where $$\mathcal{M}=E\cap\mathcal{E}_{t^{}}.\label{eq:M_event}$$ Given $E$, (\[eq:double\]) becomes $$\tilde{\alpha}\in\textrm{arg}\min_{\alpha\in\mathbb{R}}\frac{1}{2n}\left\|Y-D\alpha\right\|_{2}^{2},\qquad\textrm{while }\tilde{\beta}=0_{p}.\label{eq:double-1}$$ As a result, we obtain $\mathbb{E}\left(\tilde{\alpha}-\alpha^{}\vert\mathcal{M}\right)=\mathbb{E}\left(\frac{\frac{1}{n}D^{T}X_{K}}{\frac{1}{n}D^{T}D}\beta_{K}^{}\vert\mathcal{M}\right)+\mathbb{E}\left(\frac{\frac{1}{n}D^{T}\eta}{\frac{1}{n}D^{T}D}\vert\mathcal{M}\right)$ and $$\begin{aligned} \mathbb{E}\left(\frac{\frac{1}{n}D^{T}\eta}{\frac{1}{n}D^{T}D}\vert\mathcal{M}\right) & = & \frac{1}{\mathbb{P}\left(\mathcal{M}\right)}\mathbb{E}\left[\frac{\frac{1}{n}D^{T}\eta}{\frac{1}{n}D^{T}D}1_{\mathcal{M}}\left(D,\eta\right)\right]\nonumber \\ & = & \frac{1}{\mathbb{P}\left(\mathcal{M}\right)}\mathbb{E}_{D}\left\{ \mathbb{E}_{\eta}\left[\frac{\frac{1}{n}D^{T}\eta}{\frac{1}{n}D^{T}D}1_{\mathcal{M}}\left(D,\eta\right)\vert D\right]\right\} \nonumber \\ & = & \frac{1}{\mathbb{P}\left(\mathcal{M}\right)}\mathbb{E}_{D}\left\{ \frac{\frac{1}{n}\sum_{i=1}^{n}D_{i}\mathbb{E}_{\eta}\left[\eta_{i}1_{\mathcal{M}}\left(D,\eta\right)\vert D\right]}{\frac{1}{n}D^{T}D}\right\} \nonumber \\ & = & 0\label{eq:60}\end{aligned}$$ where $1_{\mathcal{M}}\left(D,\eta\right)=1\left\{ \left(v,\eta\right):\,\hat{I}_{1}=\hat{I}_{2}=\emptyset,\,\left\|\frac{X_{K}^{T}v}{n}\right\|_{\infty}\leq t^{}\right\} $ (recall $X$ is a fixed design); the last line follows from $\frac{1}{n}\sum_{i=1}^{n}D_{i}=0$, the distributional identicalness of $\left(\eta_{i}\right)_{i=1}^{n}$ and that $\mathbb{E}_{\eta}\left[\eta_{i}1_{\mathcal{M}}\left(D,\eta\right)\vert D\right]$ is a constant over $i$s. It remains to bound $\mathbb{E}\left(\frac{\frac{1}{n}D^{T}X_{K}}{\frac{1}{n}D^{T}D}\beta_{K}^{}\vert\mathcal{M}\right)=\mathbb{E}\left(\frac{\frac{1}{n}D^{T}X_{K}}{\frac{1}{n}D^{T}D}\beta_{K}^{}\vert\hat{I}_{2}=\emptyset,\,\mathcal{E}_{t^{}}\right)$. Note that conditioning on $\mathcal{E}_{t^{}}$, $\frac{\frac{1}{n}D^{T}X_{K}}{\frac{1}{n}D^{T}D}\beta_{K}^{}$ is positive by (\[eq:pos\]). Applying a Markov inequality yields $$\mathbb{E}\left(\frac{\frac{1}{n}D^{T}X_{K}}{\frac{1}{n}D^{T}D}\beta_{K}^{}\vert\hat{I}_{2}=\emptyset,\,\mathcal{E}_{t^{}}\right)\geq T_{1}\left(r\right)\mathbb{P}\left(\frac{D^{T}X_{K}}{D^{T}D}\beta_{K}^{}\geq T_{1}\left(r\right)\vert\hat{I}_{2}=\emptyset,\,\mathcal{E}_{t^{}}\right)\geq T_{1}\left(r\right)T_{2}\left(r\right).$$ Combining the result above with (\[eq:60\]) and maximizing over $r\in(0,\,1]$ gives the claim. We now show the case where $ab<0$. The argument is almost similar. In particular, we use $$\begin{aligned} \frac{s}{n}\gamma_{K}^{T}\beta_{K}^{}+\frac{1}{n}v^{T}X_{K}\beta_{K}^{} & \leq & \frac{s}{n}\gamma_{K}^{T}\beta_{K}^{}+\left\|\frac{1}{n}v^{T}X_{K}\right\|_{\infty}\left\|\beta_{K}^{}\right\|_{1}\\ & \leq & -\left(1+\tau\right)\left\|ab\right\|\phi^{-2}\sigma_{\eta}\sigma_{v}\frac{k\log p}{n}<0\end{aligned}$$ and replace (\[eq:pos\]) with $$-\frac{s}{n}\gamma_{K}^{T}\beta_{K}^{}-\frac{1}{n}v^{T}X_{K}\beta_{K}^{}\geq\left(1+\tau\right)\left\|ab\right\|\phi^{-2}\sigma_{\eta}\sigma_{v}\frac{k\log p}{n}.$$ Note that $$\mathbb{E}\left(\alpha^{}-\tilde{\alpha}\vert\mathcal{M}\right)=\mathbb{E}\left(\frac{-\frac{1}{n}D^{T}X_{K}}{\frac{1}{n}D^{T}D}\beta_{K}^{}\vert\mathcal{M}\right)-\mathbb{E}\left(\frac{\frac{1}{n}D^{T}\eta}{\frac{1}{n}D^{T}D}\vert\mathcal{M}\right).$$ So the rest of the proof follows from the argument for the case where $ab>0$. Proposition \[prop:bias\_post\_double\_selection\] {#appendix:bias_post_double_lower_scaling} -------------------------------------------------- Part (i) of Proposition \[prop:bias\_post\_double\_selection\] follows immediately from the proof for Proposition \[prop:bias\_post\_double\_formula\]. It remains to establish part (ii) where $\alpha^{}\neq0$, $\alpha^{}\gamma_{j}^{}\in(0,\,-\beta_{j}^{}]$, $\beta_{j}^{}<0$ for all $j\in K$ (or, $\alpha^{}\gamma_{j}^{}\in[-\beta_{j}^{},\,0)$, $\beta_{j}^{}>0$ for all $j\in K$). Because of these conditions, we have $$\left\|\pi_{j}^{}\right\|=\left\|\beta_{j}^{}+\alpha^{}\gamma_{j}^{}\right\|<\left\|\beta_{j}^{}\right\|\quad\forall j\in K.$$ Note that $\left\|\alpha^{}\right\|\leq\max_{j\in K}\frac{\left\|\beta_{j}^{\ast}\right\|}{\left\|\gamma_{j}^{\ast}\right\|}\asymp\frac{\sigma_{\eta}}{\sigma_{v}}$ and $$\begin{aligned} \left\|\frac{X^{T}u}{n}\right\|_{\infty} & = & \left\|\frac{X^{T}\left(\eta+\alpha^{}v\right)}{n}\right\|_{\infty}\\ & \leq & \left\|\frac{X^{T}\eta}{n}\right\|_{\infty}+\left\|\frac{\alpha^{}X^{T}v}{n}\right\|_{\infty}\\ & \precsim & \sigma_{\eta}\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}}+\frac{\sigma_{\eta}}{\sigma_{v}}\sigma_{v}\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}}\\ & \precsim & \phi^{-1}\sigma_{\eta}\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}}\end{aligned}$$ with probability at least $1-c_{1}^{'}\exp\left(-c_{2}^{'}\log p\right)$. The fact above justifies the choice of $\lambda_{1}$ stated in \[prop:bias\_post\_double\_selection\]. We can then apply (\[eq:nec\]) in Lemma \[prop:fixed\_design\] to show that $\hat{I}_{1}=\emptyset$ with probability at least $1-c_{5}\exp\left(-c_{6}\log p\right)$. Furthermore, under the conditions on $X$ and $\gamma_{K}^{}$, (\[eq:nec\]) in Lemma \[prop:fixed\_design\] implies that $\hat{I}_{2}=\emptyset$ with probability at least $1-c_{0}\exp\left(-c_{0}^{'}\log p\right)$. Therefore, we have $$\mathbb{P}\left(\hat{I}_{1}=\hat{I}_{2}=\emptyset\right)\geq\mathbb{P}\left(\hat{I}_{1}=\emptyset\right)+\mathbb{P}\left(\hat{I}_{2}=\emptyset\right)-1\geq1-c_{1}^{''}\exp\left(-c_{2}^{''}\log p\right).$$ Given $u=\eta+\alpha^{}v$, when $\alpha^{}\neq0$, the event $\left\{ \hat{I}_{1}=\emptyset\right\} $ is not independent of $D$, so $\mathbb{E}\left(\frac{\frac{1}{n}D^{T}X_{K}}{\frac{1}{n}D^{T}D}\beta_{K}^{}\vert E,\,\mathcal{E}_{t^{}}\right)\neq\mathbb{E}\left(\frac{\frac{1}{n}D^{T}X_{K}}{\frac{1}{n}D^{T}D}\beta_{K}^{}\vert\hat{I}_{2}=\emptyset,\,\mathcal{E}_{t^{}}\right)$ (recalling $E=\left\{ \hat{I}_{1}=\hat{I}_{2}=\emptyset\right\} $). Instead of (\[eq:58\]), we apply $$\begin{aligned} \mathbb{P}\left(\mathcal{E}_{t_{2}}^{'}\vert E,\,\mathcal{E}_{t_{1}}\right) & \geq & \mathbb{P}\left(\mathcal{E}_{t_{2}}^{'}\cap\mathcal{E}_{t_{1}}\cap E\right)\\ & \geq & \mathbb{P}\left(\mathcal{E}_{t_{1}}\right)+\mathbb{P}\left(\mathcal{E}_{t_{2}}^{'}\right)+\mathbb{P}\left(E\right)-2\\ & \geq & 1-c_{1}^{''}\exp\left(-c_{2}^{''}\log p\right)-k\exp\left(\frac{-nt_{1}^{2}}{2\frac{s}{n}\sigma_{v}^{2}}\right)\\ & & -\exp\left(\frac{-nt_{2}^{2}}{8\sigma_{v}^{4}}\right),\qquad\textrm{for any }t_{2}\in(0,\,\sigma_{v}^{2}].\end{aligned}$$ The rest of the proof follows from the argument for Proposition \[prop:bias\_post\_double\_formula\] and the bounds above. Proposition \[prop:bias\_post\_double\_selection-1\] ---------------------------------------------------- Note that we have $$\begin{aligned} \left\|\frac{X^{T}u}{n}\right\|_{\infty} & = & \left\|\frac{X^{T}\left(\eta+\alpha^{}v\right)}{n}\right\|_{\infty}\\ & \leq & \left\|\frac{X^{T}\eta}{n}\right\|_{\infty}+\left\|\frac{\alpha^{}X^{T}v}{n}\right\|_{\infty}\\ & \precsim & \sigma_{\eta}\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}}+\left\|\alpha^{}\right\|\sigma_{v}\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}},\end{aligned}$$ which justifies the choice of $\lambda_{1}$ stated in Proposition \[prop:bias\_post\_double\_selection-1\]. For part (i), recall that $\pi_{K}^{}=0_{k}$. By (i) of the intermediate results in “Preliminary” of Section \[appendix a1\], $\mathbb{P}\left(\hat{\pi}=0_{p}\right)=\mathbb{P}\left(\hat{I}_{1}=\emptyset\right)\geq1-c\exp\left(-c^{'}\log p\right)$. Substituting $\beta_{K}^{}=-\alpha^{}\gamma_{K}^{}$ in $\frac{\frac{1}{n}D^{T}X_{K}}{\frac{1}{n}D^{T}D}\beta_{K}^{}$ and following the rest of proof for Proposition \[prop:bias\_post\_double\_selection\] yields the claim. For part (ii), for all $j\in K$, note that $\beta_{j}^{}>-\alpha^{}\gamma_{j}^{}>0$ in case (1) and $\beta_{j}^{}<-\alpha^{}\gamma_{j}^{}<0$ in case (2). Under the conditions on $X$ and $\pi_{K}^{}$, (\[eq:nec\]) in Lemma \[prop:fixed\_design\] implies that $\mathbb{P}\left(\hat{\pi}=0_{p}\right)=\mathbb{P}\left(\hat{I}_{1}=\emptyset\right)\geq1-c\exp\left(-c^{'}\log p\right)$. Substituting $\beta_{K}^{}>-\alpha^{}\gamma_{K}^{}>0$ for case (1) and $\beta_{j}^{}<-\alpha^{}\gamma_{j}^{}<0$ for case (2) in the derivation of the bounds for $\frac{1}{n}D^{T}X_{K}\beta_{K}^{}$ and following the rest of proof for Proposition \[prop:bias\_post\_double\_selection\] yields the claim. Proposition \[prop:bias\_post\_double\_selection-upper\] \[appendix:Proposition-OVB-upper-proof\] ------------------------------------------------------------------------------------------------- Given $\left\{ \hat{I}_{1}=\hat{I}_{2}=\emptyset\right\} $, note that $\left\|\tilde{\alpha}-\alpha^{}\right\|\leq\left\|\frac{\frac{1}{n}D^{T}X_{K}}{\frac{1}{n}D^{T}D}\beta_{K}^{}\right\|+\left\|\frac{\frac{1}{n}D^{T}\eta}{\frac{1}{n}D^{T}D}\right\|$. We make use of the following bound on Chi-Square variables: $$\mathbb{P}\left[\frac{1}{n}\sum_{i=1}^{n}v_{i}^{2}-\mathbb{E}\left(\frac{1}{n}\sum_{i=1}^{n}v_{i}^{2}\right)\leq-\sigma_{v}^{2}r^{'}\right]\leq\exp\left(\frac{-nr^{'2}}{16}\right)\label{eq:41}$$ for all $r^{'}\geq0$. On the event $\left\{ \hat{I}_{1}=\hat{I}_{2}=\emptyset\right\} $, choosing $t_{1}=t^{}=\frac{\left\|\gamma^{}\right\|s}{4n}$ in (\[eq:50\]) and $r^{'}=\frac{1}{2}$ in (\[eq:41\]) yields $$\begin{aligned} \left\|\frac{\frac{1}{n}D^{T}X_{K}}{\frac{1}{n}D^{T}D}\beta_{K}^{}\right\| & \leq & \frac{\frac{s}{n}\left\|\gamma_{K}^{T}\beta_{K}^{}\right\|+\left\|\beta_{K}^{}\right\|_{1}t^{}}{\frac{s}{n}\gamma_{K}^{T}\gamma_{K}^{}-2\left\|\gamma_{K}^{}\right\|_{1}t^{}+\frac{1}{2}\sigma_{v}^{2}}\\ & \leq & \frac{c_{3}\sigma_{\eta}\sigma_{v}\frac{k\log p}{n}}{c_{1}\frac{k\log p}{n}\sigma_{v}^{2}+c_{2}\sigma_{v}^{2}}\\ & \leq & c_{4}\frac{\sigma_{\eta}}{\sigma_{v}}\left(\frac{k\log p}{n}\wedge1\right)\end{aligned}$$ with probability at least $1-c_{5}k\exp\left(-c_{6}\log p\right)-\exp\left(\frac{-n}{64}\right)$. We can also show that $$\begin{aligned} & & \mathbb{P}\left(\left\|\frac{1}{n}D^{T}\eta\right\|\leq t\vert\hat{I}_{1}=\hat{I}_{2}=\emptyset\right)\\ & \geq & \mathbb{P}\left(\left\{ \left(\eta,v\right):\,\left\|\frac{1}{n}D^{T}\eta\right\|\leq t\right\} \cap\left\{ \hat{I}_{1}=\hat{I}_{2}=\emptyset\right\} \right)\\ & \geq & \mathbb{P}\left(\left\|\frac{1}{n}D^{T}\eta\right\|\leq t\right)+\mathbb{P}\left(\hat{I}_{1}=\hat{I}_{2}=\emptyset\right)-1\\ & \geq & 1-c_{5}\exp\left(-c_{6}\log p\right)-\mathbb{P}\left(\left\|\frac{1}{n}D^{T}\eta\right\|>t\right).\end{aligned}$$ Note that $\left\|\frac{1}{n}D^{T}\eta\right\|\leq k\left\|\gamma^{}\right\|\left\|\frac{1}{n}X^{T}\eta\right\|_{\infty}+\left\|\frac{1}{n}v^{T}\eta\right\|$ where $$\begin{aligned} \mathbb{P}\left(\left\|\frac{1}{n}v^{T}\eta\right\|\precsim\sigma_{v}\sigma_{\eta}r\right) & \geq & 1-2\exp\left(-c_{7}nr^{2}\right)\quad\textrm{for any }r\in(0,\,1],\\ \mathbb{P}\left(\left\|\frac{1}{n}X^{T}\eta\right\|_{\infty}\precsim\sigma_{\eta}\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}}\right) & \geq & 1-2\exp\left(-c_{8}\log p\right).\end{aligned}$$ The inequalities above yield $$\begin{aligned} & \mathbb{P}\left\{ \left\|\frac{1}{n}D^{T}\eta\right\|\precsim\left[\left(\sigma_{v}\sigma_{\eta}r\right)\vee\underset{\asymp\sigma_{v}\sigma_{\eta}\frac{k\log p}{n}}{\underbrace{\left(k\left\|\gamma^{}\right\|\sigma_{\eta}\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}}\right)}}\right]\vert\hat{I}_{1}=\hat{I}_{2}=\emptyset\right\} \\ \geq & 1-c_{5}\exp\left(-c_{6}\log p\right)-2\exp\left(-c_{7}nr^{2}\right)-2\exp\left(-c_{8}\log p\right).\end{aligned}$$ We have already shown that, conditioning on $\left\{ \hat{I}_{1}=\hat{I}_{2}=\emptyset\right\} $, $\frac{1}{n}D^{T}D\succsim\left(\frac{k\log p}{n}\vee1\right)\sigma_{v}^{2}$ with probability at least $1-c_{5}k\exp\left(-c_{6}\log p\right)-\exp\left(\frac{-n}{64}\right)$. As a consequence, $$\begin{aligned} & & \mathbb{P}\left\{ \left\|\frac{\frac{1}{n}D^{T}\eta}{\frac{1}{n}D^{T}D}\right\|\precsim\frac{\sigma_{v}\sigma_{\eta}\left(r\vee\frac{k\log p}{n}\right)}{\left(\frac{k\log p}{n}\vee1\right)\sigma_{v}^{2}}\vert\hat{I}_{1}=\hat{I}_{2}=\emptyset\right\} \\ & \geq & \mathbb{P}\left\{ \left\|\frac{1}{n}D^{T}\eta\right\|\precsim\sigma_{v}\sigma_{\eta}\left(r\vee\frac{k\log p}{n}\right)\,\,and\,\,\frac{1}{n}D^{T}D\succsim\left(\frac{k\log p}{n}\vee1\right)\sigma_{v}^{2}\vert\hat{I}_{1}=\hat{I}_{2}=\emptyset\right\} \\ & \geq & \mathbb{P}\left\{ \left\|\frac{1}{n}D^{T}\eta\right\|\precsim\sigma_{v}\sigma_{\eta}\left(r\vee\frac{k\log p}{n}\right)\vert\hat{I}_{1}=\hat{I}_{2}=\emptyset\right\} \\ & & +\mathbb{P}\left\{ \frac{1}{n}D^{T}D\succsim\left(\frac{k\log p}{n}\vee1\right)\sigma_{v}^{2}\vert\hat{I}_{1}=\hat{I}_{2}=\emptyset\right\} -1\\ & \geq & 1-c_{9}k\exp\left(-c_{10}\log p\right)-c_{12}\exp\left(-c_{11}nr^{2}\right).\end{aligned}$$ Putting the pieces above together yields $$\mathbb{P}\left(\left\|\tilde{\alpha}-\alpha^{}\right\|\leq\overline{OVB}\vert\hat{I}_{1}=\hat{I}_{2}=\emptyset\right)\geq1-c_{1}^{'}k\exp\left(-c_{2}^{'}\log p\right)-c_{4}^{'}\exp\left(-c_{3}^{'}nr^{2}\right)$$ where $\overline{OVB}\asymp\max\left\{ \frac{\sigma_{\eta}}{\sigma_{v}}\left(\frac{k\log p}{n}\wedge1\right),\,\frac{\sigma_{v}\sigma_{\eta}\left(r\vee\frac{k\log p}{n}\right)}{\left(\frac{k\log p}{n}\vee1\right)\sigma_{v}^{2}}\right\} $. Proposition \[prop:bias\_post\_double\_selection-upper-1\] {#appendix:Proposition-OVB-upper-proof-1} ----------------------------------------------------------- For part (i), substituting $\beta_{K}^{}=-\alpha^{}\gamma_{K}^{}$ in $\frac{\frac{1}{n}D^{T}X_{K}}{\frac{1}{n}D^{T}D}\beta_{K}^{}$ and following the rest of proof for Proposition \[prop:bias\_post\_double\_selection-upper\] yields the claim. For part (ii), note that $\left\|\beta_{j}^{}\right\|\leq\left\|\pi_{j}^{}\right\|+\left\|\alpha^{}\gamma_{j}^{}\right\|\precsim\left(\sigma_{\eta}+\left\|\alpha^{}\right\|\sigma_{v}\right)\sqrt{\frac{n}{s}}\sqrt{\frac{\log p}{n}}$ for all $j\in K$. Substituting $\left\|\beta_{j}^{}\right\|\precsim\left(\sigma_{\eta}+\left\|\alpha^{}\right\|\sigma_{v}\right)\sqrt{\frac{n}{s}}\sqrt{\frac{\log p}{n}}$ in the derivation of the upper bound for $\left\|\frac{\frac{1}{n}D^{T}X_{K}}{\frac{1}{n}D^{T}D}\beta_{K}^{}\right\|$ and following the rest of proof for Proposition \[prop:bias\_post\_double\_selection-upper\] yields the claim. Debiased Lasso =============== In this section, we present theoretical and simulation results on the OVB of the debiased Lasso proposed by @vandergeer2014asymptotically. Theoretical results {#sec:bias_debiased_lasso} ------------------- The idea of debiased Lasso is to start with an initial Lasso estimate $\hat{\theta}=\left(\hat{\alpha},\,\hat{\beta}\right)$ of $\theta^{}=\left(\alpha^{},\,\beta^{}\right)$ in equation (\[eq:main-y\]), where $$\left(\hat{\alpha},\,\hat{\beta}\right)\in\textrm{arg}\min_{\alpha\in\mathbb{R},\beta\in\mathbb{R}^{p}}\frac{1}{2n}\left\|Y-D\alpha-X\beta\right\|_{2}^{2}+\lambda_{1}\left(\left\|\alpha\right\|+\left\|\beta\right\|_{1}\right).\label{eq:21-2}$$ Given the initial Lasso estimator $\hat{\alpha}$, the debiased Lasso adds a correction term to $\hat{\alpha}$ to reduce the bias introduced by regularization. In particular, the debiased Lasso takes the form $$\tilde{\alpha}=\hat{\alpha}+\frac{\hat{\Omega}_{1}}{n}\sum_{i=1}^{n}Z_{i}^{T}\left(Y_{i}-Z_{i}\hat{\theta}\right),\label{eq:5-3}$$ where $Z_{i}=\left(D_{i},\,X_{i}\right)$ and $\hat{\Omega}_{1}$ is the first row of $\hat{\Omega}$, which is an approximate inverse of $\frac{1}{n}Z^{T}Z$, $Z=\left\{ Z_{i}\right\} _{i=1}^{n}$. Several different strategies have been proposed for constructing the approximate inverse $\hat{\Omega}$; see, for example, @javanmard2014confidence, @vandergeer2014asymptotically, and @Zhang_Zhang. We will focus on the widely used method proposed by @vandergeer2014asymptotically, which sets $$\begin{aligned} \hat{\Omega}_{1} & := & \hat{\tau}_{1}^{-2}\left(\begin{array}{cccc} 1 & -\hat{\gamma}_{1} & \cdots & -\hat{\gamma}_{p}\end{array}\right),\\ \hat{\tau}_{1}^{2} & := & \frac{1}{n}\left\|D-X\hat{\gamma}\right\|_{2}^{2}+\lambda_{2}\left\|\hat{\gamma}\right\|_{1},\end{aligned}$$ where $\hat{\gamma}$ is defined in (\[eq:las-2\]). \[prop:bias\_debiased\_lasso\] Let Assumption \[ass:inference\_normality\] and part (i) and (iii) of Assumption \[ass:inference\_sparsity\_incoherence\] hold. Suppose: with probability at least $1-\kappa$, $\left\Vert \left(Z_{-K}^{T}X_{K}\right)\left(X_{K}^{T}X_{K}\right)^{-1}\right\Vert _{\infty}\leq1-\frac{\phi}{2}$ for some $\phi\in(0,\,1]$ such that $\phi^{-1}\precsim1$, where $Z_{-K}$ denotes the columns in $Z=\left(D,\,X\right)$ excluding $X_{K}$; the regularization parameters in (\[eq:las-2\]) and (\[eq:21-2\]) are chosen in a similar fashion as in Lemma \[prop:fixed\_design\] such that $\lambda_{1}\asymp\phi^{-1}\left(\sqrt{\frac{s}{n}}\vee\sigma_{v}\right)\sigma_{\eta}\sqrt{\frac{\log p}{n}}$ and $\lambda_{2}\asymp\phi^{-1}\sigma_{v}\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}}$; for all $j\in K$, $\left\|\beta_{j}^{}\right\|\leq\frac{\lambda_{1}n}{2s}$ and $\left\|\gamma_{j}^{}\right\|\leq\frac{\lambda_{2}n}{2s}$, but $\left\|\beta_{j}^{}\right\|\asymp\left[\sqrt{\frac{n}{s}}\vee\frac{n\sigma_{v}}{s}\right]\sigma_{\eta}\sqrt{\frac{\log p}{n}}$ and $\left\|\gamma_{j}^{}\right\|\asymp\sigma_{v}\sqrt{\frac{n}{s}}\sqrt{\frac{\log p}{n}}$. Let us consider $\tilde{\alpha}$ obtained from (\[eq:5-3\]). If $\alpha^{}=0$, then there exist positive universal constants $c^{\dagger},c_{1},c_{2},c_{3},c^{},c_{0}^{}$ such that $$\left\|\mathbb{E}\left(\tilde{\alpha}-\alpha^{}\vert\mathcal{M}^{'}\right)\right\|\geq c^{\dagger}\frac{\sigma_{\eta}}{\sigma_{v}}\left(\frac{k\log p}{n}\wedge1\right)\left[1-2\kappa-c_{1}k\exp\left(-c_{2}\log p\right)-\exp\left(-c_{3}n\right)\right],$$ where $\mathcal{M}^{'}$ is an event with $\mathbb{P}\left(\mathcal{M}^{'}\right)\geq1-2\kappa-c^{}k\exp\left(-c_{0}^{}\log p\right)$. \[prop:bias\_debiased\_lasso-upper\] Let Assumption \[ass:inference\_normality\] and part (i) and (iii) of Assumption \[ass:inference\_sparsity\_incoherence\] hold. Suppose: with probability at least $1-\kappa$, $\left\Vert \left(Z_{-K}^{T}X_{K}\right)\left(X_{K}^{T}X_{K}\right)^{-1}\right\Vert _{\infty}\leq1-\frac{\phi}{2}$ for some $\phi\in(0,\,1]$ such that $\phi^{-1}\precsim1$, where $Z_{-K}$ denotes the columns in $Z=\left(D,\,X\right)$ excluding $X_{K}$; the regularization parameters in (\[eq:las-2\]) and (\[eq:21-2\]) are chosen in a similar fashion as in Proposition \[prop:bias\_debiased\_lasso\] such that $\lambda_{1}\asymp\phi^{-1}\left(\sqrt{\frac{s}{n}}\vee\sigma_{v}\right)\sigma_{\eta}\sqrt{\frac{\log p}{n}}$ and $\lambda_{2}\asymp\phi^{-1}\sigma_{v}\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}}$; for all $j\in K$, $\gamma_{j}^{}=\gamma^{}$, $\left\|\beta_{j}^{}\right\|\leq\frac{\lambda_{1}n}{2s}$ and $\left\|\gamma^{}\right\|\leq\frac{\lambda_{2}n}{2s}$, but $\left\|\beta_{j}^{}\right\|\asymp\left[\sqrt{\frac{n}{s}}\vee\frac{n\sigma_{v}}{s}\right]\sigma_{\eta}\sqrt{\frac{\log p}{n}}$ and $\left\|\gamma^{}\right\|\asymp\sigma_{v}\sqrt{\frac{n}{s}}\sqrt{\frac{\log p}{n}}$. Let us consider $\tilde{\alpha}$ obtained from (\[eq:5-3\]). If $\alpha^{}=0$, then there exist positive universal constants $c_{1},c_{2},c_{3},c_{4},c^{},c_{0}^{}$ such that $$\mathbb{P}\left(\left\|\tilde{\alpha}-\alpha^{}\right\|\leq\overline{OVB}\vert \hat{\theta}=0_{p+1},\,\hat{\gamma}=0_{p} \right)\geq1-c_{1}k\exp\left(-c_{2}\log p\right)-c_{3}\exp\left(-c_{4}nr^{2}\right)$$ where $\mathbb{P}\left( \hat{\theta}=0_{p+1},\,\hat{\gamma}=0_{p} \right)\geq1-2\kappa-c^{}k\exp\left(-c_{0}^{}\log p\right)$ and $$\overline{OVB}\asymp\max\left\{ \frac{\sigma_{\eta}}{\sigma_{v}}\left(\frac{k\log p}{n}\wedge1\right),\,\frac{\sigma_{v}\sigma_{\eta}\left(r\vee\frac{k\log p}{n}\right)}{\left(\frac{k\log p}{n}\vee1\right)\sigma_{v}^{2}}\right\}$$ for any $r\in(0,\,1]$. One can show that a population version of the mutual incoherence condition, $\left\Vert \left[\mathbb{E}\left(Z_{-K}^{T}\right)X_{K}\right]\left(X_{K}^{T}X_{K}\right)^{-1}\right\Vert _{\infty}=1-\phi$, implies $\left\Vert \left(Z_{-K}^{T}X_{K}\right)\left(X_{K}^{T}X_{K}\right)^{-1}\right\Vert _{\infty}\leq1-\frac{\phi}{2}$ with high probability (that is, $\kappa$ is small and vanishes polynomially in $p$). For example, we can apply (\[eq:77\]) in Lemma \[lem:A3\] with slight notational changes. The event $\mathcal{M}^{'}$ in Proposition \[prop:bias\_debiased\_lasso\] is the intersection of $\left\{ \hat{\theta}=0_{p+1},\,\hat{\gamma}=0_{p}\right\}$ and an additional event, both of which occur with high probabilities. The additional event is needed in our analyses for technical reasons. See Appendix \[sec:appendix\_debiased\_lasso\] for details. Proof for Propositions \[prop:bias\_debiased\_lasso\] and \[prop:bias\_debiased\_lasso-upper\] {#sec:appendix_debiased_lasso} ---------------------------------------------------------------------------------------------- Under the conditions in Proposition \[prop:bias\_debiased\_lasso\], (\[eq:nec\]) in Lemma \[prop:fixed\_design\] implies that $\hat{\gamma}=0_{p}$ with probability at least $1-c_{0}\exp\left(-c_{0}^{'}\log p\right)$. Conditioning on this event, $\hat{\Omega}_{1}=\left(\frac{1}{n}D^{T}D\right)^{-1}e_{1}$ where $e_{1}=\left(\begin{array}{cccc} 1 & 0 & \cdots & 0\end{array}\right)$. If $\alpha^{}=0$, under the conditions in Proposition \[prop:bias\_debiased\_lasso\], we show that $\hat{\theta}=0_{p+1}$ with probability at least $1-2\kappa-c_{5}\exp\left(-c_{6}\log p\right)$. To achieve this goal, we slightly modify the argument for (\[eq:nec\]) in Lemma \[prop:fixed\_design\] by replacing (\[eq:event\]) with $\mathcal{E}=\mathcal{E}_{1}\cap\mathcal{E}_{2}$, where $$\begin{aligned} \mathcal{E}_{1} & = & \left\{ \left\|\frac{Z^{T}\eta}{n}\right\|_{\infty}\precsim\phi^{-1}\left(\sqrt{\frac{s}{n}}\vee\sigma_{v}\right)\sigma_{\eta}\sqrt{\frac{\log p}{n}}\right\} ,\\ \mathcal{E}_{2} & = & \left\{ \left\Vert \left(Z_{-K}^{T}X_{K}\right)\left(X_{K}^{T}X_{K}\right)^{-1}\right\Vert _{\infty}\leq1-\frac{\phi}{2}\right\} ,\end{aligned}$$ and $Z_{-K}$ denotes the columns in $Z$ excluding $X_{K}$. Note that by (\[eq:A37-1\]), $\mathbb{P}\left(\mathcal{E}_{1}\right)\geq1-c_{1}^{'}\exp\left(-c_{2}^{'}\log p\right)$ and therefore, $\mathbb{P}\left(\mathcal{E}\right)\geq1-\kappa-c_{1}^{'}\exp\left(-c_{2}^{'}\log p\right)$. We then follow the argument used in the proof for Lemma \[prop:fixed\_design\] to show $\mathbb{P}\left(E_{1}\cap\mathcal{E}\right)=0$ and $\mathbb{P}\left(E_{2}\cap\mathcal{E}\right)=0$, where $$\begin{aligned} E_{1} & = & \left\{ \textrm{sgn}\left(\hat{\beta}_{j}\right)=-\textrm{sgn}\left(\beta_{j}^{}\right),\,\textrm{for some }j\in K\right\} ,\\ E_{2} & = & \left\{ \textrm{sgn}\left(\hat{\beta}_{j}\right)=\textrm{sgn}\left(\beta_{j}^{}\right),\,\textrm{for some }j\in K\right\} .\end{aligned}$$ Moreover, conditioning on $\mathcal{E}$, $\hat{\alpha}=0$ and $\hat{\beta}_{K^{c}}=0_{p-k}$. Putting these facts together yield the claim that $\hat{\theta}=0_{p+1}$ with probability at least $1-2\kappa-c_{5}\exp\left(-c_{6}\log p\right)$. Letting $E=\left\{ \hat{\theta}=0_{p+1},\,\hat{\gamma}=0_{p}\right\} $ with $\mathbb{P}\left(E\right)\geq1-2\kappa-c_{1}\exp\left(-c_{2}\log p\right)$ and recalling the event $\mathcal{E}_{t^{}}$ in the proof for Proposition \[prop:bias\_post\_double\_formula\], we can then show $$\begin{aligned} \mathbb{E}\left(\tilde{\alpha}-\alpha^{}\vert\mathcal{M}^{'}\right) & = & \frac{1}{n}\mathbb{E}\left(\hat{\Omega}_{1}Z^{T}\eta\vert\mathcal{M}^{'}\right)+\mathbb{E}\left[\frac{D^{T}X_{K}}{D^{T}D}\left(\beta_{K}^{}-\hat{\beta}_{K}\right)\vert\mathcal{M}^{'}\right]\\ & = & \mathbb{E}\left(\frac{\frac{1}{n}D^{T}\eta}{\frac{1}{n}D^{T}D}\vert\mathcal{M}^{'}\right)+\mathbb{E}\left(\frac{D^{T}X_{K}}{D^{T}D}\beta_{K}^{}\vert\mathcal{M}^{'}\right)\\ & = & \mathbb{E}\left(\frac{D^{T}X_{K}}{D^{T}D}\beta_{K}^{}\vert\mathcal{M}^{'}\right)\end{aligned}$$ where $\mathcal{M}^{'}=E\cap\mathcal{E}_{t^{}}$ such that $\mathbb{P}\left(\mathcal{M}^{'}\right)\geq1-2\kappa-c_{3}^{}k\exp\left(-c_{4}^{}\log p\right)$ and the last line follows from the argument used to show (\[eq:60\]). The rest of argument is similar to what is used in showing Proposition \[prop:bias\_post\_double\_formula\]. However, because (\[eq:21-2\]) involves $D$, $\mathbb{E}\left(\frac{\frac{1}{n}D^{T}X_{K}}{\frac{1}{n}D^{T}D}\beta_{K}^{}\vert E,\,\mathcal{E}_{t^{}}\right)\neq\mathbb{E}\left(\frac{\frac{1}{n}D^{T}X_{K}}{\frac{1}{n}D^{T}D}\beta_{K}^{}\vert\hat{\gamma}=0_{p},\,\mathcal{E}_{t^{}}\right)$. Instead of (\[eq:50\]) and (\[eq:58\]), we apply $$\begin{aligned} \mathbb{P}\left(\mathcal{E}_{t_{2}}^{'}\vert E,\,\mathcal{E}_{t_{1}}\right) & \geq & \mathbb{P}\left(\mathcal{E}_{t_{2}}^{'}\cap\mathcal{E}_{t_{1}}\cap E\right)\\ & \geq & \mathbb{P}\left(\mathcal{E}_{t_{1}}\right)+\mathbb{P}\left(\mathcal{E}_{t_{2}}^{'}\right)+\mathbb{P}\left(E\right)-2\\ & \geq & 1-2\kappa-c_{1}\exp\left(-c_{2}\log p\right)-k\exp\left(\frac{-nt_{1}^{2}}{2\frac{s}{n}\sigma_{v}^{2}}\right)\\ & & -\exp\left(\frac{-nt_{2}^{2}}{8\sigma_{v}^{4}}\right),\qquad\textrm{for any }t_{2}\in(0,\,\sigma_{v}^{2}].\end{aligned}$$ Consequently, we have the claim in Proposition \[prop:bias\_debiased\_lasso\]. Following the argument used to show Proposition \[prop:bias\_post\_double\_selection-upper\], we also have the claim in Proposition \[prop:bias\_debiased\_lasso-upper\]. Simulations evidence -------------------- Here we evaluate the performance of the debiased Lasso proposed by @vandergeer2014asymptotically based on the simulation setting of Section \[sec:simulation\_evidence\]. We use cross-validation to choose the regularization parameters as it is the most commonly-used method in this literature. Figures \[fig:db\_bias\_std\] and \[fig:db\_cov\] present the results. Debiased Lasso exhibits substantial biases (relative to the standard deviation) and under-coverage for all values of $\sigma_x$ and its performance is very sensitive to the regularization choice. The performance tends to get worse as $\sigma_x$ increases because the bias decays slower than the standard deviation. A comparison to the results in Section \[sec:simulation\_evidence\] shows that post double Lasso performs better than debiased Lasso.[^27] \[fig:db\_bias\_std\] \[fig:db\_cov\] Additional simulations for post double Lasso {#app:additional_simulations} ============================================ In the main text, we consider a setting with $k=5$, normally distributed control variables, normally distributed homoscedastic errors terms, and $\alpha^\ast=0$. Here we provide additional simulation evidence based on a more general model:[^28] $$\begin{aligned} Y_{i} & = & D_{i}\alpha^{}+X_{i}\beta^{}+\sigma_{y}(D_{i},X_{i})\eta_{i},\label{eq:dgp_illustration_inference1_general}\\ D_{i} & = & X_{i}\gamma^{}+\sigma_{d}(X_{i})v_{i},\label{eq:dgp_illustration_inference2_general}\end{aligned}$$ where $\eta_{i}$ and $v_{i}$ are independent of each other and $\left\{ X_{i},\eta_{i},v_{i}\right\} _{i=1}^{n}$ consists of i.i.d. entries. The object of interest is $\alpha^{}$. We set $n=500$, $p=200$, and consider a sparse setting where $\beta^{}=\gamma^\ast=(\underbrace{1,\dots,1}_{k},0,\dots,0)^{T}$. We consider six DGPs that differ with respect to $k$, the distributions of $X_i$, $\eta_{i}$, and $v_{i}$, the specifications of $\sigma_{y}(D_{i},X_{i})$ and $\sigma_{d}(X_{i})$, as well as $\alpha^\ast$. For DGP A1, we do not report the results for $\sigma_x< 0.2$ due to numerical issues with the computation of standard errors. The results are based on 1,000 simulation repetitions. $X_i$ $k$ $\sigma_y(D_i,X_i)$ $\sigma_d(X_i)$ $\eta_i$ $v_i$ $\alpha^\ast$ -------- ------------------------------------------------------------------- ----- ------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------- ----------------------------- ------------------------------ --------------- -- -- DGP A1 Indep. $\text{Bern}\left(\frac{1-\sqrt{1-4\sigma_x^2}}{2}\right)$ 5 1 1 $ \mathcal{N}(0,1)$ $ \mathcal{N}(0,1)$ 0 DGP A2 $\mathcal{N}(0,\sigma_x^2I_p)$ 5 1 1 $\frac{t(5)}{\sqrt{(5/3)}}$ $ \frac{t(5)}{\sqrt{(5/3)}}$ 0 DGP A3 $\mathcal{N}(0,\sigma_x^2I_p)$ 5 $\sqrt{\frac{(1+D_{i}\alpha^{}+X_{i}\beta^{})^{2}}{\frac{1}{n}\sum_{i}(1+D_{i}\alpha^{}+X_{i}\beta^{})^{2}}}$ $\sqrt{\frac{(1+X_{i}\gamma^{})^{2}}{\frac{1}{n}\sum_{i}(1+X_{i}\gamma^{})^{2}}}$ $ \mathcal{N}(0,1)$ $ \mathcal{N}(0,1)$ 0 DGP A4 $\mathcal{N}(0,\sigma_x^2I_p)$ 10 1 1 $ \mathcal{N}(0,1)$ $ \mathcal{N}(0,1)$ 0 DGP A5 $\mathcal{N}(0,\sigma_x^2I_p)$ 5 1 1 $ \mathcal{N}(0,1)$ $ \mathcal{N}(0,1)$ 1 DGP A6 $\mathcal{N}(0,\sigma_x^2I_p)$ 5 1 1 $ \mathcal{N}(0,1)$ $ \mathcal{N}(0,1)$ -1 Figures \[fig:dml\_bias\_std\_app\]–\[fig:dml\_ci\_app\] present the results. The two most important determinants of the performance of post double Lasso are $k$ and $\alpha^{}$. To see why $\alpha^\ast$ is important, recall that the reduced form parameter and the error term in the first step of post double Lasso (i.e., program (\[eq:las-1\])) are $\pi^{}=\alpha^{}\gamma^{}+\beta^{}$ and $u_{i}=\eta_{i}+\alpha^{}v_{i}$. This implies that the magnitude of $\pi^{}$ as well as the variance of $u_i$ depend on $\alpha^{}$. Consequently, the selection performance of Lasso in the first step is directly affected by $\alpha^{}$. In the extreme case where $\alpha^{}$ is such that $\pi^{}=0_{p}$, Lasso does not select any controls with high probability if the regularization parameter is chosen according to the standard recommendations. The simulation results further show that there is no practical recommendation for choosing the regularization parameters. While $\lambda_{\text{min}}$ leads to the best performance when $\alpha^\ast=0$, this choice can yield poor performances when $\alpha^\ast> 0$. Finally, across all DGPs, OLS outperforms post double Lasso in terms of bias and coverage accuracy, but yields somewhat wider confidence intervals. \[fig:dml\_bias\_std\_app\] \[fig:dml\_cov\_app\] \[fig:dml\_ci\_app\] Theoretical results for random designs \[sec:Random-design\] ============================================================ Results ------- In this section, we provide some results in Lemma \[prop:random\_design\] for the Lasso with a random design $X$. The necessary result on the Lasso’s inclusion established in Lemma \[prop:random\_design\] can be adopted in a similar fashion as in Propositions \[prop:bias\_post\_double\_selection\]-\[prop:bias\_post\_double\_selection-upper-1\] and \[prop:bias\_debiased\_lasso\]-\[prop:bias\_debiased\_lasso-upper\] to establish the OVBs. We make the following assumption about (\[eq:1\]). \[ass:random\_design\] Each row of $X$ is sampled independently; for all $i=1,\dots,n$ and $j=1,\dots,p$, $\sup_{r\geq1}r^{-\frac{1}{2}}\left(\mathbb{E}\left\|X_{ij}\right\|^{r}\right)^{\frac{1}{r}}\leq\alpha<\infty$; for any unit vector $a\in\mathbb{R}^{k}$ and $i=1,\dots,n$, $\sup_{r\geq1}r^{-\frac{1}{2}}\left(\mathbb{E}\left\|a^{T}X_{i,K}^{T}\right\|^{r}\right)^{\frac{1}{r}}\leq\tilde{\alpha}<\infty$, where $X_{i,K}$ is the $i$th row of $X_{K}$ and $K=\left\{ j:\,\theta_{j}^{}\neq0\right\} $. Moreover, the error terms $\varepsilon_{1},\dots,\varepsilon_{n}$ are independent such that $\sup_{r\geq1}r^{-\frac{1}{2}}\left(\mathbb{E}\left\|\varepsilon_{i}\right\|^{r}\right)^{\frac{1}{r}}\leq\sigma<\infty$ and $\mathbb{E}\left(X_{i}\varepsilon_{i}\right)=0_{p}$ for all $i=1,\dots,n$. Assumption \[ass:random\_design\] is known as the sub-Gaussian tail condition defined in @vershynin2012. Examples of sub-Gaussian variables include Gaussian mixtures and distributions with bounded support. The first and last part of Assumption \[ass:random\_design\] imply that $X_{ij},j=1,\dots,p,$ and $\varepsilon_{i}$ are sub-Gaussian variables and is used in deriving the lower bounds on the regularization parameters. The second part of Assumption \[ass:random\_design\] is only used to establish some eigenvalue condition on $\frac{X_{K}^{T}X_{K}}{n}$. \[ass:sparsity\_incoherence\_random\_design\] The following conditions are satisfied: (i) $\theta^{}$ is exactly sparse with at most $k$ non-zero coefficients and $K\neq\emptyset$; (ii) $$\left\Vert \left[\mathbb{E}\left(X_{K^{c}}^{T}X_{K}\right)\right]\left[\mathbb{E}\left(X_{K}^{T}X_{K}\right)\right]^{-1}\right\Vert _{\infty}=1-\phi\label{eq:14-1}$$ for some $\phi\in(0,\,1]$ such that $\phi^{-1}\precsim1$; (iii) $\mathbb{E}\left(X_{ij}\right)=0$ for all $j\in K$ and $\mathbb{E}\left(X_{j}^{T}X_{j}\right)\leq s$ for all $j=1,\dots,p$; (iv) $$\begin{aligned} \max\left\{ \frac{\phi}{12(1-\phi)k^{\frac{3}{2}}},\,\frac{\phi}{6k^{\frac{3}{2}}},\,\frac{\phi}{k}\right\} \sqrt{\frac{\log p}{n}} & \leq & \alpha^{2}\quad\textrm{if }\phi\in\left(0,\,1\right),\label{eq:7-2}\\ \max\left\{ \frac{1}{6k^{\frac{3}{2}}},\,\frac{1}{k}\right\} \sqrt{\frac{\log p}{n}} & \leq & \alpha^{2}\quad\textrm{if }\phi=1,\label{eq:7-4}\\ \max\left\{ 2\tilde{\alpha}^{2},\,12\alpha^{2},\,1\right\} \sqrt{\frac{\log p}{n}} & \leq & \lambda_{\min}\left(\mathbb{E}\left[\frac{1}{n}X_{K}^{T}X_{K}\right]\right).\label{eq:7-3} \end{aligned}$$ Part (iv) of Assumption \[ass:sparsity\_incoherence\_random\_design\] is imposed to ensure that $$\begin{aligned} \left\Vert \left(\frac{1}{n}X_{K}^{T}X_{K}\right)^{-1}-\left[\mathbb{E}\left(\frac{1}{n}X_{K}^{T}X_{K}\right)\right]^{-1}\right\Vert _{\infty} & \precsim & \frac{1}{\lambda_{\min}\left(\mathbb{E}\left[\frac{1}{n}X_{K}^{T}X_{K}\right]\right)},\\ \left\Vert \frac{1}{n}X_{K^{c}}^{T}X_{K}\left(\frac{1}{n}X_{K}^{T}X_{K}\right)^{-1}\right\Vert _{\infty} & \leq & 1-\frac{\phi}{2},\end{aligned}$$ with high probability. To gain some intuition for , let us further assume $k\asymp1$, $X_{i}$ is normally distributed for all $i=1,\dots,n$, and $\mathbb{E}\left(X_{K}^{T}X_{K}\right)$ is a diagonal matrix with the diagonal entries $\mathbb{E}\left(X_{j}^{T}X_{j}\right)=s\neq0$. As a result, $\tilde{\alpha}=\alpha\asymp\sqrt{\frac{s}{n}}$ by the definition of a sub-Gaussian variable (e.g., @vershynin2012) and essentially require $\sqrt{\frac{\log p}{n}}\precsim\frac{s}{n}$. Given $$\mathbb{P}\left(\left\|\frac{X^{T}\varepsilon}{n}\right\|_{\infty}\geq t\right)\leq2\exp\left(\frac{-nt^{2}}{c_{0}\sigma^{2}\alpha^{2}}+\log p\right).\label{eq:38}$$ and $\lambda\geq\frac{c\alpha\sigma\left(2-\frac{\phi}{2}\right)}{\phi}\sqrt{\frac{\log p}{n}}$ for some sufficiently large universal constant $c>0$, we have $$\lambda\geq2\left\|\frac{X^{T}\varepsilon}{n}\right\|_{\infty}\label{eq:lambda-1}$$ with probability at least $1-c^{'}\exp\left(-c^{''}\log p\right)$. Define the following events $$\begin{aligned} \mathcal{E}_{1} & = & \left\{ \left\|\frac{X^{T}\varepsilon}{n}\right\|_{\infty}\precsim\frac{\alpha\sigma\left(2-\frac{\phi}{2}\right)}{\phi}\sqrt{\frac{\log p}{n}}\right\} ,\\ \mathcal{E}_{2} & = & \left\{ \lambda_{\max}(\hat{\Sigma}_{KK})\leq\frac{3}{2}\lambda_{\max}(\Sigma_{KK})\right\} ,\\ \mathcal{E}_{3} & = & \left\{ \left\Vert \left(\frac{1}{n}X_{K}^{T}X_{K}\right)^{-1}-\left[\mathbb{E}\left(\frac{1}{n}X_{K}^{T}X_{K}\right)\right]^{-1}\right\Vert _{\infty}\precsim\frac{1}{\lambda_{\min}\left(\mathbb{E}\left[\frac{1}{n}X_{K}^{T}X_{K}\right]\right)}\right\} ,\\ \mathcal{E}_{4} & = & \left\{ \left\Vert \frac{1}{n}X_{K^{c}}^{T}X_{K}\left(\frac{1}{n}X_{K}^{T}X_{K}\right)^{-1}\right\Vert _{\infty}\leq1-\frac{\phi}{2}\right\} .\end{aligned}$$ By (\[eq:38\]), $\mathbb{P}\left(\mathcal{E}_{1}\right)\geq1-c^{'}\exp\left(-c^{''}\log p\right)$; by (\[eq:S7\]), $\mathbb{P}\left(\mathcal{E}_{2}\right)\geq1-c_{1}^{'}\exp\left(-c_{1}^{''}\log p\right)$; by (\[eq:S4\]), $\mathbb{P}\left(\mathcal{E}_{3}\right)\geq1-c_{2}^{'}\exp\left(-c_{2}^{''}\left(\frac{\log p}{k^{3}}\right)\right)$; by (\[eq:77\]), $\mathbb{P}\left(\mathcal{E}_{4}\right)\geq1-c_{3}^{''}\exp\left(-b\left(\frac{\log p}{k^{3}}\right)\right)$, where $b$ is some positive constant that only depends on $\phi$ and $\alpha$. \[prop:random\_design\] Let Assumptions \[ass:random\_design\] and \[ass:sparsity\_incoherence\_random\_design\] hold. We solve the Lasso (\[eq:las\]) with $\lambda\geq\frac{c\alpha\sigma\left(2-\frac{\phi}{2}\right)}{\phi}\sqrt{\frac{\log p}{n}}$ for some sufficiently large universal constant $c>0$. Suppose $\mathbb{E}\left[X_{K}^{T}X_{K}\right]$ is a positive definite matrix. \(i) Then, conditioning on $\mathcal{E}_{1}\cap\mathcal{E}_{4}$ (which holds with probability at least $1-c_{1}\exp\left(-b\frac{\log p}{k^{3}}\right)$), (\[eq:las\]) has a unique optimal solution $\hat{\theta}$ such that $\hat{\theta}_{j}=0$ for $j\notin K$. \(ii) With probability at least $1-c_{1}\exp\left(-b\frac{\log p}{k^{3}}\right)$, $$\left\|\hat{\theta}_{K}-\theta_{K}^{}\right\|_{2}\leq\frac{3\lambda\sqrt{k}}{\lambda_{\min}\left(\mathbb{E}\left[\frac{1}{n}X_{K}^{T}X_{K}\right]\right)}\label{eq:5gen-1}$$ where $\theta_{K}=\left\{ \theta_{j}\right\} _{j\in K}$ and $b$ is some positive constant that only depends on $\phi$ and $\alpha$; if $\mathbb{P}\left(\left\{ \textrm{supp}(\hat{\theta})=K\right\} \cap\mathcal{E}_{1}\cap\mathcal{E}_{2}\right)>0$, conditioning on $\left\{ \textrm{supp}(\hat{\theta})=K\right\} \cap\mathcal{E}_{1}\cap\mathcal{E}_{2}$, we must have $$\left\|\hat{\theta}_{K}-\theta_{K}^{}\right\|_{2}\geq\frac{\lambda\sqrt{k}}{3\lambda_{\max}\left(\mathbb{E}\left[\frac{1}{n}X_{K}^{T}X_{K}\right]\right)}\geq\frac{\lambda\sqrt{k}}{3\sum_{j\in K}\left(\mathbb{E}\left[\frac{1}{n}X_{j}^{T}X_{j}\right]\right)}.\label{eq:4gen-1}$$ \(iii) If $\mathbb{E}\left(X_{K}^{T}X_{K}\right)$ is a diagonal matrix with the diagonal entries $\mathbb{E}\left(X_{j}^{T}X_{j}\right)=s\neq0$, then $$\left\|\hat{\theta}_{j}-\theta_{j}^{}\right\|\leq\frac{7\lambda n}{4s}\qquad\forall j\in K\label{eq:5-1}$$ with probability at least $1-c_{1}\exp\left(-b\frac{\log p}{k^{3}}\right)$; if $\mathbb{P}\left(\left\{ \hat{\theta}_{j}\neq0,\,j\in K\right\} \cap\mathcal{E}_{1}\cap\mathcal{E}_{3}\cap\mathcal{E}_{4}\right)>0$, conditioning on $\left\{ \hat{\theta}_{j}\neq0,\,j\in K\right\} \cap\mathcal{E}_{1}\cap\mathcal{E}_{3}\cap\mathcal{E}_{4}$, we must have $$\left\|\hat{\theta}_{j}-\theta_{j}^{}\right\|\geq\frac{\lambda n}{4s}\geq c_{0}\frac{\sigma}{\phi}\sqrt{\frac{n}{s}}\sqrt{\frac{\log p}{n}}.\label{eq:4-1}$$ \(iv) Suppose $K=\left\{ 1\right\} $ and $\mathbb{E}\left(X_{1}^{T}X_{1}\right)=s\neq0$. If $$\left\|\theta_{1}^{}\right\|\leq\frac{\lambda n}{4s},\label{eq:min-1}$$ then we must have $$\mathbb{P}\left(\hat{\theta}=0_{p}\right)\geq1-c\exp\left(-b\log p\right).\label{eq:nec1}$$ The part $\frac{\lambda n}{4s}\geq c_{0}\frac{\sigma}{\phi}\sqrt{\frac{n}{s}}\sqrt{\frac{\log p}{n}}$ in bound (\[eq:4-1\]) follows from the fact that $\alpha\succsim\sqrt{\frac{s}{n}}=\sqrt{\mathbb{E}\left(\frac{1}{n}\sum X_{ij}^{2}\right)}$ where $j\in K$. Main proof for Lemma \[prop:random\_design\] -------------------------------------------- In what follows, we let $\Sigma_{KK}:=\mathbb{E}\left[\frac{1}{n}X_{K}^{T}X_{K}\right]$, $\hat{\Sigma}_{KK}:=\frac{1}{n}X_{K}^{T}X_{K}$, and $\lambda_{\min}\left(\Sigma\right)$ denote the minimum eigenvalue of the matrix $\Sigma$. The proof for Proposition \[prop:random\_design\](i) follows similar argument as before but requires a few extra steps. In applying Lemma 7.23 from Chapter 7.5 of @wainwright_2019 to establish the uniqueness of $\hat{\theta}$ upon the success of PDW construction, it suffices to show that $\lambda_{\min}(\hat{\Sigma}_{KK})\geq\frac{1}{2}\lambda_{\min}(\Sigma_{KK})$ and this fact is verified in (\[eq:S7-1\]) in the appendix. As a consequence, the subproblem (\[eq:sub\]) is strictly convex and has a unique minimizer. The details that show the PDW construction succeeds conditioning on $\mathcal{E}_{1}\cap\mathcal{E}_{4}$ (which holds with probability at least $1-c_{1}\exp\left(-b\frac{\log p}{k^{3}}\right)$) can be found in Lemma \[lem:A4\] (where $b$ is some positive constant that only depends on $\phi$ and $\alpha$). To show (\[eq:5gen-1\]), note that our choice of $\lambda$ and $\left\|\hat{\delta}_{K}\right\|\leq1$ yield $$\left\|\Delta\right\|\leq\left\|\lambda\hat{\delta}_{K}\right\|+\left\|\frac{X_{K}^{T}\varepsilon}{n}\right\|\leq\frac{3\lambda}{2}1_{k},$$ which implies that $\left\|\Delta\right\|_{2}\leq\frac{3\lambda}{2}\sqrt{k}$. Moreover, we can show $$\begin{aligned} \left\|\hat{\theta}_{K}-\theta_{K}^{}\right\|_{2} & = & \frac{\left\|\left(\frac{X_{K}^{T}X_{K}}{n}\right)^{-1}\Delta\right\|_{2}}{\left\|\Delta\right\|_{2}}\left\|\Delta\right\|_{2}\label{eq:30-1}\\ & \leq & \frac{1}{\lambda_{\min}\left(\frac{1}{n}X_{K}^{T}X_{K}\right)}\frac{3\lambda}{2}\sqrt{k}.\nonumber \end{aligned}$$ Applying (\[eq:S7-1\]) and the bound $\left\|\Delta\right\|_{2}\leq\frac{3\lambda}{2}\sqrt{k}$ yields the claim. In showing (\[eq:4gen-1\]) in (ii) and (\[eq:4-1\]) in (iii), we will condition on $\left\{ \textrm{supp}(\hat{\theta})=K\right\} \cap\mathcal{E}_{1}\cap\mathcal{E}_{2}$ and $\left\{ \hat{\theta}_{j}\neq0,\,j\in K\right\} \cap\mathcal{E}_{1}\cap\mathcal{E}_{3}\cap\mathcal{E}_{4}$, respectively. To show (\[eq:4gen-1\]), note that in Step 2 of the PDW procedure, $\hat{\delta}_{K}$ is chosen such that $\left\|\hat{\delta}_{j}\right\|=1$ for any $j\in K$ whenever $\textrm{supp}(\hat{\theta})=K$. Given the choice of $\lambda$, we are ensured to have $$\left\|\Delta\right\|\geq\left\|\left\|\lambda\hat{\delta}_{K}\right\|-\left\|\frac{X_{K}^{T}\varepsilon}{n}\right\|\right\|\geq\frac{\lambda}{2}1_{k},$$ which implies that $\left\|\Delta\right\|_{2}\geq\frac{\lambda}{2}\sqrt{k}$. Moreover, we can show $$\left\|\hat{\theta}_{K}-\theta_{K}^{}\right\|_{2}=\frac{\left\|\left(\frac{X_{K}^{T}X_{K}}{n}\right)^{-1}\Delta\right\|_{2}}{\left\|\Delta\right\|_{2}}\left\|\Delta\right\|_{2}\geq\frac{1}{\lambda_{\max}\left(\frac{1}{n}X_{K}^{T}X_{K}\right)}\frac{\lambda}{2}\sqrt{k}.\label{eq:30}$$ It remains to bound $\lambda_{\max}\left(\hat{\Sigma}_{KK}\right)$. We first write $$\begin{aligned} \lambda_{\max}(\Sigma_{KK}) & = & \max_{\|\|h^{'}\|\|_{2}=1}\mu^{'T}\Sigma_{KK}\mu^{'}\\ & = & \max_{\|\|h^{'}\|\|_{2}=1}\left[\mu^{'T}\hat{\Sigma}_{KK}\mu^{'}+\mu^{'T}(\Sigma_{KK}-\hat{\Sigma}_{KK})\mu^{'}\right]\\ & \geq & \mu^{T}\hat{\Sigma}_{KK}\mu+\mu^{T}(\Sigma_{KK}-\hat{\Sigma}_{KK})\mu\end{aligned}$$ where $\mu\in\mathbb{R}^{k}$ is a unit-norm maximal eigenvector of $\hat{\Sigma}_{KK}$. Applying Lemma \[lem:A1\](b) with $t=\tilde{\alpha}^{2}\sqrt{\frac{\log p}{n}}$ yields $$\mu^{T}\left(\Sigma_{KK}-\hat{\Sigma}_{KK}\right)\mu\geq-\tilde{\alpha}^{2}\sqrt{\frac{\log p}{n}}$$ with probability at least $1-c_{1}\exp\left(-c_{2}\log p\right)$, provided that $\sqrt{\frac{\log p}{n}}\leq1$; therefore, $\lambda_{\max}(\Sigma_{KK})\geq\lambda_{\max}(\hat{\Sigma}_{KK})-\tilde{\alpha}^{2}\sqrt{\frac{\log p}{n}}$. Because $\tilde{\alpha}^{2}\sqrt{\frac{\log p}{n}}\leq\frac{\lambda_{\max}(\Sigma_{KK})}{2}$ (implied by (\[eq:7-3\])), we have $$\lambda_{\max}(\hat{\Sigma}_{KK})\leq\frac{3}{2}\lambda_{\max}(\Sigma_{KK})\label{eq:S7}$$ with probability at least $1-c_{1}\exp\left(-c_{2}\log p\right)$. As a consequence, $$\left\|\hat{\theta}_{K}-\theta_{K}^{}\right\|_{2}\geq\frac{1}{\lambda_{\max}\left(\frac{1}{n}X_{K}^{T}X_{K}\right)}\frac{\lambda}{2}\sqrt{k}\geq\frac{1}{\lambda_{\max}(\Sigma_{KK})}\frac{\lambda}{3}\sqrt{k}.$$ The second inequality in (\[eq:4gen-1\]) simply follows from the fact $\lambda_{\max}\left(\mathbb{E}\left[\frac{1}{n}X_{K}^{T}X_{K}\right]\right)\leq\sum_{j\in K}\left(\mathbb{E}\left[\frac{1}{n}X_{j}^{T}X_{j}\right]\right)$. To show (\[eq:5-1\]), note that $$\begin{aligned} \left\|\hat{\theta}_{K}-\theta_{K}^{}\right\|_{\infty} & \leq & \left\|\hat{\Sigma}_{KK}^{-1}\frac{X_{K}^{T}\varepsilon}{n}\right\|_{\infty}+\lambda\left\Vert \hat{\Sigma}_{KK}^{-1}\right\Vert _{\infty}\nonumber \\ & \leq & \left\Vert \hat{\Sigma}_{KK}^{-1}\right\Vert _{\infty}\left\|\frac{X_{K}^{T}\varepsilon}{n}\right\|_{\infty}+\lambda\left\Vert \hat{\Sigma}_{KK}^{-1}\right\Vert _{\infty}\nonumber \\ & \leq & \frac{3\lambda}{2}\left\Vert \hat{\Sigma}_{KK}^{-1}\right\Vert _{\infty}.\label{eq:10-1}\end{aligned}$$ We then apply (\[eq:S4\]) of Lemma \[lem:A2\] in the appendix, and the fact $\left\Vert \hat{\Sigma}_{KK}^{-1}\right\Vert _{\infty}-\left\Vert \Sigma_{KK}^{-1}\right\Vert _{\infty}\leq\left\Vert \hat{\Sigma}_{KK}^{-1}-\Sigma_{KK}^{-1}\right\Vert _{\infty}$ (so that $\left\Vert \hat{\Sigma}_{KK}^{-1}\right\Vert _{\infty}\leq\frac{7n}{6s}$); putting everything yields the claim. To show (\[eq:4-1\]), we again carry over the argument in the proof for Lemma \[prop:fixed\_design\]. Letting $M=\hat{\Sigma}_{KK}^{-1}-\Sigma_{KK}^{-1}$, we have $$\begin{aligned} \left\|\hat{\theta}_{K}-\theta_{K}^{}\right\| & = & \left\|\left(\Sigma_{KK}^{-1}+M\right)\left[\left(\frac{X_{K}^{T}\varepsilon}{n}\right)-\lambda\hat{\delta}_{K}\right]\right\|\\ & \geq & \left\|\Sigma_{KK}^{-1}\left[\left(\frac{X_{K}^{T}\varepsilon}{n}\right)+\lambda\hat{\delta}_{K}\right]\right\|-\left\|M\left[\left(\frac{X_{K}^{T}\varepsilon}{n}\right)-\lambda\hat{\delta}_{K}\right]\right\|\\ & \geq & \left\|\Sigma_{KK}^{-1}\right\|\left\|\left\|\lambda\hat{\delta}_{K}\right\|-\left\|\frac{X_{K}^{T}\varepsilon}{n}\right\|\right\|-\left\Vert M\right\Vert _{\infty}\left\|\left(\frac{X_{K}^{T}\varepsilon}{n}\right)-\lambda\hat{\delta}_{K}\right\|_{\infty}1_{k},\end{aligned}$$ where the third line uses the fact that $\Sigma_{KK}^{-1}$ is diagonal. Note that as before, the choice of $\lambda$ stated in Lemma \[prop:random\_design\] and the fact $\Sigma_{KK}^{-1}=\frac{n}{s}I_{k}$ yield $$\begin{aligned} \left\|\hat{\theta}_{j}-\theta_{j}^{}\right\| & \geq & \frac{\lambda n}{2s}-\left\Vert M\right\Vert _{\infty}\left\|\left(\frac{X_{K}^{T}\varepsilon}{n}\right)-\lambda\hat{\delta}_{K}\right\|_{\infty}\\ & \geq & \frac{\lambda n}{2s}-\frac{3}{2}\lambda\left\Vert M\right\Vert _{\infty}.\end{aligned}$$ By (\[eq:S4\]) of Lemma \[lem:A2\] in the appendix, with probability at least $1-c_{1}\exp\left(-b\frac{\log p}{k^{3}}\right)$, $\left\Vert M\right\Vert _{\infty}\leq\frac{1}{6}\lambda_{\min}^{-1}(\Sigma_{KK})=\frac{n}{6s}$. As a result, we have (\[eq:4-1\]). The part $\frac{\lambda n}{4s}\geq c_{0}\frac{\sigma}{\phi}\sqrt{\frac{n}{s}}\sqrt{\frac{\log p}{n}}$ in bound (\[eq:4-1\]) follows from the fact that $\alpha\succsim\sqrt{\frac{s}{n}}=\sqrt{\mathbb{E}\left(\frac{1}{n}\sum X_{ij}^{2}\right)}$ where $j\in K$. To establish (\[eq:nec1\]), we adopt argument similar to what is used in showing (\[eq:nec\]) by applying the KKT condition $$\left(\frac{1}{n}X_{1}^{T}X_{1}\right)\left(\theta_{1}^{}-\hat{\theta}_{1}\right)=\lambda\textrm{sgn}\left(\hat{\theta}_{1}\right)-\frac{X_{1}^{T}\varepsilon}{n}$$ and defining $\mathcal{E}=\mathcal{E}_{1}\cap\mathcal{E}_{4}$. Additional technical lemmas and proofs -------------------------------------- In this section, we show that the PDW construction succeeds with high probability in Lemma \[lem:A4\], which is proved using results from Lemmas \[lem:A1\]\[lem:A3\]. The derivations for Lemmas \[lem:A2\] and \[lem:A3\] modify the argument in @wainwright2009sharp and @ravikumar2010 to make it suitable for our purposes. In what follows, we let $\Sigma_{K^{c}K}:=\mathbb{E}\left[\frac{1}{n}X_{K^{c}}^{T}X_{K}\right]$ and $\hat{\Sigma}_{K^{c}K}:=\frac{1}{n}X_{K^{c}}^{T}X_{K}$. Similarly, let $\Sigma_{KK}:=\mathbb{E}\left[\frac{1}{n}X_{K}^{T}X_{K}\right]$ and $\hat{\Sigma}_{KK}:=\frac{1}{n}X_{K}^{T}X_{K}$. \[lem:A1\] (a) Let $\left(W_{i}\right)_{i=1}^{n}$ and $\left(W_{i}^{'}\right)_{i=1}^{n}$ consist of independent components, respectively. Suppose there exist parameters $\alpha$ and $\alpha^{'}$ such that $$\begin{aligned} \sup_{r\geq1}r^{-\frac{1}{2}}\left(\mathbb{E}\left\|W_{i}\right\|^{r}\right)^{\frac{1}{r}} & \leq & \alpha,\\ \sup_{r\geq1}r^{-\frac{1}{2}}\left(\mathbb{E}\left\|W_{i}^{'}\right\|^{r}\right)^{\frac{1}{r}} & \leq & \alpha^{'}, \end{aligned}$$ for all $i=1,\dots,n$. Then $$\mathbb{P}\left[\left\|\frac{1}{n}\sum_{i=1}^{n}\left(W_{i}W_{i}^{'}\right)-\mathbb{E}\left[\frac{1}{n}\sum_{i=1}^{n}\left(W_{i}W_{i}^{'}\right)\right]\right\|\geq t\right]\leq2\exp\left(-cn\left(\frac{t^{2}}{\alpha^{2}\alpha^{'2}}\wedge\frac{t}{\alpha\alpha^{'}}\right)\right).\label{eq:A37-1}$$ (b) For any unit vector $v\in\mathbb{R}^{d}$, suppose there exists a parameter $\tilde{\alpha}$ such that $$\sup_{r\geq1}r^{-\frac{1}{2}}\left(\mathbb{E}\left\|a^{T}Z_{i}^{T}\right\|^{r}\right)^{\frac{1}{r}}\leq\tilde{\alpha},$$ where $Z_{i}$ is the $i$th row of $Z\in\mathbb{R}^{n\times d}$, then we have $$\mathbb{P}(\left\|\left\|Zv\right\|_{2}^{2}-\mathbb{E}\left(\left\|Zv\right\|_{2}^{2}\right)\right\|\geq nt)\leq2\exp\left(-c^{'}n\left(\frac{t^{2}}{\tilde{\alpha}^{4}}\wedge\frac{t}{\tilde{\alpha}^{2}}\right)\right).$$ Lemma \[lem:A1\] is based on Lemma 5.14 and Corollary 5.17 in @vershynin2012. \[lem:A2\] Suppose Assumption \[ass:random\_design\] holds. For any $t>0$ and some constant $c>0$, we have $$\mathbb{P}\left\{ \left\Vert \hat{\Sigma}_{K^{c}K}-\Sigma_{K^{c}K}\right\Vert _{\infty}\geq t\right\} \leq2(p-k)k\exp\left(-cn\left(\frac{t^{2}}{k^{2}\alpha^{4}}\wedge\frac{t}{k\alpha^{2}}\right)\right),\label{eq:S2}$$ $$\mathbb{P}\left\{ \left\Vert \hat{\Sigma}_{KK}-\Sigma_{KK}\right\Vert _{\infty}\geq t\right\} \leq2k^{2}\exp\left(-cn\left(\frac{t^{2}}{k^{2}\alpha^{4}}\wedge\frac{t}{k\alpha^{2}}\right)\right).\label{eq:S3}$$ Furthermore, if $k\geq1$, $\frac{\log p}{n}\leq1$, $\tilde{\alpha}^{2}\sqrt{\frac{\log p}{n}}\leq\frac{\lambda_{\min}(\Sigma_{KK})}{2}$, and $\alpha^{2}\sqrt{\frac{\log p}{n}}\leq\frac{\lambda_{\min}(\Sigma_{KK})}{12}$, we have $$\begin{aligned} \mathbb{P}\left\{ \left\Vert \hat{\Sigma}_{KK}^{-1}\right\Vert _{2}\leq\frac{2}{\lambda_{\min}(\Sigma_{KK})}\right\} & \geq & 1-c_{1}^{'}\exp\left(-c_{2}^{'}\log p\right),\label{eq:S8}\\ \mathbb{P}\left\{ \left\Vert \hat{\Sigma}_{KK}^{-1}-\Sigma_{KK}^{-1}\right\Vert _{\infty}\leq\frac{1}{6\lambda_{\min}(\Sigma_{KK})}\right\} & \geq & 1-c_{1}\exp\left(-c_{2}\left(\frac{\log p}{k^{3}}\right)\right).\label{eq:S4} \end{aligned}$$ Let $u_{j^{'}j}$ denote the element $(j^{'},\,j)$ of the matrix difference $\hat{\Sigma}_{K^{c}K}-\Sigma_{K^{c}K}$. The definition of the $l_{\infty}$matrix norm implies that $$\begin{aligned} \mathbb{P}\left\{ \left\Vert \hat{\Sigma}_{K^{c}K}-\Sigma_{K^{c}K}\right\Vert _{\infty}\geq t\right\} & = & \mathbb{P}\left\{ \max_{j^{'}\in K^{c}}\sum_{j\in K}\|u_{j^{'}j}\|\geq t\right\} \\ & \leq & (p-k)\mathbb{P}\left\{ \sum_{j\in K}\|u_{j^{'}j}\|\geq t\right\} \\ & \leq & (p-k)\mathbb{P}\left\{ \exists j\in K\,\vert\,\|u_{j^{'}j}\|\geq\frac{t}{k}\right\} \\ & \leq & (p-k)k\mathbb{P}\left\{ \|u_{j^{'}j}\|\geq\frac{t}{k}\right\} \\ & \leq & (p-k)k\cdot2\exp\left(-cn\left(\frac{t^{2}}{k^{2}\alpha^{4}}\wedge\frac{t}{k\alpha^{2}}\right)\right), \end{aligned}$$ where the last inequality follows Lemma \[lem:A1\](a). Bound (\[eq:S3\]) can be derived in a similar fashion except that the pre-factor $(p-k)$ is replaced by $k$. To prove (\[eq:S4\]), note that $$\begin{aligned} \left\Vert \hat{\Sigma}_{KK}^{-1}-\Sigma_{KK}^{-1}\right\Vert _{\infty} & = & \left\Vert \Sigma_{KK}^{-1}\left[\Sigma_{KK}-\hat{\Sigma}_{KK}\right]\hat{\Sigma}_{KK}^{-1}\right\Vert _{\infty}\nonumber \\ & \leq & \sqrt{k}\left\Vert \Sigma_{KK}^{-1}\left[\Sigma_{KK}-\hat{\Sigma}_{KK}\right]\hat{\Sigma}_{KK}^{-1}\right\Vert _{2}\nonumber \\ & \leq & \sqrt{k}\left\Vert \Sigma_{KK}^{-1}\right\Vert _{2}\left\Vert \Sigma_{KK}-\hat{\Sigma}_{KK}\right\Vert _{2}\left\Vert \hat{\Sigma}_{KK}^{-1}\right\Vert _{2}\nonumber \\ & \leq & \frac{\sqrt{k}}{\lambda_{\min}(\Sigma_{KK})}\left\Vert \Sigma_{KK}-\hat{\Sigma}_{KK}\right\Vert _{2}\left\Vert \hat{\Sigma}_{KK}^{-1}\right\Vert _{2}.\label{eq:S5} \end{aligned}$$ To bound $\left\Vert \Sigma_{KK}-\hat{\Sigma}_{KK}\right\Vert _{2}$ in (\[eq:S5\]), we apply (\[eq:S3\]) with $t=\frac{\alpha^{2}}{\sqrt{k}}\sqrt{\frac{\log p}{n}}$ and obtain $$\left\Vert \hat{\Sigma}_{KK}-\Sigma_{KK}\right\Vert _{2}\leq\frac{\alpha^{2}}{\sqrt{k}}\sqrt{\frac{\log p}{n}},$$ with probability at least $1-c_{1}\exp\left(-c_{2}\frac{\log p}{k^{3}}\right)$, provided that $k^{-3}\frac{\log p}{n}\leq1$. To bound $\left\Vert \hat{\Sigma}_{KK}^{-1}\right\Vert _{2}$ in (\[eq:S5\]), let us write $$\begin{aligned} \lambda_{\min}(\Sigma_{KK}) & = & \min_{\|\|\mu^{'}\|\|_{2}=1}\mu^{'T}\Sigma_{KK}\mu^{'}\nonumber \\ & = & \min_{\|\|\mu^{'}\|\|_{2}=1}\left[\mu^{'T}\hat{\Sigma}_{KK}\mu^{'}+\mu^{'T}(\Sigma_{KK}-\hat{\Sigma}_{KK})\mu^{'}\right]\nonumber \\ & \leq & \mu^{T}\hat{\Sigma}_{KK}\mu+\mu^{T}(\Sigma_{KK}-\hat{\Sigma}_{KK})\mu\label{eq:S6} \end{aligned}$$ where $\mu\in\mathbb{R}^{k}$ is a unit-norm minimal eigenvector of $\hat{\Sigma}_{KK}$. We then apply Lemma \[lem:A1\](b) with $t=\tilde{\alpha}^{2}\sqrt{\frac{\log p}{n}}$ to show $$\left\|\mu^{T}\left(\Sigma_{KK}-\hat{\Sigma}_{KK}\right)\mu\right\|\leq\tilde{\alpha}^{2}\sqrt{\frac{\log p}{n}}$$ with probability at least $1-c_{1}^{'}\exp\left(-c_{2}^{'}\log p\right)$, provided that $\sqrt{\frac{\log p}{n}}\leq1$. Therefore, $\lambda_{\min}(\Sigma_{KK})\leq\lambda_{\min}(\hat{\Sigma}_{KK})+\tilde{\alpha}^{2}\sqrt{\frac{\log p}{n}}$. As long as $\tilde{\alpha}^{2}\sqrt{\frac{\log p}{n}}\leq\frac{\lambda_{\min}(\Sigma_{KK})}{2}$, we have $$\lambda_{\min}(\hat{\Sigma}_{KK})\geq\frac{1}{2}\lambda_{\min}(\Sigma_{KK}),\label{eq:S7-1}$$ and consequently (\[eq:S8\]), $$\left\Vert \hat{\Sigma}_{KK}^{-1}\right\Vert _{2}\leq\frac{2}{\lambda_{\min}(\Sigma_{KK})}$$ with probability at least $1-c_{1}^{'}\exp\left(-c_{2}^{'}\log p\right)$. Putting the pieces together, as long as $\frac{\alpha^{2}}{\lambda_{\min}(\Sigma_{KK})}\sqrt{\frac{\log p}{n}}\leq\frac{1}{12}$, $$\left\Vert \hat{\Sigma}_{KK}^{-1}-\Sigma_{KK}^{-1}\right\Vert _{\infty}\leq\frac{\sqrt{k}}{\lambda_{\min}(\Sigma_{KK})}\frac{\alpha^{2}}{\sqrt{k}}\sqrt{\frac{\log p}{n}}\frac{2}{\lambda_{\min}(\Sigma_{KK})}\leq\frac{1}{6\lambda_{\min}(\Sigma_{KK})}\label{eq:19}$$ with probability at least $1-c_{1}\exp\left(-c_{2}\frac{\log p}{k^{3}}\right)$. \[lem:A3\] Let Assumption \[ass:random\_design\] hold. Suppose $$\left\Vert \mathbb{E}\left[X_{K^{c}}^{T}X_{K}\right]\left[\mathbb{E}(X_{K}^{T}X_{K})\right]^{-1}\right\Vert _{\infty}=1-\phi\label{eq:S1}$$ for some $\phi\in(0,\,1]$. If $k\geq1$ and $$\begin{aligned} \max\left\{ \frac{\phi}{12(1-\phi)k^{\frac{3}{2}}},\,\frac{\phi}{6k^{\frac{3}{2}}},\,\frac{\phi}{k}\right\} \sqrt{\frac{\log p}{n}} & \leq & \alpha^{2}\quad\textrm{if }\phi\in\left(0,\,1\right),\label{eq:7-2-1}\\ \max\left\{ \frac{1}{6k^{\frac{3}{2}}},\,\frac{1}{k}\right\} \sqrt{\frac{\log p}{n}} & \leq & \alpha^{2}\quad\textrm{if }\phi=1,\label{eq:7-4-1}\\ \max\left\{ 2\tilde{\alpha}^{2},\,12\alpha^{2},\,1\right\} \sqrt{\frac{\log p}{n}} & \leq & \lambda_{\min}(\Sigma_{KK}),\label{eq:7-3-1} \end{aligned}$$ then for some positive constant $b$ that only depends on $\phi$ and $\alpha$, we have $$\mathbb{P}\left[\left\Vert \frac{1}{n}X_{K^{c}}^{T}X_{K}\left(\frac{1}{n}X_{K}^{T}X_{K}\right)^{-1}\right\Vert _{\infty}\geq1-\frac{\phi}{2}\right]\leq c^{'}\exp\left(-b\left(\frac{\log p}{k^{3}}\right)\right).\label{eq:77}$$ Using the decomposition in @ravikumar2010, we have $$\hat{\Sigma}_{K^{c}K}\hat{\Sigma}_{KK}^{-1}-\Sigma_{K^{c}K}\Sigma_{KK}^{-1}=R_{1}+R_{2}+R_{3},$$ where ** $$\begin{aligned} R_{1} & = & \Sigma_{K^{c}K}\left[\hat{\Sigma}_{KK}^{-1}-\Sigma_{KK}^{-1}\right],\\ R_{2} & = & \left[\hat{\Sigma}_{K^{c}K}-\Sigma_{K^{c}K}\right]\Sigma_{KK}^{-1},\\ R_{3} & = & \left[\hat{\Sigma}_{K^{c}K}-\Sigma_{K^{c}K}\right]\left[\hat{\Sigma}_{KK}^{-1}-\Sigma_{KK}^{-1}\right]. \end{aligned}$$ By (\[eq:S1\]), we have $\left\Vert \Sigma_{K^{c}K}\Sigma_{KK}^{-1}\right\Vert _{\infty}=1-\phi$. It suffices to show $\left\Vert R_{i}\right\Vert _{\infty}\leq\frac{\phi}{6}$ ** for $i=1,...,3$. For $R_{1}$, note that $$R_{1}=-\Sigma_{K^{c}K}\Sigma_{KK}^{-1}[\hat{\Sigma}_{KK}-\Sigma_{KK}]\hat{\Sigma}_{KK}^{-1}.$$ Applying the facts $\left\Vert AB\right\Vert _{\infty}\leq\left\Vert A\right\Vert _{\infty}\left\Vert B\right\Vert _{\infty}$ and $\left\Vert A\right\Vert _{\infty}\leq\sqrt{a}\left\Vert A\right\Vert _{2}$ for any symmetric matrix $A\in\mathbb{R}^{a\times a}$, we can bound $R_{1}$ in the following fashion: $$\begin{aligned} \left\Vert R_{1}\right\Vert _{\infty} & \leq & \left\Vert \Sigma_{K^{c}K}\Sigma_{KK}^{-1}\right\Vert _{\infty}\left\Vert \hat{\Sigma}_{KK}-\Sigma_{KK}\right\Vert _{\infty}\left\Vert \hat{\Sigma}_{KK}^{-1}\right\Vert _{\infty}\\ & \leq & (1-\phi)\left\Vert \hat{\Sigma}_{KK}-\Sigma_{KK}\right\Vert _{\infty}\sqrt{k}\left\Vert \hat{\Sigma}_{KK}^{-1}\right\Vert _{2}, \end{aligned}$$ where the last inequality uses (\[eq:S1\]). If $\phi=1$, then $\left\Vert R_{1}\right\Vert _{\infty}=0$ so we may assume $\phi<1$ in the following. Bound (\[eq:S8\]) from the proof for Lemma \[lem:A2\] yields $$\left\Vert \hat{\Sigma}_{KK}^{-1}\right\Vert _{2}\leq\frac{2}{\lambda_{\min}(\Sigma_{KK})}$$ with probability at least $1-c_{1}\exp\left(-c_{2}\log p\right)$. Now, we apply bound (\[eq:S3\]) from Lemma \[lem:A2\] with $t=\frac{\phi}{12(1-\phi)}\sqrt{\frac{\log p}{kn}}$ and obtain $$\mathbb{P}\left[\left\Vert \hat{\Sigma}_{KK}-\Sigma_{KK}\right\Vert _{\infty}\geq\frac{\phi}{12(1-\phi)}\sqrt{\frac{\log p}{kn}}\right]\leq2\exp\left(-c\left(\frac{\phi^{2}\log p}{\alpha^{4}(1-\phi)^{2}k^{3}}\right)\right),$$ provided $\frac{\phi}{12(1-\phi)\alpha^{2}k}\sqrt{\frac{\log p}{kn}}\leq1$. Then, if $\sqrt{\frac{\log p}{n}}\leq\lambda_{\min}(\Sigma_{KK})$, we are guaranteed that $$\mathbb{P}\left[\left\Vert R_{1}\right\Vert _{\infty}\geq\frac{\phi}{6}\right]\leq2\exp\left(-c\left(\frac{\phi^{2}\log p}{\alpha^{4}(1-\phi)^{2}k^{3}}\right)\right)+c_{1}\exp\left(-c_{2}\log p\right).$$ For $R_{2}$, note that $$\begin{aligned} \left\Vert R_{2}\right\Vert _{\infty} & \leq & \sqrt{k}\left\Vert \Sigma_{KK}^{-1}\right\Vert _{2}\left\Vert \hat{\Sigma}_{K^{c}K}-\Sigma_{K^{c}K}\right\Vert _{\infty}\\ & \leq & \frac{\sqrt{k}}{\lambda_{\min}(\Sigma_{KK})}\left\Vert \hat{\Sigma}_{K^{c}K}-\Sigma_{K^{c}K}\right\Vert _{\infty}. \end{aligned}$$ If $\frac{\phi}{6\alpha^{2}k}\sqrt{\frac{\log p}{kn}}\leq1$ and $\sqrt{\frac{\log p}{n}}\leq\lambda_{\min}(\Sigma_{KK})$, applying bound (\[eq:S2\]) from Lemma \[lem:A2\] with $t=\frac{\phi}{6}\sqrt{\frac{\log p}{kn}}$ yields $$\mathbb{P}\left[\left\Vert R_{2}\right\Vert _{\infty}\geq\frac{\phi}{6}\right]\leq2\exp\left(-c\left(\frac{\phi^{2}\log p}{\alpha^{4}k^{3}}\right)\right).$$ For $R_{3}$, applying (\[eq:S2\]) with $t=\phi\sqrt{\frac{\log p}{n}}$ to bound $\left\Vert \hat{\Sigma}_{K^{c}K}-\Sigma_{K^{c}K}\right\Vert _{\infty}$ and (\[eq:S4\]) to bound $\left\Vert \hat{\Sigma}_{KK}^{-1}-\Sigma_{KK}^{-1}\right\Vert _{\infty}$ yields $$\mathbb{P}\left[\left\Vert R_{3}\right\Vert _{\infty}\geq\frac{\phi}{6}\right]\leq c^{'}\left[\exp\left(-c\left(\frac{\phi^{2}\log p}{\alpha^{4}k^{3}}\right)\right)+\exp\left(-c\left(\frac{\log p}{k^{3}}\right)\right)\right],$$ provided that $\frac{\phi}{\alpha^{2}k}\sqrt{\frac{\log p}{n}}\leq1$ and $\sqrt{\frac{\log p}{n}}\leq\lambda_{\min}(\Sigma_{KK})$. Putting everything together, we conclude that $$\mathbb{P}\left[\left\Vert \hat{\Sigma}_{K^{c}K}\hat{\Sigma}_{KK}^{-1}\right\Vert _{\infty}\geq1-\frac{\phi}{2}\right]\leq c^{'}\exp\left(-b\left(\frac{\log p}{k^{3}}\right)\right)$$ for some positive constant $b$ that only depends on $\phi$ and $\alpha$. \[lem:A4\] Let the assumptions in Lemmas \[lem:A2\] and \[lem:A3\] hold. Suppose $\theta^{}$ is exactly sparse with at most $k$ non-zero coefficients and $K=\left\{ j:\,\theta_{j}^{}\neq0\right\} \neq\emptyset$. If we choose $\lambda\geq\frac{c\alpha\sigma\left(2-\frac{\phi}{2}\right)}{\phi}\sqrt{\frac{\log p}{n}}$ for some sufficiently large universal constant $c>0$, $\left\|\hat{\delta}_{K^{c}}\right\|_{\infty}\leq1-\frac{\phi}{4}$ with probability at least $1-c_{1}\exp\left(-b\frac{\log p}{k^{3}}\right)$, where $b$ is some positive constant that only depends on $\phi$ and $\alpha$. By construction, the subvectors $\hat{\theta}_{K}$, $\hat{\delta}_{K}$, and $\hat{\delta}_{K^{c}}$ satisfy the zero-subgradient condition in the PDW construction. With the fact that $\hat{\theta}_{K^{c}}=\theta_{K^{c}}^{}=0_{p-k}$, we have $$\begin{aligned} \hat{\Sigma}_{KK}\left(\hat{\theta}_{K}-\theta_{K}^{}\right)-\frac{1}{n}X_{K}^{T}\varepsilon+\lambda\hat{\delta}_{K} & = & 0_{k},\\ \hat{\Sigma}_{K^{c}K}\left(\hat{\theta}_{K}-\theta_{K}^{}\right)-\frac{1}{n}X_{K^{c}}^{T}\varepsilon+\lambda\hat{\delta}_{K^{c}} & = & 0_{p-k}. \end{aligned}$$ The equations above yields $$\begin{aligned} \hat{\delta}_{K^{c}} & = & -\frac{1}{\lambda}\hat{\Sigma}_{K^{c}K}\left(\hat{\theta}_{K}-\theta_{K}^{}\right)+X_{K^{c}}^{T}\frac{\varepsilon}{n\lambda},\\ \hat{\theta}_{K}-\theta_{K}^{} & = & \hat{\Sigma}_{KK}^{-1}\frac{X_{K}^{T}\varepsilon}{n}-\lambda\hat{\Sigma}_{KK}^{-1}\hat{\delta}_{K}, \end{aligned}$$ which yields $$\hat{\delta}_{K^{c}}=\left(\hat{\Sigma}_{K^{c}K}\hat{\Sigma}_{KK}^{-1}\right)\hat{\delta}_{K}+\left(X_{K^{c}}^{T}\frac{\varepsilon}{n\lambda}\right)-\left(\hat{\Sigma}_{K^{c}K}\hat{\Sigma}_{KK}^{-1}\right)X_{K}^{T}\frac{\varepsilon}{n\lambda}.$$ Using elementary inequalities and the fact that $\left\|\hat{\delta}_{K}\right\|_{\infty}\leq1$, we obtain $$\left\|\hat{\delta}_{K^{c}}\right\|_{\infty}\leq\left\Vert \hat{\Sigma}_{K^{c}K}\hat{\Sigma}_{KK}^{-1}\right\Vert _{\infty}+\left\|X_{K^{c}}^{T}\frac{\varepsilon}{n\lambda}\right\|_{\infty}+\left\Vert \hat{\Sigma}_{K^{c}K}\hat{\Sigma}_{KK}^{-1}\right\Vert _{\infty}\left\|X_{K}^{T}\frac{\varepsilon}{n\lambda}\right\|_{\infty}.$$ By Lemma \[lem:A3\], $\left\Vert \hat{\Sigma}_{K^{c}K}\hat{\Sigma}_{KK}^{-1}\right\Vert _{\infty}\leq1-\frac{\phi}{2}$ with probability at least $1-c^{'}\exp\left(-\frac{b\log p}{k^{3}}\right)$; as a result, $$\begin{aligned} \left\|\hat{\delta}_{K^{c}}\right\|_{\infty} & \leq & 1-\frac{\phi}{2}+\left\|X_{K^{c}}^{T}\frac{\varepsilon}{n\lambda}\right\|_{\infty}+\left\Vert \hat{\Sigma}_{K^{c}K}\hat{\Sigma}_{KK}^{-1}\right\Vert _{\infty}\left\|X_{K}^{T}\frac{\varepsilon}{n\lambda}\right\|_{\infty}\\ & \leq & 1-\frac{\phi}{2}+\left(2-\frac{\phi}{2}\right)\left\|\hat{X}^{T}\frac{\varepsilon}{n\lambda}\right\|_{\infty}. \end{aligned}$$ It remains to show that $\left(2-\frac{\phi}{2}\right)\left\|X^{T}\frac{\varepsilon}{n\lambda}\right\|_{\infty}\leq\frac{\phi}{4}$ with high probability. This result holds if $\lambda\geq\frac{4\left(2-\frac{\phi}{2}\right)}{\phi}\left\|X^{T}\frac{\varepsilon}{n}\right\|_{\infty}$. In particular, Lemma \[lem:A1\](a) and a union bound imply that $$\mathbb{P}\left(\left\|\frac{X^{T}\varepsilon}{n}\right\|_{\infty}\geq t\right)\leq2\exp\left(\frac{-nt^{2}}{c_{0}\sigma^{2}\alpha^{2}}+\log p\right).$$ Thus, under the choice of $\lambda$ in Lemma \[lem:A4\], we have $\left\|\hat{\delta}_{K^{c}}\right\|_{\infty}\leq1-\frac{\phi}{4}$ with probability at least $1-c_{1}\exp\left(-b\frac{\log p}{k^{3}}\right)$. [^1]: Department of Economics, University of California, San Diego, 9500 Gilman Dr. La Jolla, CA 92093. Email: [email protected] [^2]: Department of Economics, University of California, San Diego, 9500 Gilman Dr. La Jolla, CA 92093. Email: [email protected]. [^3]: Alphabetical ordering; both authors contributed equally to this work. We would like to thank Stéphane Bonhomme, Gordon Dahl, Graham Elliott, Michael Jansson, Ulrich Müller, Andres Santos, Azeem Shaikh, Aman Ullah, Jeffrey Wooldridge, and seminar participants for their comments. We are especially grateful to Yixiao Sun for providing extensive feedback on an earlier draft. This paper was previously circulated as “Behavior of Lasso and Lasso-based inference under limited variability” and “Omitted variable bias of Lasso-based inference methods under limited variability: A finite sample analysis”. Zhu acknowledges a start-up fund from the Department of Economics at UCSD and the Department of Statistics and the Department of Computer Science at Purdue University, West Lafayette. [^4]: A natural idea to avoid OVBs due to the Lasso not selecting relevant controls is to choose a regularization parameter smaller than the recommended ones [e.g., @belloni2014inference]. However, as we show in simulations, this idea does not work in general and can lead to substantial biases. [^5]: The existing theoretical framework for the Lasso-based inference methods makes it difficult to derive an informative $\text{constant}$. To our knowledge, the literature provides no such derivations. [^6]: Note that the existing Lasso theory requires the regularization parameter to exceed a certain threshold, which depends on the standard deviations of the noise and the covariates. [^7]: Such a comparison is useful as the standard deviation in @belloni2014inference is the basis for their recommended inference procedures. We would also like to compare the OVBs to the conditional or unconditional finite sample standard deviation. However, this would require exact formulas of various selection probabilities (more than just bounds on the probabilities), which are impossible to derive for the Lasso in general. [^8]: Many applications of Lasso-based inference procedures feature such specifications [e.g., @belloni2014high; @belloni2014inference; @chen2015can; @decker2016health; @fremstad2017does; @knaus2018heterogenous; @jones2018what; @schmitz2017informal]. [^9]: In our theoretical results, however, $p$ is allowed to exceed $n$. [^10]: Note that traditional robust standard errors are inconsistent in settings with many controls [@cattaneo2018inference]. [^11]: Specifically, we set $\lambda=2\sigma\sqrt{\frac{2b(1+\tau)\log p}{n}}$, assuming that $\sigma$ is known. In practice, we first normalize $X_{i}$ such that $b=1$, run Lasso using $\lambda=2\sigma\sqrt{\frac{2(1+\tau)\log p}{n}}$, and then rescale the coefficients. [^12]: For all four values of $\sigma_x$, none of the controls with zero coefficients get selected with high probability. [^13]: Note that exact formulas of these mixture probabilities cannot be derived. [^14]: For both values of $\sigma_x$, none of the controls with zero coefficients get selected with high probability. [^15]: This approach is based on the following modified Lasso program: $$\hat{\theta}\in\arg\min_{\theta\in\mathbb{R}^{p}}\frac{1}{n}\sum_{i=1}^n\left(Y_i-X_i\theta \right)^2+\frac{\lambda}{n}\sum_{j=1}^p\|\hat{l}_j\theta_j \| \label{eq:general_lasso}$$ where $(\hat{l}_1,\dots,\hat{l}_p)$ are penalty loadings obtained using the iterative post Lasso-based algorithm developed in @belloni2012sparse. Our implementation is based on the Matlab code provided on the authors’ webpage: <https://voices.uchicago.edu/christianhansen/code-and-data/>. We set $\lambda=2c\sqrt{n}\Phi^{-1}(1-\varsigma/(2p))$, where $c=1.1$ and $\varsigma=0.1/\log n$ as recommended by @belloni2014inference [@belloni2017program]. Under homoscedasticity and with nearly orthogonal $X$, this regularization choice has a similar magnitude as the one in @bickel2009simultaneous.\[footnote\_bcch\] [^16]: \[footnote:over-selection\] A similar phenomenon arises with $0.5\lambda_{\text{BCCH}}$. This choice works well when $\alpha^\ast=0$, but yields biases when $\alpha^\ast\ne 0$. We found that, under our DGPs, this is related to the fact that when $\alpha^\ast\ne 0$, and differ in terms of the underlying coefficients and the noise level, which leads to differences in the (over-)selection behavior of the Lasso. [^17]: Here focus on the intention to treat effect of 401(k) eligibility on assets as in @Poterbaetal1994 [@Poterbaetal1995; @Poterbaetal1998] and @Benjamin2003. Some studies use 401(k) eligibility as an instrument for 401(k) participation [e.g., @CH2004; @belloni2017program]. [^18]: @fryerlevitt2013 also analyze test scores at the age of four years as well as eight months. [^19]: With interviewer fixed effects and based on the sample of all children, we were able to exactly replicate the results in Table 3 of @fryerlevitt2013. [^20]: The normality of $\varepsilon_{i}$ can be relaxed without changing the essence of our results. [^21]: Here (and similarly in the rest of propositions), we implicitly assume $p$ is sufficiently large such that $1-k\exp\left(\frac{-b^{2}\left(1+\tau\right)\log p}{4\phi^{2}}\right)-\frac{2}{p^{\tau}}>0$. Indeed, probabilities in such a form are often referred to as the “high probability” guarantees in the literature of (nonasymptotic) high dimensional statistics concerning large $p$ and small enough $k$. The event $\mathcal{M}$ is the intersection of $\left\{ \hat{I}_{1}=\hat{I}_{2}=\emptyset\right\}$ and an additional event; $\left\{ \hat{I}_{1}=\hat{I}_{2}=\emptyset\right\}$ occurs with probability at least $1-\frac{2}{p^{\tau}}$ and the additional event occurs with probability at least $1-k\exp\left(\frac{-b^{2}\left(1+\tau\right)\log p}{4\phi^{2}}\right)$. The additional event is needed in our analyses for technical reasons. See (\[eq:M\_event\]) of Appendix \[sec:appendix\_post\_double\_lasso\] for the definition of $\mathcal{M}$. [^22]: We thank Ulrich Müller for suggesting this comparison. [^23]: In general, $\lambda_{1}\asymp\phi^{-1}\left(\sigma_{\eta}+\left\|\alpha^{}\right\|\sigma_{v}\right)\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}}$ and the iterative algorithm for choosing $\lambda_{1}$ in @belloni2014inference described in Footnote \[footnote\_bcch\] achieves this scaling. Under the conditions on $\left(\beta_{K}^{},\,\gamma_{K}^{},\,\alpha^{}\right)$ in Proposition \[prop:bias\_post\_double\_selection\], this scaling is equivalent to $\phi^{-1}\sigma_{\eta}\sqrt{\frac{s}{n}}\sqrt{\frac{\log p}{n}}$. [^24]: In fact, when the parameter space is unrestricted, OLS-based inference exhibits desirable optimality properties [e.g., @armstrong2016 Section 4.1]. [^25]: For a convex function $f:\,\mathbb{R}^{p}\mapsto\mathbb{R}$, $\delta\in\mathbb{R}^{p}$ is a subgradient at $\theta$, namely $\delta\in\partial f(\theta)$, if $f(\theta+\triangle)\geq f(\theta)+\left\langle \delta,\,\triangle\right\rangle $ for all $\triangle\in\mathbb{R}^{p}$. [^26]: Because $v$ and $\eta$ are independent of each other, the bound $\mathbb{P}\left(E\cap\mathcal{E}_{t^{}}\right)\geq1-k\exp\left(\frac{-b^{2}\left(1+\tau\right)\log p}{4\phi^{2}}\right)-\frac{2}{p^{\tau}}$ can be further sharpened to $\mathbb{P}\left(E\cap\mathcal{E}_{t^{*}}\right)\geq\left(1-\frac{1}{p^{\tau}}\right)^{2}-k\exp\left(\frac{-b^{2}\left(1+\tau\right)\log p}{4\phi^{2}}\right)$. [^27]: We found that one of the reasons for the relatively poor performance of debiased Lasso is that $D$ is highly correlated with the relevant controls. Unreported simulation results show that debiased Lasso exhibits a better performance when $(D,X)$ exhibit a Toeplitz dependence structure as in the simulations reported by @vandergeer2014asymptotically. [^28]: This model and the multiplicative specification of heteroscedasticity are as in the simulations of @belloni2014inference.	ArXiv
--- abstract: 'We give thirty-two diverse proofs of a small mathematical gem—the fundamental Euler sum identity $$\zeta(2,1)=\zeta(3)=8\,\zeta(\overline2,1).$$ We also discuss various generalizations for multiple harmonic (Euler) sums and some of their many connections, thereby illustrating both the wide variety of techniques fruitfully used to study such sums and the attraction of their study.' address: - \| Faculty of Computer Science\ Dalhousie University\ Halifax, Nova Scotia B3H 1W5\ Canada - \| Department of Mathematics & Statistics\ University of Maine\ 5752 Neville Hall Orono, Maine 04469-5752\ U.S.A. author: - 'Jonathan M. Borwein' - 'David M. Bradley' title: 'Thirty-Two Goldbach Variations' --- [^1] =500 Introduction {#sect:Intro} ============ There are several ways to introduce and make attractive a new or unfamiliar subject. We choose to do so by emulating Glen Gould’s passion for Bach’s Goldberg variations. We shall illustrate most of the techniques used to study Euler sums by focusing almost entirely on the identities of and $$\sum_{n=1}^\infty \frac{1}{n^2}\sum_{m=1}^{n-1}\frac1{m} = \sum_{n=1}^\infty \frac{1}{n^3} = 8\sum_{n=1}^\infty \frac{(-1)^n}{n^2}\sum_{m=1}^{n-1}\frac1{m}$$ and some of their many generalizations. Euler, Goldbach and the birth of ${\boldsymbol\zeta}$. ------------------------------------------------------ What follows is a transcription of correspondence between Euler and Goldbach [@eg1742] that led to the origin of the zeta-function and multi-zeta values, see also [@exp1; @exp2; @dunham]. > [59. Goldbach an Euler, Moskau, 24. Dez. 1742.]{}[^2] \[…\]Als ich neulich die vermeinten summas der beiden letzteren serierum in meinem vorigen Schreiben wieder betrachtet, habe ich alsofort wahrgenommen, daß selbige aus einem bloßem Schreibfehler entstanden, von welchem es aber in der Tat heißet: Si non errasset, fecerat ille minus.[^3] This is the letter in which Goldbach precisely formulates the series which sparked Euler’s further investigations into what would become the zeta-function. These investigations were apparently due to a serendipitous mistake. The above translates as follows: > When I recently considered further the indicated sums of the last two series in my previous letter, I realized immediately that the same series arose due to a mere writing error, from which indeed the saying goes, “Had one not erred, one would have achieved less."[^4] Goldbach continues > Ich halte dafür, daß es ein problema problematum ist, die summam huius: $$\begin{aligned} > 1+ \frac{1}{2^n}\left(1+ \frac{1}{2^m}\right) + > \frac{1}{3^n}\left(1+ \frac{1}{2^m} +\frac{1}{3^m}\right) > +\frac{1}{4^n}\left(1+ \frac{1}{2^m} > +\frac{1}{3^m}+\frac{1}{4^m}\right)+ etc.\end{aligned}$$ in den casibus zu finden, wo $m$ et $n$ nicht numeri integri pares et sibi aequales sind, doch gibt es casus, da die summa angegeben werden kann, exempli gr\[atia\], si $m=1$, $n=3$, denn es ist $$\begin{aligned} > 1+ \frac{1}{2^3}\left(1+ \frac{1}{2}\right) + \frac{1}{3^3}\left(1+ > \frac{1}{2} +\frac{1}{3}\right) + \frac{1}{4^3}\left(1+ > \frac{1}{2} +\frac{1}{3}+\frac{1}{4}\right)+ etc. = > \frac{\pi^4}{72}.\end{aligned}$$ The Modern Language of Euler Sums --------------------------------- For positive integers $s_1,\dots,s_m$ and signs $\sigma_j = \pm 1$, consider [@BBB] the $m$-fold Euler sum $$\zeta(s_1,\dots,s_m;\sigma_1,\dots,\sigma_m) := \sum_{k_1>\cdots>k_m>0}\;\prod_{j=1}^m \frac{\sigma_j^{k_j}}{k_j^{s_j}}.$$ As is now customary, we combine strings of exponents and signs by replacing $s_j$ by $\overline s_j$ in the argument list if and only if $\sigma_j=-1$, and denote $n$ repetitions of a substring $S$ by $\{S\}^n$. Thus, for example, $\zeta(\overline1)=-\log 2$, $\zeta(\{2\}^3)=\zeta(2,2,2)=\pi^6/7!$ and $$\zeta(s_1,\dots,s_m) = \sum_{k_1>\cdots>k_m>0}\;\prod_{j=1}^m k_j^{-s_j}. \label{mzvdef}$$ The identity $$\label{z21} \zeta(2,1) = \zeta(3)$$ goes back to Euler [@LE1] [@LE2 p. 228] and has since been repeatedly rediscovered, (see, e.g., [@Briggs; @Bruckman; @Farnum; @Klamkin]). In this language Goldbach had found $$\zeta(3,1)+\zeta(4) =\frac{\pi^4}{72}.$$ The more general formula $$\label{EulerReduction} 2\zeta(m,1)= m\zeta(m+1)-\sum_{j=1}^{m-2}\zeta(j+1)\zeta(m-j), \qquad 2\le m\in{{\mathbf Z}}$$ is also due to Euler [@LE1] [@LE2 p. 266]. Nielsen [@Niels1 p. 229] [@Niels2 p. 198] [@Niels3 pp.47–49] developed a method for obtaining and related results based on partial fractions. Formula has also been rediscovered many times [@Williams; @State; @SitSarm; @GP; @Bracken; @Vowe]. Crandall and Buhler [@CranBuh] deduced from their general infinite series formula which expresses $\zeta(s,t)$ for real $s>1$ and $t\ge 1$ in terms of Riemann zeta values. Study of the the multiple zeta function led to the discovery of a new generalization of , involving nested sums of arbitrary depth: $$\label{z21^n} \zeta(\{2,1\}^n ) = \zeta(\{3\}^n), \qquad n\in{{\mathbf Z}}^{+}.$$ Although numerous proofs of and are known (we give many in the sequel), the only proof of of which we are aware involves making a simple change of variable in a multiple iterated integral (see [@BBB; @BBBLa; @BowBradSurvey] and below). An alternating version of is $$\label{2bar1} 8\zeta(\overline{2}, 1) = \zeta(3),$$ which has also resurfaced from time to time [@Niels3 p. 50] [@Sub85 (2.12)] [@Butzer p. 267] and hints at the generalization $$\label{2bar1^n} 8^n\zeta(\{\overline 2, 1\}^n) \stackrel{?}{=} \zeta(\{3\}^n),\qquad n\in{{\mathbf Z}}^{+},$$ originally conjectured in [@BBB], and which still remains open—despite abundant, even overwhelming, evidence [@DJD]. Hilbert and Hardy Inequalities {#sect:hardy} ------------------------------ Much of the early 20th century history—and philosophy—of the “ ‘bright’ and amusing" subject of inequalities charmingly discussed in G.H. Hardy’s retirement lecture as London Mathematical Society Secretary, [@ghh]. He comments [@ghh p. 474] that Harald Bohr is reported to have remarked “Most analysts spend half their time hunting through the literature for inequalities they want to use, but cannot prove." Central to Hardy’s essay are: [(Hilbert)]{} For non-negative sequences $(a_n)$ and $(b_n)$, not both zero, and for\ $1 \le p, q \le \infty$ with $1/p+1/q=1$ one has $$\begin{aligned} \label{hilbert-p}\sum_{n=1}^\infty\sum_{m=1}^\infty \frac{a_n\,b_m}{n+m} < \pi\,{\rm csc}\left(\frac{\pi}p\right) \\|a_n\\|_p\,\\|b_n\\|_q.\end{aligned}$$ [(Hardy)]{} For a non-negative sequence $(a_n)$ and for $p>1$ $$\begin{aligned} \label{h-ineq}\sum_{n=1}^\infty\left(\frac{a_1+a_2+\cdots +a_n}n\right)^p \le \left(\frac{p}{p-1}\right)^p \,\sum_{n=1}^\infty a_n^p.\end{aligned}$$ We return to these inequalities in Section \[sect:witten\]. Hardy [@ghh p. 485] remarks that his [“own theorem was discovered as a by-product of my own attempt to find a really simple and elementary proof of Hilbert’s."]{} He reproduces Elliott’s proof of (\[h-ineq\]), writing “it can hardly be possible to find a proof more concise or elegant" and also “I have given nine \[proofs\] in a lecture in Oxford, and more have been found since then." (See [@ghh p. 488].) Our Motivation and Intentions ----------------------------- We wish to emulate Hardy and to present proofs that are either elementary, bright and amusing, concise or elegant— ideally all at the same time! In doing so we note that: 1. $\zeta(3)$, while provably irrational, is still quite mysterious, see [@bt; @agm] and [@exp2]. Hence, exposing more relationships and approaches can only help. We certainly hope one of them will lead to a proof of conjecture (\[2bar1\^n\]). 2. Identities for $\zeta(3)$ are abundant and diverse. We give three each of which is the entry-point to a fascinating set: - Our first favourite is a binomial sum [@ag] that played a role in Apéry’s 1976 proof, see [@agm; @bt] and [@exp2 Chapter 3], of the irrationality of $\zeta(3)$: $$\begin{aligned} \label{z3} \zeta(3) &=& \frac5 2\,\sum_{k=1}^{\infty} \frac{(-1)^{k+1}} {k^3\, {2k \choose k}}.\end{aligned}$$ - Our second is Broadhurst’s binary BBP formula [@broad]: $$\zeta(3)=\frac{48}7\,\mathcal{S}_{1}(1,-7,-1,10,-1,-7,1,0 )+\frac{32}7\,\mathcal{S}_{3}(1,1,-1,-2,-1,1,1,0),$$ where $\mathcal{S}_{p}(a_1,a_2,\ldots, a_8):=\sum_{k=1}^\infty {a_k}2^{-\lfloor{p(k+1)/2\rfloor}}k^{-3},$ and the coefficients $a_k$ repeat modulo 8. We refer to [@exp1 Chapter 3] for the digit properties of such formulae. Explicitly, $$\begin{aligned} \zeta(3) &=& \frac{1}{672} \sum_{k=0}^\infty \frac{1}{2^{12 k}} \left[\frac{2048}{(24 k + 1)^3} - \frac{11264}{(24 k + 2)^3} - \frac{1024}{(24 k + 3)^3} + \frac{11776}{(24 k + 4)^3} \right. \\ && \left. - \frac{512}{(24 k + 5)^3} + \frac{4096}{(24 k + 6)^3} + \frac{256}{(24 k + 7)^3} + \frac{3456}{(24 k + 8)^3} + \frac{128}{(24 k + 9)^3} \right. \\ && \left. - \frac{704}{(24 k + 10)^3} - \frac{64}{(24 k + 11)^3} - \frac{128}{(24 k + 12)^3} - \frac{32}{(24 k + 13)^3} - \frac{176}{(24 k + 14)^3} \right. \\ && \left. + \frac{16}{(24 k + 15)^3} + \frac{216}{(24 k + 16)^3} + \frac{8}{(24 k + 17)^3} + \frac{64}{(24 k + 18)^3} - \frac{4}{(24 k + 19)^3} \right. \\ && \left. + \frac{46}{(24 k + 20)^3} - \frac{2}{(24 k + 21)^3} - \frac{11}{(24 k + 22)^3} + \frac{1}{(24 k + 23)^3} \right].\end{aligned}$$ It was this discovery that lead Bailey and Crandall to their striking recent work on normality of BBP constants [@exp1 Chapter 4]. - Our third favourite due to Ramanujan [@exp2 p. 138] is the hyperbolic series approximation $$\zeta \left( 3 \right) ={\frac {7\,{\pi }^{3}}{180}}-2\,\sum _{k=1}^{\infty }{\frac {1}{{k}^{3} \left( {e^{2\,\pi \,k}}-1 \right) }},$$ in which the ‘error’ is $\zeta(3)-7\,{\pi }^{3}/180 \approx -0.003742745$, and which to our knowledge is the ‘closest’ one gets to writing $\zeta(3)$ as a rational multiple of $\pi^3$. 3. Often results about $\zeta(3)$ are more precisely results about $\zeta(2,1)$ or $\zeta(\overline{2},1)$, as we shall exhibit. 4. Double and multiple sums are still under-studied and under-appreciated. We should like to partially redress that. 5. One can now prove these seemingly analytic facts in an entirely finitary manner via words over alphabets, dispensing with notions of infinity and convergence; 6. Many subjects are touched upon—from computer algebra, integer relation methods, generating functions and techniques of integration to polylogarithms, hypergeometric and special functions, non-commutative rings, combinatorial algebras and Stirling numbers—so that most readers will find a proof worth showing in an undergraduate class; 7. For example, there has been an explosive recent interest in q-analogues, see §\[sect:zud\], and in quantum field theory, algebraic K-theory and knot theory, see [@exp1; @zagier]. For some of the broader issues relating to Euler sums, we refer the reader to the survey articles [@exp1; @BowBradSurvey; @Cartier; @Wald; @Wald2; @zagier; @Zud]. Computational issues are discussed in [@exp2; @Cran] and to an extent in [@BBBLa]. Further Notation ---------------- For positive integer $N$, denote the $N$th partial sum of the harmonic series by $H_N := \sum_{n=1}^N 1/n$. We also use $\psi = \Gamma'/\Gamma$ to denote the logarithmic derivative of the Euler gamma-function (also referred to as the digamma function), and recall the identity $\psi(N+1)+\gamma=H_N$, where $\gamma=0.5772156649\ldots$ is Euler’s constant. Where convenient, we employ the Pochhammer symbol $(a)_n= a(a+1)\cdots(a+n-1)$ for complex $a$ and non-negative integer $n$. As usual, the Kronecker $\delta_{m,n}$ is 1 if $m=n$ and $0$ otherwise. We organize our proofs by technique, although clearly this is somewhat arbitrary as many proofs fit well within more than one category. Broadly their sophistication increases as we move through the paper. In some of the later sections the proofs become more schematic. We invite readers to send additional selections for our collection, a collection which for us has all the beauty of Blake’s grain of sand[^5]: “To see a world in a grain of sand\ And a heaven in a wild flower,\ Hold infinity in the palm of your hand\ And eternity in an hour." Telescoping and Partial Fractions {#sect:telescope} ================================= For a quick proof of , consider $$S:= \sum_{n,k>0} \frac{1}{nk(n+k)} = \sum_{n,k>0}\frac1{n^2} \left(\frac1k-\frac1{n+k}\right) = \sum_{n=1}^\infty \frac1{n^2} \sum_{k=1}^n \frac1k = \zeta(3)+\zeta(2,1).$$ On the other hand, $$S = \sum_{n,k>0} \left(\frac1n+\frac1k\right)\frac{1}{(n+k)^2} = \sum_{n,k>0}\frac1{n(n+k)^2}+\sum_{n,k>0}\frac{1}{k(n+k)^2} = 2\zeta(2,1),$$ by symmetry. The above argument goes back at least to Steinberg [@Klamkin]. See also [@Klamkin3]. For , first consider $$\begin{aligned} \zeta(\overline 2,\overline 1)+\zeta(3) &= \sum_{n=1}^\infty \frac{(-1)^n}{n^2}\sum_{k=1}^n \frac{(-1)^k}{k} = \sum_{n=1}^\infty \frac{(-1)^n}{n^2}\sum_{k=1}^\infty \bigg(\frac{(-1)^k}{k}-\frac{(-1)^{n+k}}{n+k}\bigg)\nonumber\\ &=\sum_{n=1}^\infty\frac{(-1)^n}{n^2}\sum_{k=1}^\infty (-1)^k\bigg(\frac{n+k-(-1)^nk}{k(n+k)}\bigg)\nonumber\\ &=\sum_{n,k>0}\frac{(-1)^{n+k}}{nk(n+k)}+\sum_{n,k>0}\frac{(-1)^{n+k}}{n^2(n+k)} -\sum_{n,k>0}\frac{(-1)^k}{n^2(n+k)}\nonumber\\ &=\sum_{n,k>0}\bigg(\frac1n+\frac1k\bigg)\frac{(-1)^{n+k}}{(n+k)^2} +\zeta(\overline 1,2)-\sum_{n,k>0}\frac{(-1)^n(-1)^{n+k}}{n^2(n+k)}\nonumber\\ &=\sum_{n,k>0}\frac{(-1)^{n+k}}{n(n+k)^2}+\sum_{n,k>0}\frac{(-1)^{n+k}}{k(n+k)^2} +\zeta(\overline1,2)-\zeta(\overline1,\overline2)\nonumber\\ &=2\zeta(\overline2,1)+\zeta(\overline1,2)-\zeta(\overline1,\overline2). \label{alt3}\end{aligned}$$ Similarly, $$\begin{aligned} \zeta(2,\overline1)+\zeta(\overline3) &=\sum_{n=1}^\infty\frac{1}{n^2}\sum_{k=1}^n\frac{(-1)^k}{k} =\sum_{n=1}^\infty\frac1{n^2}\sum_{k=1}^\infty\bigg(\frac{(-1)^k}{k}-\frac{(-1)^{n+k}}{n+k}\bigg)\nonumber\\ &=\sum_{n=1}^\infty\frac1{n^2}\sum_{k=1}^\infty(-1)^k\bigg(\frac{n+k-(-1)^nk}{k(n+k)}\bigg)\nonumber\\ &=\sum_{n,k>0}\frac{(-1)^k}{nk(n+k)}+\sum_{n,k>0}\frac{(-1)^k}{n^2(n+k)}-\sum_{n,k>0}\frac{(-1)^{n+k}}{n^2(n+k)}\nonumber\\ &=\sum_{n,k>0}\bigg(\frac1n+\frac1k\bigg)\frac{(-1)^k}{(n+k)^2}+\sum_{n,k>0}\frac{(-1)^n(-1)^{n+k}}{n^2(n+k)} -\zeta(\overline1,2)\nonumber\\ &=\sum_{n,k>0}\frac{(-1)^n(-1)^{n+k}}{n(n+k)^2}+\sum_{n,k>0}\frac{(-1)^k}{k(n+k)^2}+\zeta(\overline1,\overline2) -\zeta(\overline1,2)\nonumber\\ &=\zeta(\overline2,\overline1)+\zeta(2,\overline1)+\zeta(\overline1,\overline2)-\zeta(\overline1,2). \label{alt3bar}\end{aligned}$$ Adding equations and now gives $$\label{dejavu} 2\zeta(\overline2,1)=\zeta(3)+\zeta(\overline3),$$ i.e.$$8\zeta(\overline2,1)=4\sum_{n=1}^\infty\frac{1+(-1)^n}{n^3}=4\sum_{m=1}^\infty \frac{2}{(2m)^3} = \zeta(3),$$ which is . Finite Series Transformations {#sect:finite} ============================= For any positive integer $N$, we have $$\label{z21finite} \sum_{n=1}^N \frac1{n^3} -\sum_{n=1}^N \frac1{n^2}\sum_{k=1}^{n-1}\frac1k =\sum_{n=1}^N \frac1{n^2}\sum_{k=1}^n\frac{1}{N-k+1}$$ by induction. Alternatively, consider $$T := \sum_{\substack{n,k=1\\k\ne n}}^N \frac{1}{nk(k-n)} = \sum_{\substack{n,k=1\\k\ne n}}^N \bigg(\frac1n-\frac1k\bigg) \frac{1}{(k-n)^2} =0.$$ On the other hand, $$\begin{aligned} T &= \sum_{\substack{n,k=1\\k\ne n}}^N \frac{1}{n^2} \bigg(\frac{1}{k-n}-\frac1k\bigg)\\ &= \sum_{n=1}^N\frac{1}{n^2}\bigg(\sum_{k=1}^{n-1}\frac{1}{k-n} +\sum_{k=n+1}^N\frac1{k-n}-\sum_{k=1}^N \frac1k+\frac1n\bigg)\\ &= \sum_{n=1}^N\frac1{n^3} -\sum_{n=1}^N\frac1{n^2}\sum_{k=1}^{n-1}\frac{1}{n-k} +\sum_{n=1}^N\frac1{n^2} \bigg(\sum_{k=n+1}^N\frac1{k-n}-\sum_{k=1}^N\frac1k\bigg).\end{aligned}$$ Since $T=0$, this implies that $$\sum_{n=1}^N\frac1{n^3}-\sum_{n=1}^N\frac1{n^2}\sum_{k=1}^{n-1}\frac1k =\sum_{n=1}^N\frac1{n^2}\bigg(\sum_{k=1}^N \frac1k-\sum_{k=1}^{N-n}\frac1k\bigg) = \sum_{n=1}^N\frac1{n^2}\sum_{k=1}^n \frac{1}{N-k+1},$$ which is . But the right hand side satisfies $$\begin{aligned} \frac{H_N}{N} = \sum_{n=1}^N \frac1{n^2}\cdot\frac{n}{N} & \le \sum_{n=1}^N \frac1{n^2} \sum_{k=1}^n \frac1{N-k+1}\\ & \le \sum_{n=1}^N \frac{1}{n^2}\cdot\frac{n}{N-n+1} =\frac1{N+1}\sum_{n=1}^N\bigg(\frac1{n}+\frac1{N-n+1}\bigg) =\frac{2H_N}{N+1}.\end{aligned}$$ Letting $N$ grow without bound now gives , since ${\displaystyle}\lim_{N\to\infty}\frac{H_N}{N}=0$. [$\square$]{} Geometric Series ================ Convolution of Geometric Series {#sect:Williams} ------------------------------- The following argument is suggested in [@Williams]. A closely related derivation, in which our explicit consideration of the error term is suppressed by taking $N$ infinite, appears in [@Bracken]. Let $2\le m\in{{\mathbf Z}}$, and consider $$\begin{aligned} \sum_{j=1}^{m-2}\zeta(j+1)\zeta(m-j) &= \lim_{N\to\infty} \sum_{n=1}^N\sum_{k=1}^N \sum_{j=1}^{m-2} \frac{1}{n^{j+1}}\frac{1}{k^{m-j}}\\ &= \lim_{N\to\infty} \bigg\{ \sum_{\substack{n,k=1\\k\ne n}}^N \bigg(\frac{1}{n^{m-1}(k-n)k}-\frac{1}{n(k-n)k^{m-1}}\bigg) +\sum_{n=1}^N\frac{m-2}{n^{m+1}}\bigg\}\\ &=(m-2)\zeta(m+1)+2\lim_{N\to\infty}\sum_{\substack{n,k=1\\k\ne n}}^N\frac{1}{n^{m-1}k(k-n)}.\end{aligned}$$ Thus, we find that $$\begin{aligned} & (m-2)\zeta(m+1)-\sum_{j=1}^{m-2}\zeta(j+1)\zeta(m-j)\\ &= 2\lim_{N\to\infty}\sum_{n=1}^N\frac{1}{n^m}\sum_{\substack{k=1\\ k\ne n}}^N \bigg(\frac1k-\frac1{k-n}\bigg)\\ &= 2\lim_{N\to\infty}\sum_{n=1}^N\frac1{n^m}\bigg\{\sum_{k=1}^{n-1} \frac1k-\frac1n+\sum_{k=1}^n\frac1{N-k+1}\bigg\}\\ &= 2\zeta(m,1)-2\zeta(m+1)+2\lim_{N\to\infty}\sum_{n=1}^N \frac1{n^m}\sum_{k=1}^n\frac1{N-k+1},\end{aligned}$$ and hence $$2\zeta(m,1)= m\zeta(m+1)-\sum_{j=1}^{m-2}\zeta(j+1)\zeta(m-j) -2\lim_{N\to\infty}\sum_{n=1}^N\frac1{n^m}\sum_{k=1}^n\frac1{N-k+1}.$$ But, in light of $$\sum_{n=1}^N\frac1{n^m}\sum_{k=1}^n\frac1{N-k+1} \le \sum_{n=1}^N \frac1{n^m}\cdot\frac{n}{N-n+1} \le\frac1{N+1}\sum_{n=1}^N\bigg(\frac1{N-n+1}+\frac1n\bigg) =\frac{2H_N}{N+1},$$ the identity now follows. A Sum Formula {#sect:SumFormula} ------------- Equation is the case $n=3$ of the following result. See [@Briggs0]. \[briggs\] If $3\le n\in{{\mathbf Z}}$ then $$\zeta(n) = \sum_{j=1}^{n-2}\zeta(n-j,j). \label{sumdepth2}$$ We discuss a generalization of the sum formula to arbitrary depth in §\[sect:SumGF\]. [Proof.]{} Summing the geometric series on the right hand side gives $$\begin{aligned} \sum_{j=1}^{n-2}\sum_{h=1}^\infty\sum_{m=1}^\infty \frac1{h^j(h+m)^{n-j}} &= \sum_{h,m=1}^\infty \bigg[\frac{1}{h^{n-2}m(h+m)}-\frac{1}{m(h+m)^{n-1}}\bigg]\\ &=\sum_{h=1}^\infty \frac1{h^{n-1}}\sum_{m=1}^\infty \bigg( \frac1m-\frac1{h+m}\bigg)-\zeta(n-1,1)\\ &= \sum_{h=1}^\infty \frac1{h^{n-1}}\sum_{k=1}^{h}\frac1k -\zeta(n-1,1)\\ &= \sum_{h=1}^\infty \frac1{h^n} +\sum_{h=1}^\infty \frac1{h^{n-1}}\sum_{k=1}^{n-1}\frac1k-\zeta(n-1,1)\\ &= \zeta(n).\end{aligned}$$ A $q$-Analogue {#sect:zud} -------------- The following argument is based on an idea of Zudilin [@Zud]. We begin with the finite geometric series identity $$\frac{uv}{(1-u)(1-uv)^s}+\frac{uv^2}{(1-v)(1-uv)^s} = \frac{uv}{(1-u)(1-v)^s} - \sum_{j=1}^{s-1}\frac{uv^2}{(1-v)^{j+1}(1-uv)^{s-j}},$$ valid for all positive integers $s$ and real $u$, $v$ with $u\ne 1$, $uv\ne 1$. We now assume $s>1$, $q$ is real and $0<q<1$. Put $u=q^m$, $v=q^n$ and sum over all positive integers $m$ and $n$. Thus, $$\begin{aligned} &\sum_{m,n>0} \frac{q^{m+n}}{(1-q^m)(1-q^{m+n})^s} +\sum_{m,n>0}\frac{q^{m+2n}}{(1-q^n)(1-q^{m+n})^s}\\ &= \sum_{m,n>0}\frac{q^{m+n}}{(1-q^m)(1-q^n)^s} -\sum_{m,n>0}\frac{q^{m+2n}}{(1-q^n)^s(1-q^{m+n})}\\ &\qquad -\sum_{j=1}^{s-2}\sum_{m,n>0} \frac{q^{m+2n}}{(1-q^n)^{j+1}(1-q^{m+n})^{s-j}}\\ &= \sum_{m,n>0}\frac{q^n}{(1-q^n)^s}\bigg[\frac{q^m}{1-q^m}- \frac{q^{m+n}}{1-q^{m+n}}\bigg]-\sum_{j=1}^{s-2}\sum_{m,n>0} \frac{q^{m+2n}}{(1-q^n)^{j+1}(1-q^{m+n})^{s-j}}\\ &= \sum_{n>0}\frac{q^n}{(1-q^n)^s}\sum_{m=1}^n\frac{q^m}{1-q^m} -\sum_{j=1}^{s-2}\sum_{m,n>0} \frac{q^{m+2n}}{(1-q^n)^{j+1}(1-q^{m+n})^{s-j}}\\ &=\sum_{n>0}\frac{q^{2n}}{(1-q^n)^{s+1}}+\sum_{n>m>0} \frac{q^{n+m}}{(1-q^n)^s(1-q^m)}-\sum_{j=1}^{s-2}\sum_{m,n>0} \frac{q^{m+2n}}{(1-q^n)^{j+1}(1-q^{m+n})^{s-j}}\\\end{aligned}$$ Cancelling the second double sum on the left with the corresponding double sum on the right and replacing $m+n$ by $k$ in the remaining sums now yields $$\sum_{k>m>0}\frac{q^{k}}{(1-q^{k})^s(1-q^m)} = \sum_{n>0}\frac{q^{2n}}{(1-q^n)^{s+1}}- \sum_{j=1}^{s-2}\sum_{k>m>0} \frac{q^{k+m}}{(1-q^m)^{j+1}(1-q^{k})^{s-j}},$$ or equivalently, that $$\sum_{k>0}\frac{q^{2k}}{(1-q^k)^{s+1}} = \sum_{k>m>0}\frac{q^k}{(1-q^k)^s(1-q^m)} +\sum_{j=1}^{s-2}\sum_{k>m>0}\frac{q^{k+m}}{(1-q^k)^{s-j}(1-q^m)^{j+1}}. \label{general}$$ Multiplying through by $(1-q)^{s+1}$ and letting $q\to 1$ gives $$\zeta(s+1)=\zeta(s,1)+\sum_{j=1}^{s-2} \zeta(s-j,j+1),$$ which is just a restatement of . Taking $s=2$ gives again. As in [@DBq], define the $q$-analog of a non-negative integer $n$ by $$[n]_q := \sum_{k=0}^{n-1} q^k = \frac{1-q^n}{1-q},$$ and the multiple $q$-zeta function $$\zeta[s_1,\dots,s_m] := \sum_{k_1>\cdots >k_m>0} \; \prod_{j=1}^m \frac{q^{(s_j-1)k_j}}{[k_j]_q^{s_j}}, \label{qMZVdef}$$ where $s_1,s_2,\dots,s_m$ are real numbers with $s_1>1$ and $s_j\ge 1$ for $2\le j\le m$. Then multiplying by $(1-q)^{s+1}$ and then setting $s=2$ gives $\zeta[2,1]=\zeta[3]$, which is a $q$-analog of . That is, the latter may be obtained from the former by letting $q\to 1-$. On the other hand, $s=3$ in gives $$\zeta[4] +(1-q)\zeta[3]=\zeta[3,1]+(1-q)\zeta[2,1]+\zeta[2,2],$$ which, in light of $\zeta[2,1]=\zeta[3]$ implies $\zeta[3,1] = \zeta[4]-\zeta[2,2]$. By Theorem 1 of [@DBq], we know that $\zeta[2,2]$ reduces to depth 1 multiple $q$-zeta values. Indeed, by the $q$-stuffle multiplication rule [@DBq], $ \zeta[2]\zeta[2] = 2\zeta[2,2] + \zeta[4] +(1-q)\zeta[3]. $ Thus, $$\zeta[3,1] = \zeta[4]-\zeta[2,2]= \tfrac32\zeta[4]-\tfrac12\left(\zeta[2]\right)^2+\tfrac12(1-q)\zeta[3],$$ which is a $q$-analog of the evaluation [@BBB; @BBBLa; @BBBLc; @BowBrad1; @BowBradSurvey; @BowBrad3; @BowBradRyoo] $$\zeta(3,1) = \frac{\pi^4}{360}.$$ Additional material concerning $q$-analogs of multiple harmonic sums and multiple zeta values can be found in [@DBq; @DBqKarl; @DBqSum; @DBqDecomp]. Integral Representations {#sect:integrals} ======================== Single Integrals I {#sect:single1} ------------------ We use the fact that $$\label{naive} \int_0^1 u^{k-1}(-\log u)\,du = \frac1{k^2},\qquad k>0.$$ Thus $$\begin{aligned} \label{logs} \sum_{k>n>1}\frac{1}{k^2n} &= \sum_{n=1}^\infty \frac1n\sum_{k>n}\int_0^1 u^{k-1}(-\log u)\,du \nonumber\\ &= \sum_{n=1}^\infty\frac1n \int_0^1 (-\log u)\sum_{k>n} u^{k-1}\,du \nonumber\\ &= \sum_{n=1}^\infty\frac1n \int_0^1 (-\log u) \frac{u^n}{1-u}\,du \nonumber\\ &= -\int_0^1 \frac{\log u}{1-u}\sum_{n=1}^\infty \frac{u^n}{n}\,du \nonumber\\ &= \int_0^1 (-\log u)(1-u)^{-1}\log(1-u)^{-1}\,du.\end{aligned}$$ The interchanges of summation and integration are in each case justified by Lebesgue’s monotone convergence theorem. After making the change of variable $t=1-u$, we obtain $$\label{morelogs} \sum_{k>n>1}\frac{1}{k^2n} = \int_0^1 \log(1-t)^{-1} (-\log t)\,\frac{dt}{t} = \int_0^1 (-\log t) \sum_{n=1}^\infty \frac{t^{n-1}}{n}\,dt.$$ Again, since all terms of the series are positive, Lebesgue’s monotone convergence theorem permits us to interchange the order of summation and integration. Thus, invoking again, we obtain $$\sum_{k>n>1}\frac{1}{k^2n} = \sum_{n=1}^\infty \frac1n\int_0^1 (-\log t)\, t^{n-1}\,dt = \sum_{n=1}^\infty \frac1{n^3},$$ which is . Single Integrals II {#sect:single2} ------------------- The Laplace transform $$\int_0^1 x^{r-1} (-\log x)^\sigma\,dx = \int_0^{\infty} e^{-ru}\, u^\sigma\,du = \frac{\Gamma(\sigma+1)}{r^{\sigma+1}},\qquad r>0,\quad \sigma>-1, \label{Laplace}$$ generalizes and yields the representation $$\zeta(m+1) = \frac{1}{m!}\sum_{r=1}^\infty\frac{\Gamma(m+1)}{r^{m+1}} = \frac{1}{m!}\sum_{r=1}^\infty \int_0^1 x^{r-1}(-\log x)^m \,dx = \frac{(-1)^m}{m!}\int_0^1 \frac{\log^m x}{1-x}\,dx.$$ The interchange of summation and integration is valid if $m>0$. The change of variable $x\mapsto 1-x$ now yields $$\zeta(m+1) = \frac{(-1)^m}{m!}\int_0^1 \log^m(1-x)\,\frac{dx}{x}, \qquad 1\le m\in{{\mathbf Z}}. \label{ZetaStirling}$$ In [@Farnum], equation in conjunction with clever use of change of variable and integration by parts, is used to prove the identity $$k!\zeta(k+2) = \sum_{n_1=1}^\infty\, \sum_{n_2=1}^\infty \cdots \sum_{n_k=1}^\infty \frac{1}{n_1n_2\cdots n_k}\; \sum_{p=1+n_1+n_2+\cdots+n_k}\frac1{p^2}, \qquad 0\le k\in{{\mathbf Z}}. \label{Tissier}$$ The case $k=1$ of is precisely . We give here a slightly simpler proof of , dispensing with the integration by parts. From , $$\begin{aligned} k!\zeta(k+2) &= \sum_{r=1}^\infty \frac1{r}\cdot\frac{\Gamma(k+1)}{r^{k+1}} = \sum_{r=1}^\infty \frac1r\int_0^1 x^{r-1}(-\log x)^k\,dx\\ &= \int_0^1 (-\log x)^k \log(1-x)^{-1}\,\frac{dx}{x}\\ &= \int_0^1 \log^k(1-x)^{-1} (-\log x)\frac{dx}{1-x}\\ &=\sum_{n_1=1}^\infty\, \sum_{n_2=1}^\infty \cdots\sum_{n_k=1}^\infty \frac{1}{n_1n_2\cdots n_k}\; \int_0^1 \frac{x^{n_1+n_2+\cdots+n_k}}{1-x}(-\log x)\,dx\\ &= \sum_{n_1=1}^\infty\, \sum_{n_2=1}^\infty \cdots\sum_{n_k=1}^\infty \frac{1}{n_1n_2\cdots n_k}\; \sum_{p>n_1+n_2+\cdots+n_k} \int_0^1 x^{p-1}(-\log x)\,dx\\ &= \sum_{n_1=1}^\infty\, \sum_{n_2=1}^\infty \cdots\sum_{n_k=1}^\infty \frac{1}{n_1n_2\cdots n_k}\; \sum_{p>n_1+n_2+\cdots+n_k}\frac1{p^2}.\end{aligned}$$ Double Integrals I {#sect:double1} ------------------ Write $$\begin{aligned} \zeta(2,1) =\sum_{k,m>0}\frac{1}{k(m+k)^2} &= \int_0^1\int_0^1 \sum_{k>0}\frac{(xy)^k}{k}\sum_{m>0} (xy)^{m-1}\,dx\,dy\\ &= -\int_0^1\int_0^1 \frac{\log(1-xy)}{1-xy}\,dx\,dy.\end{aligned}$$ Now make the change of variable $u=xy$, $v=x/y$ with Jacobian $1/(2v)$, obtaining $$\begin{aligned} \zeta(2,1) = -\frac12\int_0^1 \frac{\log(1-u)}{1-u}\int_u^{1/u}\frac{dv}{v}\,du = \int_0^1 \frac{(\log u)\log(1-u)}{1-u}\,du,\end{aligned}$$ which is . Now continue as in §\[sect:single1\]. Double Integrals II {#sect:double2} ------------------- The following is reconstructed from a phone conversation with Krishna Alladi. See also [@Beukers]. Let ${\varepsilon}>0$. By expanding the integrand as a geometric series, one sees that $$\sum_{n=1}^\infty \frac1{(n+{\varepsilon})^2} = \int_0^1\int_0^1 \frac{(xy)^{{\varepsilon}}}{1-xy}\,dx\,dy.$$ Differentiating with respect to ${\varepsilon}$ and then letting ${\varepsilon}=0$ gives $$\zeta(3) = -\frac12\int_0^1\int_0^1 \frac{\log(xy)}{1-xy}\,dx\,dy = -\frac12\int_0^1\int_0^1 \frac{\log x+\log y}{1-xy}\,dx\,dy = -\int_0^1\int_0^1\frac{\log x}{1-xy}\,dx$$ by symmetry. Now integrate with respect to $y$ to get $$\label{parts} \zeta(3) = \int_0^1 (\log x)\log(1-x)\frac{dx}{x}.$$ Comparing with completes the proof of . Integration by Parts {#sect:parts} -------------------- Start with and integrate by parts, obtaining $$2\zeta(3) = \int_0^1\frac{\log^2 x}{1-x}\,dx = \int_0^1 \log^2(1-x)\frac{dx}{x} = \sum_{n,k>0} \int_0^1 \frac{x^{n+k-1}}{nk}\,dx = \sum_{n,k>0}\frac{1}{nk(n+k)}.$$ Now see §\[sect:telescope\]. Triple Integrals I {#sect:iterint1} ------------------ This time, instead of we use the elementary identity $$\frac{1}{k^2n} = \int_0^1 y_1^{-1}\int_0^{y_1}y_2^{k-n-1} \int_0^{y_2} y_3^{n-1}\,dy_3\,dy_2\,dy_1,\qquad k>n>0.$$ This yields $$\label{irep} \sum_{k>n>0}\frac{1}{k^2n} = \int_0^1 y_1^{-1}\int_0^{y_1}(1-y_2)^{-1}\int_0^{y_2}(1-y_3)^{-1} \,dy_3\,dy_2\,dy_1.$$ Now make the change of variable $y_i=1-x_i$ for $i=1,2,3$ to obtain $$\begin{aligned} \sum_{k>n>0}\frac{1}{k^2n} &= \int_0^1 (1-x_1)^{-1}\int_{x_1}^1 x_2^{-1}\int_{x_2}^1 x_3^{-1} \,dx_3\,dx_2\,dx_1\\ &= \int_0^1 x_3^{-1}\int_0^{x_3}x_2^{-1}\int_0^{x_2}(1-x_1)^{-1} \,dx_1\,dx_2\,dx_3.\end{aligned}$$ After expanding $(1-x_1)^{-1}$ into a geometric series and interchanging the order of summation and integration, one arrives at $$\sum_{k>n>0}\frac{1}{k^2n} = \sum_{n=1}^\infty \int_0^1 x_3^{-1}\int_0^{x_3}x_2^{-1}\int_0^{x_2} x_1^{n-1}\,dx_1\,dx_2\,dx_3 = \sum_{n=1}^\infty \frac1{n^3},$$ as required. More generally [@BBB; @BBBLa; @BBBLc; @BowBradSurvey; @BowBrad3; @CK], $$\zeta(s_1,\dots,s_k) = \sum_{n_1>\cdots>n_k>0}\;\prod_{j=1}^k n_j^{-s_j} =\int \prod_{j=1}^k \bigg(\prod_{r=1}^{s_j-1} \frac{dt_r^{(j)}}{t_r^{(j)}}\bigg) \frac{dt_{s_j}^{(j)}}{1-t_{s_j}^{(j)}}, \label{iterint1}$$ where the integral is over the simplex $$1>t_1^{(1)}>\cdots>t_{s_1}^{(1)}>\cdots>t_1^{(k)}>\cdots>t_{s_k}^{(k)}>0,$$ and is abbreviated by $$\int_0^1 \prod_{j=1}^k a^{s_j-1}b, \qquad a=\frac{dt}{t},\qquad b=\frac{dt}{1-t}. \label{shortiterint1}$$ The change of variable $t\mapsto 1-t$ at each level of integration switches the differential forms $a$ and $b$, thus yielding the duality formula [@BBB] [@CK p. 483] (conjectured in [@Hoff92]) $$\label{duality} \zeta(s_1+2,{\{1\}}^{r_1},\dots,s_n+2,{\{1\}}^{r_n}) = \zeta(r_n+2,{\{1\}}^{s_n},\dots,r_1+2,{\{1\}}^{s_1}),$$ which is valid for all nonnegative integers $s_1,r_1,\ldots,s_n,r_n$. The case $s_1=0$, $r_1=1$ of is . More generally, can be restated as $$\int_0^1 (ab^2)^n = \int_0^1 (a^2b)^n$$ and thus is recovered by taking each $s_j=0$ and each $r_j=1$ in . For further generalizations and extensions of duality, see [@BBBLa; @DBq; @DBqKarl]. For alternations, we require in addition the differential form $c:=-dt/(1+t)$ with which we may form the generating function $$\sum_{n=1}^\infty z^{3n} \zeta(\{\overline2,1\}^n) = \sum_{n=0}^\infty \bigg\{ z^{6n+3}\int_0^1 (ac^2ab^2)^n ac^2 + z^{6n+6} \int_0^1 (ac^2ab^2)^{6n+6}\bigg\}.$$ A lengthy calculation verifies that the only changes of variable that preserve the unit interval and send the non-commutative polynomial ring ${{\mathbf Q}}\langle a,b\rangle$ into ${{\mathbf Q}}\langle a,b,c\rangle$ are $$\begin{aligned} {2} S(a,b) &= S(a,b),\qquad\qquad & t &\mapsto t,\label{identity}\\ S(a,b) &= R(b,a),\qquad\qquad & t &\mapsto 1-t,\label{tau}\\ S(a,b) &= S(2a,b+c),\qquad\qquad & t &\mapsto t^2,\label{sumsigns}\\ S(a,b) &= S(a+c,b-c),\qquad\qquad & t &\mapsto \frac{2t}{1+t},\label{Landen}\\ S(a,b) &= S(a+2c,2b-2c),\qquad\qquad & t &\mapsto \frac{4t}{(1+t)^2},\label{quadLanden}\end{aligned}$$ and compositions thereof, such as $t\mapsto 1-2t/(1+t)=(1-t)/(1+t)$, etc. In –, $S(a,b)$ denotes a non-commutative word on the alphabet $\{a,b\}$ and $R(b,a)$ denotes the word formed by switching $a$ and $b$ and then reversing the order of the letters. Now view $a$, $b$ and $c$ as indeterminates. In light of the polynomial identity $$ab^2-8ac^2 = 2[ab^2-2a(b+c)^2] + 8[ab^2-(a+c)(b-c)^2] + [(a+2c)(2b-2c)^2-ab^2]$$ in the non-commutative ring ${{\mathbf Z}}\langle a,b,c\rangle$ and the transformations , and above, each bracketed term vanishes when we make the identifications $a=dt/t$, $b=dt/(1-t)$, $c=-dt/(1+t)$ and perform the requisite iterated integrations. Thus, $$\zeta(2,1)-8\zeta(\overline2,1)=\int_0^1 ab^2-8\int_0^1 ac^2 =0,$$ which in light of proves . Triple Integrals II {#sect:iterint2} ------------------- First, note that by expanding the integrands in geometric series and integrating term by term, $$\zeta(2,1) = 8\int_0^1 \frac{dx}{x}\int_0^x \frac{y\,dy}{1-y^2}\int_0^y\frac{z\,dz}{1-z^2}.$$ Now make the change of variable $$\frac{x\,dx}{1-x^2}=\frac{du}{1+u}, \qquad \frac{y\,dy}{1-y^2}=\frac{dv}{1+v}, \qquad \frac{z\,dz}{1-z^2}=\frac{dw}{1+w}$$ to obtain the equivalent integral $$\zeta(2,1) = 8\int_0^\infty \bigg(\frac{du}{2u}+\frac{du}{2(2+u)} -\frac{du}{1+u}\bigg)\int_0^u\frac{dv}{1+v} \int_0^v\frac{dw}{1+w}.$$ The two inner integrals can be directly performed, leading to $$\zeta(2,1) = 4\int_0^\infty\frac{\log^2(u+1)}{u(u+1)(u+2)}\,{du}.$$ Finally, make the substitution $u+1=1/\sqrt{1-x}$ to obtain $$\zeta(2,1) = \frac12\int_0^1 \frac{\log^2(1-x)}{x}\,dx = \zeta(3),$$ by . Complex Line Integrals I {#sect:Perron} ------------------------ Here we apply the Mellin inversion formula [@Apostol p.243], [@Tenenbaum pp. 130–132 ] $$\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} y^z\,\frac{dz}{z} = \ \begin{cases} 1,\quad y>1 \\ 0, \quad y<1\\ \tfrac12,\quad y=1\end{cases}$$ which is valid for fixed $c>0$. It follows that if $c>0$ and $s-1>c>1-t$ then the Perron-type formula $$\begin{aligned} \zeta(s,t) +\frac12\zeta(s+t) &= \sum_{n=1}^\infty n^{-s} \sum_{k=1}^\infty k^{-t} \frac1{2\pi i} \int_{c-i\infty}^{c+i\infty} \bigg(\frac{n}{k}\bigg)^z\frac{dz}{z}\nonumber\\ &=\frac1{2\pi i}\int_{c-i\infty}^{c+i\infty} \zeta(s-z)\zeta(t+z)\,\frac{dz}{z} \label{Perron}\end{aligned}$$ is valid. (Interchanging the order of summation and integration is permissible by absolute convergence.) Although we have not yet found a way to exploit in proving identities such as , we note that by integrating around the rectangular contour with corners $(\pm c\pm iM)$ and then letting $M\to+\infty$, one can readily establish the stuffle [@BBBLa; @BowBradSurvey; @DBPrtn; @DBq] formula in the form $$\zeta(s,t)+\frac12\zeta(s+t)+\zeta(t,s)+\frac12\zeta(t+s) = \zeta(s)\zeta(t), \qquad s,t>1+c.$$ The right hand side arises as the residue contribution of the integrand at $z=0$. One can also use to establish $$\sum_{s=2}^\infty\left[\zeta(s,1)+\tfrac12\zeta(s+1)\right]x^{s-1} = \sum_{n>m>0}\frac{x}{mn(n-x)} +\frac12\sum_{n=1}^\infty \frac{x}{n(n-x)},$$ but this is easy to prove directly. Complex Line Integrals II {#sect:Dirichlet} ------------------------- We let $\lambda(s) :=\sum_{n>0} \lambda_n\, n^{-s}$ represent a formal Dirichlet series, with real coefficients $\lambda_n$ and we set $s:=\sigma+i\,\tau$ with $\sigma=\Re(s)>0$, and consider the following integral: $$\begin{aligned} \label{int1} \iota_\lambda(\sigma):=\int_{0}^\infty \left\|\frac{\lambda(s)}s\right\|^2\,d\tau = \frac 12\, \int_{-\infty}^\infty \left\|\frac{\lambda(s)}s\right\|^2\,d\tau,\end{aligned}$$ as a function of $\lambda$. We begin with a useful variant of the Mellin inversion formula, namely $$\begin{aligned} \label{intc}\int _{-\infty}^{\infty }\!{\frac {\cos \left( at \right) }{{t}^{2}+{ u}^{2}}}{dt}={\frac {\pi }{u}}\,e^{-au},\end{aligned}$$ for $u, a>0$, as follows by contour integration, from a computer algebra system, or otherwise. This leads to \[dir-thm\] [(Theorem 1 of [@JB]).]{} [For $\lambda(s)=\sum_{n=1}^\infty \,\lambda_n\,n^{-s}$ and $s=\sigma+i\,\tau$ with fixed $\sigma=\Re(s)>0$ such that the Dirichlet series is absolutely convergent it is true that $$\begin{aligned} \label{ans1} \iota_\lambda(\sigma) = \int_{0}^\infty \left\|\frac{\lambda(s)}s\right\|^2\,d\tau =\frac{\pi}{2\sigma}\,\sum_{n=1}^\infty\frac{\Lambda_n^2- \Lambda_{n-1}^2}{n^{2\sigma}},\end{aligned}$$ where $\Lambda_n:=\sum_{k=1}^{n}\lambda_k$ and $\Lambda_0:=0$.]{} More generally, for given absolutely convergent Dirichlet series $\alpha(s):=\sum_{n=1}^\infty\,{\alpha_n}\,{n^{-s}}$ and $\beta(s):=\sum_{n=1}^\infty\,{\beta_n}\,{n^{-s}}$ $$\begin{aligned} \label{ans-gen} \frac12\int_{-\infty}^\infty \frac{\alpha(s)\,\overline{\beta}(s)}{\sigma^2+\tau^2}\,d\tau = \frac{\pi}{2\sigma}\,\sum_{n=1}^\infty \frac{A_n\,\overline{B_n}-A_{n-1}\,\overline{B_{n-1}}}{n^{2\sigma}},\label{ans3}\end{aligned}$$ in which $A_n=\sum_{k=1}^{n}\alpha_k$ and $B_n=\sum_{k=1}^{n}\beta_k.$ Note that the righthand side of is always a generalized Euler sum. For the Riemann zeta function, and for $\sigma>1$, Theorem \[dir-thm\] applies and yields $$\frac{\sigma}{\pi}\,\iota_\zeta(\sigma)= \zeta(2\sigma-1)-\frac12\,\zeta(2\sigma),$$ as $\lambda_n=1$ and $\Lambda_n=n-1/2$. By contrast it is known that on the critical line $$\frac{1/2}{\pi}\,\iota_\zeta\left(\frac 12\right)=\log(\sqrt{2\,\pi})-\frac 12\,\gamma.$$ There are similar formulae for $s \mapsto \zeta(s-k)$ with $k$ integral. For instance, applying the result in (\[ans1\]) with $\zeta_1:=t \mapsto \zeta(t+1)$ yields $$\frac{1}{\pi}\,\int_{0}^\infty \frac{\|\zeta(3/2+i\tau)\|^2}{1/4+\tau^2}\,d\tau = \frac{1}{\pi}\,\iota_{\zeta_1}\left(\frac 1 2\right)= 2\, \zeta(2,1)+\zeta(3)=3\,\zeta(3),$$ on using . For the alternating zeta function, $\alpha:=s\mapsto(1-2^{1-s})\zeta(s)$, the same approach via (\[ans-gen\]) produces $$\frac{1}{\pi}\,\int_{0}^\infty \frac{\alpha(3/2+i\tau)\,\overline{\alpha(3/2+i\tau)}}{1/4+\tau^2}\,d\tau = 2\,\zeta(\overline{2},\overline{1})+\zeta(3)=3\, \zeta(2)\,\log(2)-\frac 94 \zeta(3),$$ and $$\frac{1}{2\pi}\,\int_{-\infty}^\infty \frac{\alpha(3/2+i\tau)\,\overline{\zeta(3/2+i\tau)}}{1/4+\tau^2}\,d\tau = \zeta(\overline{2},1)+\zeta(2,\overline{1})+\alpha(3)=\frac 98\, \zeta(2)\,\log(2)-\frac {3}4 \zeta(3),$$ since as we have seen repeatedly $\zeta(\overline{2},1) =\zeta(3)/8$; while $\zeta(2,\overline{1})=\zeta(3)-3/2\,\zeta(2)\log(2)$ and $\zeta(\overline{2},\overline{1})= 3/2\,\zeta(2)\log(2)-13/8\,\zeta(3),$ (e.g., [@BZB]). As in the previous subsection we have not been able to directly obtain or even , but we have connected them to quite difficult line integrals. Contour Integrals and Residues {#sect:residue} ------------------------------ Following [@State], let $\mathscr{C}_n$ $(n\in{{\mathbf Z}}^{+})$ be the square contour with vertices $(\pm 1\pm i)(n+1/2)$. Using the asymptotic expansion $$\psi(z) \sim \log z - \frac1{2z}-\sum_{r=1}^\infty\frac{B_{2r}}{2rz^{2r}}, \qquad \|\arg z\|<\pi$$ in terms of the Bernoulli numbers $$\frac{t}{1-e^{-t}}=1+\frac{t}{2}+\sum_{r=1}^\infty \frac{B_{2r}}{(2r)!}t^{2r},\qquad \|t\|<2\pi$$ and the identity $$\psi(z)=\psi(-z) - \frac1{z}-\pi\cot \pi z,$$ we can show that for each integer $k\ge 2$, $$\lim_{n\to\infty} \int_{\mathscr{C}_n} z^{-k}\,\psi^2(-z)\,dz = 0.$$ Then by the residue theorem, we obtain \[thm:State\] For every integer $k\ge 2$, $$2\sum_{n=1}^\infty n^{-k}\,\psi(n) = k\zeta(k+1)-2\gamma \zeta(k)-\sum_{j=1}^{k-1}\zeta(j)\zeta(k-j+1),$$ where $\gamma=0.577215664\dots$ is Euler’s constant. In light of the identity $$\psi(n)+\gamma= H_{n-1}=\sum_{k=1}^{n-1}\frac1k,\qquad n\in{{\mathbf Z}}^{+},$$ Theorem \[thm:State\] is equivalent to . The case $k=2$ thus gives . Flajolet and Salvy [@Flaj] developed the residue approach more systematically, and applied it to a number of other Euler sum identities in addition to . Witten Zeta-functions {#sect:witten} ===================== We recall that for $r,s>1/2$: $$\mathcal{W}(r,s,t):= \sum_{n=1}^\infty\sum_{m=1}^\infty \frac{1}{n^r\,m^s\,(n+m)^t}$$ is a Witten $\zeta$-function, [@zagier; @moll; @CranBuh]. We refer to [@zagier] for a description of the uses of more general Witten $\zeta$-functions. Ours are also called Tornheim double sums, [@moll]. There is a simple algebraic relation $$\begin{aligned} \label{w-alg}\mathcal{W}(r,s,t)=\mathcal{W}(r-1,s,t+1)+\mathcal{W}(r,s-1,t+1).\end{aligned}$$ This is based on writing $$\frac{m+n}{(m+n)^{t+1}}=\frac{m}{(m+n)^{t+1}}+\frac{n}{(m+n)^{t+1}}.$$ Also $$\begin{aligned} \label{w-alg1}\mathcal{W}(r,s,t) = \mathcal{W}(s,r,t),\end{aligned}$$ and $$\begin{aligned} \label{w-alg2}\mathcal{W}(r,s,0)=\zeta(r)\,\zeta(s)\quad \mbox{while} \quad \mathcal{W}(r,0,t) =\zeta(t,r).\end{aligned}$$ Hence, $\mathcal{W}(s,s,t)=2\,\mathcal{W}(s,s-1,t+1)$ and so $$\mathcal{W}(1,1,1)=2\,\mathcal{W}(1,0,2)=2\,\zeta(2,1)=2\,\zeta(3).$$ Note the analogue to (\[w-alg\]), viz. $\zeta(s,t)+\zeta(t,s)=\zeta(s)\,\zeta(t)-\zeta(s+t)$, shows $\mathcal{W}(s,0,s)=2\,\zeta(s,s)=\zeta^2(s)-\zeta(2s).$ Thus, $\mathcal{W}(2,0,2)=2\,\zeta(2,2)=\pi^4/36-\pi^4/90=\pi^4/72$. More generally, recursive use of (\[w-alg\]) and (\[w-alg1\]), along with initial conditions (\[w-alg2\]) shows that all integer $\mathcal{W}(s,r,t)$ values are expressible in terms of double (and single) Euler sums. If we start with $\Gamma(s)/(m+n)^{t} = \int _0^1\! (-\log \sigma)^{t-1}\,\sigma^{m+n-1}\,d \sigma $ we obtain $$\begin{aligned} \label{gamma}\mathcal{W}(r,s,t)= \frac{1}{\Gamma(t)}\,\int _0^1\! {\rm Li}_r(\sigma)\,{\rm Li}_s(\sigma)\,\frac{\left(-\log \sigma\right)^{t-1}}{\sigma}\,d \sigma.\end{aligned}$$ For example, we recover an analytic proof of $$\begin{aligned} \label{z21-w}2\,\zeta(2,1)=\mathcal{W}(1,1,1)= \int _0^1\!\frac{\ln^2(1-\sigma)}{\sigma}\,d\sigma =2\,\zeta(3),\end{aligned}$$ Indeed $S$ in the proof of §\[sect:telescope\] is precisely $\mathcal{W}(1,1,1)$. We may now discover analytic as opposed to algebraic relations. Integration by parts yields $$\begin{aligned} \label{w-parts}\mathcal{W}(r,s+1,1) +\mathcal{W}(r+1,s,1)={\rm Li}_{r+1}(1)\,{\rm Li}_{s+1}(1)=\zeta(r+1)\,\zeta(s+1),\end{aligned}$$ So, in particular, $\mathcal{W}(s+1,s,1)=\zeta^2(s+1)/2$. Symbolically, Maple immediately evaluates $\mathcal{W}(2,1,1)=\pi^4/72,$ and while it fails directly with $\mathcal{W}(1,1,2)$, we know it must be a multiple of $\pi^4$ or equivalently $\zeta(4)$; and numerically obtain $\mathcal{W}(1,1,2)/\zeta(4)=.49999999999999999998\ldots$. The Hilbert Matrix ------------------ Letting $a_n:=1/n^r$ and $b_n:=1/n^s$, inequality (\[hilbert-p\]) of Section \[sect:hardy\] yields $$\begin{aligned} \label{wittenp}\mathcal{W}(r,s,1) \le \pi\,{\rm csc}\left(\frac{\pi}p\right)\,\sqrt[p]{\zeta(pr)}\,\sqrt[q]{\zeta(qs)}.\end{aligned}$$ Indeed, the constant in (\[hilbert-p\]) is best possible [@ghh; @steele]. We consider $$\mathcal{R}_p(s):=\frac{\mathcal{W}((p-1)s,s,1)}{\pi\,\zeta(ps)},$$ and observe that with $\sigma_n^p(s):=\sum_{m=1}^\infty (n/m)^{-(p-1)s}/(n+m) \to \pi\,{\rm csc}\left(\frac{\pi}q \right),$ we have $$\begin{aligned} \mathcal{L}_p:&=&\lim_{s\to 1/p}(ps-1)\,\sum_{n=1}^\infty\sum_{m=1}^\infty \frac{n^{-s}\,m^{-(p-1)s}}{n+m}= \lim_{s\to 1/p}(ps-1)\,\sum_{n=1}^\infty\frac{1}{n^{ps}}\,\sigma_n^p(s)\\&=&\lim_{s\to 1/p}\,(ps-1) \,\sum_{n=1}^\infty \,\frac{\left\{\sigma_n^p(s)-\pi\,{\rm csc}\left(\pi/q) \right)\right\}}{n^{ps}}+\lim_{s\to 1/p}\,(2s-1)\zeta(ps)\,\pi\,{\rm csc}\left(\frac{\pi}q \right)\\&=&0+\pi\,{\rm csc}\left(\frac{\pi}q \right).\end{aligned}$$ Setting $r:=(p-1)s,s \to 1/p^+$ we check that $\zeta(ps)^{1/p}\,\zeta(qr)^{1/q}=\zeta(ps)$ and hence the best constant in (\[wittenp\]) is the one given. To recapitulate in terms of the celebrated infinite Hilbert matrix, $\mathcal{H}_0:=\left\{1/(m+n)\right\}_{m,n=1}^\infty$, [@exp2 pp. 250–252], we have actually proven: \[hmat\] Let $1<p,q < \infty$ be given with $1/p+1/q=1$. The Hilbert matrice $\mathcal{H}_0$ determines a bounded linear mappings from the sequence space $\ell^p$ to itself such that $$\\|\mathcal{H}_0\\|_{p,p}=\lim_{s \to 1/p}\frac{\mathcal{W}(s,(p-1)s,1)}{\zeta(ps)}=\pi\,{\rm csc}\left(\frac{\pi}p \right).$$ [Proof.]{} Appealing to the isometry between $(\ell^p)^$ and $\ell^q$, and given the evaluation $\mathcal{L}_p$ above, we directly compute the operator norm of $\mathcal{H}_0$ as $$\begin{aligned} \\|\mathcal{H}_0\\|_{p,p} = \sup_{\\|x\\|_p=1} \\|\mathcal{H}_0 x\\|_p=\sup_{\\|y\\|_q=1}\sup_{\\|x\\|_p=1} \langle \mathcal{H}_0 x, y\rangle=\pi\,{\rm csc}\left(\frac{\pi}p \right).\end{aligned}$$ A delightful operator-theoretic introduction to the Hilbert matrix $\mathcal{H}_0$ is given by Choi in his Chauvenet prize winning article [@choi]. One may also study the corresponding behaviour of Hardy’s inequality (\[h-ineq\]). For example, setting $a_n:=1/n$ in (\[h-ineq\]) and denoting $H_n:=\sum_{k=1}^n 1/k$ yields $$\sum_{n=1}^\infty \left(\frac{H_n}{n}\right)^p \le \left(\frac{p}{p-1}\right)^p\,\zeta(p).$$ Application of the integral test and the evaluation $$\int_1^\infty\,\left(\frac{\log x}{x}\right)^p\,dx = \frac{\Gamma \left( 1+p \right)}{ \left( p-1 \right) ^{p+1}},$$ for $p>1$ easily shows the constant is again best possible. A Stirling Number Generating Function {#sect:Stirling} ===================================== Following [@Butzer], we begin with the integral representation of §\[sect:single2\]. In light of the expansion $$\frac{(-1)^m}{m!}\log^m(1-x) = \sum_{n=0}^\infty u(n,m) \frac{x^n}{n!},\qquad 0\le m\in{{\mathbf Z}},$$ in terms of the unsigned Stirling numbers of the first kind (also referred to as the Stirling cycle numbers in [@GKP]), we have $$\zeta(m+1) = \int_0^1 \bigg\{\sum_{n=1}^\infty u(n,m)\frac{x^n}{n!}\bigg\}\frac{dx}{x} = \sum_{n=1}^\infty \frac{u(n,m)}{n!\, n},\qquad 1\le m\in{{\mathbf Z}}.$$ Telescoping the known recurrence $$\label{StirlingRecur} u(n,m) = u(n-1,m-1)+(n-1)u(n-1,m),\qquad 1\le m\le n,$$ yields $$\label{StirlingSumRecur} u(n,m) = (n-1)! \left\{ \delta_{m,1}+ \sum_{j=1}^{n-1} \frac{u(j,m-1)}{j!}\right\}.$$ Iterating this gives the representation $$\zeta(m+1) = \zeta(2,\{1\}^{m-1}), \qquad 1\le m\in{{\mathbf Z}},$$ the $m=2$ case of which is . See also $n=0$ in below. For the alternating case, we begin by writing the recurrence in the form $$u(n+1,k) +(j-n)u(n,k) = u(n,k-1) + j\, u(n,k).$$ Following [@Butzer], multiply both sides by $(-1)^{n+k+1}j^{k-m-1}/(j-n)_n$, where $1\le n\le j-1$ and $k, m\in{{\mathbf Z}}^{+}$, yielding $$\begin{gathered} (-1)^k \left\{\frac{(-1)^{n+1}\,u(n+1,k)}{(j-n)_n} - \frac{(-1)^n \,u(n,k)}{(j-n+1)_{n-1}}\right\} j^{k-m-1}\\ = \frac{(-1)^n}{(j-n)_n}\left\{(-1)^{k-1}\,u(n,k-1)j^{k-m-1} - (-1)^k \,u(n,k) j^{k-m}\right\}.\end{gathered}$$ Now sum on $1\le k\le m$ and $1\le n\le j-1$, obtaining $$\sum_{k=1}^m \frac{(-1)^{k+j} \,u(j,k)}{j!\,j^{m-k}} -\frac1{j^m} = \frac{(-1)^{m+1}}{(j-1)!}\sum_{n=m}^{j-1}(-1)^n (j-n-1)!\,u(n,m).$$ Finally, sum on $j\in{{\mathbf Z}}^{+}$ to obtain $$\zeta(m) = \sum_{k=1}^m \sum_{j=k}^\infty \frac{(-1)^{k+j}\,u(j,k)}{j!\,j^{m-k}} + \sum_{n=m}^\infty (-1)^{n+m}\,u(n,m)\sum_{j=n+1}^\infty \frac{(j-1-n)!}{(j-1)!}.$$ Noting that $$\sum_{j=n+1}^\infty \frac{(j-1-n)!}{(j-1)!} = \sum_{k=0}^\infty \frac{k!}{(k+n)!} = \frac1{n!} \;{}_2F_1(1,1;n+1;1) = \frac{1}{(n-1)!\,(n-1)},$$ we find that $$\zeta(m) = \sum_{k=1}^m \sum_{j=k}^\infty \frac{(-1)^{j+k}\,u(j,k)}{j!\,j^{m-k}} + \sum_{n=m}^\infty \frac{(-1)^{n+m}\,u(n,m)}{(n-1)!\,(n-1)}.$$ Now employ the recurrence again to get $$\begin{aligned} \zeta(m) &= \sum_{k=1}^{m-2}\sum_{j=k}^\infty \frac{(-1)^{j+k}\,u(j,k)}{j!\,j^{m-k}}+\sum_{j=m-1}^\infty \frac{(-1)^{j+m-1}\,u(j,m-1)}{j!\,j}+\sum_{j=m}^\infty \frac{(-1)^{j+m}\,u(j,m)}{j!}\nonumber\\ &\qquad + \sum_{n=m}^\infty \frac{(-1)^{n+m}\,u(n-1,m)}{(n-1)!} + \sum_{n=m}^\infty \frac{(-1)^{n+m}\,u(n-1,m-1)}{(n-1)!\,(n-1)}\nonumber\\ &=\sum_{k=1}^{m-2}\sum_{j=k}^\infty \frac{(-1)^{j+k}\,u(j,k)}{j!\,j^{m-k}}+2\sum_{j=m-1}^\infty \frac{(-1)^{j+m-1}\,u(j,m-1)}{j!\,j}. \label{Butzer2bar}\end{aligned}$$ Using again, we find that the case $m=3$ of gives $$\begin{aligned} \zeta(3) &= \sum_{j=1}^\infty \frac{(-1)^j\,u(j,1)}{j!\,j^2}+2\sum_{j=2}^\infty \frac{(-1)^j \,u(j,2)}{j!\,j}\\ &= \sum_{j=1}^\infty \frac{(-1)^{j+1}}{j^3}+2\sum_{j=2}^\infty \frac{(-1)^j}{j!\,j}(j-1)!\sum_{k=1}^{j-1}\frac{u(k,1)}{k!}\\ &= \sum_{j=1}^\infty \frac{(-1)^{j+1}}{j^3}+2\sum_{j=2}^\infty \frac{(-1)^j}{j^2}\sum_{k=1}^{j-1}\frac{1}{k}\\ &= 2\zeta(\overline2,1)-\zeta(\overline3),\end{aligned}$$ which easily rearranges to give , shown in §\[sect:telescope\] to be trivially equivalent to . Polylogarithm Identities {#sect:dilog} ======================== Dilogarithm and Trilogarithm {#sect:dilog3} ---------------------------- Consider the power series $$J(x):= \zeta_x(2,1)=\sum_{n>k>0} \frac{x^n}{n^2 k}, \qquad 0\le x\le 1.$$ In light of , we have $$J(x) = \int_0^x \frac{dt}{t}\int_0^t \frac{du}{1-u}\int_0^v\frac{dv}{1-v} = \int_0^x\frac{\log^2(1-t)}{2t}\,dt.$$ The computer algebra package [Maple]{} readily evaluates $$\label{maple} \int_0^x\frac{\log^2(1-t)}{2t}\,dt =\zeta(3)+\frac{1}{2} \log^2(1-x) \log(x) + \log (1-x){\rm Li}_2(1-x)-{\rm Li}_3(1-x)$$ where $${\rm Li}_s(x) := \sum_{n=1}^\infty \frac{x^n}{n^s}$$ is the classical polylogarithm [@Lewin1; @Lewin2]. (One can also readily verify the identity by differentiating both sides by hand, and then checking trivially holds as $x\to0+$. See also [@Berndt1 p. 251, Entry 9].) Thus, $$J(x) =\zeta(3)+\frac{1}{2} \log^2(1-x) \log(x) + \log (1-x){\rm Li}_2(1-x)-{\rm Li}_3(1-x).$$ Letting $x\to1-$ gives again. In [@Berndt1 p. 251, Entry 9], we also find that $$J(-z)+J(-1/z) = -\tfrac16\log^3 z-\mathrm{Li}_2(-z)\log z+\mathrm{Li}_3(-z)+\zeta(3)\label{Jinversion}$$ and $$J(1-z) = \tfrac12\log^2 z\log(z-1)-\tfrac13\log^3 z-\mathrm{Li}_2(1/z)\log z-\mathrm{Li}_3(1/z)+\zeta(3)\label{Jreflection}.$$ Putting $z=1$ in and employing the well-known dilogarithm evaluation [@Lewin1 p. 4] $$\mathrm{Li}_{2}(-1) = \sum_{n=1}^\infty \frac{(-1)^n}{n^2} = -\frac{\pi^2}{12}$$ gives . Putting $z=2$ in and employing the dilogarithm evaluation [@Lewin1 p. 6] $$\mathrm{Li}_{2}\bigg(\frac12\bigg) = \sum_{n=1}^\infty \frac1{n^2\, 2^n} = \frac{\pi^2}{12}-\frac12\log^2 2$$ and the trilogarithm evaluation [@Lewin1 p. 155] $$\mathrm{Li}_{3}\bigg(\frac12\bigg) = \sum_{n=1}^\infty \frac{1}{n^3\, 2^n} = \frac78\zeta(3)-\frac{\pi^2}{12}\log 2+\frac16\log^3 2$$ gives again. Finally, as in [@BBBLa Lemma 10.1], differentiation shows that $$\label{feq-J} J(-x) = -J(x)+\frac14J(x^2)+J\!\left(\frac{2x}{x+1}\right) -\frac18J\!\left(\frac{4x}{(x+1)^2}\right).$$ Putting [@BBBLa Theorem 10.3] $x=1$ gives $8J(-1)=J(1)$ immediately, i.e. . In [@BBBLa], it is noted that once the component functions in are known, the coefficients can be deduced by computing each term to high precision with a common transcendental value of $x$ and then employing a linear relations finding algorithm. We note here a somewhat more satisfactory method for arriving at . First, as in §\[sect:iterint1\] one must determine the fundamental transformations –. While this is not especially difficult, as the calculations are somewhat lengthy, we do not include them here. By performing these transformations on the function $J(x)$, one finds that $$\begin{aligned} {2} J(x) & = \int_0^x ab^2,\qquad\qquad & J\bigg(\frac{2x}{1+x}\bigg) &= \int_0^x (a+c)(b-c)^2,\\ J(-x) & = \int_0^x ac^2, \qquad\qquad &{} &{}\\ J(x^2) &= \int_0^x 2a(b+c)^2, \qquad\qquad & J\bigg(\frac{4x}{(1+x)^2}\bigg) &= \int_0^x (a+2c)4(b-c)^2.\end{aligned}$$ It now stands to reason that we should seek rational numbers $r_1$, $r_2$, $r_3$ and $r_4$ such that $$ac^2 = r_1 ab^2 + 2r_2 \, a(b+c)^2 + r_3(a+c)(b-c)^2 +r_4(a+2c)4(b-c)^2$$ is an identity in the non-commutative polynomial ring ${{\mathbf Q}}\langle a,b,c\rangle$. The problem of finding such rational numbers reduces to solving a finite set of linear equations. For example, comparing coefficients of the monomial $ab^2$ tells us that $r_1+2r_2+r_3+4r_4=0$. Coefficients of other monomials give us additional equations, and we readily find that $r_1=-1$, $r_2=1/4$, $r_3=1$ and $r_4=-1/8$, thus proving as expected. Convolution of Polylogarithms {#sect:dilogp} ----------------------------- Motivated by [@Boy2001; @Boy2002], for real $0<x<1$ and integers $s$ and $t$, consider $$\begin{aligned} T_{s,t}(x) &:= \sum_{\substack{m,n=1\\ m\ne n}}^\infty \frac{x^{n+m}}{n^s\, m^t(m-n)} =\sum_{\substack{m,n=1\\ m\ne n}}^\infty \frac{x^{n+m}(m-n+n)}{n^s\, m^{t+1}(m-n)}\\ &=\sum_{\substack{m,n=1\\ m\ne n}}^\infty \frac{x^{n+m}}{n^s \, m^{t+1}}+\sum_{\substack{m,n=1\\ m\ne n}}^\infty \frac{x^{n+m}}{n^{s-1}\, m^{t+1}(m-n)}\\ &=\sum_{n=1}^\infty \frac{x^n}{n^s}\sum_{m=1}^\infty \bigg(\frac{x^m}{m^{t+1}}-\frac{x^n}{n^{t+1}}\bigg)+T_{s-1,t+1}(x)\\ &=\mathrm{Li}_s(x)\mathrm{Li}_{t+1}(x)-\mathrm{Li}_{s+t+1}(x^2) +T_{s-1,t+1}(x).\end{aligned}$$ Telescoping this gives $$\begin{aligned} T_{s,t}(x) &= T_{0,s+t}(x)-s\,\mathrm{Li}_{s+t+1}(x^2) +\sum_{j=1}^s \mathrm{Li}_j(x)\mathrm{Li}_{s+t+1-j}(x), \qquad 0\le s\in{{\mathbf Z}}.\nonumber\\ \intertext{With $t=0$, this becomes} T_{s,0}(x) &= T_{0,s}(x) - s\,\mathrm{Li}_{s+1}(x^2) +\sum_{j=1}^{s}\mathrm{Li}_j(x)\mathrm{Li}_{s+1-j}(x), \qquad 0\le s\in{{\mathbf Z}}.\nonumber\\ \intertext{But for any integers $s$ and $t$, there holds} T_{s,t}(x) &= \sum_{\substack{m,n=1\\ m\ne n}}^\infty \frac{x^{n+m}}{n^t m^s(m-n)} = -\sum_{\substack{m,n=1\\ m\ne n}}^\infty \frac{x^{n+m}}{m^s n^t(n-m)} = - T_{s,t}(x).\nonumber\\ \intertext{Therefore,} T_{s,0}(x) &= \frac12 \sum_{j=1}^s \mathrm{Li}_j(x)\mathrm{Li}_{s+1-j}(x) - \frac{s}{2}\,\mathrm{Li}_{s+1}(x^2), \qquad 0\le s\in{{\mathbf Z}}.\label{Tp0}\\ \intertext{On the other hand,} T_{s,0}(x) &= \sum_{n=1}^\infty\frac{x^n}{n^s} \sum_{\substack{m=1\\m\ne n}}^\infty\frac{x^m}{m-n} = \sum_{n=1}^\infty\frac{x^{2n}}{n^s}\sum_{m=n+1}^\infty \frac{x^{m-n}}{m-n} - \sum_{n=1}^\infty \frac{x^n}{n^s}\sum_{m=1}^{n-1}\frac{x^m}{n-m}\nonumber\\ &= \mathrm{Li}_s(x^2)\mathrm{Li}_1(x)-\sum_{n=1}^\infty \frac{x^n}{n^s}\sum_{j=1}^{n-1}\frac{x^{n-j}}{j}.\nonumber\\ \intertext{Comparing this with~\eqref{Tp0} gives} \sum_{n=1}^\infty\frac{x^n}{n^s}\sum_{j=1}^{n-1}\frac{x^{n-j}}{j} & =\frac{s}2\,\mathrm{Li}_{s+1}(x^2) -\left[\mathrm{Li}_s(x)-\mathrm{Li}_s(x^2)\right]\mathrm{Li}_1(x) -\frac12\sum_{j=2}^{s-1}\mathrm{Li}_j(x)\mathrm{Li}_{s+1-j}(x),\label{TakeLimit}\end{aligned}$$ where in and what follows, we now require $2\le s\in{{\mathbf Z}}$ because the terms $j=1$ and $j=s$ in the sum were separated, and assumed to be distinct. Next, note that if $n$ is a positive integer and $0<x<1$, then $$1-x^n = (1-x)\sum_{j=0}^{n-1} x^j < (1-x)n.$$ Thus, if $2\le s\in{{\mathbf Z}}$ and $0<x<1$, then $$\begin{aligned} 0<\left[\mathrm{Li}_s(x)-\mathrm{Li}_s(x^2)\right]\mathrm{Li}_1(x) &= \mathrm{Li}_1(x)\sum_{n=1}^\infty \frac{x^n(1-x^n)}{n^s} < (1-x)\mathrm{Li}_1(x)\sum_{n=1}^\infty \frac{x^n}{n^{s-1}}\\ &< (1-x)\log^2(1-x).\end{aligned}$$ Since the latter expression tends to zero in the limit as $x\to 1-$, taking the limit in gives $$\zeta(s,1) = \frac12\, s\,\zeta(s+1)-\frac12\sum_{j=1}^{s-2} \zeta(j+1)\zeta(s-j), \qquad 2\le s\in{{\mathbf Z}},$$ which is . Fourier Series {#sect:fourier} ============== The Fourier expansions $$\sum_{n=1}^{\infty}\frac{{\displaystyle}\sin(nt)}{n}=\frac{{\displaystyle}\pi-t}{2}\qquad\text{and}\qquad \sum_{n=1}^{\infty} \frac{{\displaystyle}\cos(nt)}{n}= -\log\|2\sin(t/2)\|$$ are both valid in the open interval $0<t<2\pi$. Multiplying these together, simplifying, and doing a partial fraction decomposition gives $$\begin{aligned} \sum_{n=1}^{\infty}\frac{\sin(nt)}{n} \sum_{k=1}^{n-1} \frac 1 k & = \frac12\sum_{n=1}^\infty\frac{\sin(nt)}{n} \sum_{k=1}^{n-1}\bigg(\frac1k+\frac1{n-k}\bigg) = \frac12\sum_{n>k>0} \frac{\sin(nt)}{k(n-k)}\nonumber\\ &= \frac12\sum_{m,n=1}^\infty \frac{\sin(m+n)t}{mn}= \sum_{m,n=1}^\infty \frac{\sin(mt)\cos(nt)}{mn}\nonumber\\ &= -\frac{\pi-t}2 \log \left \|2\sin(t/2) \right\|, \label{sin-ser} \intertext{again for $0<t<2\pi$. Integrating (\ref{sin-ser}) term by term yields} \sum_{n=1}^{\infty}\frac{\cos(n \theta)}{n^2} \sum_{k=1}^{n-1}\frac1k &= \zeta(2,1)+\frac12\int_0^\theta (\pi -t)\log \left \|2\sin(t/2)\right\|\,dt, \label{z21f}\\ \intertext{valid for $0\le \theta\le 2\pi$. Likewise for $0\le \theta\le 2\pi$,} \sum_{n=1}^{\infty}\frac{ \cos(n \theta)}{n^3} &= \zeta(3)+\int_0^\theta (\theta-t)\log \left \|2\sin(t/2) \right\|\,dt.\label{z3f}\end{aligned}$$ Setting $\theta=\pi$ in and produces $$\zeta(2,1)-\zeta(\overline{2},1) = -\frac{1}{2}\,\int_0^{\pi}(\pi-t) \log \left \|2\sin(t/2) \right\| \,dt = \frac{\zeta(3)-\zeta(\overline{3})}2.$$ In light of , this implies $$\zeta(\overline2,1)=\frac{\zeta(3)+\zeta(\overline3)}{2}= \frac12\sum_{n=1}^\infty\frac{1+(-1)^n}{n^3}=\sum_{m=1}^\infty\frac{1}{(2m)^3} =\frac18\zeta(3),$$ which is . Applying Parseval’s equation to gives (via [@BB; @BBG; @Flaj]) the integral evaluation $$\frac{1}{4\pi} \int_{0}^{2\pi} (\pi-t)^2 \log^2(2 \sin(t/2))\, dt = \sum_{n=1}^{\infty} \frac{H_n^2}{(n+1)^2} = \frac{11}4\, \zeta(4).$$ A reason for valuing such integral representations is that they are frequently easier to use numerically. Further Generating Functions {#sect:gfs} ============================ Hypergeometric Functions {#sect:hyper} ------------------------ Note that in the notation of , $\zeta(2,1)$ is the coefficient of $xy^2$ in $$\label{xyy} G(x,y) := \sum_{m=0}^\infty\sum_{n=0}^\infty x^{m+1}y^{n+1}\zeta(m+2,{\{1\}}^n) = y\sum_{m=0}^\infty x^{m+1}\sum_{k=1}^\infty \frac1{k^{m+2}}\prod_{j=1}^{k-1} \left(1+\frac{y}j\right).$$ Now recall the notation $(y)_k := y(y+1)\cdots(y+k-1)$ for the rising factorial with $k$ factors. Thus, $$\frac{y}{k}\prod_{j=1}^{k-1} \left(1+\frac{y}j\right) = \frac{(y)_k}{k!}.$$ Substituting this into , interchanging order of summation, and summing the resulting geometric series yields the hypergeometric series $$\begin{aligned} G(x,y) &= \sum_{k=1}^\infty\frac{(y)_k}{k! }\left(\frac{x}{k-x}\right) = -\sum_{k=1}^\infty\frac{(y)_k(-x)_k}{k! (1-x)_k} = 1-{}_2F_1\left(\begin{array}{cc} y, -x\\ 1-x\end{array}\bigg\|1\right).\end{aligned}$$ But, Gauss’s summation theorem for the hypergeometric function [@AS p. 557] [@Bailey p. 2] and the power series expansion for the logarithmic derivative of the gamma function [@AS p. 259] imply that $${}_2F_1\left(\begin{array}{cc} y, -x\\ 1-x\end{array}\bigg\|1\right) =\frac{{\Gamma}(1-x){\Gamma}(1-y)}{{\Gamma}(1-x-y)} =\exp\bigg\{\sum_{k=2}^\infty\left(x^k+y^k-(x+y)^k\right) \frac{\zeta(k)}{k}\bigg\}.$$ Thus, we have derived the generating function equality [@BBB] (see [@DBq] for a $q$-analog) $$\label{DrinGF} \sum_{m=0}^\infty\sum_{n=0}^\infty x^{m+1}y^{n+1}\zeta(m+2,{\{1\}}^n) = 1-\exp\bigg\{\sum_{k=2}^\infty \left(x^k+y^k-(x+y)^k\right)\frac{\zeta(k)}{k}\bigg\}.$$ Extracting coefficients of $xy^2$ from both sides of yields . The generalization can be similarly derived: extract the coefficient of $x^{m-1}y^2$ from both sides of . In fact, it is easy to see that provides a formula for $\zeta(m+2,{\{1\}}^n)$ for all nonnegative integers $m$ and $n$ in terms of sums of products of values of the Riemann zeta function at the positive integers. In particular, Markett’s formula [@Mark] (cf. also [@BBG]) for $\zeta(m,1,1)$ for positive integers $m>1$ is most easily obtained in this way. Noting symmetry between $x$ and $y$ in gives Drinfeld’s duality formula [@Drin] $$\label{DrinDuality} \zeta(m+2,{\{1\}}^n) = \zeta(n+2,{\{1\}}^m)$$ for non-negative integers $m$ and $n$, a special case of the more general duality formula . Note that is just the case $m=n=0$. Similarly [@Chu 2.1b] equating coefficients of $xy^2$ in Kummer’s summation theorem [@Kummer p. 53] [@Bailey p.9] $${}_2F_1\left(\begin{array}{cc}x,y\\1+x-y\end{array}\bigg\|-1\right) = \frac{\Gamma(1+x/2)\Gamma(1+x-y)}{\Gamma(1+x)\Gamma(1+x/2-y)}$$ yields . A Generating Function for Sums {#sect:SumGF} ------------------------------ The identity can also be recovered by setting $x=0$ in the following result: \[thm:SumGF\] If $x$ is any complex number not equal to a positive integer, then $$\sum_{n=1}^\infty\frac1{n(n-x)}\sum_{m=1}^{n-1}\frac1{m-x} =\sum_{n=1}^\infty\frac1{n^2(n-x)}.$$ [Proof.]{} Fix $x\in{{\mathbf C}}\setminus{{\mathbf Z}}^{+}$. Let $S$ denote the left hand side. By partial fractions, $$\begin{aligned} S & =\sum_{n=1}^\infty \sum_{m=1}^{n-1}\bigg(\frac{1}{n(n-m)(m-x)} -\frac{1}{n(n-m)(n-x)}\bigg)\\ &=\sum_{m=1}^\infty\frac1{m-x}\sum_{n=m+1}^\infty\frac1{n(n-m)} -\sum_{n=1}^\infty \frac1{n(n-x)}\sum_{m=1}^{n-1}\frac1{n-m}\\ &=\sum_{m=1}^\infty\frac1{m(m-x)}\sum_{n=m+1}^\infty\bigg(\frac1{n-m} -\frac1n\bigg)-\sum_{n=1}^\infty\frac1{n(n-x)}\sum_{m=1}^{n-1}\frac1m.\end{aligned}$$ Now for fixed $m\in{{\mathbf Z}}^{+}$, $$\begin{aligned} \sum_{n=m+1}^\infty \bigg(\frac1{n-m}-\frac1n\bigg) &= \lim_{N\to\infty}\sum_{n=m+1}^N \bigg(\frac1{n-m}-\frac1n\bigg) = \sum_{n=1}^m\frac1n-\lim_{N\to\infty}\sum_{n=1}^m\frac1{N-n+1}\\ &= \sum_{n=1}^m \frac1n,\end{aligned}$$ since $m$ is fixed. Therefore, we have $$\begin{aligned} S &=\sum_{m=1}^\infty\frac1{m(m-x)}\sum_{n=1}^m\frac1n - \sum_{n=1}^\infty\frac1{n(n-x)}\sum_{m=1}^{n-1}\frac1m =\sum_{n=1}^\infty\frac1{n(n-x)}\bigg(\sum_{m=1}^n\frac1m- \sum_{m=1}^{n-1}\frac1m\bigg)\\ &=\sum_{n=1}^\infty\frac1{n^2(n-x)}.\end{aligned}$$ [$\square$]{} Theorem \[thm:SumGF\] is in fact equivalent to the sum formula [@Gran; @Ohno] $$\label{sum} \sum_{\substack{\sum a_i = s\\a_i\ge 0}} \zeta(a_1+2,a_2+1,\dots,a_r+1) = \zeta(r+s+1),$$ valid for all integers $s\ge 0$, $r\ge 1$, and which generalizes Theorem \[briggs\] to arbitrary depth. The identity is simply the case $r=2$, $s=0$. A $q$-analog of the sum formula is derived as a special case of more general results in [@DBq]. See also [@DBqSum]. An Alternating Generating Function {#sect:SumAGF} ---------------------------------- An alternating counterpart to Theorem \[thm:SumGF\] is given below. [(Theorem 3 of [@DJD]).]{} For all non-integer $x$ $$\begin{aligned} \sum_{n=1}^\infty \frac{(-1)^n}{n^2-x^2} \bigg\{H_n +\sum_{n=1}^\infty \frac{x^2}{n(n^2-x^2)}\bigg\} &= \sum_{n=1}^\infty \frac{(-1)^n}{n^2-x^2}\bigg\{\psi(n)-\psi(x) - \frac{\pi}{2}\cot(\pi x) -\frac1{2x}\bigg\}\\ &= \sum _{o>0\,\mbox{{\small odd}}}^{\infty }{\frac {1}{ o \left( o^{2 }-x^2 \right) }}+\sum _{n=1}^{\infty }{\frac { \left( -1 \right) ^{n}\,n}{ \left( {n}^{2}-{x}^{2} \right) ^{2}}}\\ &=\sum _{e>0\,\mbox{{\small even}}}^{\infty }{\frac {e}{ \left( {x}^{2}-e^{2} \right) ^{2}}}-{x}^{2}\sum _{o>0\,\mbox{{\small odd}}}^{\infty }{\frac {1}{ o \left( {x}^{2}- o^{2} \right) ^{2}}}. \end{aligned}$$ Setting $x=0$ reproduces in the form $\zeta(\overline{2},1)=\sum _{n>0}^{\infty }(2n)^{-3}$. We record that $$\sum _{n=1}^{\infty}\frac{(-1)^n}{n^2-x^2} =\frac1{2 x^2}-{\frac {\pi }{2 x\sin(\pi x)}},$$ while $$\begin{gathered} \sum_{n=1}^\infty \frac{(-1)^n}{n^2-x^2} \bigg\{\psi(n)-\psi(x)-\frac\pi2\cot(\pi x)-\frac1{2x}\bigg\} =\sum_{n=1}^\infty \frac{(-1)^n}{n^2-x^2}\bigg\{H_n +\sum_{n=1}^\infty \frac{x^2}{n(n^2-x^2)} \bigg\}\\ = \sum_{n=1}^\infty\frac{1}{(2n-1)((2n-1)^2-x^2)} +\sum_{n=1}^\infty \frac{n (-1)^n}{(n^2-x^2)^2}.\end{gathered}$$ The Digamma Function {#susbsect:DPsiGF} -------------------- Define an auxiliary function $\Lambda$ by $$\begin{aligned} x \Lambda(x):= \tfrac12\psi'(1-x) -\tfrac12\left(\psi(1-x)+\gamma\right)^2 -\tfrac12\zeta(2).\end{aligned}$$ We note, but do not use, that $$x \Lambda(x)=\frac12\int_0^\infty \frac{t \left(e^{-t}+e^{-t (1-x)}\right)}{1-e^{-t}}\,dt -\frac12\left(\int_0^\infty \frac{e^{-t}-e^{-t ( 1-x)}}{1-{e^{-t}}}\,dt\right)^2-\zeta(2).$$ It is easy to verify that $$\begin{aligned} \label{2ids} \psi(1-x)+\gamma &= \sum_{n=1}^\infty\frac{x}{n(x-n)},\nonumber\\ \psi'(1-x)-\zeta(2) & =\sum_{n=1}^\infty\bigg(\frac{1}{(x-n)^2}-\frac1{n^2}\bigg) =\sum_{n=1}^\infty \frac{2nx-x^2}{n^2(n-x)^2},\end{aligned}$$ and $$\sum_{n=0}^\infty\zeta(n+2,1)x^n =\sum_{n=1}^\infty\frac1{n(n-x)}\sum_{m=1}^{n-1}\frac1m.$$ Hence, $$\Lambda(x) =\sum_{n=1}^\infty\frac{1}{n^2(n-x)} -x\sum_{n=1}^\infty\frac1{n(n-x)}\sum_{m=1}^{n-1}\frac{1}{m(m-x)}.$$ Now, $$\sum_{n=1}^\infty\frac{1}{n^2(n-x)} - x\sum_{n=1}^\infty\frac{1}{n(n-x)}\sum_{m=1}^{n-1}\frac{1}{m(m-x)} =\sum_{n=1}^\infty\frac1{n(n-x)}\sum_{m=1}^{n-1}\frac1m$$ is directly equivalent to Theorem \[thm:SumGF\] of §\[sect:SumGF\]—see [@DJD Section 3]—and we have proven $$\Lambda(x)= \sum_{n=0}^\infty \zeta(n+2,1)\,x^n,$$ so that comparing coefficients yields yet another proof of Euler’s reduction . In particular, setting $x=0$ again produces . The Beta Function ----------------- Recall that the beta function is defined for positive real $x$ and $y$ by $$B(x,y) := \int_0^1 t^{x-1}(1-t)^{y-1}\,dt = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}.$$ We begin with the following easily obtained generating function: $$\sum_{n=1}^\infty t^n H_n = -\frac{\log(1-t)}{1-t}.$$ For $m\ge 2$ the Laplace integral now gives $$\begin{aligned} \label{zm1=b1} \zeta(m,1) & = \frac{(-1)^m}{(m-1)!} \int_0^1\frac{\log^{m-1}(t)\log(1-t)}{1-t}\, dt\nonumber\\ & = \frac{(-1)^m}{2(m-1)!} \int_0^1 (m-1) \log^{m-2}(t)\log^2(1-t)\,\frac{dt}{t}\nonumber\\ & = \frac{(-1)^m}{2(m-2)!}\, b_1^{(m-2)}(0),\end{aligned}$$ where $$b_1(x) := \left. \frac{\partial^{2}}{\partial y^2} B(x,y) \right\|_{y=1} = 2\Lambda(-x)$$ (cf. §\[susbsect:DPsiGF\]). Since $$\frac{\partial^2}{\partial y^2} B(x,y) = B(x,y) \left[(\psi(y)-\psi(x+y))^2 + \psi'(y) - \psi'(x+y)\right],$$ we derive $$b_1(x) = \frac{(\psi(1) - \psi(x+1))^2 + \psi'(1) - \psi'(x+1)}{x}.$$ Now observe that from , $$\begin{aligned} \zeta(2,1) &= \frac12 \, b_1(0) = \lim_{x \downarrow0} \frac{(-\gamma - \psi(x+1))^2 }{2x} -\lim_{x \downarrow0} \frac{\psi'(x+1)-\psi'(1)}{2x} =-\frac12\,\psi''(1)\\ &=\zeta(3).\end{aligned}$$ Continuing, from the following two identities, cognate to , $$\begin{aligned} (-\gamma-\psi(x+1))^2 & = \bigg(\sum_{m=1}^\infty (-1)^m\zeta(m+1)\, x^m \bigg)^2\\ & = \sum_{m=1}^\infty (-1)^m \sum_{k=1}^{m-1}\zeta(k+1)\zeta(m-k+1)\, x^m,\\ \zeta(2) -\psi'(x+1) & = \sum_{m=1}^\infty (-1)^{m+1} (m+1) \zeta(m+2)\, x^m,\end{aligned}$$ we get $$2\sum_{m=2}^\infty (-1)^m \zeta(m,1) \,x^{m-2} = \sum_{m=2}^\infty \frac{b_1^{(m-2)}(0)}{(m-2)!}\, x^{m-2}\, = b_1(x)$$ $$= \sum_{m=1}^\infty (-1)^{m-1} \left( (m+1) \zeta(m+2) - \sum_{k=1}^{m-1} \zeta(k+1) \zeta(m-k+1)\right) x^{m-1},$$ from which Euler’s reduction follows—indeed this is close to Euler’s original path. Observe that is especially suited to symbolic computation. We also note the pleasing identity $$\begin{aligned} \psi'(x)=\frac{\Gamma''(x)}{\Gamma(x)}-\psi^2(x)\label{DPsiGF} .\end{aligned}$$ In some informal sense (\[DPsiGF\]) generates , but we have been unable to make this sense precise. A Decomposition Formula of Euler {#sect:parfracs} ================================ For positive integers $s$ and $t$ and distinct non-zero real numbers $\alpha$ and $x$, the partial fraction expansion $$\label{parfrac} \frac{1}{x^s(x-\alpha)^t} = (-1)^t\sum_{r=0}^{s-1}\binom{t+r-1}{t-1}\frac{1}{x^{s-r}\alpha^{t+r}} +\sum_{r=0}^{t-1}\binom{s+r-1}{s-1} \frac{(-1)^r}{\alpha^{s+r}(x-\alpha)^{t-r}}$$ implies [@Niels3 p. 48] [@Mark] Euler’s decomposition formula $$\begin{gathered} \label{niels} \zeta(s,t) = (-1)^t\sum_{r=0}^{s-2}\binom{t+r-1}{t-1}\zeta(s-r,t+r) +\sum_{r=0}^{t-2}(-1)^r\binom{s+r-1}{s-1}\zeta(t-r)\zeta(s+r)\\ -(-1)^t\binom{s+t-2}{s-1}\big\{\zeta(s+t)+\zeta(s+t-1,1)\big\}.\end{gathered}$$ The depth-2 sum formula is obtained by setting $t=1$ in . If we also set $s=2$, the identity results. To derive from we follow [@Mark], separating the last term of each sum on the right hand side of , obtaining $$\begin{aligned} \frac{1}{x^s(x-\alpha)^t} &= (-1)^t\sum_{r=0}^{s-2}\binom{t+r-1}{t-1}\frac{1}{x^{s-r}\alpha^{t+r}} +\sum_{r=0}^{t-2}\binom{s+r-1}{s-1} \frac{(-1)^r}{\alpha^{s+r}(x-\alpha)^{t-r}}\\ &-(-1)^{t}\binom{s+t-2}{s-1} \frac{1}{\alpha^{s+t-1}}\bigg(\frac{1}{x-\alpha} -\frac1x\bigg).\end{aligned}$$ Now sum over all integers $0<\alpha <x<\infty$. Nielsen states without proof [@Niels3 p. 48, eq.(9)]. Markett proves by induction [@Mark Lemma 3.1], which is the proof technique suggested for the $\alpha=1$ case of in . However, it is easy to prove directly by expanding the left hand side into partial fractions with the aid of the residue calculus. Alternatively, as in [@DBqDecomp] note that is an immediate consequence of applying the partial derivative operator $$\frac{1}{(s-1)!}\bigg(-\frac{\partial}{\partial x}\bigg)^{s-1} \frac{1}{(t-1)!}\bigg(-\frac{\partial}{\partial y}\bigg)^{t-1}$$ to the identity $$\frac{1}{xy} = \frac{1}{(x+y)x} + \frac{1}{(x+y)y},$$ and then setting $y=\alpha-x$. This latter observation is extended in [@DBqDecomp] to establish a $q$-analog of another of Euler’s decomposition formulas for $\zeta(s,t)$. Equating Shuffles and Stuffles {#sect:shtuff} ============================== We begin with an informal argument. By the stuffle multiplication* rule [@BBBLa; @BowBradSurvey; @DBPrtn; @DBq] $$\label{divergentstuff} \zeta(2)\zeta(1)= \zeta(2,1)+\zeta(1,2)+\zeta(3).$$ On the other hand, the shuffle multiplication rule [@BBBLa; @BBBLc; @BowBradSurvey; @BowBrad3; @BowBradRyoo] gives $ab{ \setlength{\unitlength}{.4pt} \begin{picture}(40,20) \put(10,2){\line(1,0){20}} \put(10,2){\line(0,1){10}} \put(20,2){\line(0,1){10}} \put(30,2){\line(0,1){10}} \end{picture}}b = 2abb+bab$, whence $$\label{divergentshuff} \zeta(2)\zeta(1) = 2\zeta(2,1)+\zeta(1,2).$$ The identity now follows immediately on subtracting from . Of course, this argument needs justification, because it involves cancelling divergent series. To make the argument rigorous, we introduce the multiple polylogarithm [@BBBLa; @BowBrad1; @BowBradSurvey]. For real $0\le x\le 1$ and positive integers $s_1,\dots,s_k$ with $x=s_1=1$ excluded for convergence, define $$\zeta_x(s_1,\dots,s_k) := \sum_{n_1>\cdots>n_k>0}\; x^{n_1}\prod_{j=1}^k n_j^{-s_j} =\int \prod_{j=1}^k \bigg(\prod_{r=1}^{s_j-1} \frac{dt_r^{(j)}}{t_r^{(j)}}\bigg) \frac{dt_{s_j}^{(j)}}{1-t_{s_j}^{(j)}}, \label{iterint}$$ where the integral is over the simplex $$x>t_1^{(1)}>\cdots>t_{s_1}^{(1)}>\cdots>t_1^{(k)}>\cdots>t_{s_k}^{(k )}>0,$$ and is abbreviated by $$\int_0^x \prod_{j=1}^k a^{s_j-1}b, \qquad a=\frac{dt}{t},\qquad b=\frac{dt}{1-t}. \label{shortiterint}$$ Then $$\zeta(2)\zeta_x(1) = \sum_{n>0}\frac1{n^2}\sum_{k>0}\frac{x^k}{k} = \sum_{n>k>0}\frac{x^k}{n^2k}+\sum_{k>n>0}\frac{x^k}{kn^2} +\sum_{k>0}\frac{x^k}{k^3},$$ and $$\zeta_x(2)\zeta_x(1) = \int_0^x ab \int_0^x b = \int_0^x \left(2abb+bab\right) = 2\zeta_x(2,1)+\zeta_x(1,2).$$ Subtracting the two equations gives $$\big[\zeta(2)-\zeta_x(2)\big]\zeta_x(1) = \zeta_x(3)-\zeta_x(2,1)+\sum_{n>k>0}\frac{x^k-x^n}{n^2k}.$$ We now take the limit as $x\to 1-.$ Uniform convergence implies the right hand side tends to $\zeta(3)-\zeta(2,1)$. That the left hand side tends to zero follows immediately from the inequalities $$\begin{aligned} 0\le x \big[\zeta(2)-\zeta_x(2)\big]\zeta_x(1) &= x\int_x^1\log(1-t)\log(1-x)\frac{dt}{t}\\ & \le \int_x^1 \log^2(1-t)\,dt\\ & =(1-x)\left\{1+(1-\log(1-x))^2\right\}.\end{aligned}$$ The alternating case is actually easier using this approach, since the role of the divergent sum $\zeta(1)$ is taken over by the conditionally convergent sum $\zeta(\overline 1)=-\log 2$. By the stuffle multiplication rule, $$\begin{aligned} \zeta(\overline 2)\zeta(\overline 1) &= \zeta(\overline 2, \overline 1)+\zeta(\overline 1, \overline 2) + \zeta(3) \label{convergentstuff1},\\ \zeta(2)\zeta(\overline 1) &= \zeta(2, \overline 1)+\zeta(\overline 1, 2) + \zeta(\overline 3)\label{convergentstuff2}.\end{aligned}$$ On the other hand, the shuffle multiplication rule gives $ac{ \setlength{\unitlength}{.4pt} \begin{picture}(40,20) \put(10,2){\line(1,0){20}} \put(10,2){\line(0,1){10}} \put(20,2){\line(0,1){10}} \put(30,2){\line(0,1){10}} \end{picture}}c=2ac^2+cac$ and $ab{ \setlength{\unitlength}{.4pt} \begin{picture}(40,20) \put(10,2){\line(1,0){20}} \put(10,2){\line(0,1){10}} \put(20,2){\line(0,1){10}} \put(30,2){\line(0,1){10}} \end{picture}}c=abc+acb+cab$, whence $$\begin{aligned} \zeta(\overline 2)\zeta(\overline 1) &= 2\zeta(\overline 2, 1) + \zeta(\overline 1, 2)\label{convergentshuff1},\\ \zeta(2)\zeta(\overline 1) &= \zeta(2, \overline 1) + \zeta(\overline 2, \overline 1) + \zeta(\overline 1, \overline 2)\label{convergentshuff2}.\end{aligned}$$ Comparing with and with yields the two equations $$\begin{aligned} \zeta(\overline 2, \overline 1) &= \zeta(\overline 1, 2) +2\zeta(\overline 2, 1) -\zeta(\overline 1, \overline 2) -\zeta(3),\\ \zeta(\overline 2, \overline 1) &= \zeta(\overline 1, 2)-\zeta( \overline 1, \overline 2) +\zeta(\overline 3).\end{aligned}$$ Subtracting the latter two equations yields $2\zeta(\overline 2, 1) = \zeta(3)+\zeta(\overline 3)$, i.e. , which was shown to be trivially equivalent to in §\[sect:telescope\]. 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--- abstract: 'H emission is a well-known indicator of magnetic activity in the Sun and other stars. It is also viewed as an important signature of chromospheric heating. However, the H line has not been used as a diagnostic of magnetic flux emergence from the solar interior. Here we report on Hinode observations of chromospheric H brightenings associated with a repeated, small-scale flux emergence event. We describe this process and investigate the evolution of the magnetic flux, G-band brightness, and H intensity in the emerging region. Our results suggest that energy is released in the chromosphere as a consequence of interactions between the emerging flux and the pre-existing magnetic field, in agreement with recent 3D numerical simulations.' author: - 'S. L. Guglielmino and F. Zuccarello' - 'P. Romano' - 'L. R. Bellot Rubio' title: \| HINODE Observations of Chromospheric Brightenings in the\ H Line during small-scale Flux Emergence Events --- Introduction ============ Numerical simulations predict that magnetic flux emerging into the solar atmosphere interact and reconnect with the pre-existing chromospheric and coronal field. This suggests that flux emergence is a relevant source of energy for the chromosphere [@Archontis:04; @Archontis:05]. The efficiency of the interaction and the consequent heating seem to depend on the geometry of the two flux systems [@Galsgaard:07]. While at large scale these results have been confirmed by high-resolution observations [@Moreno:08; @Zuccarello:08], the role of small-scale emergence events in the heating of the upper atmospheric layers, as described by @Isobe:08, is still lacking observational confirmation. In the absence of other diagnostics, heating events in the chromosphere can be detected through the intensity profiles of the H and K lines. The correlation between flux emergence and chromospheric emission was analyzed in detail by @Balasu:01, who observed anomalous profiles in the K line in an emerging active region. However, very few examples of small-scale transient brightenings have been reported in the literature, and most of them are related to flux cancellation events [e.g. @Bellot:05]. The Hinode satellite, with its unprecedented spatial resolution, offers for the first time the possibility to investigate the processes that occur in the chromosphere during the emergence of magnetic flux at small spatial scales. In this Letter we analyze simultaneous chromospheric and photospheric observations of an emerging flux region taken with the Solar Optical Telescope aboard Hinode. During the emergence event, strong brightenings were detected in the H line core without relevant counterparts in G-band intensity, which suggests that chromospheric heating did occur at the site of flux emergence. Observations and data reduction =============================== On 2007 September 30, as part of the Hinode Operation Plan 14 (Hinode/Canary Islands Campaign), the active region NOAA 10971 was observed by the Solar Optical Telescope [SOT; @Tsuneta:08] onboard Hinode [@Kosugi:07]. The field of view (FoV) was centered at solar coordinates ($174\arcsec$, $-79\arcsec$), i.e., $11^{\circ}$ away from disk center. The SOT spectro-polarimeter [SP; @Tsuneta:08] performed six raster scans of the active region from 08:00 to 14:00 UT, acquiring the Stokes I, Q, U, and V profiles of the photospheric lines at 630.15 nm and 630.25 nm. The FoV covered by the SP observations is $164\arcsec \times 164\arcsec$, with an effective pixel size of $0.32\arcsec$ (Fast Map mode). Simultaneously, the SOT Broadband Filter Imager (BFI) acquired filtergrams in the core of the H line ($396.85 \pm 0.3 \;\textrm{nm}$) and in the G band ($430.5 \pm 0.8 \;\textrm{nm}$), while the Narrowband Filter Imager (NFI) obtained shuttered Stokes I and V filtergrams in the wings of the D1 line at $589.6 \;\textrm{nm}$. The BFI images have a spatial sampling of $0.05\arcsec$/pixel (G-band) and $0.1\arcsec$/pixel ( H), while that of the NFI filtergrams is $0.16\arcsec$/pixel. The BFI and NFI time series have a cadence of one minute and extend from 07:00 to 17:00 UT, with a small gap between 10:05 and 10:20 UT. We have corrected the SOT/SP and SOT/FG images for dark current, flat field, and cosmic rays with standard SolarSoft routines. Besides obtaining photospheric and chromospheric information through corrected G-band and H filtergrams, we have constructed magnetograms from the D1 Stokes I and V images acquired $\pm 156$ mÅ off the line center. From the ratio $$\frac{V}{I}=\frac{1}{2}\left( \frac{V_{\rm blue}}{I_{\rm blue}} + \frac{V_{\rm red}}{I_{\rm red}} \right)$$ we calculate the magnetic flux density $\Phi_{\rm d}$ using the weak fied approximation [@Stix:02] as $$\Phi_{\rm d}= 8 \times 10^3 \: \frac{V}{I} \;\;\; [\textrm{Mx cm}^{-2}]$$ [see @Guglielmino:08]. To first order, the magnetograms computed in this way are not affected by Doppler shifts. We remind the reader that, at disk center, the D1 line refers to the upper photospheric layers and not to the chromosphere. For each raster scan of the SP, the profile with the minimum total polarization degree, $P=\left[\left(Q^{2}+U^{2}+V^{2}\right)/I^{2}\right]^{1/2}$, was selected as a reference profile. All the spectra in the scan were normalized to the continuum of this profile and corrected for limb darkening. Also, a stray light profile was computed by averaging the reference profiles of the six scans. Adopting a grid paradigm, we have inverted the spectra with $P > 2\%$ using the SIR code [@RuizIniesta:92]. The inversion yields the temperature stratification in the range $-4.0 < \log \, \tau_{5} < 0$ ($\tau_{5}$ is the optical depth of the continuum at $500 \;\textrm{nm}$), together with the magnetic field strength, inclination and azimuth angles in the line-of-sight (los) reference frame, the los velocity, and the magnetic filling factor, assuming these quantities to be constant with height. Azimuth and inclination angles have been transformed to the local solar frame, whereas the los velocity has been calibrated using the mean quiet-Sun intensity profile computed from pixels with $P < 0.5\%$, following the procedure of @Marti:97. Finally, all the SOT/FG and SOT/SP images have been aligned through cross-correlation algorithms. Results ======= NOAA 10971 has a classical bipolar $\beta$ configuration, as can be seen in the D1 magnetogram of Fig. \[fig1\]. The various SOT instruments recorded the emergence of a small bipolar region, which appeared at the internal edge of the main negative polarity. Figure \[fig2\] displays a temporal sequence of D1 magnetograms with a cadence of about 20 minutes for the $8 \times 8$ Mm$^2$ area marked in Fig. \[fig1\]. The first magnetogram of the sequence, acquired at 07:50 UT, shows the presence of a positive-polarity knot. In the following magnetograms we clearly recognize an increase in its area, as well as the appearance of a negative-polarity patch. The subsequent evolution is characterized by the separation of the opposite magnetic polarities, as indicated by the arrows. The corresponding temporal sequence of H filtergrams is also displayed in Fig. \[fig2\] and shows transient brightness enhancements at the location of the positive footpoint. This emergence event led to the appearance of bright points in the G band (Fig. \[fig3\], left panel) and to intensity enhancements in the H line core (Fig. \[fig3\], right panel). Maps of the physical parameters derived from the SP raster scans are displayed in Fig. \[fig4\]. They demonstrate the rapid evolution of the small bipolar region: the changes in, e.g., the magnetic field distribution (second and third rows) indicate a very dynamic phase. The small bipolar region shows an emergence zone, i.e., a region between the two main polarities with horizontal fields [@Lites:98] in which upflows of $\sim 1 \;\textrm{km s}^{-1}$ can be seen at 9:18 UT and 12:23 UT. The footpoints of the emerging region exhibit vertical fields and downflows of $1.5 - 2 \;\textrm{km s}^{-1}$. The initial photospheric total flux content of the emerging region is $1.4 \times 10^{19} \;\textrm{Mx}$, classifying as a small ephemeral region. The bipole axis was inclined about $45 \degr$ to the north-south direction in the first raster scan, but this angle varied with time. The negative-polarity footpoint soon merged with the dominant negative flux of the active region, disappearing as an individual feature. In this area we have detected chromospheric H brightness enhancements with two main peaks during the observations, each one preceded by a minor peak. As can be seen in Fig. \[fig2\], the brightenings are associated with the positive-polarity footpoint. The duration of each peak is about half an hour, with an enhancement of $\sim 80\%$ with respect to the “quiet” level. The presence of these peaks points to interactions between the new emerging and the old pre-existing flux systems. We have computed the average intensity of the four most luminous pixels within the $8 \times 8 \;\textrm{Mm}^{2}$ FoV for the H and G-band filtergrams, respectively. Figure \[fig5\] shows the trend of brightness in H and G band in normalized units. The different behaviour indicates that they are not correlated, as the chromospheric brightness enhancements are much more intense than the increase observed in the G band at the same times. Thus, the observed H enhancements are genuinely due to photons coming from the chromosphere, and not to the significant photospheric contribution included in the passband of the SOT H filter [@Carlsson:07]. We have calculated the positive flux in the $8 \times 8$ Mm$^2$ area using the D1 magnetograms, which have a noise level of approximately $6.75 \times 10^{14} \;\textrm{Mx/pixel}$. The main contribution to the positive flux comes from the positive polarities of the emerging region. In Fig. \[fig6\] we show the flux evolution with time: the chromospheric brightness enhancements clearly correspond to an increase of positive magnetic flux in the upper photosphere. Interestingly, in both cases the maximum Ca II H intensities are reached some 30 minutes after the positive flux starts to increase. Taking into account the temporal coincidence between the chromospheric brightenings and the positive flux increase, as well as the spatial coincidence between the location and morphology of the H brightenings and the emerging bipolar region (Fig. \[fig2\]; compare also the second, third, and fifth rows of Fig. \[fig4\]), we conclude that the localized chromospheric heating is a consequence of the emergence and subsequent interaction of the positive flux of the new bipole, which cancels with the negative ambient magnetic field. Conclusions =========== Using Hinode filtergrams and spectropolarimetric measurements, we have studied a small-scale flux emergence event. Two peaks of chromospheric origin have been detected in the H line-core intensity almost simultaneously to a magnetic flux increase in the upper photosphere. We suggest that the chromospheric brightness enhancements may be indication that two different flux systems undergo magnetic reconnection: the old flux system belonging to the active region and the emerging magnetic field. The energy released in the process heats the chromosphere. The observed H brightenings are associated with a relatively modest amount of emerged magnetic flux (only $\sim 4 \times 10^{18} \;\textrm{Mx}$ compared with the total negative flux in the region of $\sim 2.5 \times 10^{19} \;\textrm{Mx}$), which points to a highly efficient heating mechanism. We conjecture that, while the positive flux increases, part of it cancels with the pre-existing negative flux, very likely in a process of magnetic reconnection. Our result suggests that Joule dissipation may be a significant source of chromospheric heating during the reconnection of an emerging flux system with a pre-existing magnetic field. This would confirm the predictions of recent numerical simulations [e.g. @Galsgaard:05], also at small scales. Moreover, our work suggests that H brightness enhancements can be used as a valuable diagnostics of flux emergence. Further investigations should put this result into a firm observational and theoretical basis. Financial support by the European Commission through the SOLAIRE Network (MTRN-CT-2006-035484) is gratefully acknowledged. This work has been partly funded by the Spanish Ministerio de Educación y Ciencia through projects ESP2006-13030-C06-02, PCI2006-A7-0624, and Programa de Acceso a Infraestructuras Científicas y Tecnológicas Singulares. Hinode is a Japanese mission developed and launched by ISAS/JAXA, with NAOJ as domestic partner and NASA and STFC (UK) as international partners. It is operated by these agencies in co-operation with ESA and NSC (Norway). 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--- abstract: 'We construct integral bases for the $SO(3)$-TQFT-modules of surfaces in genus one and two at roots of unity of prime order and show that the corresponding mapping class group representations preserve a unimodular Hermitian form over a ring of algebraic integers. For higher genus surfaces the Hermitian form sometimes must be non-unimodular. In one such case, genus $3$ and $p=5,$ we still give an explicit basis.' address: - \| Department of Mathematics\ Louisiana State University\ Baton Rouge, LA 70803\ USA - \| Institut de Mathématiques de Jussieu (UMR 7586 du CNRS)\ Université Paris 7 (Denis Diderot)\ Case 7012\ 2, place Jussieu\ 75251 Paris Cedex 05\ FRANCE - \| Department of Mathematics\ Louisiana State University\ Baton Rouge, LA 70803\ USA author: - 'Patrick M. Gilmer' - Gregor Masbaum - Paul van Wamelen date: 'July 8, 2002' title: Integral bases for TQFT Modules and unimodular representations of mapping class groups --- Introduction ============ Integrality properties of Witten-Reshetikhin-Turaev quantum invariants of $3$-manifolds have been studied intensively in the last several years. H. Murakami [@Mu1; @Mu2] showed that the $SU(2)$- and $SO(3)$-invariants at a root of unity $q$ of prime order are algebraic integers. This was reproved in [@MR] and generalised to all classical Lie types in [@MW; @TY] and then to all Lie types in [@Le]. These integrality properties are crucial for establishing the relationship of the invariants with the Casson invariant [@Mu1; @Mu2] and with the perturbative invariants or Ohtsuki series [@OhCambridge; @Oh1; @Le]. Quantum invariants fit into Topological Quantum Field Theories (TQFT). This means in particular that there are representations of mapping class groups associated with them. (Actually the representations are usually only projective-linear; equivalently, one has to consider certain central extensions of mapping class groups here.) If a $3$-manifold $M$ is presented as a Heegaard splitting where two handlebodies are glued together by a diffeomorphism $\varphi$ along their boundary, the quantum invariant of $M$ can be recovered from the representation of $\varphi$ on the TQFT-vector space $V({{\Sigma}})$ associated to the boundary surface ${{\Sigma}}$. The TQFT-representations are finite-dimensional and can be defined over a finite extension of the cyclotomic number field $\mathbb{Q}(q)$, where the quantum parameter $q$ is a root of unity. They also preserve a non-degenerate Hermitian form $\langle \ , \ \rangle_{{\Sigma}}$ on $V({{\Sigma}})$ (which may or may not be unitary; this usually depends on the choice of the embedding of the cyclotomic field into $\mathbb{C}$). A quite striking result was recently announced by Andersen [@An] who proved that in the $SU(n)$ case the representations are asymptotically faithful (here asymptotically means letting the order of $q$ go to infinity). At a fixed root of unity, they are certainly not faithful, as Dehn twists are always represented by matrices of finite order. Roberts [@R] showed that the representations are irreducible in the $SU(2)$-case if the order of $q$ is prime. An interesting question is to determine the image of the mapping class group in the TQFT-representations. For the $SU(2)$ and $SO(3)$-theories, the first author proved that in the genus one case the image is a finite group [@G1]. However in higher genus, the image is not finite [@Fu]; in fact, it contains elements of infinite order [@Madeira]. One might hope that this image is equal to the linear transformations which are automorphisms of some (yet to be found) structure, just as a linear transformation of the homology group $H_1({{\Sigma}};\mathbb{Z})$ is represented by a mapping class if and only if it preserves the intersection form. In this paper we are concerned with integrality properties of the TQFT representations. For simplicity, we restrict ourselves to the $SO(3)$ case; specifically, we use a variant of the $V_p$-theories of [@BHMV2] with $p$ an odd prime. Here, $p$ is the order of the root of unity $q$. Let ${{\mathcal{O}}}$ denote the ring of algebraic integers in the cyclotomic ground field. The main idea to obtain an integral structure on the TQFT already appears in [@G]. Namely, we define an ${{\mathcal{O}}}$-submodule ${{\mathcal{S}_p}}({{\Sigma}})$ of the TQFT-vector space ${{V_p}}({{\Sigma}})$ as the ${{\mathcal{O}}}$-span of vectors represented by connected $3$-manifolds with boundary ${{\Sigma}}$. The point of this definition is that the submodule ${{\mathcal{S}_p}}({{\Sigma}})$ is clearly preserved under the mapping class group. It was shown in [@G] that ${{\mathcal{S}_p}}({{\Sigma}})$ is always a free finitely generated ${{\mathcal{O}}}$-module. One can also rescale the Hermitian form $\langle \ , \ \rangle_{{\Sigma}}$ on ${{V_p}}({{\Sigma}})$ to obtain a non-degenerate ${{\mathcal{O}}}$-valued form $(\ ,\ )_{{\Sigma}}$ on ${{\mathcal{S}_p}}({{\Sigma}})$. This relies on the integrality results for the invariants mentioned above. The form $(\ ,\ )_{{\Sigma}}$ is again preserved by the mapping class group. In particular, the image of the mapping class group in the TQFT-representation $V_p({{\Sigma}})$ lies in the subgroup preserving a lattice defined over ${{\mathcal{O}}}$ and a non-degenerate Hermitian form on it. In what sense is ${{\mathcal{S}_p}}$ a TQFT defined over ${{\mathcal{O}}}$? For instance one might hope that the form $(\ ,\ )_{{\Sigma}}$ was unimodular. Here, we show that this is indeed the case in genus one and two. This is a consequence of our main result which is to describe explicit bases of ${{\mathcal{S}_p}}({{\Sigma}})$ in genus one and two. In fact, we will describe two quite different bases in genus one. The first basis is given in Theorem \[firstbasis\]. It is $$\{ \omega, t(\omega) , t^2(\omega) , \ldots , t^{d-1}(\omega)\}$$ where $\omega$ is the element appearing in the surgery axiom of the ${{V_p}}$-theory and $t$ is the twist map. It is easy to see that these elements lie in ${{\mathcal{S}_p}}(S^1\times S^1)$ and we use a Vandermonde matrix argument to show that they form a basis. A crucial step is to show that the Hermitian form $(\ ,\ )_{S^1\times S^1}$ is unimodular with respect to this basis. The second basis given in Theorem \[secondbasis\] is of a quite different nature. It is $$\{ 1, v ,v^2 , \ldots , v^{d-1}\}$$ where $v=(z+2)/(1+A)$ (here $z$ is represented by the core of the solid torus). We call it the $v$-basis. This time, it is not even obvious [a priori]{} that its elements lie in ${{\mathcal{S}_p}}(S^1\times S^1)$. The proof involves two different arguments. One is to show that the ${{\mathcal{O}}}$-span of the $v$-basis is stable under the twist map. This is shown in Section \[Kauf\]. In fact, we prove it in the more general context where the skein variable $A$ is an indeterminate rather than a root of unity. The second ingredient is to express $\omega$ in the $v$-basis and thereby relate the $v$-basis to the first basis. This is done in Section \[2nd\]. The $v$-basis lends itself nicely to finding bases in higher genus. In Section \[g2\], we describe a basis of ${{\mathcal{S}_p}}(\Sigma)$ in genus two consisting of $v$-colored links in a genus two handlebody. These links are described by arrangements of curves in a twice punctured disk. Again, the unimodularity of the Hermitian form with respect to this basis is a crucial step in the argument. In principle this method can be used to study ${{\mathcal{S}_p}}(\Sigma)$ in higher genus as well. It turns out, however, that the Hermitian form $(\ ,\ )_{{{\Sigma}}}$ is not always unimodular. For example, a simple argument given in Section \[non-uni\] shows that it cannot be unimodular for surfaces of genus 3 and 5, assuming $p \equiv 5 \pmod{8}.$ In this paper we will not attempt to deal with the higher genus case in general. We only give in Section \[g3\] a basis of ${{\mathcal{S}_p}}(\Sigma)$ for a surface of genus three when $p=5.$ Although in this case the Hermitian form is not unimodular, it is nearly so. This allows us to find a basis easily in this one case. Note that our definition of ${{\mathcal{S}_p}}({{\Sigma}})$ is analogous to the construction of integral modular categories in [@MW]; in both cases one constructs integral structures by considering the span, over the subring of algebraic integers of the coefficient field, of the morphisms of the geometrically defined category (tangles in the case of [@MW], $3$-dimensional cobordisms in the case at hand). It might be that this is not always enough: It is conceivable that one might be able to enlarge ${{\mathcal{S}_p}}({{\Sigma}})$ in some way to make the form always unimodular; however this enlargement would not be generated by $3$-cobordisms anymore. We conclude the paper by showing how the ${{\mathcal{S}_p}}$-theory defined over ${{\mathcal{O}}}$ can be used to prove a divisibility result for the Kauffman bracket of links in $S^3$. This generalizes a result of Cochran and Melvin [@CM] for zero framed links (see also [@OhCambridge; @KS]). [Notational conventions.]{} Throughout the paper, $p\geq 3$ will be an odd integer, and we put $d=(p-1)/2$. From Section \[elem\] onwards, $p$ is supposed to be prime. The twist map on the Kauffman Bracket module of a solid torus {#Kauf} ============================================================= In this section we define a sequence of submodules $K(n)$ of the Kauffman Bracket skein module of the solid torus $S^1 \times D^2$ and show that they are preserved under the twist map. We use the notations of [@BHMV1]. Suppose $R$ is a commutative ring with identity and an invertible element $A.$ The universal example is $R={{\mathbb{Z}}}[A,A^{-1}]$ which we also denote by ${{\mathbb{Z}}}[A^{\pm}]$. Recall that the Kauffman bracket skein module $K(M,R)$ of a $3$-manifold $M$ is the free $R$ module generated by isotopy classes of banded links in $M$ modulo the submodule generated by the Kauffman relations. We let $z$ denote the skein element of $K(S^1 \times D^2,R)$ given by the banded link $S^1\times J,$ where $J$ is a small arc in the interior of $D^2.$ As is well known, $K(S^1 \times D^2,R)$ is a free $R$-module on the nonnegative powers of $z,$ where $z^n$ means $n$ parallel copies of $z$. This also makes $K(S^1 \times D^2,R)$ into an $R$-algebra isomorphic to the polynomial ring $ R [z].$ Let $t:K (S^1 \times D^2,R) \rightarrow K (S^1 \times D^2,R)$ denote the twist map induced by a full right handed twist on the solid torus. It is well known (see [e.g.]{} [@BHMV1]) that there is a basis $ \{e_i\}_{i\ge0}$ of eigenvectors for the twist map. It is defined recursively by $$\label{ei} e_0=1, \quad e_1=z, \quad e_i= z e_{i-1}- e_{i-2} ~.$$ The eigenvalues are given by $$\label{mui}t(e_i)=\mu_i e_i, \text{ where } \mu_i=(-1)^i A^{i^2+2i} ~.$$ [Let ${K}(n)$ denote the ${{\mathbb{Z}}}[A^{\pm}]$-submodule of $K (S^1 \times D^2, {{\mathbb{Z}}}[A^{\pm},\frac 1 {1+A}])$ generated by $\{1,v,v^2,\ldots,v^n\}$, where $$v= \frac {z+2}{1+A}~.$$ ]{} \[tK\] The twist map $t$ sends $K(n)$ to itself. Consider the basis $ \{ (z+2)^i\}_{i\ge0}$ of $K (S^1 \times D^2,{{\mathbb{Z}}}[A^{\pm}]).$ The following Lemma gives the change of basis formulas. \[changeofbasis\] For each $n \ge 1,$ $$(z+2)^{n-1}= \sum_{k=1}^n \binom{2n}{n-k} \frac k n e_{k-1}$$ $$e_{n-1}= \sum_{i=1}^n (-1)^{n-i} \binom{n+i-1}{n-i} (z+2)^{i-1}$$ Prove each separately by induction on $n$ using the recursion formula (\[ei\]). [It follows that $\binom{2n}{n-k} \frac k n \in {{\mathbb{Z}}},$ which can also be seen directly:]{} $$\binom{2n}{n-k} \frac k n= \binom{2n}{n-k} (1-\frac {n-k} n) = \binom{2n}{n-k}- 2\binom{2n-1}{n-k-1}~.$$ It is enough to show Theorem \[tK\] for the endomorphism $-At$ in place of $t$. Let us compute $-At$ in the basis $(z+2)^n.$ Note that $-At(e_{i-1})=(-A)^{i^2} e_{i-1}$. $$\begin{aligned} -At\left((z+2)^{n-1}\right) & = \sum_{k=1}^n \binom{2n}{n-k} \frac k n (-A) t(e_{k-1}), \\ &= \sum_{k=1}^n \binom{2n}{n-k} \frac k n (-A)^{k^2} \sum_{i=1}^k (-1)^{k-i} \binom {k+i-1}{k-i} (z+2)^{i-1},\\ &= \sum_{i=1}^n (-1)^i \left( \frac 1 n \sum_{k=i}^n k \binom {2n}{n-k} \binom {k+i-1}{k-i} A^{k^2} \right)(z+2)^{i-1},\\ &= \sum_{i=1}^n (-1)^{i} S_{1,i,n}(A) (z+2)^{i-1}, \end{aligned}$$ Here, for $ m \ge 1,$ we define $$S_{m,i,n}(A)= \frac 1 n \sum_{k=i}^n k^m \binom {2n} {n-k} \binom {k+i-1}{k-i} A^{k^2}\in {{\mathbb{Z}}}[A].$$ \[at -1\] $$S_{1,i,n}(-1) = \begin{cases} (-1)^n, &i=n\\ 0 &i \ne n \end{cases}$$ If we put $A=-1,$ then all $\mu_i=1$ and hence $-At$ is the identity. The following formula is a very special case of a transformation formula for terminating hypergeometric series due to Bailey [@B Formula 4.3.1]. This was pointed out to us by Krattenthaler’s [HYP]{} package, [@hyp]. \[recurs\] $$S_{m,i,n}= i^2 S_{m-2,i,n} + 2i (2i+1) S_{m-2,i+1,n}$$ $$\begin{aligned} S_{m,i,n} &=\frac 1 n \sum_{k=i}^n k^{m-2} \left( (k+i) (k-i) +i^2 \right) \binom {2n} {n-k} \binom {k+i-1}{k-i} A^{k^2} \\ &= i^2 S_{m-2,i,n} + \frac 1 n \sum_{k=i+1}^n k^{m-2} (k+i) (k-i) \binom {2n} {n-k} \binom {k+i-1}{k-i} A^{k^2} \\ & \text{ (the term with $k=i$ is zero) }\\ &= i^2 S_{m-2,i,n} + \frac 1 n 2 i (2i+1)\sum_{k=i+1}^n k^{m-2} \binom {2n} {n-k} \binom {k+i}{k-i-1} A^{k^2} \\ &= i^2 S_{m-2,i,n} + 2i (2i+1) S_{m-2,i+1,n}\end{aligned}$$ Here we use the simple identity: $$(k+i)(k-i) \binom {k+i-1}{k-i} = 2i (2i+1) \binom {k+i}{k-i-1}$$ \[eq2.7\] \[divis\] $S_{1,i,n}(A)$ is divisible by $(1+A)^{n-i}$ in ${{\mathbb{Z}}}[A]$ for $i \le n.$ It suffices to show: $$\label{eq2} \left[ \left( \frac {d} {dA} \right) ^{k} S_{1,i,n}(A) \right]_{A=-1} =0$$ for all $k=0,1,\ldots , n-i-1$. Note that $\frac {d} {dA} S_{m,i,n}= A^{-1} S_{m+2,i,n} \in {{\mathbb{Z}}}[A].$ Thus $$\left( \frac {d} {dA} \right) ^k S_{1,i,n} \in \text{ Span}_{{{\mathbb{Z}}}[A^{\pm}]}\{ S_{m,i,n} \,\|\, m \text{ odd}, \, 1 \le m \le 2n-2i-1\}$$ for all $k$ in the required range. But using Lemma \[recurs\] one may decrease $m$ at the cost of increasing $i,$ and see that $$\text{ Span}_{{{\mathbb{Z}}}[A^{\pm}]}\{ S_{m,i,n} \,\| \, m \text{ odd},\, 1 \le m \le 2n-2i-1\} \subseteq \text{ Span}_{{{\mathbb{Z}}}[A^{\pm}]}\{ S_{1,j,n} \,\|\, j < n\}.$$ By Lemma \[at -1\], $S_{1,j,n}(-1) =0,$ for $ j < n,$ and (\[eq2\]) follows. [Proof of Theorem \[tK\].]{} We have $$t(v^{n-1}) = - A^{-1} \sum_{i=1}^n s_{i,n}(A) v^{i-1},$$ where $s_{i,n}(A)= (-1)^i (1+A)^{i-n}S_{1,i,n}(A)$ lies in ${{\mathbb{Z}}}[A]$ by Proposition \[eq2.7\]. [Theorem \[tK\] remains valid if we take $A$ to be a root of unity, other than $-1,$ rather than an indeterminant. ]{} [Let $\tilde {K}(n)$ be defined as $K(n)$ but with $v=({z+2})/({1+A})$ replaced with $\tilde v= ({z+2})/({1-A^2})$. Then a similar argument shows that $\tilde {K}(n)$ is stable under $t^2$, the square of the twist map. (To see this, one should replace $-At$ with $A^2 t^2$ in the above and express everything in terms of $q=A^2$. This leads to polynomials $\tilde S_{m,i,n}(q)$ defined similarly as the $ S_{m,i,n}(A)$ except that $A$ is replaced with $q$ and an extra factor of $(-1)^k$ is inserted in the sum. The remainder of the argument is the same.) ]{} The $SO(3)$-TQFTs ================= Let $p\geq 3$ be an odd integer. (In this section, $p$ need not be prime.) We consider a variation of the $2+1$ dimensional cobordism category considered in [@BHMV2] whose objects are closed oriented surfaces (with extra structure) with a (possibly empty) collection of banded points ($=$ small oriented arcs) colored by integers in the range $[0,p-2].$ The morphisms are (equivalence classes of) oriented 3-dimensional manifolds (with extra structure) with $p$-admissibly colored banded trivalent graphs. (Two morphisms are considered equivalent if they are related by a homeomorphism respecting the boundary identifications.) For the definition of $p$-admissibility in the $p$-odd case see [@BHMV2 Theorem 1.15]; see also Section \[g2\]. The variation consists of replacing the $p_1$-structures of [@BHMV2] with structures put forward by Walker [@W] and Turaev [@Tu]. Surfaces are equipped with a Lagrangian subspace of their first homology. We use homology with rational coefficients when considering Lagrangian subspaces. Cobordisms are equipped with integer weights, as well as Lagrangian subspaces for the target and source. This is also described in [@G]. We will denote this category by $\mathcal{C}.$ We call the objects of this category e-surfaces, and call the morphisms 3-e-manifolds. The procedure of [@BHMV2] defines a TQFT-functor ${{V_p}}$ on $\mathcal{C}$ over a commutative ring $R$ containing $p^{-1}$, a primitive 2pth root of unity $A$ and a solution of $\kappa^2= A^{ -6- p(p+1)/2 }.$ The number $\kappa$ here plays the role of $\kappa^3$ in [@BHMV2]. Here we use the term TQFT slightly loosely as the tensor product axiom does not hold unless only even colors are used in the cobordism category. The even colors correspond to irreducible representations of $SU(2)$ which lift to $SO(3)$. Therefore the ${{V_p}}$-theory for odd $p$ is considered a $SO(3)$ variant of the Witten-Reshetikhin-Turaev $SU(2)$-TQFT. For us it is convenient to use odd colors as well as even colors. However, if we insist that only even colors be used in coloring the banded points on the surfaces, then we do obtain an honest TQFT with the tensor product axiom, but we are still allowed us to use the language of odd colors to describe states. This will be useful in Sections \[g2\] and \[g3\]. If $M$ is a 3-e-manifold viewed as morphism from ${{\Sigma}}$ to ${{\Sigma}}'$ in $\mathcal{C}$, we denote the associated endomorphism from ${{V_p}}({{\Sigma}})$ to ${{V_p}}({{\Sigma}}')$ by ${{Z_p}}(M)$. (It is denoted by $(Z_p)_M$ in [@BHMV2]). If $M$ is a closed 3-e-manifold viewed as morphism from $\emptyset$ to $\emptyset,$ ${{Z_p}}(M)$ induces multiplication by a scalar from $R= V(\emptyset).$ This scalar is denoted by $\langle M\rangle.$ If $M$ is a 3-e-manifold viewed as morphism from $\emptyset$ to $\Sigma,$ let $[M]$ denote ${{Z_p}}(M)(1) \in V(\Sigma).$ ($[M]$ is denoted by $Z_p(M)$ in [@BHMV2]). We call such an element $[M]$ a [vacuum state.]{} If $M$ is connected, $[M]$ is called a connected vacuum state. The modules ${{V_p}}({{\Sigma}})$ are always free over $R$. They also carry a nonsingular Hermitian form [@BHMV2] : $$\langle \ ,\ \rangle _{\Sigma}: {{V_p}}(\Sigma) \times {{V_p}}(\Sigma) \rightarrow R$$ given by $$\langle [N_1],[N_2]\rangle _{\Sigma}= \langle N_1 \cup_{\Sigma} -N_2\rangle~.$$ Here $-N_2$ is the 3-e-manifold obtained by reversing the orientation, multiplying the weight by $-1, $ and leaving the Lagrangian on the boundary alone. If $\Sigma$ is an e-surface with no colored points, and $H$ is a handlebody (weighted zero) with boundary $\Sigma,$ then ${{V_p}}(\Sigma)$ has a specified isomorphism to a quotient of the skein module $K(H,R)$ [@BHMV2 p. 891]. In fact if $H$ is a subset of $S^3$ then two skeins represent the same element if and only if they are equal as “maps of outsides” in Lickorish’s phrase [@L]. Let $S^1 \times S^1$ denote an e-surface of genus one with no colored points. Let $d$ denote $(p-1)/ 2.$ It turns out that $d$ is the dimension or rank of ${{V_p}}(S^1 \times S^1).$ In fact, the module ${{V_p}}(S^1 \times S^1) $ is isomorphic as an $R$-module to the quotient of $K(S^1 \times D^2,R)=R[z]$ by the ideal generated by $e_d- e_{d-1}\in R[z].$ It follows [@BHMV1 p.696] that $e_{p-1}=0$ in ${{V_p}}(S^1 \times S^1) $ and $e_{d+i}= e_{d-1-i}.$ Thus the module ${{V_p}}(S^1 \times S^1) $ has indeed rank $d$ with the basis $\{e_0,e_1,\ldots ,e_{d-1}\}$. Note that this basis is the same, up to reordering, as the even basis $\{e_0,e_2,\ldots, e_{p-3}\}.$ Let $\Sigma_g$ denote an e-surface of genus $g$ with no colored points on the boundary. The rank of the free module ${{V_p}}(\Sigma_g)$ is given by the formula [@BHMV2 Cor. 1.16] $$\text{rank}\left({{V_p}}(\Sigma_g)\right) = \left( \frac p 4\right)^{g-1} \sum_{j=1}^{d} \left( \sin \frac {2 \pi j}{p} \right)^{2-2g} .$$ This is the same as $2^{-g}$ times the dimension of $V_{2p}({{\Sigma}}_g)$ (this fact comes from a tensor product formula, see [@BHMV2 Thm. 1.5]). Note that $V_{2p}({{\Sigma}}_g)$ is an $SU(2)$-TQFT module, with dimension given by the $SU(2)$ Verlinde formula at level $p-2$ (where the colors are again the set of integers in the range $[0,p-2]$). In genus 2, we have $$\text{rank}\left({{V_p}}(\Sigma_2)\right)= \frac {d (d+1)(2d+1)}{6}$$ as will be seen by an explicit counting argument in Section \[g2\]. Some facts from elementary number theory {#elem} ======================================== In the remainder of this paper, we assume $p$ is an odd prime. We continue to use the notation $d=(p-1)/2$. In this section, we collect some notation and a few elementary number-theoretical facts. To be specific we pick particular values for $A$ and $\kappa$. We put $A= \zeta_{2p}$ where $\zeta_n = e^{2 \pi i/{n}},$ and also use the notation[^1] $$q=A^2.$$ We may then take $\kappa = A^{-3}(-i)^{({p+1})/{2}}.$ Note that $\zeta_{2p}\in {{\mathbb{Z}}}[\zeta_p]$. Thus the coefficient ring is $R ={{\mathbb{Z}}}[\zeta_p, \frac 1 p]$ if $p \equiv -1 \pmod{4},$ and $R ={{\mathbb{Z}}}[\zeta_{p},i, \frac 1 p]={{\mathbb{Z}}}[\zeta_{4p}, \frac 1 p]$ if $p \equiv 1 \pmod{4}.$ Of course, the coefficient ring remains unchanged if $A$ is replaced by another primitive $2p$-th root of unity, and $\kappa$ is changed accordingly. We let $\eta$ denote $\langle S^3\rangle$, the invariant of $S^3$ with weight zero, and put $\mathcal{D}= \eta^{-1}$. Then using equations on [@BHMV2 p.897] $$\label{D} \mathcal{D} = \frac{i\ \sqrt{p} }{q-q^{-1}} = \frac{i^{\frac {p+1} 2} }{q-q^{-1}} \left(\frac 1 2 \sum _{m=1}^{2p}(-1)^m A^{m^2}\right)$$ In particular $$\label{DD} \mathcal{D}^2= \frac {-p}{ ({q-q^{-1}})^2}~.$$ We denote by $\mathcal{O} $ the ring of integers in $R.$ Note that $\mathcal{O}={{\mathbb{Z}}}[\zeta_p]$ if $p \equiv -1 \pmod{4},$ and $\mathcal{O} ={{\mathbb{Z}}}[\zeta_{p},i]={{\mathbb{Z}}}[\zeta_{4p}]$ if $p \equiv 1 \pmod{4}.$ The following notation will be useful. If $x,y$ are elements of ${{\mathcal{O}}}$ (or, more generally, of its quotient field), we write $x\sim y$ if there exists a unit $u\in {{\mathcal{O}}}$ such that $x=uy$. \[num\] - $1-A$ is a unit in ${{\mathcal{O}}}$, and $1-q\sim 1+A$. - One has $\mathcal{D}\in {{\mathcal{O}}}$. Moreover, $\mathcal{D}\sim (1-q)^{({p-3})/2}=(1-q)^{d-1}$. - The quantum integers $[n]=(q^n-q^{-n})/(q-q^{-1})$ are units for $1\leq n\leq p-1$. - If $0\leq i,j \leq d-1$ and $i\neq j$, then the twist coefficients $\mu_i$ (see (\[mui\])) satisfy $\mu_i-\mu_j\sim 1-q$. - Put $\lambda_i=-q^{i+1}-q^{-i-1}$. If $0\leq i \leq d-1$, then $\lambda_0-\lambda_i\sim (1-q)^2$. The fact that $1-A$ is a unit follows easily from the fact that $A$ is a zero of the $2p$-th cyclotomic polynomial $1-X+X^2-\ldots +X^{p-1}$. This proves (i). It is well-known that $p\sim (1-q)^{p-1}$ (see [e.g.]{} [@MR Lemma 3.1]). Together with Formulas (\[D\]) and (\[DD\]), this shows (ii). Observing that $[n]\sim 1+q^2+\ldots +q^{2n-2}$, (iii) is also shown in [@MR Lemma 3.1]. For (iv), observe that $\mu_i=\mu_{p-2-i}$ so that the set of $\mu_i$ in question is equal to the set of $\mu_{2i}=q^{ 2 i^2+2i}$ for $i=0,1,\ldots d-1$. These powers of $q$ are all distinct, which implies $$\mu_i-\mu_j \sim 1-q^n \sim 1-q$$ for some $0<n<p$. This proves (iv). Similarly, (v) follows from $\lambda_0-\lambda_i\sim(1-q^{i+2})(1-q^i)$. [It is well-known that $1-q$ is a prime in ${{\mathbb{Z}}}[q]={{\mathbb{Z}}}[\zeta_{p}]$. But if $p \equiv 1 \pmod{4},$ then $1-q$ is not a prime in ${{\mathcal{O}}}={{\mathbb{Z}}}[\zeta_{4p}]$ (it splits as a product of two conjugate prime ideals). ]{} Associated Integral Cobordism Functors ====================================== In [@G], a cobordism functor from a restricted cobordism category to the category of free finitely generated $\mathcal{O}$-modules is described. Let $\mathcal{C''}$ denote the subcategory of $\mathcal{C}$ defined by considering only nonempty connected surfaces and connected morphisms between such surfaces. This represents a further restriction of $\mathcal{C}$ than that considered in [@G], but it suffices for our purposes. [If $\Sigma$ is a connected e-surface, define ${{\mathcal{S}_p}}(\Sigma)$ to be the $\mathcal{O}$-submodule of ${{V_p}}(\Sigma)$ generated by connected vacuum states. If $N: \Sigma \rightarrow \Sigma'$ is a morphism of $\mathcal{C''}$ then ${{Z_p}}(N)$ sends $ [M] \in {{\mathcal{S}_p}}(\Sigma)$ to $[M \cup_{\Sigma}N] \in {{\mathcal{S}_p}}(\Sigma').$ In this way we get a functor from $\mathcal{C''}$ to the category of finitely generated $\mathcal{O}$-modules. We also rescale the Hermitian form on ${{V_p}}(\Sigma)$ to obtain an $\mathcal{O}$-valued Hermitian form $$(\ ,\ )_{\Sigma}: {{\mathcal{S}_p}}(\Sigma) \otimes_{{\mathcal{O}}}{{\mathcal{S}_p}}(\Sigma) \rightarrow \mathcal{O},$$ defined by $$([N_1],[N_2])_{\Sigma}= \mathcal{D} \langle [N_1],[N_2]\rangle_{\Sigma} = \mathcal{D} \langle N_1 \cup_{\Sigma} -N_2\rangle .$$ ]{}** This form takes values in $\mathcal{O}$ by the integrality result for closed 3-e-manifolds [@Mu2; @MR]. These theorems are also used in proving that ${{\mathcal{S}_p}}(\Sigma)$ is finitely generated [@G]. [Over a Dedekind domain such as ${{\mathcal{O}}}$, a finitely generated torsion-free module is always projective, but it need not be free. (The typical examples are non-principal ideals in ${{\mathcal{O}}}$.) Somewhat surprisingly, however, it turns out that the modules ${{\mathcal{S}_p}}(\Sigma)$ are always free. This is proved in [@G]. We will not actually make use of this fact in genus $1$ and $2$: freeness will follow from the construction of explicit bases. ]{} [A Hermitian form on a projective ${{\mathcal{O}}}$-module $S$ is called [non-degenerate]{} (or [non-singular]{}) if its adjoint map $S\rightarrow S^$ is injective. It is called [unimodular]{} if the adjoint map is an isomorphism.* ]{} Note that if $S$ is free and $M$ is the matrix of the Hermitian form in some basis, then the form is non-degenerate (resp. unimodular) if $\det M$ is non-zero (resp. a unit in ${{\mathcal{O}}}$). In our situation, the form $ (\ ,\ )_{\Sigma}$ is always non-degenerate (since the original form $\langle \ ,\ \rangle_{{\Sigma}}$ on ${{V_p}}({{\Sigma}})$ is). We will show that $ (\ ,\ )_{\Sigma}$ is unimodular in genus $1$ and $2$. There is a standard basis $\{u_\sigma\}$ of ${{V_p}}(\Sigma_g)$ given by p-admissible [ even]{} colorings $\sigma$ of the graph (-.5,-.2)(5,1) (0,.5)[.35]{} (1,.5)[.35]{} (.35,.5)(.65,.5) (1.35,.5)(1.65,.5) (3.35,.5)(3.65,.5) (4,.5)[.35]{} (2,.45)[$\cdots$]{} (where there are $g$ loops) embedded in a 3-e-handlebody $H_g$ of genus $g$ with boundary the e-surface ${{\Sigma}}_g$ (see [@BHMV2 4.11]). One may actually use any trivalent graph in $H_g$ to which $H_g$ deformation retracts. (In the case $g=1$, this is the same as the basis given by the elements $e_i$.) These basis elements lie in ${{\mathcal{S}_p}}(\Sigma_g)$ because the denominators appearing in the Jones-Wenzl idempotents needed to expand colored graphs into skein elements are invertible in ${{\mathcal{O}}}$ (see [@MR]). Warning: the $u_\sigma$ do [not]{} generate ${{\mathcal{S}_p}}(\Sigma_g)$ over ${{\mathcal{O}}}$. \[detb\] The elements $u_\sigma$ are orthogonal for the form $(\ ,\ )_{\Sigma_g}.$ Moreover, one has $$(u_\sigma,u_\sigma)_{\Sigma_g}\sim {{\mathcal{D}}}^g\sim (1-q)^{(d-1)g}.$$ By [@BHMV2 Theorem 4.11] one has that $\langle u_\sigma,u_\sigma\rangle_{\Sigma_g}$ is equal to $\eta^{1-g}={{\mathcal{D}}}^{g-1}$ times a product of non-zero quantum integers or their inverses, which are units in ${{\mathbb{Z}}}[q]$ by Lemma \[num\]. Since the form $(\ ,\ )_{\Sigma_g} $ is just a rescaling of the form $\langle \ ,\ \rangle_{\Sigma_g} $, the result follows. One of the reasons to study the form $(\ ,\ )_{\Sigma_g} $ is that it is preserved by the TQFT-action of the mapping class group. More precisely, let $\tilde{\Gamma}(\Sigma)$ denote the central extension of the mapping class group $\Gamma(\Sigma)$ of $\Sigma$ realized by the subcategory of $C''$ consisting of e-manifolds homeomorphic to $\Sigma \times I$ such that the colored graph is given by $I$ times the colored banded points of $\Sigma.$ This homeomorphism need not respect the boundary identification at $\Sigma \times \{1\},$ but should respect the boundary identification at $\Sigma \times \{0\}.$ In fact considering this boundary identification at $\Sigma \times \{1\},$ defines the quotient homomorphism from $\tilde{\Gamma}(\Sigma)$ to $\Gamma(\Sigma),$ which has kernel ${{\mathbb{Z}}}$ given by the integral weights on $\Sigma \times I$ with standard boundary identifications. The group $\tilde \Gamma(\Sigma)$ is isomorphic to the signature extension (see [e.g.]{} Atiyah [@At], Turaev [@Tu].) This extension can be described nicely using skein theory [@MR1]. The group $\tilde \Gamma(\Sigma)$ acts on ${{V_p}}(\Sigma)$ preserving the $\mathcal{O}$-lattice ${{\mathcal{S}_p}}(\Sigma)$ and the ${{\mathcal{O}}}$-valued Hermitian form $ (\ ,\ )_{\Sigma}$. This follows from the definition of ${{\mathcal{S}_p}}(\Sigma)$ and the fact that the group $\tilde \Gamma(\Sigma)$ preserves the original Hermitian form $ \langle\ ,\ \rangle_{\Sigma}$. The module ${{\mathcal{S}_p}}(\Sigma)$ can be described using the notion of ‘mixed graph’. Recall the element $$\omega= {{\mathcal{D}}}^{-1} \sum_{i=0}^{d-1} \langle e_i\rangle e_i \in K(S^1 \times D^2,R).$$ Here $\langle e_i\rangle=(-1)^i [i+1]$. It plays an important role in the surgery axioms of the ${{V_p}}$-theory. By a [mixed graph]{} in a weighted 3-manifold $M, $ we mean a trivalent banded graph in $M$ whose simple closed curve components may possibly be colored $\omega$ or by integer colors in the range $[ 0,p-2]$ and whose other edges are colored p-admissibly by integers in the range $[ 0,p-2]$. A mixed graph can be expanded multilinearly into a $R$-linear combination of colored graphs. The result should be thought of as a superposition of e-morphisms. If the graph is a link and every component is colored $\omega,$ we say the link is $\omega$-colored. A mixed graph in a handlebody $H$ specifies an element in ${{V_p}}(\partial H).$ \[ml\] A mixed graph in a connected 3-e-manifold $M$ with boundary $\Sigma$ represents an element of ${{\mathcal{S}_p}}(\Sigma).$ If $H$ is a 3-e-handlebody with boundary e-surface $\Sigma$ then ${{\mathcal{S}_p}}(\Sigma)$ is generated over $\mathcal{O}$ by elements specified by mixed graphs in $H.$ The first statement follows from the fact that ${{V_p}}$ satisfies the surgery axiom (S2) [@BHMV2 p 889]. The second statement follows from the fact that any connected 3-manifold with boundary $\Sigma$ can be obtained by a sequence of 2-surgeries to $H$ [@BHMV2 Proof of Lemma p. 891]. \[5.7\] [Suppose that we know that some collection $T$ of elements of ${{V_p}}(\Sigma)$ lie in the $\mathcal{O}$-lattice ${{\mathcal{S}_p}}(\Sigma).$ Then $\text{Span}_\mathcal{O}(T)$ is a $\mathcal{O}$-sublattice of ${{\mathcal{S}_p}}(\Sigma).$ This sublattice might not be invariant under $\tilde{\Gamma}(\Sigma)$. Let $G=\{g_i\} \in \tilde \Gamma(\Sigma)$ be a finite set of elements whose image in $\Gamma(\Sigma)$ generate. The sequence of submodules of ${{\mathcal{S}_p}}(\Sigma)$: $\text{Span}_\mathcal{O}(T),$ $\text{Span}_\mathcal{O}(T\cup G(T)),$ $\text{Span}_\mathcal{O}(T\cup G(T) \cup G(G(T))),$ , $ \ldots $ etc. must stabilize in an $\mathcal{O}$-sublattice of ${{\mathcal{S}_p}}(\Sigma)$ which is invariant under the mapping class group. This procedure is well suited to computer investigation. The basis given in Section \[2nd\] was originally found by this procedure. We used the computer program Kant [@D] starting with $T= \{e_0,e_1,\ldots, e_{d-1}\}$ in ${{\mathcal{S}_p}}(S^1\times S^1).$ ]{} First integral basis in genus 1 {#1st} =============================== By a slight abuse of notation, we let $\omega$ denote the element in ${{\mathcal{S}_p}}(S^1 \times S^1)$ given by coloring the core of $S^1 \times D^2$ with $\omega.$ Let $t$ also denote the induced map on ${{V_p}}(S^1 \times S^1)$ given by giving $S^1 \times D^2$ a full right handed twist. Note that $t^n(\omega)\in {{\mathcal{S}_p}}(S^1 \times S^1)$ for all $n$. \[firstbasis\] $\{ \omega, t(\omega) , t^2(\omega) , \ldots , t^{d-1}(\omega)\}$ is a basis for the module ${{\mathcal{S}_p}}(S^1 \times S^1).$ The form $( \ ,\ )_{S^1 \times S^1}$ is unimodular. Note that it follows in particular that the ${{\mathcal{O}}}$-span of $\{ \omega, t(\omega) , t^2(\omega) , \ldots , t^{d-1}(\omega)\}$ is stable under the action of the mapping class group $\tilde{\Gamma}( S^1 \times S^1) $. Recall that $\mu_i=(-1)^{i} A^{i^2+2i}$ denotes the eigenvalue of $e_i$ under the twist map $t$. We have that $$t^j (\omega)= {{\mathcal{D}}}^{-1} \sum_{i=0}^{d-1} \langle e_i\rangle \mu_i^j e_i~.$$ Note that $\langle e_i\rangle=(-1)^i [i+1]$ is a unit by Lemma \[num\](iii). The matrix $W$ which expresses $\{ \omega, t(\omega) , t^2(\omega) , \ldots , t^{d-1}(\omega)\}$ in terms of $\{e_0,e_1,\ldots e_{d-1}\}$ has as determinant a unit (the product of the $\langle e_i\rangle$) times ${{\mathcal{D}}}^{-d}$ times the determinant of the Vandermonde matrix $[ \mu_i^j ]$ where $0 \le i, j\le d-1.$ Moreover by Lemma \[num\](iv) $$\det [ \mu_i^j ] = \pm \prod_{i<j} (\mu_i-\mu_j)\sim (1-q)^{{d(d-1)}/2}~.$$ As ${{\mathcal{D}}}\sim (1-q)^{d-1},$ we conclude that $$\label{detW} \det W\sim (1-q)^{- {d(d-1)}/2}~.$$ In particular, this determinant is non-zero, hence the $t^j(\omega)$ are linearly independent. Let $\mathcal {W}$ denote the $\mathcal{O}$-module spanned by the $t^j(\omega)$. Clearly $\mathcal W \subset {{\mathcal{S}_p}}(S^1 \times S^1)$. Now we know by Proposition \[detb\] that $(e_i,e_i)\sim (1-q)^{d-1}$ (here we simply write $(\ ,\ )$ for the Hermitian form $(\ ,\ )_{S^1 \times S^1} $). Therefore the matrix for $(\ ,\ )$ with respect to the orthogonal basis $\{e_0,e_1,\ldots, e_{d-1}\}$ has determinant $(1-q)^{d(d-1)}$. By (\[detW\]) it follows that the matrix for $(\ ,\ )$ with respect to $\{ \omega, t(\omega) , t^2(\omega) , \ldots , t^{d-1}(\omega)\}$ has unit determinant. (Here we use that $\overline{1-q}=1-q^{-1}\sim 1-q$.) In other words, the form $(\ ,\ )$ restricted to $\mathcal {W}$ is unimodular. But then $\mathcal {W}$ must be equal to ${{\mathcal{S}_p}}(S^1 \times S^1)$. This completes the proof. \[wc\] If $H$ is a 3-e-handlebody with boundary the e-surface of $\Sigma$ and $\Sigma$ has no colored points in the boundary, then $ {{\mathcal{S}_p}}(\Sigma)$ is generated over $\mathcal{O}$ by elements represented by $\omega$-colored banded links in $H.$ By the above theorem, each $e_i$ (in particular $e_1=z$) can be expressed as an $\mathcal{O}$-linear combinations of the elements $t^j(\omega)$. Therefore every mixed graph can be written as an $\mathcal{O}$-linear combination of $\omega$-colored banded links in $H.$ The result now follows from Theorem \[ml\]. Second integral basis in genus 1 {#2nd} ================================ Consider $K(d-1)$ in the notation of Section \[Kauf\], now taking $A= \zeta_{2p}.$ Let $\mathcal{V}$ denote its image in ${{V_p}}(S^1 \times S^1).$ In other words $\mathcal{V}$ is the $\mathcal{O}$-submodule of ${{V_p}}(S^1 \times S^1)$ generated by $\{ 1, v ,v^2 , \ldots , v^{d-1}\},$ where $v=(z+2)/(1+A)$. \[secondbasis\] One has $\mathcal{V}={{\mathcal{S}_p}}(S^1 \times S^1).$ In particular, $\{ 1, v ,v^2 , \ldots , v^{d-1}\}$ is a basis for the free module ${{\mathcal{S}_p}}(S^1 \times S^1).$ We refer to this basis as the $v$-basis of ${{\mathcal{S}_p}}(S^1 \times S^1).$ We originally found it by the procedure outlined in Remark \[5.7\]. Since $\tilde{\Gamma}(\Sigma)$ preserves ${{\mathcal{S}_p}}(S^1 \times S^1)$, we have the following Corollary. $\mathcal{V}=\text{\em Span}_{{\mathcal{O}}}\{1,v,v^2,\ldots,v^{d-1}\}$ is stable under the action of the mapping class group $\tilde{\Gamma}(S^1 \times S^1)$. [The mapping class group $\tilde{\Gamma}(S^1 \times S^1)$ is a central extension of $SL(2,{{\mathbb{Z}}})$. Its image in $GL({{V_p}}(S^1 \times S^1))$ is generated by $\kappa $ times the identity matrix (the central generator acts as multiplication by $\kappa$), the twist map $t$, and the so-called $S$-matrix. The entries of the $S$-matrix in the $e_i$-basis are well-known. One can therefore write down its entries in the $v$-basis (using the change of basis formulas in Lemma \[changeofbasis\]). The fact that these entries lie in ${{\mathcal{O}}}$ is by no means obvious. We originally proved this fact using some identities involving binomial coefficients. The argument is similar to the proof that the $v$-basis is stable under the twist map $t$ given in Section \[Kauf\], but considerably more complicated. We found proofs of these identities using Zeilberger’s algorithm together with some identities from [@B] as above. In particular the [Gosper]{} command in the Mathematica package “Fast Zeilberger” (V 2.61) by Peter Paule and Markus Schorn, [@Zeil] was used. As the proof we give below is much simpler, we omit the details of this computation. ]{} One has $\omega \in {{\mathcal{V}}}~.$ Let $\lambda_i= -q^{i+1}- q^{-i-1}$. Recall [@BHMV1] that $e_i$ is an eigenvector with eigenvalue $\lambda_i$ for the endomorphism $c$ of $K (S^1 \times D^2,{{\mathbb{Z}}}[A^{\pm}]) $ given by sending a skein in $S^1 \times D^2$ to the skein circled by a meridian. Let $\langle\ ,\ \rangle_H$ be the Hopf pairing ([i.e.]{} the symmetric bilinear form on ${{V_p}}(S^1 \times S^1)$ which sends two elements $x,y$ to the bracket of the zero-framed Hopf link with one component cabled by $x$, and the other component cabled by $y$). Then $$\label{omip} \langle \omega,e_i\rangle_H = \begin{cases} \langle\omega\rangle={{\mathcal{D}}}, &\text{if } i=0\\ \ \ \ 0 &\text{if } 1\leq i\leq d-1 \end{cases}$$ Note that $\langle z-\lambda_i,e_i\rangle_H=0$ for $1\leq i\leq d-1$, and $\langle z-\lambda_i,e_0\rangle_H=\lambda_0-\lambda_i$ (since $\langle z,e_0\rangle = \langle z\rangle =-q-q^{-1}=\lambda_0$). It follows that $$\label{omi} \omega={{\mathcal{D}}}\prod_{i=1}^{d-1} \frac {z-\lambda_i} {\lambda_0-\lambda_i}$$ since the pairing $\langle\ ,\ \rangle_H$ is non-degenerate. (Note the similarity with the polynomials $Q_n$ of [@BHMV1].) Since ${{\mathcal{D}}}\sim (1-q)^{d-1}$ and $\lambda_0-\lambda_i\sim (1-q)^2$ by Lemma \[num\], it follows that $$\label{omi2} \omega \sim \prod_{i=1}^{d-1} \frac {z-\lambda_i} {1-q}\sim \prod_{i=1}^{d-1} \frac {z-\lambda_i} {1+A}$$ (where $\sim$ means equality up to multiplication by a unit). Now $$\begin{aligned} z-\lambda_i &= (z+2)-(2 +\lambda_i)\\ &= (z+2)-(1-q^{i+1})(1-q^{-i-1})\\ &= (z+2)+ u_i (1+A)^2\end{aligned}$$ where $u_i \in {{\mathcal{O}}}.$ It follows that $$(z-\lambda_i)/(1+A) \in \text{Span}_{{\mathcal{O}}}\{1,v\}~,$$ and so (\[omi2\]) implies $\omega \in {{\mathcal{V}}}~,$ proving the lemma. By Theorem \[tK\], $K(n)$ hence ${{\mathcal{V}}}$ is stable under the twist map $t$. It follows that $$\mathcal{W}=\text{Span}_{{\mathcal{O}}}\{\omega, t(\omega),\ldots, t^{d-1}(\omega)\} \subseteq \mathcal{V}~.$$ Now recall from the proof of Theorem \[firstbasis\] that the matrix $W$ which expresses $\{\omega, t(\omega),\ldots, t^{d-1}(\omega)\}$ in terms of $\{e_0,e_1,\ldots,e_{d-1}\}$ has determinant $\det W \sim (1-q)^{-d(d-1)/2}$. Remembering $v=(z+2)/(1+A)$ and $1+A\sim 1-q$, it is easy to see that the same is true for the matrix which expresses $\{ 1,v,\ldots,v^{d-1}\}$ in terms of $\{e_0,e_1,\ldots e_{d-1}\}$. Since $\mathcal{W}\subset\mathcal{V}$, it follows that actually $\mathcal{W}= \mathcal{V}$. By Theorem \[firstbasis\] we conclude $\mathcal {V}={{\mathcal{S}_p}}(S^1 \times S^1)$. This completes the proof. By a $v$-colored banded link in a 3-manifold, we mean a banded link whose components are colored $v.$ As before this should be interpreted as the linear combination (superposition) of the colored banded links that one obtains by expanding multilinearly. We note that $i$ parallel strands colored $v$ is the same as one strand colored $v^i.$ \[v\] If $H$ is a 3-e-handlebody with boundary the e-surface $\Sigma$ and $\Sigma$ has no colored points in the boundary, then ${{\mathcal{S}_p}}(\Sigma)$ is generated over $\mathcal{O}$ by elements represented by $v$-colored banded links in $H.$ The matrix of the Hermitian form $(\ ,\ )_{S^1\times S^1}$ in the $v$-basis is easily computed. One has for $0\leq i,j\leq d-1$ $$\begin{aligned} (v^i,v^j)_{S^1\times S^1}&={{\mathcal{D}}}\langle v^i,v^j\rangle_{S^1\times S^1} = \langle v^{i+j},\omega \rangle_H ={(1+A)^{-(i+j)}} \langle (z+2)^{i+j},\omega \rangle_H\\ &=\frac {1} {(1+A)^{i+j}} \binom{2i+2j+2} {i+j} \frac 1 {i+j+1} \langle e_0,\omega \rangle_H\\ &=\frac {{{\mathcal{D}}}} {(1+A)^{i+j}} \binom{2i+2j+2} {i+j} \frac 1 {i+j+1}\end{aligned}$$ Here we have used Lemma \[changeofbasis\] to express $(z+2)^{i+j}$ in terms of the $e_n$, and then retained only the $e_0$ term. Indeed, the others are annihilated by the Hopf pairing with $\omega$ since $0\leq i+j\leq 2d-2$ (see (\[omip\]) and remember that $e_{d+i}=e_{d-1-i}$ in ${{V_p}}(S^1\times S^1)$). It is instructive to check directly that the expression above lies in ${{\mathcal{O}}}$ (use that $p$ divides the binomial coefficient $\binom{2i+2j+2} {i+j}$ if $d\leq i+j\leq 2d-2$). Integral basis in genus 2 {#g2} ========================= Let ${{\Sigma}}_2$ be a closed surface of genus $2$. In this section, we describe a basis for the module ${{\mathcal{S}_p}}({{\Sigma}}_2)$ and show that the Hermitian form $(\ ,\ )_{{{\Sigma}}_2}$ is unimodular. Let $H_2$ be a regular neighborhood of the hand cuff graph ${ \psset{unit=.5cm} \begin{pspicture}[.4](-.5,0)(1.5,1) \pscircle(0,.5){.35} \pscircle(1,.5){.35} \psline(.35,.5)(.65,.5) \end{pspicture} }$ in $\mathbb{R}^3$. Then $H_2$ is a genus $2$ handlebody and by Corollary \[v\], ${{\mathcal{S}_p}}({{\Sigma}}_2)$ is spanned by $v$-colored banded links in $H_2$. We think of $H_2$ as $P_2 \times I$ where $P_2$ is a disk with two holes. The skein module $K(H_2,R)$ is free on the set of isotopy classes of collections of nonintersecting essential simple closed curves in $P_2$. We refer to these isotopy classes as arrangements of curves. Such arrangements can be indexed by 3-tuples of nonnegative integers. Let $C_{\alpha,\beta,\gamma}$ denote the arrangement with $\gamma$ parallel curves going around both holes, and within them $\alpha$ parallel curves going around the left hole, and $\beta$ parallel curves going around the right hole. See Figure \[fig0\] for an example. (-.6,-.8)(2.6,.8) (0,0)[.35]{}(2,0)[.35]{} (-.05,-.05)(.05,.05) (-.05,.05)(.05,-.05) (1.95,-.05)(2.05,.05) (1.95,.05)(2.05,-.05) (0,0)[.45]{}(2,0)[.45]{} (2,0)[.55]{} (0,0)[.7]{}[45]{}[315]{}(2,0)[.7]{}[225]{}[135]{} (.5,.49)(.6,.43)(.8,.4)(1.2,.4)(1.4,.43)(1.5,.49) (.5,-.49)(.6,-.43)(.8,-.4)(1.2,-.4)(1.4,-.43)(1.5,-.49) \[thg2\] Let $C_{\alpha,\beta,\gamma}(v)$ be the element of ${{\mathcal{S}_p}}({{\Sigma}}_2)$ obtained by coloring each curve of $C_{\alpha,\beta,\gamma}$ by $v=(z+2)/(1+A)$. Then the set $$\{ C_{\alpha,\beta,\gamma}(v)\,\|\, 0\le \gamma \le d-1, \quad 0\le \alpha,\beta \le d-1-\gamma \}$$ is a basis of ${{\mathcal{S}_p}}({{\Sigma}}_2).$ Moreover, the Hermitian form $(\ ,\ )_{{{\Sigma}}_2}$ is unimodular. Note that $C_{\alpha,\beta,\gamma}(v)$ lies in ${{\mathcal{S}_p}}({{\Sigma}}_2)$ because $v$ lies in ${{\mathcal{S}_p}}(S^1\times S^1)$ by Theorem \[secondbasis\]. Let us first describe a basis of ${{V_p}}({{\Sigma}}_2)$ consisting of elements represented by colorings of the hand cuff graph (-.5,0)(1.5,1) (0,.5)[.35]{} (1,.5)[.35]{} (.35,.5)(.65,.5) . Let $G(i,j,k)$ be the element defined by the colored graph (-.5,0)(2,1) (0,.5)[.35]{} (1.5,.5)[.35]{} (.35,.5)(1.15,.5) (-.7,.5)[$i$]{} (2,.5)[$j$]{} (.6,.7)[$k$]{} For this element to exist, $k$ must be even. Then the coloring is $p$-admissible if and only if $\frac k 2 \leq i,j\leq p-2-\frac k 2$ (see [@BHMV2 Thm 1.15]). The standard basis of ${{V_p}}({{\Sigma}}_2)$ would be to take the $p$-admissible $G(i,j,k)$ with both $i$ and $j$ even. It is also possible to impose that one or both of $i,j$ be odd [@BHMV2 Thm 4.14]. We will need a different basis where $i,j$ are allowed to be both even and odd, but $\leq d-1$. This is given in the following Lemma. \[lemG\] The $G(i,j,k)$ with $k$ even in the range $[0,p-3]$, and both $i$ and $j$ in the range $[\frac k 2, d-1]$ (but not necessarily even), form a basis of ${{V_p}}({{\Sigma}}_2)$. [ Let $ \mathcal{G}$ be the basis described in the above Lemma. Let $ \mathcal{G}_k$ be the subset of elements of $ \mathcal{G}$ with middle arc colored $k$. The cardinality of $ \mathcal{G}_k$ is $(d-\frac k 2)^2.$ Thus we see directly that the cardinality of this basis is $ \sum_{j=0}^{d-1} (d-j)^2= \sum_{j=1}^{d} j^2= d (d+1)(2d+1)/6 .$ ]{} Lemma \[lemG\] could be proved using the methods of [@BHMV2]. Here we give a different, more direct proof. For $i$ in the range $[0,p-2]$ we let $i'=p-2-i$. We claim that $$G(i,j,k)\sim G(i',j,k)\sim G(i,j',k)\sim G(i',j',k)$$ (where $\sim$ means equality up to multiplication by a unit in ${{\mathcal{O}}}$). It is enough to prove that $G(i,j,k)\sim G(i',j,k).$ This is done in Figure \[fig1\]. Note that if $i$ is even and $> d-1$ then $i'$ is odd and $\leq d-1$. Thus the basis of where all $i,j$ are even may be replaced by the basis of Lemma \[lemG\]. (-2,-1.5)(3.5,1.5) (-.1,-.1)(.1,.1) (-.1,.1)(.1,-.1) (0,0)[1.5]{} (1.5,0)(3,0) (-2,.1)[$i$]{} (2,.3)[$k$]{} $=$ (-2,-1.5)(3.5,1.5) (-.1,-.1)(.1,.1) (-.1,.1)(.1,-.1) (0,0)[1]{} (-.6,.25)[$\tilde p$]{} (0,0)[1.5]{} (1.5,0)(3,0) (-2,.1)[$i$]{} (2,.3)[$k$]{} \ $ = c_1$ (-1.5,-1.5)(3.5,1.5) (-.1,-.1)(.1,.1) (-.1,.1)(.1,-.1) (0,0)[1]{}[270]{}[90]{} (.35,0)[$\tilde p$]{} (0,0)[1.5]{}[270]{}[90]{} (0,0)[1.25]{}[118]{}[242]{} (0,1.5)(-.3,1.4)(-.6,1.1) (0,1)(-.3,1.02)(-.6,1.1) (0,-1.5)(-.3,-1.4)(-.6,-1.1) (0,-1)(-.3,-1.02)(-.6,-1.1) (1.5,0)(3,0) (-1.8,.1)[$i'$]{} (2,.3)[$k$]{} (1.1,1.4)[$i$]{} (1.3,-1.3)[$i$]{} $ = c_1 c_2$ (-1.5,-1.5)(3.5,1.5) (-.1,-.1)(.1,.1) (-.1,.1)(.1,-.1) (0,0)[1.5]{} (1.5,0)(3,0) (-2,.1)[$i'$]{} (2,.3)[$k$]{} Let $ \mathcal{A}(v)$ be the set of the $v$-colored elements $C_{\alpha, \beta,\gamma}(v)$ claimed to be a basis in Theorem \[thg2\], and let $ \mathcal{A}$ be the set of the uncolored ([i.e.]{} colored by $z=e_1$) elements $C_{\alpha, \beta,\gamma}$ (in the same range for $\alpha, \beta,\gamma$). The set $ \mathcal{A}$ is a basis of ${{V_p}}(\Sigma_2)$. Moreover, the basis change from $ \mathcal{G}$ to $\mathcal{A}$ has determinant $\pm 1$. Using the Wenzl recursion formula for the idempotents of the Temperley-Lieb algebra, one can expand the elements of $\mathcal{A}$ as $\mathcal{O}$-linear combinations of elements of the graph basis $\mathcal{G}.$ In fact, in the expansion of $C_{\alpha,\beta,\gamma}$, only those $G(i,j,k)$ occur where $i\leq \alpha +\gamma$, $j\leq \beta +\gamma$, and $k\leq 2\gamma$; moreover, $G(\alpha +\gamma,\beta +\gamma, 2\gamma)$ occurs with coefficient one. We can find orderings of $\mathcal{A}$ and $\mathcal{G}$ so that the matrix which expresses $\mathcal{A}$ in terms of $\mathcal{G}$ is triangular with ones on the diagonal (use the lexicographical orderings where $\gamma$ resp. $k$ is counted first). This implies the Lemma. Let $ r=d (d+1)(2d+1)/{6}$ be the rank of ${{V_p}}(\Sigma_2)$. By Proposition \[detb\], the matrix for $(\ ,\ )_{\Sigma_2} $ with respect to the orthogonal basis $\mathcal{G}$ has determinant $\sim (1-q)^{2 (d-1) r}$. The preceding Lemma shows that the same holds true for the matrix for $(\ ,\ )_{\Sigma_2} $ with respect to $\mathcal{A}$. Let $N$ denote the sum over $ \mathcal{A}$ of the number of curves appearing in each arrangement. The change of basis matrix for writing $\mathcal{A}(v)$ in terms of $\mathcal{A}$ is again triangular and has determinant $\sim (1-q)^{-N}.$ Thus the matrix for $(\ ,\ )_{\Sigma_2} $ with respect to $\mathcal{A}(v)$ has determinant $\sim (1-q)^{2 (d-1) r -2N}$. The following Lemma \[lemf\] shows that this determinant is a unit. As in the genus one case (see the proof of Theorem \[firstbasis\]), we conclude that $\mathcal{A}(v)$ is a basis for ${{\mathcal{S}_p}}({{\Sigma}}_2)$ and that the form $(\ ,\ )_{\Sigma_2} $ is unimodular on ${{\mathcal{S}_p}}({{\Sigma}}_2)$. \[lemf\] $N=(d-1)r$. To count $N, $ we write ${{\mathcal{A}}}= \cup_{ 0\le \gamma \le d-1} {{\mathcal{A}}}_\gamma,$ where $${{\mathcal{A}}}_\gamma= \{ C_{\alpha,\beta,\gamma} \| 0 \le \alpha,\beta \le d-\gamma-1 \}~.$$ Note that $\|{{\mathcal{A}}}_\gamma\|= (d-\gamma)^2.$ The total number of curves appearing in ${{\mathcal{A}}}_\gamma$ is $$\gamma (d-\gamma)^2 + \sum_{\alpha=0}^{d-\gamma-1}\sum_{\beta=0}^{d-\gamma-1} (\alpha +\beta) = (d-1) (d-\gamma)^2~.$$ Thus each ${{\mathcal{A}}}_\gamma$ contributes $d-1$ times its cardinality to the count. As $\sum_{\gamma =0}^{d-1} \|{{\mathcal{A}}}_\gamma\| =r,$ we see that $N=(d-1) r.$ This completes the proof of Theorem \[thg2\]. Non-Unimodularity {#non-uni} ================== Even without knowing an explicit basis of ${{\mathcal{S}_p}}({{\Sigma}}_g)$, it is possible to see that the form $(\ ,\ )_{{{\Sigma}}_g}$ is sometimes not unimodular. \[9.1\] If $p \equiv 1 \pmod{4}$ and both the genus $g$ and the rank of ${{V_p}}({{\Sigma}}_g)$ are odd, then the form $(\ ,\ )_{{{\Sigma}}_g}$ is not unimodular on ${{\mathcal{S}_p}}({{\Sigma}}_g)$. For example, if $g=3$ and $p=5$ then the rank is $15$ and the form $(\ ,\ )_{{{\Sigma}}_3}$ is not unimodular on $\mathcal{S}_5({{\Sigma}}_3)$. [We used Mathematica [@Wo] to calculate the rank of $V_{p}(\Sigma_g)$ for small $g$ using the formula [@BHMV2 1.16(ii)]. We found that: $$\begin{aligned} \text{rank} \left( V_{4k+1}(\Sigma_3)\right) &= (1 /{45} )(3 \ k+32 \ k^2+120 \ k^3+200 \ k^4+192 \ k^5+128 \ k^6) \\ \begin{split} \text{rank} \left(V_{4k+1}(\Sigma_5)\right) &= ( 1/ {14175} ) (45 \ k + 864 \ k^2 + 6892 \ k^3 + 30184 \ k^4\\ & + 83760 \ k^5 + 172512 \ k^6 + 304896 \ k^7 + 458112 \ k^8\\ & + 542720 \ k^9 + 487424 \ k^{10} + 294912 \ k^{11} + 98304 \ k^{12}) \end{split} \end{aligned}$$ Thus the rank of $V_p({{\Sigma}}_3)$ is odd if $p \equiv 5 \pmod{8},$ and the form $(\ ,\ )_{{{\Sigma}}_3}$ is not unimodular in this case. Similarly $(\ ,\ )_{{{\Sigma}}_5}$ is not unimodular if $p \equiv 5 \pmod{8}. $ ]{} The argument relies on the following result of [@G]. Assume $p \equiv 1 \pmod{4}$ and recall that ${{\mathcal{O}}}={{\mathbb{Z}}}[\zeta_{4p}]$ in this case. Put ${{\mathcal{O}}}^+={{\mathbb{Z}}}[\zeta_{p}]\subset {{\mathcal{O}}}$. Let the Lagrangian assigned to $\Sigma_g$ be the kernel of the map on the first homology induced by the inclusion of $ \Sigma_g$ to $H_g$ and assign $H_g$ the weight zero. Then $H_g$ is an [ even]{} (in the sense of [@G]) morphism from $\emptyset$ to $\Sigma_g.$ Note that the quantum integers $[n]$ for $1\le n \le p-1$ are units in ${{\mathcal{O}}}^+.$ [@G] If $p \equiv 1 \pmod{4}$ then ${{\mathcal{S}_p}}({{\Sigma}}_g)\simeq {{\mathcal{S}^+_p}}({{\Sigma}}_g)\otimes {{\mathcal{O}}}$ where ${{\mathcal{S}^+_p}}({{\Sigma}}_g)\subset {{\mathcal{S}_p}}({{\Sigma}}_g)$ is a free ${{\mathcal{O}}}^+$-module. Moreover, one has $\mathcal {G}\subset{{\mathcal{S}^+_p}}({{\Sigma}}_g)$, where $\mathcal {G}$ is the colored graph basis of ${{V_p}}({{\Sigma}}_g)$ (see Proposition \[detb\]). The matrix of $(\ ,\ )_{{{\Sigma}}_g}$ with respect to $\mathcal {G}$ has determinant ${{\mathcal{D}}}^{gr}\sim (1-q)^{(d-1)gr}$ where $r$ denotes the rank of ${{V_p}}({{\Sigma}}_g)$. Let $\mathcal {B}$ be a basis of the free ${{\mathcal{O}}}^+$-module ${{\mathcal{S}^+_p}}({{\Sigma}}_g)$, and let $D$ be the determinant of the matrix expressing $\mathcal {B}$ in terms of $\mathcal {G}$. The matrix of $(\ ,\ )_{{{\Sigma}}_g}$ with respect to the basis $\mathcal {B}$ has determinant $\sim \Delta$, where $$\label{Delta} \Delta = D \overline D (1-q)^{(d-1)gr}~.$$ If the form is unimodular, $\Delta$ must be a unit in ${{\mathcal{O}}}$, and since $\Delta$ lies in ${{\mathcal{O}}}^+$, it must be a unit in ${{\mathcal{O}}}^+$. But $1-q$ is a self-conjugate prime in ${{\mathcal{O}}}^+={{\mathbb{Z}}}[q]={{\mathbb{Z}}}[\zeta_{p}]$, and since $D^{-1}$ lies in ${{\mathcal{O}}}^+$ as well, $\Delta$ can be a unit only if $(d-1)gr$ is even. Thus one of $g$ and $r$ must be even (since $d-1=(p-3)/2$ is odd in our situation). This completes the proof. \[9.4\] [The use of the ${{\mathcal{O}}}^+$-module ${{\mathcal{S}^+_p}}({{\Sigma}}_g)$ can in general not be avoided in this argument. Here is why. Recall that $1-q$ splits in ${{\mathcal{O}}}={{\mathbb{Z}}}[\zeta_{4p}]$ as the product of two conjugate prime ideals $\mathfrak{p}$ and $\overline{\mathfrak{p}}$. If $\mathfrak p$ is principal (this happens for example if $p=5$), then there exists $D\in {{\mathcal{O}}}$ such that the number $\Delta$ defined as in (\[Delta\]) is a unit even when $(d-1)gr$ is odd. Of course, such a $D$ does not exist in ${{\mathcal{O}}}^+$.]{} [If we assign extra structure to ${{\Sigma}}_g$ and $H_g$ as described above in the proof of 9.1, then ${{\mathcal{O}}}^+$ linear combinations of banded links in $H$ represent elements in ${{\mathcal{S}^+_p}}({{\Sigma}}).$ Moreover the bases described in Sections \[1st\], \[2nd\], \[g2\] for ${{\mathcal{S}_p}}(S^1 \times S^1), $ and ${{\mathcal{S}_p}}({{\Sigma}}_2)$ are actually bases for ${{\mathcal{S}^+_p}}(S^1 \times S^1), $ and ${{\mathcal{S}^+_p}}({{\Sigma}}_2).$ There are also plus versions of Theorem \[ml\] and Corollaries \[wc\] and \[v\].]{} [When restricted to ${{\mathcal{S}^+_p}}(\Sigma_g)$, the Hermitian form $(\ ,\ )_{\Sigma_g}$ does not take values in ${{\mathcal{O}}}^+,$ if $g$ is odd. This follows from the proof of Proposition \[detb\], since ${{\mathcal{D}}}\not\in {{\mathcal{O}}}^+$. In the next section, we will use the sesquilinear form $$(\ ,\ )^+_{\Sigma_g}: {{\mathcal{S}^+_p}}(\Sigma_g) \times {{\mathcal{S}^+_p}}(\Sigma_g) \rightarrow \mathcal{O}^+$$ obtained by multiplying the form $(\ ,\ )_{\Sigma_g}$ by $i^{\varepsilon (g)},$ where ${\varepsilon (g)}$ is zero or one accordingly as $g$ is even, or odd. This form takes values in ${{\mathcal{O}}}^+$ since $i {{\mathcal{D}}}\in {{\mathcal{O}}}^+$. ]{} Genus three at the prime five {#g3} ============================= In genus $g\geq 3$, one can also try to find a set of banded links in a handlebody so that one obtains a basis of ${{\mathcal{S}_p}}({{\Sigma}}_g)$ by cabling each curve component with $v=(z+2)/(1+A)$. This is suggested by Corollary \[v\] and the fact that ${{\mathcal{S}_p}}({{\Sigma}}_g)$ is a free ${{\mathcal{O}}}$-module [@G]. In fact, we now find such a set of links giving a basis for $\mathcal{S}^+_5({{\Sigma}}_3)$ (and therefore also for $\mathcal{S}_5({{\Sigma}}_3)$) by adapting the above procedures. These links are described by arrangements of curves in a thrice punctured disk. Although the Hermitian form and the related ${{\mathcal{O}}}^+$-valued sesquilinear form are not unimodular, in this particular situation they are nearly so, and this is essential for our argument. It seems more difficult to find an explicit collection of banded links with this property for $\mathcal{S}_p({{\Sigma}}_3)$ for $p>5,$ and for $\mathcal{S}_p({{\Sigma}}_g)$ for $g>4.$ We plan to return to this question elsewhere. We think of the handlebody $H_3$ as $P_3 \times I$ where $P_3$ is a disk with three holes. We give $H_3$ weight zero. We equip ${{\Sigma}}_3$ with the Lagrangian given by the kernel of the map induced on the first homology by the inclusion of ${{\Sigma}}_3$ in $P_3 \times I.$ Consider the set of 15 arrangements of curves in $P_3$ $$\begin{aligned} {{\mathcal{A}}}=\{A_\emptyset,&A_1,A_2,A_3, A_1A_2,A_2A_3,A_3A_1, A_1A_2A_3,\\ &A_{12},A_{23},A_{13}, A_{12}A_3, A_{23}A_1, A_{31}A_2, A_{123}\}~.\end{aligned}$$ Here, $A_\emptyset$ is the empty arrangement, $A_i$ (resp. $A_{ij}$, resp. $A_{123}$) is a curve of the shape pictured in Figure \[fig3\] around just the $i$-th hole (resp. around both the $i$-th and $j$-th hole, resp. around all three holes), and the multiplicative notation $A_\alpha A_\beta$ means disjoint union of $A_\alpha$ and $A_\beta$. See Figure \[fig3\] for two examples. Note that the total number of curves in ${{\mathcal{A}}}$ is $22$. \[thg3\] The set ${{\mathcal{A}}}(v)=\{A_\emptyset(v)=A_\emptyset,A_1(v),A_2(v),\ldots\}$ consisting of the curve arrangements in ${{\mathcal{A}}}$ colored $v$ is a basis of $\mathcal{S}^+_5({{\Sigma}}_3), $ and thus also a basis for $\mathcal{S}_5({{\Sigma}}_3).$ Note that it follows in particular that the ${{\mathcal{O}}}^+$-span of ${{\mathcal{A}}}(v)$ is stable under the action of the index two subgroup of even morphisms in the mapping class group $\tilde{\Gamma}({{\Sigma}}_2)$. (-.6,-2.8)(2.6,.8) (-.05,-.05)(.05,.05) (-.05,.05)(.05,-.05) (1.95,-.05)(2.05,.05) (1.95,.05)(2.05,-.05) (0,0)[.6]{} (2,0)[.6]{} (1,-1.73)[.6]{} (.95,-1.78)(1.05,-1.68) (.95,-1.68)(1.05,-1.78) (1,-.577)(.51,-.29) (1,-.577)(1.49,-.29) (1,-.577)(1,-1.13) (.1,-.1)[${1}$]{} (2.1,-.1)[${2}$]{} (1.1,-1.83)[$3$]{} (-.6,-2.8)(2.6,.8) (.1,-.1)[${1}$]{} (2.1,-.1)[${2}$]{} (1.1,-1.83)[$3$]{} (1.6,-2.2)[$A_{123}$]{} (-.05,-.05)(.05,.05) (-.05,.05)(.05,-.05) (1.95,-.05)(2.05,.05) (1.95,.05)(2.05,-.05) (0,0)[.6]{}[350]{}[310]{} (2,0)[.6]{}[230]{}[190]{} (1,-1.73)[.6]{}[110]{}[70]{} (.6,-.1)(1,-.33) (1.4,-.1)(1,-.33) (.385,-.475)(.8,-.7) (1.615,-.475)(1.2,-.7) (1.2,-.7)(1.2,-1.175) (.8,-.7)(.8,-1.175) (.95,-1.78)(1.05,-1.68) (.95,-1.68)(1.05,-1.78) (-.6,-2.8)(2.6,.8) (.1,-.1)[${1}$]{} (2.1,-.1)[${2}$]{} (1.1,-1.83)[$3$]{} (1.6,-2.2)[$A_{3}$]{} (2.3,-.8)[$A_{12}$]{} (-.05,-.05)(.05,.05) (-.05,.05)(.05,-.05) (1.95,-.05)(2.05,.05) (1.95,.05)(2.05,-.05) (.95,-1.78)(1.05,-1.68) (.95,-1.68)(1.05,-1.78) (0,0)[.6]{}[20]{}[340]{} (2,0)[.6]{}[200]{}[160]{} (.56,.2)(1.44,.2) (.56,-.2)(1.44,-.2) (1,-1.73)[.6]{} The set ${{\mathcal{A}}}$ (where its elements are considered as planar banded links in $H_3$) is a basis of $V_5({{\Sigma}}_3)$. By the proof of Lemma \[lemG\], we can find a graph basis $\mathcal{G}$ for $V_5({{\Sigma}}_3)$ by 5-admissible colorings of the graph $G$ in Figure \[fig3\], where the loops are colored zero or one and the non-loop edges are colored zero or two. If a non-loop edge is colored two, then the loop at the end of the edge must be colored one. Also, the number of non-loop edges colored two must be zero, two, or three. This summarises 5-admissibility in this case. There are 15 such colorings. Again using Wenzl’s recursion formula, there is a triangular change of basis with ones on the diagonal from the basis $\mathcal{G}$ to the set $\mathcal{A}$ which is therefore also a basis of $V_5({{\Sigma}}_3)$. Recall that ${{\mathcal{A}}}(v)$ consists of the $15$ elements of ${{\mathcal{S}^+_5}}(\Sigma_3)$ obtained by replacing each of the 22 curves in ${{\mathcal{A}}}$ with $v=(z+2)/(1+A)$. Again there is a triangular change of basis matrix from ${{\mathcal{A}}}$ to ${{\mathcal{A}}}(v)$. Therefore the elements of ${{\mathcal{A}}}(v)$ span $V_5({{\Sigma}}_3)$ and hence are linearly independent over ${{\mathcal{O}}}^+$. Consider the inclusion $$\label{strict} \text{Span}_{{{\mathcal{O}}}^+}{{\mathcal{A}}}(v) \subset {{\mathcal{S}^+_5}}(\Sigma_3)~.$$ The matrix for $(\ ,\ )^+_{\Sigma}$ with respect to $\mathcal{A }(v)$ has determinant $\sim (1-q)^{3\cdot 15-2 \cdot 22}=1-q.$ Since $1-q$ is a prime in ${{\mathcal{O}}}^+ $ and ${{\mathcal{S}^+_5}}(\Sigma_3)$ is also a free ${{\mathcal{O}}}^+$-module, we conclude that the inclusion (\[strict\]) cannot be strict. Thus $\mathcal{A }(v)$ is a basis for ${{\mathcal{S}^+_5}}(\Sigma_3).$ [Theorem \[thg3\] remains true if we replace $v$ by $\omega$ throughout. The same proof works.]{} [As in Remark \[9.4\], it is crucial for this argument to use ${{\mathcal{S}^+_5}}({{\Sigma}}_3)$ rather than ${{\mathcal{S}_5}}({{\Sigma}}_3)$, since for $p=5$ there exists $a\in {{\mathcal{O}}}={{\mathbb{Z}}}[\zeta_{20}]$ such that $1-q=a\overline{a}$.]{} [Kerler has announced in [@Ke] a construction of integral bases for the Reshetikhin-Turaev $SO(3)$ TQFT at the prime $p=5$ for any genus.]{} A divisibility result for the Kauffman bracket ============================================== In this final section, we let $A$ again be an indeterminant. The fact that $v=(z+2)/(1+ \zeta_{2p} )$ lies in ${{\mathcal{S}_p}}(S^1\times S^1)$ for all odd primes $p$ has the following application to the Kauffman bracket $\langle\ \rangle$ of banded links in $S^3$. \[th11\] Let $L$ be a banded link in $S^3$ with $\mu$ components. Let $L(z+2)$ denote this link colored $z+2.$ Then the Kauffman Bracket $\langle L(z+2) \rangle \in {{\mathbb{Z}}}[A^{\pm}]$ is divisible by $(1+A)^{\mu}$. Here the Kauffman bracket is normalized so that the bracket of the empty link is $\langle \emptyset\rangle=1$. Note that $$\langle L(z+2)\rangle = \sum_{L'\subset L} 2^{\mu-\mu(L')} \langle L'\rangle~,$$ where the sum is over all sublinks $L'$ of $L$, and $\mu(L')$ denotes the number of components of $L'$. When we evaluate the Kauffman bracket $\langle J \rangle$ of a banded link $J$ in $S^3$ at $A=\zeta_{2p}$, we obtain the quantum invariant of the pair $(S^3,J)$ (where $S^3$ is given the weight zero) in the normalization $$\langle J \rangle\vert_{A=\zeta_{2p}}=I_p(S^3,J)=\frac {\langle (S^3,J) \rangle} {\langle S^3 \rangle}={{\mathcal{D}}}{\langle (S^3,J) \rangle} ~.$$ This normalization $I_p(M,J)$ of the quantum invariant is precisely the one which is always an algebraic integer [@Mu1; @MR] and which is at the basis of the integral cobordism functors ${{\mathcal{S}_p}}$. Let $f(A)$ denote the Kauffman bracket $\langle L(z+2)\rangle \in {{\mathbb{Z}}}[A^\pm].$ Since $v= (z+2)/(1+ \zeta_{2p} )\in S_p(S^1 \times S^1),$ we have $I_p(S^3,L(v)) \in {{\mathbb{Z}}}[\zeta_{2p}],$ for every odd prime $p.$ Thus $$\label{divisi} f(\zeta_{2p}) = I_p(S^3,L(z+2)) \in (1+\zeta_{2p})^\mu {{\mathbb{Z}}}[\zeta_{2p}],$$ for every odd prime $p.$ Now recall the following elementary Lemma (see [@Mu1 Lemma 5.5] and note that $-\zeta_{2p}$ is a primitive $p$-th root). Suppose $f(A) \in {{{\mathbb{Z}}}} [A^\pm].$ Let $f^{(k)}(A)$ denote the $k$-th derivative of $f(A)$. Assume $0\leq \mu<p$ where $p$ is prime. Then $f(\zeta_{2p})\in {{{\mathbb{Z}}}} [\zeta_{p} ]$ is divisible by $(1+\zeta_{2p})^\mu$ if and only if $f^{(k)}(-1)\equiv 0\pmod{p}$ for every $0\le k< \mu.$ By this lemma, (\[divisi\]) implies $f^{(k)}(-1)\equiv 0\pmod{p}$ for each $0\le k< \mu,$ provided $p$ is larger than $\mu.$ Since there are infinitely many such primes, it follows that $f^{(k)}(-1)=0$ for each $0\le k< \mu.$ But this means that $ (1+A)^{\mu}$ divides $f(A).$ \[Kcor\] If $L$ is as in the theorem, then the Kauffman Bracket $\langle L(z+[2]) \rangle \in {{\mathbb{Z}}}[A^{\pm}]$ is also divisible by $(1+A)^{\mu}$. This follows immediately from the fact that $2-[2]=2-A^2-A^{-2}=(1-A^2)(1-A^{-2})$ is divisible by $1+A.$ [In a similar way, Theorem \[tK\] remains true if we replace $v$ with $\hat v= (z+ [2])/(1+A)$ in the definition of $K(n).$ Similarly Theorems \[secondbasis\], \[thg2\], \[thg3\] remain true if we replace $v$ by $\hat v.$ ]{} [Theorem \[th11\] can also be proved by computing the Kauffman bracket from the (framed) Kontsevich integral via an appropriate weight system. Actually this proof is an adaptation of an argument going back to Kricker and Spence [@KS Proof of Thm. 2], but they only considered algebraically split links. Previously Ohtsuki [@OhCambridge Prop. 3.4] had obtained a stronger divisiblity result for $\langle L(z+[2]) \rangle$ using quantum groups, for algebraically split links satisfying some extra conditions. Later, Cochran and Melvin generalised the Kontsevich integral argument, and their result [@CM Theorem 2.5] contains Corollary \[Kcor\] for zero-framed links. (The results of [@OhCambridge; @KS; @CM] are stated in terms of the Jones polynomial, but it is well-known that the Jones polynomial and the Kauffman bracket are equivalent.) However, the restriction to zero framing is not really necessary (although a small additional argument is needed). We will not give details of this alternative proof here, as the techniques are completely different from the ones in the present paper. ]{} [BHMV2]{} Asymptotic Faithfulness of the quantum $SU(n)$ representations of the mapping class groups. arXiv:math.QA/0204084 On framings of $3$-manifolds. [Topology]{} [29]{} (1990) 1-7. . Cambridge University Press, Cambridge, 1935. Three manifold invariants derived from the Kauffman bracket, [Topology]{} [31]{} (1992), 685-699 Topological quantum field theories derived from the Kauffman bracket, [Topology]{} [34]{} (1995), 883-927 Quantum cyclotomic orders of 3-manifolds. [Topology]{} [40]{} (2001), no. 1, 95-125. , [J. Symbolic Comp,]{} [24]{} (1997), 267-283 On the TQFT representations of the mapping class groups. [Pacific J. Math.]{} [188]{} (1999), no. 2, 251–274. On the Witten-Reshetikhin-Turaev representations of mapping class groups. [Proc. AMS]{} [127]{} (1999) 2483-2488. , arXiv:math.QA/0105059. HYP and HYPQ - Mathematica packages for the manipulation of binomial sums and hypergeometric series, respectively q-binomial sums and basic hypergeometric series, [ J. Symbol. Comput. ]{}[20]{} (1995), 737-744 , arXiv:math.GT/0110007. Ohtsuki’s invariants are of finite type. [J. Knot Theory Ramifications]{} [6]{} (1997) 583-597. Quantum invariants of 3-manifolds: integrality, splitting, and perturbative expansion. arXiv:math.QA/0004099. An element of infinite order in TQFT-representations of mapping class groups. [Contemp. Math.]{} [233]{} (1999) 137-139. (1993), no. 2, 171–194. On central extensions of mapping class groups. [Math. Ann.]{} [302]{}, 131-150 (1995) , Math. Proc. Cambridge Philos. Soc. [121]{} (1997) no. 3, 443–454 $3$-valent graphs and the Kauffman bracket. [Pacific J. Math.]{} [164]{}, (1994) 361-381. Integral modular categories and integrality of quantum invariants at roots of unity of prime order. [J. reine angew. Math. (Crelle’s Journal)]{} [505]{} (1998) 209-235. (1994), no. 2, 253–281 (1995), no. 2, 237–249 A polynomial invariant of integral homology $3$-spheres. [Math. Proc. Camb. Phil. Soc.]{} [117]{} (1995) 83-112. A polynomial invariant of rational homology $3$-spheres. [Invent. Math.]{} [123]{} (1996) 241-257. Irreducibility of some quantum representations of mapping class groups. [J. Knot Theory Ramifications]{} [10]{} (2001), no. 5, 763–767. The $PSU(n)$ invariants of $3$-manifolds are polynomials. [J. Knot Theory Ramifications]{} [8]{} (1999) 521-532 de Gruyter (1994) A Mathematica version of Zeilberger’s algorithm for proving binomial coefficient identities. (1995) 673–698. Preprint 1991 Wolfram Media, Inc., Champaign, IL; Cambridge University Press, Cambridge. (1999) [^1]: Warning: In many places ([e.g.]{} in [@MR]), $q$ denotes $A^4$ rather than $A^2.$	ArXiv
--- abstract: 'We use pseudodifferential calculus and heat kernel techniques to prove a conjecture by Chamseddine and Connes on rationality of the coefficients of the polynomials in the cosmic scale factor $a(t)$ and its higher derivatives, which describe the general terms $a_{2n}$ in the expansion of the spectral action for general Robertson-Walker metrics. We also compute the terms up to $a_{12}$ in the expansion of the spectral action by our method. As a byproduct, we verify that our computations agree with the terms up to $a_{10}$ that were previously computed by Chamseddine and Connes by a different method.' author: - \| $ $\ Farzad Fathizadeh, Asghar Ghorbanpour, Masoud Khalkhali title: 'Rationality of Spectral Action for Robertson-Walker Metrics' --- Department of Mathematics, Western University\ London, Ontario, Canada, N6A 5B7 [^1]\ 0.1cm [Mathematics Subject Classification (2010).]{} 81T75, 58B34, 58J42. 0.1 cm [Keywords.]{} Robertson-Walker metrics, Dirac operator, Spectral action, Heat kernel, Local invariants, Pseudodifferential calculus. Introduction ============ Noncommutative geometry in the sense of Alain Connes [@ConBook] has provided a paradigm for geometry in the noncommutative setting based on spectral data. This generalizes Riemannian geometry [@ConReconstruct] and incorporates physical models of elementary particle physics [@ConGravity; @ConMixing; @ChaConMarGS; @ConMarBook; @ChaConConceptual; @ChaConWhy; @GraIocSch; @Sit; @Sui1; @Sui2]. An outstanding feature of the spectral action defined for noncommutative geometries is that it derives the Lagrangian of the physical models from simple noncommutative geometric data [@ConMixing; @ChaConSAP; @ChaConMarGS]. Thus various methods have been developed for computing the terms in the expansion in the energy scale $\Lambda$ of the spectral action [@ChaConUFNCG; @ChaConGravity; @ChaConUncanny; @ChaConRW; @IocLevVasGlobal; @IocLevVasTorsion]. Potential applications of noncommutative geometry in cosmology have recently been carried out in [@KolMar; @Mar; @MarPie; @MarPieTeh2012; @MarPieTeh; @NelOchSal; @NelSak1; @NelSak2; @EstMar]. Noncommutative geometric spaces are described by spectral triples $(\mathcal{A}, \mathcal{H}, D)$, where $\mathcal{A}$ is an involutive algebra represented by bounded operators on a Hilbert space $\mathcal{H}$, and $D$ is an unbounded self-adjoint operator acting in $\mathcal{H}$ [@ConBook]. The operator $D$, which plays the role of the Dirac operator, encodes the metric information and it is further assumed that it has bounded commutators with elements of $\mathcal{A}$. It has been shown that if $\mathcal{A}$ is commutative and the triple satisfies suitable regularity conditions then $\mathcal{A}$ is the algebra of smooth functions on a spin$^c$ manifold $M$ and $D$ is the Dirac operator acting in the Hilbert space of $L^2$-spinors [@ConReconstruct]. In this case, the Seeley-de Witt coefficients $a_{n}(D^2) = \int_M a_n (x, D^2) \,dv(x)$, which vanish for odd $n$, appear in a small time asymptotic expansion of the form $$\textnormal{Tr}(e^{-t D^2}) \sim t^{- \textnormal{dim} (M)/2} \sum_{n\geq 0} a_{2n} (D^2) t^n \qquad (t \to 0).$$ These coefficients determine the terms in the expansion of the spectral action. That is, there is an expansion of the form $$\textnormal{Tr} f(D^2/\Lambda^2) \sim \sum_{n \geq 0} f_{2n}\, a_{2n} (D^2/\Lambda^2),$$ where $f$ is a positive even function defined on the real line, and $f_{2n} $ are the moments of the function $f$ [@ChaConSAP; @ChaConUFNCG]. See Theorem 1.145 in [@ConMarBook] for details in a more general setup, namely for spectral triples with simple dimension spectrum. By devising a direct method based on the Euler-Maclaurin formula and the Feynman-Kac formula, Chamseddine and Connes have initiated in [@ChaConRW] a detailed study of the spectral action for the Robertson-Walker metric with a general cosmic scale factor $a(t)$. They calculated the terms up to $a_{10}$ in the expansion and checked the agreement of the terms up to $a_6$ against Gilkey’s universal formulas [@GilBook1; @GilBook2]. The present paper is intended to compute the term $a_{12}$ in the spectral action for general Robertson-Walker metrics, and to prove the conjecture of Chamseddine and Connes [@ChaConRW] on rationality of the coefficients of the polynomials in $a(t)$ and its derivatives that describe the general terms $a_{2n}$ in the expansion. In passing, we compare the outcome of our computations up to the term $a_{10}$ with the expressions obtained in [@ChaConRW], and confirm their agreement. In terms of the above aims, explicit formulas for the Dirac operator of the Robertson-Walker metric and its pseudodifferential symbol in Hopf coordinates are derived in §\[DiracinHopf\]. Following a brief review of the heat kernel method for computing local invariants of elliptic differential operators using pseudodifferential calculus [@GilBook1], we compute in §\[Termsupto10\] the terms up to $a_{10}$ in the expansion of the spectral action for Robertson-Walker metrics. The outcome of our calculations confirms the expressions obtained in [@ChaConRW]. This forms a check in particular on the validity of $a_8$ and $a_{10}$, which as suggested in [@ChaConRW] also, seems necessary due to the high complexity of the formulas. In §\[Term12\], we record the expression for the term $a_{12}$ achieved by a significantly heavier computation, compared to the previous terms. It is checked that the reduction of $a_{12}$ to the round case $a(t)=\sin t $ conforms to the full expansion obtained in [@ChaConRW] for the round metric by remarkable calculations that are based on the Euler-Maclaurin formula. In order to validate our expression for $a_{12}$, parallel but completely different computations are performed in spherical coordinates and the final results are confirmed to match precisely with our calculations in Hopf coordinates. In §\[ProofofConjecture\], we prove the conjecture made in [@ChaConRW] on rationality of the coefficients appearing in the expressions for the terms of the spectral action for Robertson-Walker metrics. That is, we show that the term $a_{2n}$ in the expansion is of the form $Q_{2n}\big(a(t),a'(t),\dots,a^{(2n)}(t)\big)/a(t)^{2n-3}$, where $Q_{2n}$ is a polynomial with rational coefficients. We also find a formula for the coefficient of the term with the highest derivate of $a(t)$ in $a_{2n}$. It is known that values of Feynman integrals for quantum gauge theories are closely related to multiple zeta values and periods in general and hence tend to be transcendental numbers [@MarBook]. In sharp distinction, the rationality result proved in this paper is valid for all scale factors $a(t)$ in Robertson-Walker metrics. Although it might be exceedingly difficult, it is certainly desirable to find all the terms $a_{2n}$ in the spectral action. The rationality result is a consequence of a certain symmetry in the heat kernel and it is plausible that this symmetry would eventually reveal the full structure of the coefficients $a_{2n}$. This is a task for a future work. Our main conclusions are summarized in §\[Conclusions\]. The Dirac Operator for Robertson-Walker Metrics {#DiracinHopf} =============================================== According to the spectral action principle [@ConGravity; @ChaConSAP], the spectral action of any geometry depends on its Dirac operator since the terms in the expansion are determined by the high frequency behavior of the eigenvalues of this operator. For spin manifolds, the explicit computation of the Dirac operator in a coordinate system is most efficiently achieved by writing its formula after lifting the Levi-Civita connection on the cotangent bundle to the spin connection on the spin bundle. In this section, we summarize this formalism and compute the Dirac operator of the Robertson-Walker metric in Hopf coordinates. Throughout this paper we use Einstein’s summation convention without any further notice. Levi-Civita connection. ----------------------- The spin connection of any spin manifold $M$ is the lift of the Levi-Civita connection for the cotangent bundle $T^M$ to the spin bundle. Let us, therefore, recall the following recipe for computing the Levi-Civita connection and thereby the spin connection of $M$. Given an orthonormal frame $\{\theta_\alpha\}$ for the tangent bundle $TM$ and its dual coframe $\{\theta^\alpha\}$, the connection 1-forms $\omega^\alpha_\beta$ of any connection $\nabla$ on $T^M$ are defined by $$\nonumber \nabla{\theta^\alpha}=\omega_\beta^\alpha \,\theta^\beta.$$ Since the Levi-Civita connection is the unique torsion free connection which is compatible with the metric, its 1-forms are uniquely determined by $$d\theta^\beta =\omega^\beta_\alpha \wedge \theta^\alpha.$$ This is justified by the fact that the compatibility with metric enforces the relations $$\nonumber \omega^\alpha_\beta=-\omega^\beta_\alpha,$$ while, taking advantage of the first Cartan structure equation, the torsion-freeness amounts to the vanishing of $$\nonumber T^\alpha = d\theta^\alpha - \omega^\alpha_\beta \wedge \theta^\beta.$$ The spin connection of Robertson-Walker metrics in Hopf coordinates. --------------------------------------------------------------------- The (Euclidean) Robertson-Walker metric with the cosmic scale factor $a(t)$ is given by $$\nonumber ds^{2}=dt^{2}+a^{2}\left( t\right) d\sigma^2,$$ where $d\sigma^2$ is the round metric on the 3-sphere $\mathbb{S}^3$. It is customary to write this metric in spherical coordinates, however, for our purposes which will be explained below, it is more convenient to use the Hopf coordinates, which parametrize the 3-sphere $S^3\subset \mathbb{C}^2$ by $$\nonumber z_1=e^{i\phi_1}\sin(\eta), \qquad z_2=e^{i\phi_2}\cos(\eta),$$ with $\eta$ ranging in $[0,\pi/2)$ and $\phi_1,\phi_2$ ranging in $ [0,2\pi)$. The Robertson-Walker metric in the coordinate system $x=(t, \eta, \phi_1, \phi_2)$ is thus given by $$\nonumber ds^{2}=dt^{2}+a^{2}\left( t\right)\left(d\eta^2+\sin^2(\eta)d\phi_1^2+\cos^2(\eta)d\phi_2^2\right).$$ An orthonormal coframe for $ds^{2}$ is then provided by $$\begin{aligned} \theta^1= dt, \qquad \theta^2 =a(t)\, d \eta, \qquad \theta^3 = a(t)\, \sin \eta \,d \phi_1, \qquad \theta^4 = a(t)\, \cos \eta \, d \phi_2. \nonumber \end{aligned}$$ Applying the exterior derivative to these forms, one can easily show that they satisfy the following equations, which determine the connection 1-forms of the Levi-Civita connection: $$\begin{aligned} && d\theta^1= 0, \nonumber \\ && d\theta^2 =\frac{a'(t)}{a(t)}\, \theta^1\wedge \theta^2, \nonumber \\ && d\theta^3 =\frac{a'(t)}{a(t)}\, \theta^1\wedge \theta^3+ \frac{\cot\eta}{a(t)}\, \theta^2\wedge \theta^3, \nonumber \\ && d\theta^4 =\frac{a'(t)}{a(t)}\, \theta^1\wedge \theta^4- \frac{\tan\eta}{a(t)}\, \theta^2\wedge \theta^4. \nonumber \end{aligned}$$ We recast the above equations into the matrix of connection 1-forms $$\omega=\frac{1}{a(t)}\left( \begin{array}{cccc} 0 & - a'(t)\, \theta ^2 & - a'(t)\, \theta ^3 & - a'(t)\, \theta ^4 \\ a'(t)\, \theta ^2 & 0 & -\cot \eta \, \theta ^3 & \tan \eta \, \theta ^4 \\ a'(t)\, \theta ^3 & \cot \eta \, \theta ^3 & 0 & 0 \\ a'(t)\, \theta ^4 & - \tan\eta \, \theta ^4 & 0 & 0 \\ \end{array} \right) \in\mathfrak{so}(4),$$ which lifts to the spin bundle using the Lie algebra isomorphism $\mu:\mathfrak{so}(4)\to \mathfrak{spin}(4)$ given by (see [@LawMic]) $$\nonumber \mu(A)= \frac{1}{4}\sum_{\alpha,\beta}\langle A\theta^\alpha,\theta^\beta\rangle c(\theta^\alpha)c(\theta^\beta), \qquad A \in \mathfrak{so}(4).$$ Since $\langle \omega\theta^\alpha,\theta^\beta\rangle =\omega^\alpha_\beta$, the lifted connection $\tilde{\omega}$ is written as $$\tilde\omega=\frac{1}{4}\sum_{\alpha,\beta}\omega^\alpha_\beta c(\theta^\alpha)c(\theta^\beta).$$ In the case of the Robertson-Walker metric we find that $$\label{exprspinconn} \tilde\omega=\frac{1}{2a(t)} \left( a'(t)\theta ^2 \gamma ^{12}+ a'(t)\theta ^3 \gamma ^{13}+ a'(t) \theta ^4\gamma ^{14}+ \cot (\eta ) \theta ^3\gamma ^{23}- \tan (\eta ) \theta ^4\gamma ^{24}\right),$$ where we use the notation $\gamma^{i j} = \gamma^i \gamma^j$ for products of pairs of the gamma matrices $\gamma^1, \gamma^2, \gamma^3, \gamma^4$, which are respectively written as $$\left( \begin{array}{cccc} 0 & 0 & i & 0 \\ 0 & 0 & 0 & i \\ i & 0 & 0 & 0 \\ 0 & i & 0 & 0 \end{array} \right), \left( \begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \end{array} \right), \left( \begin{array}{cccc} 0 & 0 & 0 & -i \\ 0 & 0 & i & 0 \\ 0 & i & 0 & 0 \\ -i & 0 & 0 & 0 \end{array} \right), \left( \begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{array} \right).$$ The Dirac Operator of Robertson-Walker metrics in Hopf coordinates. ------------------------------------------------------------------- Using the expression obtained for the spin connection and considering the predual of the orthonormal coframe $\{ \theta^\alpha \}$, $$\begin{aligned} \theta_1= \frac{\partial}{\partial t}, \qquad \theta_2=\frac{1}{a(t)}\frac{\partial}{\partial \eta}, \qquad \theta_3 = \frac{1}{a(t)\, \sin \eta} \frac{\partial}{\partial \phi_1}, \qquad \theta_4 = \frac{1}{a(t)\, \cos \eta} \frac{\partial}{\partial \phi_2}, \nonumber \end{aligned}$$ we compute the Dirac operator for the Robertson-Walker metric explicitly: $$\begin{aligned} D&=c(\theta^\alpha)\nabla_{\theta_\alpha} \\ &=\gamma^\alpha\left(\theta_\alpha+\tilde \omega(\theta_\alpha)\right)\\ &=\gamma^1\left(\frac{\partial}{\partial t}\right)+ \gamma^2\left(\frac{1}{a}\frac{\partial}{\partial \eta}+\frac{a'}{2a}\gamma^{12}\right) +\gamma^3\left(\frac{1}{a\sin(\eta)}\frac{\partial}{\partial \phi_1}+\frac{a'}{2a}\gamma^{13}+\frac{\cot(\eta)}{2a}\gamma^{23}\right)\\ & \quad +\gamma^4\left(\frac{1}{a\cos(\eta)}\frac{\partial}{\partial \phi_2}+\frac{a'}{2a}\gamma^{14}-\frac{\tan(\eta)}{2a}\gamma^{24}\right)\\ &=\gamma^1 \frac{\partial}{\partial t}+\gamma^2 \frac{1}{a}\frac{\partial}{\partial \eta}+\gamma^3 \frac{1}{a\, \sin \eta} \frac{\partial}{\partial \phi_1}+\gamma^4 \frac{1}{a\, \cos \eta } \frac{\partial}{\partial \phi_2} +\frac{3a'}{2a}\gamma^1+\frac{\cot(2\eta)}{a}\gamma^2. \end{aligned}$$ Thus the pseudodifferential symbol of $D$ is given by $$\begin{aligned} \nonumber \sigma_D({ x,\xi}) = i\xi_1\gamma^1+ \frac{i\xi_2}{a}\gamma^2+ \frac{i\xi_3}{a\, \sin \eta}\gamma^3+ \frac{i\xi_4}{a\, \cos \eta } \gamma^4 +\frac{3a'}{2a}\gamma^1+\frac{\cot(2\eta)}{a}\gamma^2. \end{aligned}$$ For the purpose of employing pseudodifferential calculus in the sequel to compute the heat coefficients, we record in the following proposition the pseudodifferential symbol of $D^2$. This can be achieved by a straightforward computation to find an explicit expression for $D^2$, or alternatively, one can apply the composition rule for symbols, $\sigma_{P_1 P_2}({ x,\xi})=\sum_\alpha \frac{(-i)^{\|\alpha\|}}{\alpha !}\partial^\alpha_\xi\sigma_{P_1}\partial^\alpha_{ x}\sigma_{P_2}$, to the symbol of $D$. The pseudodifferential symbol of $D^2$, where $D$ is the Dirac operator for the Robertson-Walker metric, is given by $$\sigma(D^2)= p_2 + p_1 + p_0,$$ where the homogeneous components $p_i$ of order $i$ are written as $$\begin{aligned} \label{symbolHopf} p_2&=&\xi _1^2+\frac{1}{a^2}\xi _2^2+\frac{1}{a^2 \sin ^2(\eta )}\xi _3^2+\frac{ 1}{a^2\cos ^2(\eta)}\xi _4^2, \nonumber \\ p_1&=& \frac{-3 i a a'}{a^2}\xi _1+\frac{-i a' \gamma ^{12}-2i \cot (2\eta )}{a^2}\xi _2 -\frac{i a' \csc (\eta ) \gamma ^{13}+i \cot (\eta ) \csc (\eta ) \gamma ^{23}}{a^2}\xi _3 \nonumber \\ && +\frac{i \tan (\eta ) \sec (\eta ) \gamma ^{24}-i a' \sec (\eta ) \gamma ^{14} }{a^2}\xi _4 , \nonumber \\ p_0&=&\frac{1}{4 a(t)^2}\Big(-6 a(t) a''(t)-3 a'(t)^2+\csc ^2(\eta )+\sec ^2(\eta ) \nonumber \\ &&+4+2 a'(t) (\cot (\eta )-\tan (\eta ))\gamma ^{12}\Big). \end{aligned}$$ Terms up to $a_{10}$ and their Agreement with Chamseddine-Connes’ Result {#Termsupto10} ======================================================================== The computation of the terms in the expansion of the spectral action for a spin manifold, or equivalently the calculation of the heat coefficients, can be achieved by recursive formulas while working in the heat kernel scheme of local invariants of elliptic differential operators and index theory [@GilBook1]. Pseudodifferential calculus is an effective tool for dealing with the necessary approximations for deriving the small time asymptotic expansions in which the heat coefficients appear. Universal formulas in terms of the Riemann curvature operator and its contractions and covariant derivatives are written in the literature only for the terms up to $a_{10}$, namely Gilkey’s formulas up to $a_6$ [@GilBook1; @GilBook2] and the formulas in [@AmsBerOc; @Avr; @Van] for $a_8$ and $a_{10}$. Small time heat kernel expansions using pseudodifferential calculus. {#heatcoefsbypesudo} -------------------------------------------------------------------- In [@GilBook1], by appealing to the Cauchy integral formula and using pseudodifferential calculus, recursive formulas for the heat coefficients of elliptic differential operators are derived. That is, one writes [^2] $$e^{-tD^2}=-\frac{1}{2\pi i}\int_\gamma e^{-t\lambda}(D^2-\lambda)^{-1}d\lambda,$$ where the contour $\gamma$ goes around the non-negative real axis in the counterclockwise direction, and one uses pseudodifferential calculus to approximate $(D^2-\lambda)^{-1}$ via the homogeneous terms appearing in the expansion of the symbol of the parametrix of $D^2-\lambda$. Although left and right parametrices have the same homogeneous components, for the purpose of finding recursive formulas for the coefficients appearing in each component, which will be explained shortly, it is more convenient for us to consider the right parametrix $\tilde{R}(\lambda)$. Therefore, the next task is to compute recursively the homogeneous pseudodifferential symbols $r_j$ of order $-2-j$ in the expansion of $\sigma(\tilde{R}(\lambda))$. Using the calculus of symbols, with the crucial nuance that $\lambda$ is considered to be of order 2, one finds that $$r_0=(p_2-\lambda)^{-1},$$ and for any $n>1$ $$\begin{aligned} \label{recursive1} r_n=-r_0\sum_{\begin{array}{c}\|\alpha\|+j+2-k=n\\ j<n\end{array}} \frac{(-i)^{\|\alpha\|}}{\alpha!}d^\alpha_\xi p_k\,d_x^\alpha r_j.\end{aligned}$$ We summarize the process of obtaining the heat coefficients by explaining that one then uses these homogeneous terms in the Cauchy integral formula to approximate the integral kernel of $e^{-t D^2}.$ Integration of the kernel of this operator on the diagonal yields a small time asymptotic expansion of the form $$\nonumber {\rm Tr}(e^{-tD^2})\sim \sum_{n=0}^\infty \frac{t^{(n-4)/2}}{16\pi^4}\int {\rm tr}(e_n(x)) \,dvol_g \qquad (t\to 0),$$ where $$\label{engeneralform} e_n(x) \sqrt{\det g}=\frac{-1}{2\pi i}\int \int_\gamma e^{-\lambda}r_n(x,\xi,\lambda)\,d\lambda \,d\xi.$$ For detailed discussions, we refer the reader to [@GilBook1]. It is clear from that cross derivatives of $p_2$ vanish and $d_\xi^\alpha p_k=0$ if $\|\alpha\|>k$. Furthermore, $\frac{\partial}{\partial\phi_k}r_n=0$ for $n \geq 0$, and the summation is written as $$\begin{aligned} \label{rnshort} r_n&=&-r_0\,p_0\,r_{n-2} -r_0\,p_1\,r_{n-1} +ir_0\frac{\partial}{\partial\xi_1}p_1 \frac{\partial}{\partial t }r_{n-2} +ir_0\frac{\partial}{\partial\xi_2}p_1 \frac{\partial}{\partial \eta}r_{n-2} \nonumber \\ &&+ir_0\frac{\partial}{\partial\xi_1}p_2 \frac{\partial}{\partial t }r_{n-1} +ir_0\frac{\partial}{\partial\xi_2}p_2 \frac{\partial}{\partial \eta }r_{n-1} +\frac{1}{2}r_0\frac{\partial^2}{\partial\xi_1^2}p_2 \frac{\partial^2}{\partial t^2 }r_{n-2} \nonumber \\ &&+\frac{1}{2}r_0\frac{\partial^2}{\partial\xi_2^2}p_2 \frac{\partial^2}{\partial \eta^2 }r_{n-2}. \end{aligned}$$ Using induction, we find that $$\label{rnja} r_n=\sum_{ \begin{array}{c} 2j-2-\|\alpha\|=n \\ n/2+1 \leq j \leq 2n+1\end{array}}r_{n,j,\alpha }(x)\, r_0^j \,\xi^\alpha.$$ For example, one can see that for $n=0$ the only non-zero $r_{0,j,\alpha}$ is $r_{0,1,\bf{0}}=1$, and for $n=1$ the non-vanishing terms are $$r_{1,2,{\bf e}_k}=\frac{\partial p_1}{\partial \xi_k}, \qquad r_{1,3,2{\bf e}_l+{\bf e}_k}=-2ig^{kk}\frac{\partial g^{ll}}{\partial x_k},$$ where ${\bf e}_j$ denotes the $j$-th standard unit vector in $\mathbb{R}^4$. It then follows from the equations , and that $$\begin{aligned} \label{en} e_n(x) \,a(t)^{3}\sin(\eta)\cos(\eta)&=\frac{-1}{2\pi i}\int_{\mathbb{R}^4}\int_\gamma e^{-t\lambda} r_n(x,\xi,\lambda)\, d\lambda \,d\xi\nonumber\\ &=\sum r_{n,j,\alpha}(x)\int_{\mathbb{R}^4}\xi^\alpha \frac{-1}{2\pi i}\int_\gamma e^{-t\lambda}r_0^j\,d\lambda \,d\xi\\ &=\sum \frac{c_\alpha}{(j-1)!} r_{n,j,\alpha} \,a(t)^{\alpha_2+\alpha_3+\alpha_4+3}\sin(\eta)^{\alpha_3+1}\cos(\eta)^{\alpha_4+1},\nonumber \end{aligned}$$ where $$c_\alpha=\prod_k \Gamma\left(\frac{\alpha_k+1}{2}\right)\frac{(-1)^{\alpha_k}+1}{2}.$$ It is straightforward to justify the latter using these identities: $$\begin{aligned} \frac{1}{2\pi i}\int_\gamma e^{-\lambda}r_0^jd\lambda&=&(-1)^{j}\frac{(-1)^{j-1}}{(j-1)!}e^{-\|\|\xi\|\|^2}=\frac{-1}{(j-1)!}\prod_{k=1}^4 e^{-g^{kk}\xi_k^2}, \nonumber \\ \int_\mathbb{R}x^ne^{-bx^2}dx&=&\frac{1}{2} \left((-1)^n+1\right) b^{-\frac{n}{2}-\frac{1}{2}} \Gamma \left(\frac{n+1}{2}\right).\nonumber\end{aligned}$$ A key point that facilitates our calculations and the proof of our main theorem presented in §\[proofofrationality\] is the derivation of recursive formulas for the coefficients $r_{n, j, \alpha}$ as follows. By substitution of into we find a recursive formula of the form $$\begin{aligned} \label{rnjarec} r_{n,j,\alpha}&= -p_0r_{n-2,j-1,\alpha}-\sum_{k}\frac{\partial p_1}{\partial\xi_k} r_{n-1,j-1,\alpha-{\bf e}_k}\nonumber\\ &\qquad +i\sum_{k}\frac{\partial p_1}{\partial \xi_k}\frac{\partial}{\partial x_k}r_{n-2,j-1,\alpha}+i(2-j)\sum_{k,l}\frac{\partial g^{ll}}{\partial x_k}\frac{\partial p_1}{\partial \xi_k}r_{n-2,j-2,\alpha-2{\bf e}_l}\nonumber\\ &\qquad+2i\sum_k g^{kk} \frac{\partial}{\partial x_k}r_{n-1,j-1,\alpha-{\bf e}_k}+i (4-2j)\sum_{k,l}g^{kk}\frac{\partial g^{ll}}{\partial x_k} r_{n-1,j-2,\alpha-2{\bf e}_l-{\bf e}_k}\\ &\qquad +\sum_k g^{kk} \frac{\partial^2}{\partial x_k^2}r_{n-2,j-1,\alpha}+(4-2j)\sum_{k,l}g^{kk}\frac{\partial g^{ll}}{\partial x_k} \frac{\partial}{\partial x_k}r_{n-2,j-2,\alpha-2{\bf e}_l}\nonumber\\ &\qquad +(2-j)\sum_{k,l}g^{kk} \frac{\partial^2 g^{ll}}{\partial x_k^2} r_{n-2,j-2,\alpha-2{\bf e}_l}\nonumber \\ &\qquad +(3-j)(2-j)\sum_{k,l,l'}g^{kk} \frac{\partial g^{ll}}{\partial x_k}\frac{\partial g^{l'l'}}{\partial x_k} r_{n-2,j-3,\alpha-2{\bf e}_l-2{\bf e}_{l'}}.\nonumber\end{aligned}$$ It is undeniable that the mechanism described above for computing the heat coefficients involves heavy computations which need to be overcome by computer programming. Calculating explicitly the functions $e_n(x)$, $n=0, 2, \dots, 12$, and computing their integrals over $\mathbb{S}_a^3$ with computer assistance, we find the explicit polynomials in $a(t)$ and its derivatives recorded in the sequel, which describe the corresponding terms in the expansion of the spectral action for the Robertson-Walker metric. That is, each function $a_n$ recorded below is the outcome of $$\begin{aligned} a_n &=&\frac{1}{16\pi^4}\int_{\mathbb{S}_a^3}{\rm tr}(e_n) \,dvol_g \nonumber \\ &=&\frac{1}{16\pi^4}\int_0^{2\pi}\int_0^{2\pi}\int_0^{\pi/2}{\rm tr}(e_n) \, a(t)^{3}\sin(\eta)\cos(\eta) \,d\eta \,d\phi_1 \,d\phi_2. \nonumber\end{aligned}$$ The terms up to $a_{6}$ ----------------------- These terms were computed in [@ChaConRW] by their direct method, which is based on the Euler-Maclaurin summation formula and the Feynman-Kac formula, and they were checked by Gilkey’s universal formulas. Our computations based on the method explained in the previous subsection also gives the same result. The first term, whose integral up to a universal factor gives the volume, is given by $$a_0=\frac{a(t)^3}{2}.$$ Since the latter appears as the leading term in the small time asymptotic expansion of the heat kernel it is related to Weyl’s law, which reads the volume from the asymptotic distribution of the eigenvalues of $D^2$. The next term, which is related to the scalar curvature, has the expression $$a_2= \frac{1}{4} a(t) \left(a(t) a''(t)+a'(t)^2-1\right).$$ The term after, whose integral is topological, is related to the Gauss-Bonnet term (cf. [@ChaConRW]) and is written as $$a_4=\frac{1}{120} \Big(3 a^{(4)}(t) a(t)^2+3 a(t) a''(t)^2-5 a''(t)+9 a^{(3)}(t) a(t) a'(t)-4 a'(t)^2 a''(t)\Big).$$ The term $a_6$, which is the last term for which Gilkey’s universal formulas are written, is given by $a_6=\frac{1}{5040 a(t)^2}\Big(9 a^{(6)}(t) a(t)^4-21 a^{(4)}(t) a(t)^2-3 a^{(3)}(t)^2 a(t)^3-56 a(t)^2 a''(t)^3+42 a(t) a''(t)^2+36 a^{(5)}(t) a(t)^3 a'(t)+6 a^{(4)}(t) a(t)^3 a''(t)-42 a^{(4)}(t) a(t)^2 a'(t)^2+60 a^{(3)}(t) a(t) a'(t)^3+21 a^{(3)}(t) a(t) a'(t)+240 a(t) a'(t)^2 a''(t)^2-60 a'(t)^4 a''(t)-21 a'(t)^2 a''(t)-252 a^{(3)}(t) a(t)^2 a'(t) a''(t)\Big).$\ The terms $a_8$ and $a_{10}$ ---------------------------- These terms were computed by Chamseddine and Connes in [@ChaConRW] using their direct method. In order to form a check on the final formulas, they have suggested to use the universal formulas of [@AmsBerOc; @Avr; @Van] to calculate these terms and compare the results. As mentioned earlier, Gilkey’s universal formulas were used in [@ChaConRW] to check the terms up to $a_6$, however, they are written in the literature only up to $a_6$ and become rather complicated even for this term. In this subsection, we pursue the computation of the terms $a_8$ and $a_{10}$ in the expansion of the spectral action for Robertson-Walker metrics by continuing to employ pseudodifferential calculus, as presented in §\[heatcoefsbypesudo\], and check that the final formulas agree with the result in [@ChaConRW]. The final formulas for $a_8$ and $a_{10}$ are the following expressions: $$a_8=$$ $-\frac{1}{10080 a(t)^4}\Big(-a^{(8)}(t) a(t)^6+3 a^{(6)}(t) a(t)^4+13 a^{(4)}(t)^2 a(t)^5-24 a^{(3)}(t)^2 a(t)^3-114 a(t)^3 a''(t)^4+43 a(t)^2 a''(t)^3-5 a^{(7)}(t) a(t)^5 a'(t)+2 a^{(6)}(t) a(t)^5 a''(t)+9 a^{(6)}(t) a(t)^4 a'(t)^2+16 a^{(3)}(t) a^{(5)}(t) a(t)^5-24 a^{(5)}(t) a(t)^3 a'(t)^3-6 a^{(5)}(t) a(t)^3 a'(t)+69 a^{(4)}(t) a(t)^4 a''(t)^2-36 a^{(4)}(t) a(t)^3 a''(t)+60 a^{(4)}(t) a(t)^2 a'(t)^4+15 a^{(4)}(t) a(t)^2 a'(t)^2+90 a^{(3)}(t)^2 a(t)^4 a''(t)-216 a^{(3)}(t)^2 a(t)^3 a'(t)^2-108 a^{(3)}(t) a(t) a'(t)^5-27 a^{(3)}(t) a(t) a'(t)^3+801 a(t)^2 a'(t)^2 a''(t)^3-588 a(t) a'(t)^4 a''(t)^2-87 a(t) a'(t)^2 a''(t)^2+108 a'(t)^6 a''(t)+27 a'(t)^4 a''(t)+78 a^{(5)}(t) a(t)^4 a'(t) a''(t)+132 a^{(3)}(t) a^{(4)}(t) a(t)^4 a'(t)-312 a^{(4)}(t) a(t)^3 a'(t)^2 a''(t)-819 a^{(3)}(t) a(t)^3 a'(t) a''(t)^2+768 a^{(3)}(t) a(t)^2 a'(t)^3 a''(t)+102 a^{(3)}(t) a(t)^2 a'(t) a''(t)\Big),$ \ and $$a_{10}=$$ $\frac{1}{665280 a(t)^6}\Big(3 a^{(10)}(t) a(t)^8-222 a^{(5)}(t)^2 a(t)^7-348 a^{(4)}(t) a^{(6)}(t) a(t)^7-147 a^{(3)}(t) a^{(7)}(t) a(t)^7-18 a''(t) a^{(8)}(t) a(t)^7+18 a'(t) a^{(9)}(t) a(t)^7-482 a''(t) a^{(4)}(t)^2 a(t)^6-331 a^{(3)}(t)^2 a^{(4)}(t) a(t)^6-1110 a''(t) a^{(3)}(t) a^{(5)}(t) a(t)^6-1556 a'(t) a^{(4)}(t) a^{(5)}(t) a(t)^6-448 a''(t)^2 a^{(6)}(t) a(t)^6-1074 a'(t) a^{(3)}(t) a^{(6)}(t) a(t)^6-476 a'(t) a''(t) a^{(7)}(t) a(t)^6-43 a'(t)^2 a^{(8)}(t) a(t)^6-11 a^{(8)}(t) a(t)^6+8943 a'(t) a^{(3)}(t)^3 a(t)^5+21846 a''(t)^2 a^{(3)}(t)^2 a(t)^5+4092 a'(t)^2 a^{(4)}(t)^2 a(t)^5+396 a^{(4)}(t)^2 a(t)^5+10560 a''(t)^3 a^{(4)}(t) a(t)^5+39402 a'(t) a''(t) a^{(3)}(t) a^{(4)}(t) a(t)^5+11352 a'(t) a''(t)^2 a^{(5)}(t) a(t)^5+6336 a'(t)^2 a^{(3)}(t) a^{(5)}(t) a(t)^5+594 a^{(3)}(t) a^{(5)}(t) a(t)^5+2904 a'(t)^2 a''(t) a^{(6)}(t) a(t)^5+264 a''(t) a^{(6)}(t) a(t)^5+165 a'(t)^3 a^{(7)}(t) a(t)^5+33 a'(t) a^{(7)}(t) a(t)^5-10338 a''(t)^5 a(t)^4-95919 a'(t)^2 a''(t) a^{(3)}(t)^2 a(t)^4-3729 a''(t) a^{(3)}(t)^2 a(t)^4-117600 a'(t) a''(t)^3 a^{(3)}(t) a(t)^4-68664 a'(t)^2 a''(t)^2 a^{(4)}(t) a(t)^4-2772 a''(t)^2 a^{(4)}(t) a(t)^4-23976 a'(t)^3 a^{(3)}(t) a^{(4)}(t) a(t)^4-2640 a'(t) a^{(3)}(t) a^{(4)}(t) a(t)^4-12762 a'(t)^3 a''(t) a^{(5)}(t) a(t)^4-1386 a'(t) a''(t) a^{(5)}(t) a(t)^4-651 a'(t)^4 a^{(6)}(t) a(t)^4-132 a'(t)^2 a^{(6)}(t) a(t)^4+111378 a'(t)^2 a''(t)^4 a(t)^3+2354 a''(t)^4 a(t)^3+31344 a'(t)^4 a^{(3)}(t)^2 a(t)^3+3729 a'(t)^2 a^{(3)}(t)^2 a(t)^3+236706 a'(t)^3 a''(t)^2 a^{(3)}(t) a(t)^3+13926 a'(t) a''(t)^2 a^{(3)}(t) a(t)^3+43320 a'(t)^4 a''(t) a^{(4)}(t) a(t)^3+5214 a'(t)^2 a''(t) a^{(4)}(t) a(t)^3+2238 a'(t)^5 a^{(5)}(t) a(t)^3+462 a'(t)^3 a^{(5)}(t) a(t)^3-162162 a'(t)^4 a''(t)^3 a(t)^2-11880 a'(t)^2 a''(t)^3 a(t)^2-103884 a'(t)^5 a''(t) a^{(3)}(t) a(t)^2-13332 a'(t)^3 a''(t) a^{(3)}(t) a(t)^2-6138 a'(t)^6 a^{(4)}(t) a(t)^2-1287 a'(t)^4 a^{(4)}(t) a(t)^2+76440 a'(t)^6 a''(t)^2 a(t)+10428 a'(t)^4 a''(t)^2 a(t)+11700 a'(t)^7 a^{(3)}(t) a(t)+2475 a'(t)^5 a^{(3)}(t) a(t)-11700 a'(t)^8 a''(t)-2475 a'(t)^6 a''(t)\Big).$ Computation of the Term $a_{12}$ in the Expansion of the Spectral Action {#Term12} ======================================================================== We pursue the computation of the term $a_{12}$ in the expansion of the spectral action for Robertson-Walker metrics by employing pseudodifferential calculus to find the term $r_{12}$ for the parametrix of $\lambda - D^2$, which is homogeneous of order $-14$, and by performing the appropriate integrations. Since there is no universal formula in the literature for this term, we have performed two heavy computations, one in Hopf coordinates and the other in spherical coordinates, to form a check on the validity of the outcome of our calculations. Another efficient way of computing the term $a_{12}$ is to use the direct method of [@ChaConRW]. The result of the computation in Hopf coordinates. {#exprfora12} -------------------------------------------------- Continuing the recursive procedure commenced in the previous section and exploiting computer assistance, while the calculation becomes significantly heavier for the term $a_{12}$, we find the following expression: $$a_{12}=$$ $\frac{1}{17297280 a(t)^{8}}\Big(3 a^{(12)}(t) a(t)^{10}-1057 a^{(6)}(t)^2 a(t)^9-1747 a^{(5)}(t) a^{(7)}(t) a(t)^9-970 a^{(4)}(t) a^{(8)}(t) a(t)^9-317 a^{(3)}(t) a^{(9)}(t) a(t)^9-34 a''(t) a^{(10)}(t) a(t)^9+21 a'(t) a^{(11)}(t) a(t)^9+5001 a^{(4)}(t)^3 a(t)^8+2419 a''(t) a^{(5)}(t)^2 a(t)^8+19174 a^{(3)}(t) a^{(4)}(t) a^{(5)}(t) a(t)^8+4086 a^{(3)}(t)^2 a^{(6)}(t) a(t)^8+2970 a''(t) a^{(4)}(t) a^{(6)}(t) a(t)^8-5520 a'(t) a^{(5)}(t) a^{(6)}(t) a(t)^8-511 a''(t) a^{(3)}(t) a^{(7)}(t) a(t)^8-4175 a'(t) a^{(4)}(t) a^{(7)}(t) a(t)^8-745 a''(t)^2 a^{(8)}(t) a(t)^8-2289 a'(t) a^{(3)}(t) a^{(8)}(t) a(t)^8-828 a'(t) a''(t) a^{(9)}(t) a(t)^8-62 a'(t)^2 a^{(10)}(t) a(t)^8-13 a^{(10)}(t) a(t)^8+45480 a^{(3)}(t)^4 a(t)^7+152962 a''(t)^2 a^{(4)}(t)^2 a(t)^7+203971 a'(t) a^{(3)}(t) a^{(4)}(t)^2 a(t)^7+21369 a'(t)^2 a^{(5)}(t)^2 a(t)^7+1885 a^{(5)}(t)^2 a(t)^7+410230 a''(t) a^{(3)}(t)^2 a^{(4)}(t) a(t)^7+163832 a'(t) a^{(3)}(t)^2 a^{(5)}(t) a(t)^7+250584 a''(t)^2 a^{(3)}(t) a^{(5)}(t) a(t)^7+244006 a'(t) a''(t) a^{(4)}(t) a^{(5)}(t) a(t)^7+42440 a''(t)^3 a^{(6)}(t) a(t)^7+163390 a'(t) a''(t) a^{(3)}(t) a^{(6)}(t) a(t)^7+35550 a'(t)^2 a^{(4)}(t) a^{(6)}(t) a(t)^7+3094 a^{(4)}(t) a^{(6)}(t) a(t)^7+34351 a'(t) a''(t)^2 a^{(7)}(t) a(t)^7+19733 a'(t)^2 a^{(3)}(t) a^{(7)}(t) a(t)^7+1625 a^{(3)}(t) a^{(7)}(t) a(t)^7+6784 a'(t)^2 a''(t) a^{(8)}(t) a(t)^7+520 a''(t) a^{(8)}(t) a(t)^7+308 a'(t)^3 a^{(9)}(t) a(t)^7+52 a'(t) a^{(9)}(t) a(t)^7-2056720 a'(t) a''(t) a^{(3)}(t)^3 a(t)^6-1790580 a''(t)^3 a^{(3)}(t)^2 a(t)^6-900272 a'(t)^2 a''(t) a^{(4)}(t)^2 a(t)^6-31889 a''(t) a^{(4)}(t)^2 a(t)^6-643407 a''(t)^4 a^{(4)}(t) a(t)^6-1251548 a'(t)^2 a^{(3)}(t)^2 a^{(4)}(t) a(t)^6-43758 a^{(3)}(t)^2 a^{(4)}(t) a(t)^6-4452042 a'(t) a''(t)^2 a^{(3)}(t) a^{(4)}(t) a(t)^6-836214 a'(t) a''(t)^3 a^{(5)}(t) a(t)^6-1400104 a'(t)^2 a''(t) a^{(3)}(t) a^{(5)}(t) a(t)^6-48620 a''(t) a^{(3)}(t) a^{(5)}(t) a(t)^6-181966 a'(t)^3 a^{(4)}(t) a^{(5)}(t) a(t)^6-18018 a'(t) a^{(4)}(t) a^{(5)}(t) a(t)^6-319996 a'(t)^2 a''(t)^2 a^{(6)}(t) a(t)^6-11011 a''(t)^2 a^{(6)}(t) a(t)^6-115062 a'(t)^3 a^{(3)}(t) a^{(6)}(t) a(t)^6-11154 a'(t) a^{(3)}(t) a^{(6)}(t) a(t)^6-42764 a'(t)^3 a''(t) a^{(7)}(t) a(t)^6-4004 a'(t) a''(t) a^{(7)}(t) a(t)^6-1649 a'(t)^4 a^{(8)}(t) a(t)^6-286 a'(t)^2 a^{(8)}(t) a(t)^6+460769 a''(t)^6 a(t)^5+1661518 a'(t)^3 a^{(3)}(t)^3 a(t)^5+83486 a'(t) a^{(3)}(t)^3 a(t)^5+13383328 a'(t)^2 a''(t)^2 a^{(3)}(t)^2 a(t)^5+222092 a''(t)^2 a^{(3)}(t)^2 a(t)^5+342883 a'(t)^4 a^{(4)}(t)^2 a(t)^5+36218 a'(t)^2 a^{(4)}(t)^2 a(t)^5+7922361 a'(t) a''(t)^4 a^{(3)}(t) a(t)^5+6367314 a'(t)^2 a''(t)^3 a^{(4)}(t) a(t)^5+109330 a''(t)^3 a^{(4)}(t) a(t)^5+7065862 a'(t)^3 a''(t) a^{(3)}(t) a^{(4)}(t) a(t)^5+360386 a'(t) a''(t) a^{(3)}(t) a^{(4)}(t) a(t)^5+1918386 a'(t)^3 a''(t)^2 a^{(5)}(t) a(t)^5+98592 a'(t) a''(t)^2 a^{(5)}(t) a(t)^5+524802 a'(t)^4 a^{(3)}(t) a^{(5)}(t) a(t)^5+55146 a'(t)^2 a^{(3)}(t) a^{(5)}(t) a(t)^5+226014 a'(t)^4 a''(t) a^{(6)}(t) a(t)^5+23712 a'(t)^2 a''(t) a^{(6)}(t) a(t)^5+8283 a'(t)^5 a^{(7)}(t) a(t)^5+1482 a'(t)^3 a^{(7)}(t) a(t)^5-7346958 a'(t)^2 a''(t)^5 a(t)^4-72761 a''(t)^5 a(t)^4-11745252 a'(t)^4 a''(t) a^{(3)}(t)^2 a(t)^4-725712 a'(t)^2 a''(t) a^{(3)}(t)^2 a(t)^4-27707028 a'(t)^3 a''(t)^3 a^{(3)}(t) a(t)^4-819520 a'(t) a''(t)^3 a^{(3)}(t) a(t)^4-8247105 a'(t)^4 a''(t)^2 a^{(4)}(t) a(t)^4-520260 a'(t)^2 a''(t)^2 a^{(4)}(t) a(t)^4-1848228 a'(t)^5 a^{(3)}(t) a^{(4)}(t) a(t)^4-205296 a'(t)^3 a^{(3)}(t) a^{(4)}(t) a(t)^4-973482 a'(t)^5 a''(t) a^{(5)}(t) a(t)^4-110136 a'(t)^3 a''(t) a^{(5)}(t) a(t)^4-36723 a'(t)^6 a^{(6)}(t) a(t)^4-6747 a'(t)^4 a^{(6)}(t) a(t)^4+17816751 a'(t)^4 a''(t)^4 a(t)^3+721058 a'(t)^2 a''(t)^4 a(t)^3+2352624 a'(t)^6 a^{(3)}(t)^2 a(t)^3+274170 a'(t)^4 a^{(3)}(t)^2 a(t)^3+24583191 a'(t)^5 a''(t)^2 a^{(3)}(t) a(t)^3+1771146 a'(t)^3 a''(t)^2 a^{(3)}(t) a(t)^3+3256248 a'(t)^6 a''(t) a^{(4)}(t) a(t)^3+389376 a'(t)^4 a''(t) a^{(4)}(t) a(t)^3+135300 a'(t)^7 a^{(5)}(t) a(t)^3+25350 a'(t)^5 a^{(5)}(t) a(t)^3-15430357 a'(t)^6 a''(t)^3 a(t)^2-1252745 a'(t)^4 a''(t)^3 a(t)^2-7747848 a'(t)^7 a''(t) a^{(3)}(t) a(t)^2-967590 a'(t)^5 a''(t) a^{(3)}(t) a(t)^2-385200 a'(t)^8 a^{(4)}(t) a(t)^2-73125 a'(t)^6 a^{(4)}(t) a(t)^2+5645124 a'(t)^8 a''(t)^2 a(t)+741195 a'(t)^6 a''(t)^2 a(t)+749700 a'(t)^9 a^{(3)}(t) a(t)+143325 a'(t)^7 a^{(3)}(t) a(t)-749700 a'(t)^{10} a''(t)-143325 a'(t)^8 a''(t))\Big).$ Agreement of the result with computations in spherical coordinates. ------------------------------------------------------------------- Taking a similar route as in §\[DiracinHopf\], we explicitly write the Dirac operator for the Roberson-Walker metric in spherical coordinates $$\nonumber ds^{2}=dt^{2}+a^{2}\left( t\right) \big ( d\chi^{2}+\sin^{2}(\chi) \left( d\theta^{2}+\sin^{2}(\theta) \, d\varphi^{2}\right) \big ).$$ Using the computations carried out in [@ChaConRW] with the orthonormal coframe $$dt, \qquad a(t)\, d \chi, \qquad a(t)\, \sin \chi \,d \theta, \qquad a(t)\, \sin \chi \, \sin \theta \,d \varphi,$$ the corresponding matrix of connection 1-forms for the Levi-Civita connection is written as $$\left ( \begin{array}{cccc} 0 &-a'(t)d\chi & -a'(t)\sin(\chi)d\theta & -a'(t)\sin(\chi)\sin(\theta)d\varphi\\ a'(t)d\chi &0 & -\cos(\chi)d\theta &-\cos(\chi)\sin(\theta)d\varphi \\ a'(t)\sin(\chi)d\theta & \cos(\chi)d\theta & 0 &-\cos(\theta)d\varphi \\ a'(t)\sin(\chi)\sin(\theta)d\varphi&\cos(\chi)\sin(\theta)d\varphi & \cos(\theta)d\varphi & 0\\ \end{array} \right ).$$ Lifting to the spin bundle by means of the Lie algebra isomorphism $\mu:\mathfrak{so}(4)\to \mathfrak{spin}(4)$ and writing the formula for the Dirac operator yield the following expression for this operator expressed in spherical coordiantes: $$\begin{aligned} D &=& \gamma^1 \frac{\partial}{\partial t}+\gamma^2 \frac{1}{a}\frac{\partial}{\partial \chi}+\gamma^3 \frac{1}{a\, \sin \chi} \frac{\partial}{\partial \theta}+\gamma^4 \frac{1}{a\, \sin \chi \, \sin \theta} \frac{\partial}{\partial \varphi} \nonumber \\ &&+\frac{3a'}{2a}\gamma^1+\frac{\cot(\chi)}{a}\gamma^2+\frac{\cot(\theta)}{2a\sin(\chi)}\gamma^3. \nonumber\end{aligned}$$ Thus the pseudodifferential symbol of $D$ is given by $$\begin{aligned} \sigma_D({ x,\xi})&=&i\gamma^1\xi_1 +\frac{i}{a}\gamma^2\xi_2+\frac{i}{a\sin(\chi)}\gamma^3\xi_3+\frac{i}{a\sin(\chi)\sin(\theta)}\gamma^4\xi_4 \nonumber \\ &&+\frac{3a'}{2a}\gamma^1+\frac{\cot(\chi)}{a}\gamma^2+\frac{\cot(\theta)}{2a\sin(\chi)}\gamma^3. \nonumber\end{aligned}$$ Accordingly, the symbol of $D^2$ is the sum $p_2'+p_1'+p_0'$ of three homogeneous components $$\begin{aligned} p_2'&=&\xi _1^2+\frac{1}{a(t)^2}\xi _2^2+\frac{1}{a(t)^2 \sin ^2(\chi )}\xi _3^2+\frac{ 1}{a(t)^2\sin ^2(\theta ) \sin ^2(\chi )}\xi _4^2, \nonumber \\ p_1'&=&-\frac{ 3 i a'(t)}{a(t)}\xi _1-\frac{i }{a(t)^2} \left(\gamma ^{12} a'(t)+2 \cot (\chi )\right)\xi _2 \nonumber \\ &&-\frac{i }{a(t)^2} \left(\gamma ^{13} \csc (\chi ) a'(t)+\cot (\theta ) \csc ^2(\chi )+\gamma ^{23} \cot (\chi ) \csc (\chi )\right)\xi _3\nonumber \\ &&-\frac{i }{a(t)^2} (\csc (\theta ) \csc (\chi ) a'(t)\gamma ^{14}+\cot (\theta ) \csc (\theta ) \csc ^2(\chi )\gamma ^{34} \nonumber \\ &&+\csc (\theta ) \cot (\chi ) \csc (\chi )\gamma ^{24} )\xi _4, \nonumber \\ p_0'&=&\frac{1}{8 a(t)^2}\left(-12 a(t) a''(t)-6 a'(t)^2+3 \csc ^2(\theta ) \csc ^2(\chi )-\cot ^2(\theta ) \csc ^2(\chi )+\right. \nonumber \\ && \left. 4 i \cot (\theta ) \cot (\chi ) \csc (\chi )-4 i \cot (\theta ) \cot (\chi ) \csc (\chi )-4 \cot ^2(\chi )+5 \csc ^2(\chi )+4\right) \nonumber \\ &&-\frac{\left(\cot (\theta ) \csc (\chi ) a'(t)\right)}{2 a(t)^2}\gamma ^{13} -\frac{ \left(\cot (\chi ) a'(t)\right)}{a(t)^2}\gamma ^{12}-\frac{ (\cot (\theta ) \cot (\chi ) \csc (\chi ))}{2 a(t)^2}\gamma ^{23}. \nonumber\end{aligned}$$ We have performed the computation of the heat coefficients up to the term $a_{12}$ using the latter symbols and have checked the agreement of the result with the computations in Hopf coordinates, presented in the previous subsections. This is in particular of great importance for the term $a_{12}$, since it ensures the validity of our computations performed in two different coordinates. Agreement with the full expansion for the round metric. ------------------------------------------------------- We first recall the full expansion for the spectral action for the round metric, namely the case $a(t) = \sin (t)$, worked out in [@ChaConRW]. Then we show that the term $a_{12}$ presented in §\[exprfora12\] reduces correctly to the round case. The method devised in [@ChaConRW] has wide applicability in the spectral action computations since it can be used for the cases when the eigenvalues of the square of the Dirac operator have a polynomial expression while their multiplicities are also given by polynomials. In the case of the round metric on $\mathbb{S}^4$, after remarkable computations based on the Euler-Maclaurin formula, this method leads to the following expression with control over the remainder term [@ChaConRW]: $$\begin{aligned} \frac{3}{4}{\rm Trace}(f(tD^2)) &=& \int_0^\infty f(tx^2)(x^3-x)dx+\frac{11 f(0)}{120}-\frac{31 f'(0) t}{2520} +\frac{41 f''(0) t^2}{10080} \nonumber \\ &&-\frac{31 f^{(3)}(0) t^3}{15840}+\frac{10331 f^{(4)}(0) t^4}{8648640}-\frac{3421 f^{(5)}(0) t^5}{3931200}+\dots +R_m. \nonumber\end{aligned}$$ This implies that the term $a_{12}$ in the expansion of the spectral action for the round metric is equal to $\frac{10331}{6486480}$. To check our calculations against this result, we find that for $a(t)=\sin(t)$ the expression for $a_{12}(t)$ reduces to $\frac{10331 \sin ^3(t)}{8648640},$ and hence $$a_{12}=\int_0^\pi a_{12}(\mathbb{S}^4)\,dt=\frac{4}{3} \frac{10331}{8648640}=\frac{10331}{6486480},$$ which is in complete agreement with the result in [@ChaConRW], mentioned above. Chameseddine-Connes’ Conjecture {#ProofofConjecture} =============================== In this section we prove a conjecture of Chamseddine and Connes from [@ChaConRW]. More precisely, we show that the term $a_{2n}$ in the asymptotic expansion of the spectral action for Robertson-Walker metrics is, up to multiplication by $a(t)^{3-2n}$, of the form $Q_{2n}(a,a',\dots,a^{(2n)})$, where $Q_{2n}$ is a polynomial with rational coefficients. Proof of rationality of the coefficients in the expressions for $a_{2n}$ {#proofofrationality} ------------------------------------------------------------------------ A crucial point that enables us to furnish the proof of our main theorem, namely the proof of the conjecture mentioned above, is the independence of the integral kernel of the heat operator of the Dirac operator of the Robertson-Walker metric from the variables $\phi_1, \phi_2, \eta$. Note that since the symbol and the metric are independent of $\phi_1, \phi_2$, the computations involved in the symbol calculus clearly imply the independence of the terms $e_n$ from these variables. However, the independence of $e_n$ from $\eta$ is not evident, which is proved as follows. \[ind\] The heat kernel $k(t, x, x)$ for the Robertson-Walker metric is independent of $\phi_1,\phi_2, \eta$. The round metric on $\mathbb{S}^3$ is the bi-invariant metric on ${\rm SU}(2)$ induced from the Killing form of its Lie algebra $\mathfrak{su}(2)$. The corresponding Levi-Civita connection restricted to the left invariant vector fields is given by $\frac{1}{2}[X,Y]$, and to the right invariant vector fields by $\frac{-1}{2}[X,Y]$. Since the Killing form is ${\rm ad}$-invariant, we have $$\langle [X,Y],Z\rangle+\langle Y,[X,Z]\rangle=0,\qquad X,Y,Z\in \mathfrak{su}(2),$$ which implies that in terms of the connection on left (right) invariant vector fields $X,Y,Z$, it can be written as $$\label{Killingequ} \langle \nabla_YX,Z\rangle+\langle Y,\nabla_ZX\rangle=0.$$ Considering the fact that $\nabla X:\mathfrak{X}(M)\to \mathfrak{X}(M)$ is an endomorphism of the tangent bundle, the latter identity holds for any $Y,Z\in\mathfrak{X}(M)$. Therefore, the equation is the Killing equation and shows that any left and right invariant vector field on ${\rm SU}(2)$ is a Killing vector field. By direct computation in Hopf coordinates, we find the following vector fields which respectively form bases for left and right invariant vector fields on ${\rm SU}(2)$: $$\begin{aligned} X^L_1&=\frac{\partial}{\partial \phi_1}+\frac{\partial}{\partial \phi_2},\\ X^L_2&=\sin (\phi_1+\phi_2)\frac{\partial}{\partial \eta}+\cot(\eta )\cos (\phi_1+\phi_2)\frac{\partial}{\partial \phi_1}-\tan (\eta )\cos (\phi_1+\phi_2) \frac{\partial}{\partial \phi_2},\\ X^L_3&=\cos (\phi_1+\phi_2)\frac{\partial}{\partial \eta}-\cot (\eta ) \sin (\phi_1+\phi_2)\frac{\partial}{\partial \phi_1}+\tan (\eta ) \sin(\phi_1+\phi_2)\frac{\partial}{\partial \phi_2},\\ X^R_1&=-\frac{\partial}{\partial \phi_1}+\frac{\partial}{\partial \phi_2},\\ X^R_2&=-\sin(\phi_1-\phi_2)\frac{\partial}{\partial \eta}-\cot (\eta ) \cos (\phi_1-\phi_2)\frac{\partial}{\partial \phi_1}-\tan(\eta )\cos(\phi_1-\phi_2)\frac{\partial}{\partial \phi_2},\\ X^R_3&=\cos(\phi_1-\phi_2)\frac{\partial}{\partial \eta}-\cot (\eta ) \sin(\phi_1-\phi_2)\frac{\partial}{\partial \phi_1}-\tan(\eta )\sin (\phi_1-\phi_2)\frac{\partial}{\partial \phi_2}.\end{aligned}$$ One can check that these vector fields are indeed Killing vector fields for the Robertson-Walker metrics on the four dimensional space. Thus, for any isometry invariant function $f$ we have: $$\begin{aligned} &&\frac{\partial}{\partial \phi_1}f=\frac{1}{2}(X^L_1-X^R_1)f=0, \nonumber \\ &&\frac{\partial}{\partial \phi_2}f=\frac{1}{2}(X^L_1+X^R_1)f=0,\nonumber \\ &&\frac{\partial}{\partial \eta}f=(\sin(\phi_1+\phi_2)X^L_2+\cos(\phi_1+\phi_2)X^L_3)f=0. \nonumber\end{aligned}$$ In particular, the heat kernel restricted to the diagonal, $k(t, x, x)$, is independent of $\phi_1, \phi_2,\eta$, and so are the coefficient functions $e_n$ in its asymptotic expansion. We stress that although $e_n (x)$ is independent of $\eta,\phi_1,\phi_2$, its components denoted by $e_{n, j, \alpha}$ in the proof of the following theorem are not necessarily independent of these variables. \[rationalitytheorem\] The term $a_{2n}$ in the expansion of the spectral action for the Robertson-Walker metric with cosmic scale factor $a(t)$ is of the form $$\frac{1}{a(t)^{2n-3}}\,Q_{2n}\left(a(t),a'(t),\dots,a^{(2n)}(t)\right),$$ where $Q_{2n}$ is a polynomial with rational coefficients. Using we can write $$\begin{aligned} \label{ensum} e_n&=\sum_{\begin{array}{c}2j-2-\|\alpha\|=n\\ n/2+1\leq j\leq 2n+1\end{array}} c_\alpha \,e_{n,j,\alpha}, \end{aligned}$$ where $$e_{n,j,\alpha}=\frac{1}{(j-1)!}r_{n,j,\alpha} \,a(t)^{\alpha_2+\alpha_3+\alpha_4}\sin(\eta)^{\alpha_3}\cos(\eta)^{\alpha_4}.$$ The recursive equation implies that $$\label{recforenja} e_{n,j,\alpha}=$$ $\frac {1}{(j-1)a(t)}\Big( (\gamma^{14} a'(t)-\tan(\eta)\gamma^{24} )e_{n-1,j-1,\alpha-{\bf e}_4} % % +(\gamma^{13}a'(t)+\cot(\eta)\gamma^{23})e_{n-1,j-1,\alpha-{\bf e}_3} % % +(\gamma^{12}a'(t)+1((2\alpha_4-1)\tan(\eta)+(1-2\alpha_3)\cot(\eta)))e_{n-1,j-1,\alpha-{\bf e}_2} % % +4a'(t)e_{n-1,j-2,\alpha-{\bf e}_1-2{\bf e}_2} % % +4a'(t)e_{n-1,j-2,\alpha-{\bf e}_1-2{\bf e}_3} % % + 4a'(t)e_{n-1,j-2,\alpha-{\bf e}_1-2{\bf e}_4} % % + (-2\alpha_2-2\alpha_3-2\alpha_4 + 3 )a' (t)e_{n-1,j-1,\alpha-{\bf e}_1} % % +2a(t)\frac{\partial} {\partial t} e_{n-1,j-1,\alpha-{\bf e}_1} % % -4\tan(\eta)e_{n-1,j-2,\alpha-{\bf e}_2-2{\bf e}_4} % % +4\cot(\eta)e_{n-1,j-2,\alpha-{\bf e}_2-2{\bf e}_3} % % + 2\frac{\partial}{\partial\eta} e_{n-1,j-1,\alpha-{\bf e}_2}\Big) \newline % % % % % % + \frac {1} {(j-1)a(t)^2} \Big( % % a(t)^2\frac {\partial^2} {\partial t^2} e_{n-2,j-1,\alpha} % % + 4a'(t)a(t)\frac {\partial}{\partial t} e_{n-2,j-2,\alpha-2{\bf e}_2} % % + 4 a'(t)a(t)\frac{\partial} {\partial t} e_{n-2,j-2,\alpha-2{\bf e}_3} % % + 4a' (t) a (t)\frac{\partial} {\partial t} e_{n-2,j-2,\alpha-2{\bf e}_4} % % + (-2\alpha_2-2\alpha_3-2\alpha_4+3) a'(t)a(t) \frac{\partial} {\partial t} e_{n-2,j-1,\alpha} % % + 4a'(t)^2 e_{n-2,j-3,\alpha-4{\bf e}_2} % % +8a'(t)^2 e_{n-2,j-3,\alpha-2{\bf e}_2-2{\bf e}_3} % % +8 a' (t)^2e_{n-2,j-3,\alpha-2{\bf e}_2-2{\bf e}_4} % % +4\cot(\eta)\frac{\partial}{\partial\eta}e_{n-2,j-2,\alpha-2{\bf e}_3} % % - 4\tan(\eta)\frac {\partial} {\partial\eta} e_ {n-2,j-2,\alpha-2{\bf e}_4} % % +\frac{\partial^2}{\partial\eta^2} e_{n-2,j-1,\alpha} % % +\big(2\cot(\eta)\gamma^{12}a'(t)+ (- 4(\alpha_2+\alpha_3+\alpha_4-2)a'(t)^2+ 4 (-(\alpha_3-1)\csc^2(\eta)+\alpha_3+\alpha_4-2) + 2 a(t)a''(t) ) \big)e_{n-2,j-2,\alpha-2{\bf e}_3} % % + \big( (\cot(\eta)(1-2\alpha_3)+(2\alpha_4-1)\tan (\eta)) + \gamma^{12} a'(t) \big)\frac {\partial}{\partial\eta} e_{n-2,j-1,\alpha} % % +\big((-4(\alpha_2+\alpha_3+\alpha_4- 2) a'(t)^2 +4(-(\alpha_4-1)\sec^2(\eta)+\alpha_3 + \alpha_4-2 )+2a(t)a''(t)) - 2\gamma^{12}\tan(\eta)a'(t) \big)e_{n-2,j-2,\alpha-2{\bf e}_4} % % +8(a'(t)^2-1) e_{n-2,j-3,\alpha-2{\bf e}_4} % % + 4(\cot^2(\eta)+a'(t)^2) e_{n-2,j-3,\alpha-4{\bf e}_3} % % +4(\tan^2(\eta)+a'(t)^2 ) e_{n-2,j-3,\alpha-4{\bf e}_4} % % + (2a(t)a''(t)-4(\alpha_2+\alpha_3+ \alpha_4-2)a'(t)^2)e_{n-2,j-2,\alpha-2{\bf e}_2} % % +\big(\frac{1}{2} (\cot(\eta)(1-2\alpha_3)+(2\alpha_4-1)\tan(\eta))\gamma^{12} a'(t)+\frac{1}{4} ((4\alpha_3^2-1)\csc^2 (\eta)-4(\alpha_3+\alpha_4-1)^2 +(2\alpha_2+2\alpha_3+2\alpha_4-3)(2\alpha_2+2\alpha_3+2\alpha_4-1) a'(t)^2 +\sec^2(\eta)(4\alpha_4^2-1)- 2(2\alpha_2+2\alpha_3+2\alpha_4-3)a(t) a''(t) ) \big)e_{n-2,j-1,\alpha} \Big). $ The functions associated with the initial indices are: $$\begin{aligned} &&e_{0,1,0,0,0,0}=1, \qquad e_ {1,2,1,0,0,0}= \frac {3ia'(t)}{a(t)}, \qquad e_ {1,3,1,2,0,0}= \frac {2ia'(t)}{a(t)}, \nonumber \\ && e_ {1,3,1,0,2,0}= \frac{2ia'(t)}{a (t)}, \qquad e_ {1,3,1,0,0,2}= \frac{2ia'(t)}{a (t)}, \qquad e_ {1,3,0,1,0,2}=-\frac{(2 i)\tan(\eta)}{a(t)}, \nonumber \\ &&e_ {1,3,0,1,2,0}=\frac{(2 i)\cot(\eta)}{a(t)}, \qquad e_ {1, 2, 0, 0, 1,0}= \frac {i\gamma ^{13} a'(t)} {a (t)}+\frac{i\gamma^{23}\cot(\eta)}{a (t)}, \nonumber \\ && e_ {1,2,0,0,0,1}=\frac{i\gamma ^{14}a'(t)}{a(t)}-\frac{i\gamma^{24}\tan(\eta)}{a (t)}, \qquad e_ {1,2,0,1,0,0}=\frac{2i\cot(2\eta)}{a (t)}+ \frac{i\gamma^{12}a'(t)}{a(t)}. \nonumber\end{aligned}$$ It is then apparent that $e_{0}$ and $e_1$ are, respectively, a polynomial in $a(t)$, and a polynomial in $a(t)$ and $a'(t)$, divided by some powers of $a(t)$. Thus, it follows from the above recursive formula that all $e_{n,j,\alpha}$ are of this form. Accordingly, we have $$e_n=\frac{P_n}{a(t)^{d_n}},$$ where $P_n$ is a polynomial in $a(t)$ and its derivatives with matrix coefficients. Writing $e_{n,j,\alpha}=P_{n,j,\alpha}/a(t)^{d_n}$, we obtain $d_n=\max\{d_{n-1}+1,d_{n-2}+2\}.$ Starting with $d_0=0$, $d_1=-1$, and following to obtain $d_n=n$, we conclude that $$e_{n,j,\alpha}=\frac{1}{a^{n}(t)}P_{n,j,\alpha}(a(t),\dots,a^{(n)}(t)),$$ where $P_{n,j,\alpha}$ is a polynomial whose coefficients are matrices with entries in the algebra generated by $\sin(\eta),\csc(\eta), \cos(\eta), \sec(\eta)$ and rational numbers. In the calculation of the even terms $a_{2n}$, only even $\alpha_k$ have contributions in the summation . This implies that the corresponding $c_\alpha$ is a rational multiple of $\pi^2$ and $P_{2n}$ is a polynomial with rational matrix coefficients, which is independent of variables $\eta,\phi_1,\phi_2$ by Lemma \[ind\]. Hence $$a_{2n}=\frac{1}{16\pi^4}\int_{\mathbb{S}_a^3}{\rm tr}(e_{2n})\,dvol_g= \frac{2\pi^2a(t)^3}{16\pi^4}\,{\rm tr}\Big(\frac{P_{2n}}{a(t)^{2n}}\Big)= \frac{Q_{2n}}{a(t)^{n-3}},$$ where $Q_{2n}$ is a polynomial in $a(t), a'(t), \dots, a^{(2n)}(t)$ with rational coefficients. The polynomials $P_{n, j, \alpha}$ also satisfy recursive relations that illuminate interesting features about their structure. \[monomialformofpnja\] Each $P_{n, j ,\alpha}$ is a finite sum of the form $$\sum c_k \, a(t)^{k_0}a'(t)^{k_1}\cdots a^{(n)}(t)^{k_n},$$ where each $c_k$ is a matrix of functions that are independent from the variable $t$, and $ \sum_{j=0}^n k_j=\sum_{j=0}^n jk_j=l,$ for some $0 \leq l \leq n$. This follows from an algebraically lengthy recursive formula for $P_{n, j ,\alpha}$ which stems from the equation , similar to the recursive formula for $e_{n, j, \alpha}$ in the proof of Theorem \[rationalitytheorem\]. In addition, one needs to find the following initial cases: $$\begin{aligned} &&P_ {0, 1, 0, 0, 0, 0} = I, \qquad P_ {1, 2, 1, 0, 0, 0} = 3 i a' (t), \quad P_ {1, 2, 0, 0, 1, 0} = i\gamma ^{13} a' (t) + i\gamma ^{23}\cot (\eta), \nonumber \\ &&P_ {1, 2, 0, 0, 0, 1} = i\gamma ^{14} a' (t) - i\gamma^{24}\tan (\eta), \qquad P_ {1, 2, 0, 1, 0, 0} = 2 i\cot (2\eta) + i\gamma^{12} a' (t), \nonumber \\ &&P_ {1, 3, 0, 1, 0, 2} = -2 i\tan (\eta),\qquad P_ {1, 3, 0, 1, 2, 0} = 2 i\cot (\eta), \qquad P_ {1, 3, 1, 2, 0, 0} = 2 i a' (t),\nonumber \\ &&P_ {1, 3, 1, 0, 2, 0} = 2 i a' (t), \qquad P_ {1, 3, 1, 0, 0, 2} = 2 i a' (t). \nonumber \end{aligned}$$ A recursive formula for the coefficient of the highest order term in $a_{2n}$ ----------------------------------------------------------------------------- The highest derivative of the cosmic scale factor $a(t)$ in the expression for $a_n$ is seen in the term $a(t)^{n-1}a^{(n)}(t)$, which has a rational coefficient based on Theorem \[rationalitytheorem\]. Let us denote the coefficient of $a(t)^{n-1}a^{(n)}(t)$ in $a_n$ by $h_n$. Since the coefficients $h_n$ are limited to satisfy the recursive relations derived in the proof of the following proposition, one can find the following closed formula for these coefficients. The coefficient $h_n$ of $a(t)^{n-1}a^{(n)}(t)$ in $a_n$ is equal to $$\sum_{\begin{array}{c} [n/2]+1\leq j\leq 2n+1\\ 0\leq k \leq j-n/2-1 \end{array}} \Gamma\left(\frac{2k+1}{2}\right)H_{n,j,2k},$$ where, starting from $$\begin{aligned} &&H_{1,2,1}=H_{1,3,1}=\frac{3 i}{2 \sqrt{\pi }},\qquad H_{2,4,2}=-\frac{1}{\sqrt{\pi }}, \nonumber \\ &&H_{2,3,0}=H_{2,2,0}=\frac{3}{4 \sqrt{\pi }},\qquad H_{2,3,2}=-\frac{3}{2 \sqrt{\pi }}, \nonumber\end{aligned}$$ the quantities $H_{n,j,\alpha}$ are computed recursively by $$H_{n,j,\alpha}=\frac{1}{j-1}(H_{n-2,j-1,\alpha}+2 i H_{n-1,j-1,\alpha-1}).$$ It follows from Proposition \[monomialformofpnja\] that the highest derivative of $a(t)$ in $a_n$ appears in the term $a(t)^{n-1}a^{(n)}(t)$. By a careful analysis of the equation we find that only the terms $$\frac{1}{j-1}\Big (a(t)^2 \frac{\partial ^2}{\partial t^2}P_{n-2,j-1,\alpha}+ 2 i a(t) \frac{\partial}{\partial t} P_{n-1,j-1,\alpha-{\bf e}_1} \Big)$$ contribute to its recursive formula. Denoting the corresponding monomial in $P_{n,j,\alpha}$ by $H_{n,j,\alpha}a(t)^{n-1}a^{(n)}(t)$ and substituting it into the above formula we obtain the equation $$H_{n,j,\alpha}=\frac{1}{j-1}(H_{n-2,j-1,\alpha}+2 i H_{n-1,j-1,\alpha-{\bf e}_1}),$$ for any $n>2$. Denoting $$H_{n,j,\alpha_1}=\sum \prod_{k=2}^4 \Gamma\left(\frac{\alpha_k+1}{2}\right)\frac{(-1)^{\alpha_k}+1}{2}\,{\rm tr}\left(\frac{ 1}{(2 \pi )^2}\int _0^{\pi/2 }H_{n,j,\alpha_1,\alpha_2,\alpha_3,\alpha_4}d\eta \right),$$ the recursive formula converts to $$H_{n,j,\alpha}=\frac{1}{j-1}(H_{n-2,j-1,\alpha}+2 i H_{n-1,j-1,\alpha-1}).$$ Thus, the coefficient of $a(t)^{n-1}a^{(n)}(t)$ in $a_n$ is given by the above expression. Using the above proposition we find that: $$\begin{aligned} && h_2 = \frac{1}{4}, \qquad h_4= \frac{1}{40}, \qquad h_6= \frac{1}{560}, \qquad h_8= \frac{1}{10080}, \qquad h_{10}=\frac{1}{221760}, \nonumber \\ && h_{12}=\frac{1}{5765760}, \qquad h_{14} = \frac{1}{172972800}, \qquad h_{16} = \frac{1}{5881075200}, \nonumber \\ && h_{18}=\frac{1}{223480857600}, \qquad h_{20}= \frac{1}{9386196019200}. \nonumber\end{aligned}$$ Conclusions {#Conclusions} =========== Pseudodifferential calculus is an effective tool for applying heat kernel methods to compute the terms in the expansion of a spectral action. We have used this technique to derive the terms up to $a_{12}$ in the expansion of the spectral action for the Robertson-Walker metric on a 4-dimensional geometry with a general cosmic scale factor $a(t)$. Performing the computations in Hopf coordinates, which reflects the symmetry of the space more conveniently at least from a technical point of view, we proved the independence of the integral kernel of the corresponding heat operator from three coordiantes of the space. This allowed us to furnish the proof of the conjecture of Chamseddine and Connes on rationality of the coefficients of the polynomials in $a(t)$ and its derivatives that describe the general terms $a_{2n}$ in the expansion. The terms up to $a_{10}$ were previously computed in [@ChaConRW] using their direct method, where the terms up to $a_6$ were checked against Gilkey’s universal formulas [@GilBook1; @GilBook2]. The outcome of our computations confirms the previously computed terms. Thus, we have formed a check on the terms $a_8$ and $a_{10}$. In order to confirm our calculation for the term $a_{12}$, we have performed a completely different computation in spherical coordinates and checked its agreement with our calculation in Hopf coordinates. It is worth emphasizing that the high complexity of the computations, which is overcome by computer assistance, raises the need to derive the expressions at least in two different ways to ensure their validity. We have found a formula for the coefficient of the term with the highest derivative of $a(t)$ in $a_{2n}$ for all $n$ and make the following observation. The polynomials $Q_{2n}$ in $a_{2n}=Q_{2n}\left(a(t),a'(t),\dots,a^{(2n)}(t)\right)/a(t)^{2n-3}$ are of the following form up to $Q_{12}$: $$Q_{2n}(x_0, x_1, \dots, x_{2n}) = \sum c_k\, x_0^{k_0} x_1^{k_1} \cdots x_{2n}^{k_{2n}}, \qquad c_k \neq0,$$ where the summation is over all tuples of non-negative integers $k=(k_0,k_1, \dots, k_{2n})$ such that either $ \sum k_j = 2n$ while $ \sum j k_j = 2n$, or $ \sum k_j = 2n-2$ while $ \sum j k_j = 2n-2$. This provides enough evidence and hope to shed more light on general structure of the terms $a_{2n}$ by further investigations, which are under way. Acknowledgments {#acknowledgments .unnumbered} =============== We are indebted to Alain Connes for helpful discussions and encouragements on the present topic. F.F. thanks the Institut des Hautes Études Scientifiques (I.H.E.S.) and its IT department, in particular Francois Bachelier, for their support and the excellent environment and facilities during his visit in the Fall of 2013. [99]{} P. 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--- abstract: 'The validity of Hubble’s law defies the determination of the center of the big bang expansion, even if it exists. Every point in the expanding universe looks like the center from which the rest of the universe flies away. In this article, the author shows that the distribution of apparently circular galaxies is not uniform in the sky and that there exists a special direction in the universe in our neighborhood. The data is consistent with the assumption that the tidal force due to the mass distribution around the universe center causes the deformation of galactic shapes depending on its orientation and location relative to the center and our galaxy. Moreover, the cmb dipole data can also be associated with the center of the universe expansion, if the cmb dipole at the center of our supercluster is assumed to be due to Hubble flow. The location of the center is estimated from the cmb dipole data. The direction to the center from both sets of data is consistent and the distance to the center is computed from the cmb dipole data.' author: - Yukio Tomozawa title: The CMB Dipole and Circular Galaxy Distribution --- \[sec:level1\]Introduction ========================== Since the discovery of Hubble’s law in 1929 and the big bang interpretation of its data, the question lingers whether a center for the expansion exists and if so where. The Hubble law, v = H$_{\text{0}}$ r, yields the relationship, v$_{\text{2}}$ - v$_{\text{1}}$= H$_{\text{0}}$ ( r$_{\text{2}}$ - r$_{\text{1}}$) for any two galaxies with positions and velocities, r$_{\text{1}}$, v$_{\text{1}}$ and r$_{\text{2}}$ , v$_{\text{2}}$ respectively, where H$_{\text{0}}$ = 100 h km/s Mpc is the Hubble constant (with h = 0.5 0.85). For convenience of discussion, we assume the value of H$_{\text{0}}$ to be 70.0 km/s Mpc in this article. The last equation implies that every point appears to be the center of the expansion. Besides, the distribution of galaxies surrounding us seems to be isotropic on some large distance scale. In order to get information about the existence of a center for expansion of the universe, one has to search for observational data beyond the Hubble law. The author will show that such information can be obtained from data for the distribution of circular galixies and the cmb dipole in the temperature distribution. These independent data indicate that the direction to the center is consistent for both, and one set of data gives a determination of the distance. Distribution of Circular Galaxies ================================= Circular galaxies appear when the circular faces of spiral galaxies or elliptical galaxies are facing us; otherwise they are elliptical in shape. The fraction of circular galaxies in any direction depends on the probability of the orientation of the symmetry axis coinciding with the line of sight. If one assumes that such a probability is uniform in any direction, then one expects the distribution of circular galaxies to be isotropic. The author will show that that is not the case for the existing data. Fig. 1 (a) shows the distribution of 23,011 bright galaxies in galactic coordinates compiled in $\ $RC3[@rc3]. The galaxies in the galactic plane are missing for the obvious reason that they are prevented from observation due to our own Milky Way. Plots in equatorial coordinates and supergalactic coordinates show a similar uniform distribution. The number of galaxies on the two sides of the galactic plane is 12,323 in the north and 10,628 in the south with a ratio of 1.1651. The difference in these numbers may be partly due to a statistical fluctuation and partly due to a historical bias in past observation. Anyway, any study hereafter can be normalized with this ratio. Fig. 1 (b) shows the velocity distribution for the compiled galaxies. The number of galaxies for which the velocities, cz, are known is 10633 (5923 in the north and 4710 in the south). The approximate upper bound is $\sim$15,000 km/s and the average, $\sim$7,500 km/s, corresponds to a distance of 110 Mpc. In Fig. 2 (a), (b) and (c), we show the distribution of galaxies with circular shape in galactic, supergalactic and equatorial J2000.0 coordinates, respectively. These galaxies correspond to those with the value R25 = 0.00 (the error is in the range of $\sim$0.10) in RC3, where R25 stands for the logarithm of the ratio of the apparent major and minor axes. The number of such galaxies is 682 in the north, 416 in the south with 1098 for the total. Also shown in Fig. 2 (d) is the velocity distribution, which is similar to Fig 1 (b). The pertinent characteristics of Fig. 2 are the following: \(i) Apparently circular galaxies are not distributed uniformly in the sky. If the orientation of galaxies is distributed at random, Fig 2 (a)-(c) should show statistically uniform distributions. It definitely shows directionality. \(ii) The southern part is more compact compared to the northern part. Since a closed curve surrounding the pole is extended to all longitudes in Fig 1 (a), the views in other coordinates, supergalactic coordinates in Fig 2 (b) and equatorial coordinates J2000.0 in Fig 2 (c), provide intuitively clearer distribution plots. Fig 2 (b) shows similar shapes for both, northern and southern, distributions. The normalized ratio of the northern to the southern distribution is (682)/(416)/(1.1651) = 1.4071. The shapes in supergalactic coordinates in Fig 2 (b) match this ratio. Note that the left and right blobs in Fig 2 (b) approximately correspond to those in the north and the south in Fig 2 (a). Fig 3 (a)-(d) shows the distribution of galaxies with R25 = 0.01-0.10, 0.11-0.20, 0.21-0.40 and 0.41-1.00. They do not show a distinct directionality except for a higher density in the neighborhood of (120d, -30d). The total number of galaxies, the numbers in the north and south as well as the normalized north/south ratio are given for sliced values of R25 in Table 1. In summary, a distinct directionality exists only for R25 = 0.00, and not for the other values of R25. The meaning of this directionality in the distribution of apparently circular galaxies is an important question .The author suggests that the existence of a center for universe expansion is consistent with the presented data and suggests the direction to the center, if not the determination of the distance. \(I) Galaxies are subject to tidal forces due to the gravity of the masses which are contained in a sphere of radius equal to the distance from the center to the galaxies in question. Circular galaxies in the radial direction from the center through us stay circular in the presence of tidal forces due to axial symmetry, while those in directions perpendicular to the radial seen from us are deformed by the squeeze and stretch of tidal forces and therefore become elliptical. This explains the directionality in Fig. 2 (a)-(c). \(II) If the center lies at a large distance compared with the observed galaxies in the RC3 catalog, then the circular galaxy blobs in the both hemispheres should be approximately equal in size, since the defining cone for circular invariance constitutes parallel lines. The disparity in the south and the north suggests that the center is nearby and the direction to the center must be towards the north, since the tidal force near the center is small so that the distortation is minimal in the neighborhood. This explains the larger extension of circular galaxies in the northern hemisphere. The central value of the direction of the center is in the neighborhood of $$l_{c}=120{{}^\circ}+180{{}^\circ}=300{{}^\circ}\pm15{{}^\circ},\text{ \ \ \ }b_{c}=30{{}^\circ}\pm15{{}^\circ}. \label{center}$$ The distance to the center, $R_{c}$, is bounded by 10000km/s ==$>$143Mpc. \(III) If the matter density is constant, it produces a gravitational force that is linear in the radial direction, thus yielding a tidal force that is spherically symmetric and attractive. Obviously, such a tidal force does not explain the data of Fig.2 (a)-(c). Although the assumption of homogeneity and isotropy in the Friedman-Robertson-Walker metric yields a constant matter density at a fixed time, experimental evidence for those assumptions is yet to come. As a matter of fact, the analysis of this article finds that the matter density of the universe is not constant on this length scale. We are discussing a length scale that involves our local supercluster which is not uniform in density. In Fig 4(a), circular galaxy distribution (R25 = 0) from a newer compilation, the Hyperleda catalog (PGC 2003) with 12,008 items out of a total of 983,261 is shown. Three blobs in the equatorial slab are stars. Away from the equatorial slab, most of data points are galaxies. The data away from the equatorial slab has features similar to that of the data from the RC3 catalog, Fig 2(a). Fig 4(b) shows a spherical plot of Fig 4(a). The cmb Dipole in the Temperature Distribution ============================================== A dipole component was observed in the temperature distribution in the cmb (cosmic microwave background radiation) measurement. The blue shift dipole value for the solar system [@dipole] is v = 371$\pm$0.5 km/s, l = 264.4$\pm$0.3${{}^\circ}$, b = 48.4$\pm$0.5${{}^\circ}$. Or alternatively, the redshift dipole value is$$v=371\pm0.5\text{ }km/s,\text{ \ \ \ }l=84.4\pm0.3{{}^\circ},\text{ \ \ \ }b=-48.4\pm0.5{{}^\circ}.$$ By using the observation of a peculiar velocity for the solar system, one can compute the dipole component of the cluster (Virgo) center and that of the supercluster (Great Attractor) center. Then, one can associate the last dipole component with Hubble flow. In order to understand the relationship among Hubble flow, peculiar velocity and the cmb dipole, the author presents pertinent theorems. With the assumption of the existence of a center for expansion of the universe, Hubble flow creates a cmb dipole with red shift in the direction of the Hubble flow with the magnitude of the Hubble flow velocity. Let the velocities of the Hubble flow and the cmb emitter in the direction of the Hubble flow be $v_{H}$ and $v$, respectively. The relative velocity of the two is $v_{+}=(v-v_{H})/(1-vv_{H}/c^{2})$. The analogous velocities in the opposite direction are $-v_{H}$ and $v-2v_{H}$, and the relative velocity is $v_{-}=(v-2v_{H}+v_{H})/(1+(v-2v_{H})v_{H}/c^{2})$ in the opposite direction. The difference of these opposite relative velocities is then $$v_{+}-v_{-}=(v-v_{H})(2v-2v_{H})v_{H}/c^{2}(1-vv_{H}/c^{2})(1+(v-v_{H})v_{H}/c^{2})\approx2v_{H},\text{ \ \ \ \ \ \ \ }for\text{ \ }v\approx c\text{ and }v_{H}\ll c.$$ and$$v_{\pm}\approx v-v_{H}\pm v_{H}$$ This proves the statement. The cmb dipole vanishes at the center of the universe. The cmb dipoles for red shift at the center of a cluster and at a member galaxy with a peculiar velocity $\mathbf{v}_{p}$ are related by$$\mathbf{v}(dipole\text{ }at\text{ }the\text{ }clustercenter)=\mathbf{v}(dipole\text{ }at\text{ }the\text{ }galaxy)-\mathbf{v}_{p}+\mathbf{v}(clustercenter-galaxy),$$ where$$\mathbf{v}(A-B)=\mathbf{v}(A)-\mathbf{v}(B).$$ The peculiar velocity plays a role similar to Hubble flow, as far as the cmb dipole is concerned. This can be seen by considering the case where a peculiar velocity, $\mathbf{v}_{p}$, and a Hubble flow, $\mathbf{v}_{H}$, are parallel to each other. In this case, the galaxy is at rest with respect to a galaxy with a Hubble flow, $\mathbf{v}_{H}+\mathbf{v}_{p}$. Since both galaxies should have the same cmb dipole, a peculiar velocity yields a redshift cmb dipole with $\mathbf{v}_{p}$. That is the reason for the subtraction. The term, $\mathbf{v}(clustercenter-galaxy)=$ $\mathbf{v}(clustercenter)-\mathbf{v}(galaxy)$, is for adjustament of the Hubble law. The center of the universe should have an intrinsically vanishing value for the cmb dipole. Hereafter a cmb dipole implies a redshift dipole unless otherwise stated. The peculiar velocity of the sun relative to the Virgo center of the local cluster is estimated to be [@sciama] $$v=415\text{ }km/s,\text{ \ \ \ }l=335{{}^\circ},\text{ \ \ \ }b=7{{}^\circ} \label{sciama1}$$ or$$v=630\text{ }km/s,\text{ \ \ \ }l=330{{}^\circ},\text{ \ \ \ }b=45{{}^\circ} \label{sciama2}$$ We examine these two possibilities. The outcomes are listed in this order for each case. Using Theorem 2, one computes the cmb dipole at the Virgo center, which is located at$$v=1050\pm200\text{ }km/s,\text{ \ \ \ }l=287{{}^\circ},\text{ \ \ \ }b=72.3{{}^\circ}$$ corresponding to a distance of $15\pm3$ $Mpc$. Hereafter, all distances are expressed in the form of receding velocities due to the Hubble law, $\mathbf{v}=H_{0}\mathbf{r}$. The key formula is$$\mathbf{v}(dipole\text{ }at\text{ }Virgo)=\mathbf{v}(dipole\text{ }at\text{ }the\text{ }Sun)-\mathbf{v}_{p}(Sun/Virgo)+\mathbf{v}(Virgo). \label{dipole1}$$ For the cmb dipole at the Virgo center, one obtains $$v=728.3\pm148\text{ }km/s,\text{ \ \ \ }l=336.0\pm8.2{{}^\circ},\text{ \ \ \ }b=67.4\pm11.3{{}^\circ}$$$$v=418.8\pm47\text{ }km/s,\text{ \ \ \ }l=328.8\pm6.4{{}^\circ},\text{ \ \ \ }b=41.5\pm28.0{{}^\circ}$$ depending on the two choices of peculiar velocity. We note that the Cartesian coordinates for $(v,l,b)$ are expressed as $(v\cos(b)\cos l,v\cos(b)\sin l,v\sin(b).$ Further, the Virgo cluster is considered to be a part of a supercluster centered around the GA (great attractor). In order to compute the cmb dipole at the GA, one assumes the velocity of the Virgo center to be corresponding to a distance of $15\pm3$ $Mpc$. The position of the GA is taken to be [@ga]$$v=4200\text{ }km/s,\text{ \ \ \ }l=309{{}^\circ},\text{ \ \ \ }b=18{{}^\circ} \label{eqga1}$$ or$$v=3000\text{ }km/s,\text{ \ \ \ }l=305{{}^\circ},\text{ \ \ \ }b=18{{}^\circ} \label{eqga2}$$ and the infall velocity of the Virgo center to the GA to be $$v_{in}=1000\pm200km/s.$$ The direction of the infall is determined by$$\mathbf{v}(GA-V)=\mathbf{v}(GA)-\mathbf{v}(Virgo)$$ resulting in$$v(GA-V)=3712.3\pm75\text{ }km/s,\text{ \ \ \ }l(GA-V)=310.9\pm0.4{{}^\circ},\text{ \ \ \ }b(GA-V)=4.6\pm2.8{{}^\circ}\text{ \ \ \ } \label{infall1}$$ for Eq.(\[eqga1\]) and$$v(GA-V)=2552.5\pm59\text{ }km/s,\text{ \ \ \ }l(GA-V)=307.2\pm1.6{{}^\circ},\text{ \ \ \ }b(GA-V)=1.6\pm3.0{{}^\circ} \label{infall2}$$ for Eq. (\[eqga2\]). Using Theorem 2, one can compute the cmb dipole at the GA by$$\mathbf{v}(dipole\text{ }at\text{ }GA)=\mathbf{v}(dipole\text{ }at\text{ }Vigo)-\mathbf{v}(Virgo\text{ infall})+\mathbf{v}(GA-V), \label{dipole2}$$ where the direction of the infall is given by Eq. (\[infall1\]) or Eq. (\[infall2\]). Then, the cmb dipole at the GA is given by$$v=2609.3\pm120\text{ }km/s,\text{ \ \ \ }l=308.1\pm0.2{{}^\circ},\text{ \ \ \ }b=19.9\pm3.4{{}^\circ}$$$$v=2459.1\pm98\text{ }km/s,\text{ \ \ \ }l=308.6\pm0.6{{}^\circ},\text{ \ \ \ }b=11.6\pm3.9{{}^\circ}$$ for Eq. (\[eqga1\]), and$$v=1455.6\pm142\text{ }km/s,\text{ \ \ \ }l=301.3\pm0.6{{}^\circ},\text{ \ \ \ }b=25.5\pm5.3{{}^\circ}$$$$v=1286.6\pm104\text{ }km/s,\text{ \ \ \ }l=302.0\pm0.7{{}^\circ},\text{ \ \ \ }b=10.4\pm7.3{{}^\circ}$$ for Eq. (\[eqga2\]). Based on Theorem 1, one may assume that the cmb dipole at the GA calculated above is due to the Hubble flow of the the center of the GA. Then, one can compute the position of the center for expansion of the universe by$$\mathbf{v}(universe\text{ }center)=\mathbf{v}(GA\text{ }center)-\mathbf{v}(the\text{ }cmb\text{ }dipole\text{ }at\text{ }GA) \label{dipole3}$$ The position of the Universe center thus obtained is$$v_{c}=1595.3\pm196\text{ }km/s,\text{ \ \ \ }l_{c}=310.5\pm0.1{{}^\circ},\text{ \ \ \ }b_{c}=14.8\pm1.5{{}^\circ} \label{center1}$$$$v_{c}=1779.6\pm184km/s,\text{ \ \ \ }l_{c}=309.7\pm0.2{{}^\circ},\text{ \ \ \ }b_{c}=26.8\pm2.7{{}^\circ} \label{center2}$$ for Eq. (\[eqga1\]), and$$v_{c}=1585.4\pm196\text{ }km/s,\text{ \ \ \ }l_{c}=308.1\pm0.2{{}^\circ},\text{ \ \ \ }b_{c}=13.0\pm1.6{{}^\circ} \label{center3}$$$$v_{c}=1759.3\pm181\text{ }km/s,\text{ \ \ \ }l_{c}=307.4\pm0.0{{}^\circ},\text{ \ \ \ }b_{c}=25.3\pm2.9{{}^\circ} \label{center4}$$ for Eq. (\[eqga2\]). Conversion to the ordinary distance scale yields $22.8\pm2.8$ $Mpc$, $25.4\pm2.6$ $Mpc$ for Eq. (\[eqga1\]) and $22.6\pm2.8$ $Mpc$, $25.1\pm2.6Mpc$ for Eq. (\[eqga2\]). The angles obtained here for the direction of the universe center are quite consistent with those obtained in the previous section from the circular galaxy distribution, Eq. (\[center\]). Combining all the processes, Eq. (\[dipole1\]), Eq.(\[dipole2\]) and Eq. (\[dipole3\]), one gets$$\mathbf{v}(universe\text{ }center)=-(\mathbf{v}(dipole\text{ }at\text{ }the\text{ }Sun)-\mathbf{v}_{p}(total))$$ where$$\mathbf{v}_{p}(total)=\mathbf{v}_{p}(Sun/Virgo)+\mathbf{v}_{p}(Virgo/GA)$$ is the total peculiar velocity of the sun towards the GA. This equation gives identical results for the center of the universe, Eq. (\[center1\])-Eq. (\[center4\]). In other words, the author has arrived at the following theorem. The Hubble flow of the solar system is given by$$v_{H}(the\text{ }Sun)=\mathbf{v}(the\text{ }cmb\text{ }dipole\text{ }at\text{ }the\text{ }Sun)-\mathbf{v}_{p}(the\text{ }total\text{ peculiar }velocity\text{ }of\text{ }the\text{ }Sun)$$ The similarity of the final results for the location of the center of the universe, Eq. (\[center1\])-Eq. (\[center2\]) and Eq. (\[center3\])-Eq. (\[center4\]), is easily understood from this theorm, since the directions of the two solutions for the GA position, Eq. (\[eqga1\]) and Eq. (\[eqga2\]), are similar and therefore the peculiar velocities from Virgo towards the GA are similar, despite a large difference between the distances to the GA. Therefore, the difference comes from the peculiar velocity for the sun towaeds the Virgo center, Eq. (\[sciama1\]) or Eq. (\[sciama2\]). In order to settle on the distance to the universe center, one has to settle on one of the two choices for the peculiar velocity, Eq. (\[sciama1\]) or Eq. (\[sciama2\]). Summary and Discussion ====================== The data compiled in RC3 is an accumulation by various groups, so that the criterion for circularity may not be uniform across observational groups. Nevertheless, the consistency between the cmb dipole data and the circular galaxy distribution for the direction of the universe center suggests consistency among the many groups that contributed to the RC3 catalog in that respect. The recent measurement by SDSS increases the data on galaxies in number and quality. It is desirable, however, to enlarge the scope of sky coverage into the southern hemisphere to get improved data for the circular galaxy distribution. Finally, the author suggests that the deformation of galaxies by cosmic tidal forces discussed in this article be taken into account for the systematic study of the triaxiality of galaxies [@triax] and asymmetric spiral galaxies [@spiral]. From statistical analysis of these galaxy deformations, one may be able to reach a more accurate determination of the new cosmological parameter, the position of the center of the universe. As for the discussion of the cmb dipole, the observed cmd dipole cannot be explained merely by the presence of the peculiar velocity of the solar system. One should be involved with the Hubble flow even only from dimensional considerations. Theorem 3 suggests an important departure from normal thinking in cosmology. If further information on the peculiar velocity of the GA supercluster is found in the future, a similar procedure should be applied to the new situation. Then, the final result for the center of the universe expansion might be changed. However, the conclusion from the circular galaxy distribution gives a stringent constraint on the distance and the direction of the universe center. There is a proposal and discussion of a rotation of the universe based on data of polarization of radio waves[@birch]. In general, an intrinsic polarization of radio waves can be related to the existence of the center of the universe, since the deformation of a galaxy by tidal forces can cause an asymmetric magnetic field that creates polarization at the source which is called intrinsic. Then, some features of the observed polarization can be explained by the mere existence of the center of the universe, but it is not clear whether all the features can be explained from that or not. If not, one can ask the question whether there exists a unified description of all the data by the center and a rotation of the universe. Certainly, any rotation axis of the universe should pass through the center in a simple picture. In fact, the angles reported in the two works are close to each other: For the center of the universe, (l, b) = (310${{}^\circ}$, 30${{}^\circ}$) –$>$ (ra, dec) = (13h17m, -32d), and for the rotation axis, (ra, dec) = (14h35m,-35d) with an error bar of (30m, 30m). The proximity of these angles would make the construction of a unified picture easier. The author would like to thank the members of the Physics Department and the Astronomy Department of the University of Michigan for useful information. Correspondence should be addressed to the author at [email protected]. [9]{} de Vaucouleurs, G. et. al., Third Reference Catalogue of Bright Galaxies, Vol I-III (Spring-Verlag, New York, 1991). Melchiorri, B. et. al., New Astron. Rev. 46, 693 (2002); Fixsen, D.J. et. al., Apj. 473, 576 (1996). Sciama, D. W., Phys. Rev. Letters 18, 1065 (1967); Rees, M. and Sciama, D. W., Nature 213, 374 (1967); Stewart, J. M. and Sciama, D. W., Nature 216, 748 (1967). Faber, S. M. and Burstein, D., in Large-Scale Motions in the Universe (ed. Rubin, V. C. and Coyne, G. V., Princeton University Press 1988) p.115; Lynden-Bell, D. et.al., Apj. 326, 19 (1988). Binney, J. & Tremaine, S., Galactic Dynamics, (Princeton University Press, Princeton,1987). Schoenmakers, R. H. M., Franx, M. and de Zeeuw, P. T., Measuring non-axisymmetry in spiral galaxies. MNRAS 292, 349-364 (1997); Schoenmakers, R. H. M., Asymmetries in Spiral Galaxies, (University Press, Veenendaal, 2000) and references therein. Birch, P., Nature 298, 451 (1982); Okunov, Yu. N., Gen. Relat. Grav. 24, 121 (1992); Noland, B. and Ralston, J. P., Phys. Rev. Lett., 78, 3043 (1997); ibid. 79, 1958 (1997); and many others. Figure Caption. Fig.1 Distribution of bright galaxies compiled in RC3 [@rc3]. (a) Galaxy distribution in galactic coordinates. (b) Velocity distribution vs. galactic latitudes. Fig.2 The distribution of apparently circular galaxies which have the value R25= 0.00: (a) In galactic coordinates. (b) In supergalactic coordinates. (c) In equatorial J2000.0 coordinates. (d) Velocity distribution vs. galactic latitudes. Fig.3 Distribution of galaxies in various ranges of the value of R25: (a) 0.01-0.10, (b) 0.11-0.20, (c) 0.21-0.40 and (d) 0.41-1.00. Fig.4(a): Circular galaxy distribution (R25 = 0) from the Hyperleda catalog (PGC 2003) in galactic coordinates (12,008 items out of a total of 983,261). Fig.4(b): Spherical plot of Fig.4(a). Table Caption. Table1. The total number of galaxies, the numbers in the north and south and the normalized north/south ratio in various ranges of R25. \[c\][lllll]{}Range of R25 & Total & North & South & North/South Normalized Ratio\ 0.00-0.00 & 1098 & 682 & 416 & 1.4071\ 0.01-0.10 & 4573 & 2330 & 2243 & 0.8916\ 0.11-0.20 & 4448 & 2322 & 2126 & 0.9374\ 0.21-0.30 & 3286 & 1714 & 1572 & 0.9358\ 0.31-0.40 & 2246 & 1148 & 1098 & 0.8974\ 0.41-0.50 & 1638 & 861 & 787 & 0.9281\ 0.51-1.00 & 4050 & 2210 & 1835 & 1.0337	ArXiv
--- abstract: 'Solid-state qubits hold the promise to achieve unmatched combination of sensitivity and spatial resolution. To achieve their potential, the qubits need however to be shielded from the deleterious effects of the environment. While dynamical decoupling techniques can improve the coherence time, they impose a compromise between sensitivity and bandwidth, since to higher decoupling power correspond higher frequencies of the field to be measured. Moreover, the performance of pulse sequences is ultimately limited by control bounds and errors. Here we analyze a versatile alternative based on continuous driving. We find that continuous dynamical decoupling schemes can be used for AC magnetometry, providing similar frequency constraints on the AC field and improved sensitivity for some noise regimes. In addition, the flexibility of phase and amplitude modulation could yield superior robustness to driving errors and a better adaptability to external experimental scenarios.' author: - Masashi Hirose - 'Clarice D. Aiello' - Paola Cappellaro bibliography: - '../../Biblio.bib' title: Continuous dynamical decoupling magnetometry --- Solid-state qubits have emerged as promising quantum sensors, as they can be fabricated in small volumes and brought close to the field to be detected. Notably, Nitrogen-Vacancy (NV) centers in nano-crystals of diamond [@Jelezko02] have been applied for high sensitivity detection of magnetic [@Taylor08; @Maze08; @Balasubramanian08] and electric fields [@Dolde11] and could be used either as nano-scale scanning tips [@Maletinsky12] or even in-vivo due their small dimensions and low cytotoxicity [@McGuinness11]. Unfortunately, solid-state qubits are also sensitive probes of their environment [@Bar-Gill12; @Bylander11] and this leads to rapid signal decay, which limits the sensor interrogation time and thus its sensitivity. Dynamical decoupling (DD) methods [@Carr54; @Viola99b; @Uhrig07; @Khodjasteh07; @Biercuk11] have been adopted to prolong the coherence time of the sensor qubits [@Taylor08; @deLange11; @Bar-Gill12; @Pham12]. Although DD techniques prevent measuring constant, DC fields, they provide superior sensitivity to oscillating AC fields, as they can increase the sensor coherence time by orders of magnitude. The sensitivity is maximized by carefully matching the decoupling period to the AC field; conversely, one can study the response of a decoupling scheme to fields of various frequencies, thus mapping out their bandwidth. Still, the refocusing power of pulsed DD techniques is ultimately limited by pulse errors and bounds in the driving power. Here we investigate an alternative strategy, based on continuous dynamical decoupling (CoDD), that has the potential to overcome these limitations. We consider the problem of measuring a small external field, coupled to the sensor by a Hamiltonian: ${{\mathcal{H}}}_{b}=\gamma b(t) S_{z}$, where $S_{z}$ is the spin operator of the quantum sensor. For example, $b(t)$ can be an external magnetic field and $\gamma$ the spin’s gyromagnetic ratio. The figure of merit for a quantum sensor is the smallest field $\delta b_{min}$ that can be read out during a total time $\mathbf{t}$, that is, the sensitivity $\eta=\delta b_{min}\sqrt{\mathbf{t}}$. We use this metric to compare pulsed and continuous DD schemes and show how CoDD can offer an advantage for some noise regimes. The principle of DD schemes rests on the spin echo sequence, which refocuses unwanted phase accumulation due to a slow bath by reversing the system evolution with control pulses. More complex DD sequences can in principle extend the coherence time indefinitely, by increasing the number of pulses. In practice, however, a large number of imperfect, finite-width pulses provokes the accumulation of error and degrades DD performance [@Khodjasteh07; @Khodjasteh05; @Wang12]. CoDD has been first introduced in the context of NMR to mitigate pulse errors [@Burum81; @Boutis03] and it has then lead to many schemes, such as composite pulses [@Shaka83b; @Levitt86], dynamically corrected gates [@Khodjasteh09] and optimized modulations [@Jones12]. In general, phase and amplitude modulation of the continuous driving allows great flexibility and CoDD can achieve high decoupling power. Here we consider only two schemes, constant continuous driving (C) and Rotary Echo (RE) [@Solomon57; @Aiello12; @Laraoui11], as their periodicity allows an easier use for AC magnetometry (see Fig. \[fig:Sequence\]); we will compare these schemes to the simplest pulsed DD scheme, period dynamical decoupling (PDD). ![Pulse sequences for four AC magnetometry schemes: PDD (P), constant driving (S), RE with optimal frequency (R$_k^{\text{opt}}$) and spin-locking (S). Blue boxes represent microwave driving, with phase (x and y) as indicated.[]{data-label="fig:Sequence"}](Sequence2){width="45.00000%"} As an example, we compute the signal and sensitivity of AC magnetometry under RE, but similar derivations apply for the other schemes. The RE sequence consists of a continuous on-resonance driving field of constant amplitude $\Omega$ and phase inverted at periodic intervals (see Fig. \[fig:Sequence\]). RE is parametrized by the angle $\theta=\Omega T/2$, where $T$ is the sequence period. While RE is usually employed to refocus errors in the driving field, for $\theta=2\pi k$ the sequence also refocus dephasing noise, with performance depending on both $k$ and the Rabi frequency. We consider the evolution of a sensor qubit under a sequence of $2\pi k$-RE and in the presence of an external AC magnetic field of frequency $\omega$ whose magnitude $b$ is to be sensed: $$\mathcal{H}(t) = \Omega \mathbb{SW}(t)S_x + \gamma b\cos(\omega t + \phi)S_z,$$ where $\mathbb{SW}(t)$ is the square wave of period $T = {4\pi k}/{\Omega}$. In the toggling frame of the driving field, the Hamiltonian becomes $$\widetilde{\mathcal{H}}(t)\!=\!\frac{\gamma b\cos(\omega t+\phi)}{2}[ \cos(\Omega t)S_z-\mathbb{SW}(t)\sin(\Omega t)S_y ].$$ We consider only the cases where $\phi=0$ and $\omega T=2m\pi$, with $m$ an odd integer, since as we show below this yields good sensitivities. Under this assumption $\tilde{\mathcal{H}}(t)$ is periodic and for small fields $b$ the evolution operator can be well approximated from a first order average Hamiltonian over the period $T$, $\overline{\mathcal{H}} \approx \frac{1}{T}\int_{0}^{T}\tilde{\mathcal{H}}(t)dt=\gamma\overline b\,S_y$. If $m = 1$, we define $\omega_{low} = \frac{\Omega}{2 k}$, which, for a fixed $\Omega$, is easily adjustable by changing the echo angle $2\pi k$. Setting instead $m = (2k-1)$, we define $\omega_{opt} = \frac{\Omega (2k-1)}{2k}$, which yields $\overline b=4bk/[\pi(4k-1)]$ and attains the best sensitivity of the method. The sensitivity, obtained as $\eta(t) = \displaystyle\lim_{b \rightarrow 0}\textstyle\frac{\Delta\mathcal{S}}{\|\frac{\partial \mathcal{S}}{\partial b}\|}\sqrt{t}$, where $\mathcal{S}$ is the signal and $\Delta\mathcal{S}$ its shot-noise limited uncertainty, depends on $\overline b$, that is, on the averaging of the AC field over the sequence period due to the DD modulation. We compare the performance of both $2\pi k$-RE schemes to PDD (optimum $\omega = {2\pi}/{t}$, $\phi = {\pi}/{2}$) and a constant modulation with $\omega=\Omega$ (see Fig. \[fig:Sequence\]). We obtain for the schemes considered: $$\renewcommand{\arraystretch}{2} \begin{array}{lclc} \eta^{opt}_{R_k}=\eta\frac{4k-1}{2k} &\quad (\theequation.a) &\qquad \eta_P=\eta&\quad (\theequation.b)\\ \eta^{low}_{R_k}=\eta\frac{4k^2-1}{2k} &\quad (\theequation.c) &\qquad \eta_C=\frac4\pi\eta&\quad (\theequation.d), \end{array}\nonumber \label{eq:sensitivity} \addtocounter{equation}{1}$$ where $\eta=\frac{\pi }{2\gamma C \sqrt{t}}$, with $C$ a parameter capturing inefficiencies in the sensor readout [@Taylor08]. Here $R_k$ labels a $2k\pi$-RE scheme, $P$ the PDD scheme and $C$ the constant modulation (see Figure \[fig:Sequence\]). A fourth operating scheme can be obtained by a “spin-locking” sequence [@Redfield55], where the spin is first rotated to the transverse plane before applying a driving field in the same direction; choosing $\phi=0$ and $\omega=\Omega$ yields the same sensitivity as for the constant modulation, $\eta_S\!=\!\eta_C$, even when the driving phase is inverted periodically. We note that if the phase $\phi$ of the AC field is not optimized, the sensitivities are reduced by a factor $\Phi(\phi)$, with $\Phi_P=\Phi_C=\csc(\phi)$ and $\Phi_{R_k}=\sec\phi$. If in addition the phase of the AC field cannot be fixed, $\Phi(\phi)=\sqrt{2}$ when considering the average signal over many realizations. These ideal sensitivities are degraded in the presence of noise and whenever the frequency of the AC field is not matched to the DD period. In the following we analyze these two contributions, showing that they lead to a sensitivity $\eta\to\eta\mathcal D(t)/W(\omega)$, where $\mathcal D(t)$ describes the decay under DD sequences and $W(\omega)$ is the reduction in the accumulated phase when the field frequency $\omega$ is suboptimal. ![Bandwidth for AC magnetometry. We plot the weight functions $W(\omega)$ that scale the phase acquired during DD magnetometry for AC fields of frequency $\omega$. Left: we plot $W(\omega)$ and the envelop of its passband decay for PDD (blue dotted), RE ($k=1$, red, thick) and constant driving (green, thin line) for $n=2$ cycles, expressing the frequency in terms of the sequence period. In the inset: we compare the main peak for $n=1$ (red, thick) and $n=10$ (gray) for RE ($k=1$) showing the reduction in bandwidth. Right: we compare $W(\omega)$ for continuous driving (green) and for RE with $k=1$ (red, thick) and $k=4$ (gray, dashed), plotting as a function of $\omega$ in units of the Rabi frequency $\Omega$.](WeightT "fig:"){width="22.00000%"}![Bandwidth for AC magnetometry. We plot the weight functions $W(\omega)$ that scale the phase acquired during DD magnetometry for AC fields of frequency $\omega$. Left: we plot $W(\omega)$ and the envelop of its passband decay for PDD (blue dotted), RE ($k=1$, red, thick) and constant driving (green, thin line) for $n=2$ cycles, expressing the frequency in terms of the sequence period. In the inset: we compare the main peak for $n=1$ (red, thick) and $n=10$ (gray) for RE ($k=1$) showing the reduction in bandwidth. Right: we compare $W(\omega)$ for continuous driving (green) and for RE with $k=1$ (red, thick) and $k=4$ (gray, dashed), plotting as a function of $\omega$ in units of the Rabi frequency $\Omega$.](WeightRabi "fig:"){width="22.00000%"} \[fig:bandwidth\] Optimal sensitivities are obtained by carefully matching the period of the DD schemes to the oscillating field. In practice, however, when field frequencies are either unknown or known to a finite precision, it is of relevance to determine the bandwidth of the scheme and the deviation from optimum sensitivities. We estimate the bandwidth by calculating the phase accumulated by the sensor over the total interrogation time $t = nT$, $\overline B t= \int_{0}^{t}b(t) f(t)\mathrm{d}t$, and examining the frequency dependence of its absolute value. For PDD, the filter function is $f_{P}(t) =\mathbb{SW}_{P}(t)$, the square wave with the period of the modulation. For continuous driving schemes such as RE and Rabi, $f(t)$ is the strength of the toggling frame Hamiltonian. In particular, $f_{R_k}(t) = \mathbb{SW}(t)\sin(\Omega t)$ yielding the weight function $W_{R_k}(\omega)=\|\overline B_{R_k}(\omega)\|/\|\overline B_{R_k}(\omega^{opt})\|$: $$W_{R_k}(\omega)\!=\!\frac{(4 k-1)/n}{\left\|(4 k)^2-\left(\frac{T \omega }{\pi }\right)^2\right\|}\left\|\sin (n T \omega ) \tan \left(\frac{T \omega }{4}\right)\right\| .$$ $W_{R_k}$ has peaks (pass-bands) at $\omega = {2\pi(2(k+p)-1)}/{T}$, where $p$ is an integer satisfying $p\geq1-k$. The lowest pass-band occurs for $p = 1-k$, corresponding to $\omega_{low} = {\Omega}/{2k}$. The strongest peak is for $p=0$ at $\omega_{opt}$. Subsequent periodic peaks are attenuated from the symmetry point $\omega = \Omega$ as $\sim \frac{\Omega^2}{\|\omega^2 - \Omega^2\|}$.The FWHM of the optimum peak in $W_{R_k}(\omega)$ decays as $\approx \frac{7.58}{2nT}$, where $7.58 \approx$ FWHM of the squared sinc function, a result common to the other DD schemes. A similar calculation for the accumulated phase during a PDD sequence indicates the existence of peaks at $\omega = {m\pi}/{T}$, with $m$ odd, whose intensity decays as $1/m$. This slower decay than for the RE pass-bands could be beneficial if the goal is to detect fields of unknown frequencies. On the other hand, AC magnetometry under continuous driving or spin locking could be used for frequency-selective detection because $W_{C}(\omega)$ has a unique peak at $\omega = {2\pi}/{T}$ with FWHM on the same order of that for RE. A comparison of the different weight functions is depicted in Fig. \[fig:bandwidth\]. We note that while $W(\omega)$ describes the poor performance of DD schemes at detecting AC fields with unmatched frequencies, this property could in turn be used for frequency-selective measurements and even spectroscopy, by scanning the sequence period. While constant driving provides the best selectivity (canceling out higher octaves), RE provides more flexibility by changing both the period time and the angle $2\pi k$, which allows more uniform noise cancellation. ![Sensitivity for AC magnetometry. We compare the magnetic field sensitivity of a single NV center for PDD (left) and RE ($k\!=\!1$ center; $k\!=\!4$ right). We assumed $T_2\!=\!500\mu$s under OU noise (comparable to a $^{13}$C bath), yielding a decay $\propto\!e^{-T^3/(n^2 T_2^3)}$, and a single readout with $C=0.03$. A larger number of refocusing cycles (with shorter periods) achieves better sensitivity but can only detect higher frequencies, as shown by the color of the curves (right bar, MHz).](SensitivityACall2){width="42.00000%"} \[fig:SensitivityACtime\] The refocusing power of RE can surpass that of pulsed schemes. Consider for example a noise with long correlation time $\tau_c$: In this limit, the signals decays as ${\left\langle \mathcal{S}_{R_k}(t)\right\rangle}=e^{-(\Gamma_{2R}t)^3/n^2}$, with $\Gamma^3_{2R}=\frac{3\sigma^2}{8 k^2 \pi ^2 \tau_c}$. Using a similar derivation [@Kubo62; @Cappellaro06], the decay under a PDD sequence is instead ${\left\langle \mathcal{S}_P(t)\right\rangle}=e^{-(\Gamma_{2P}t)^3/n^2}$, with $\Gamma^3_{2P}=\frac{2\sigma^2}{3 \tau_c}$. The sensitivities in Eq. (\[eq:sensitivity\]) are further limited by the signal decay $\mathcal D(t)$ under the DD sequences. The achievable sensitivity is then a compromise between the refocusing power of the sequence used and the frequency that it allows detecting (Fig. \[fig:SensitivityACtime\]). While the decay for pulsed DD has been widely studied, evolution under continuous DD is more complex [@Dobrovitski09]. We can estimate the RE decay to first leading order using a cumulant expansion [@Kubo62; @Cappellaro06]. We assume a stochastic Hamiltonian, ${{\mathcal{H}}}(t)=\Omega {\,\mathbb{SW}}_k(t){\sigma_x}+\delta(t){\sigma_z}$, where $\delta(t)$ is an Ornstein-Uhlenbek noise with zero mean and autocorrelation function $G(\tau)=\sigma^2 e^{-\frac{\tau}{\tau_c}}$, with $\sigma$ the dispersion and $\tau_c$ the correlation time. The signal decay can be calculated from the average of the superoperator ${\left\langle \mathbb S(t)\right\rangle}={\left\langle \mathcal T e^{-i\int_0^t\widehat Hdt'}\right\rangle}$, where we indicate by a hat the superoperators $\widehat A=A\otimes{\leavevmode\hbox{\small1\normalsize\kern-.33em1}}-{\leavevmode\hbox{\small1\normalsize\kern-.33em1}}\otimes A$ and $\mathcal T$ is the time ordering operator. In turns, this can be approximated by the cumulants, ${\left\langle \mathbb S(t)\right\rangle}\approx \exp{[-(K_1+K_2+\dots)t]}$, with the first cumulant $K_1=0$ and the second given by $$K_2=\frac1{2t}\int_0^tdt_1\,\int_0^{t_1}dt_2\,\langle{\widehat H(t_1)\widehat H(t_2)}\rangle_c,$$ where the cumulant average is $$\langle{\widehat H(t_1)\widehat H(t_2)}\rangle_c=\mathcal T\langle\widehat H(t_1)\widehat H(t_2)\rangle-\langle{\widehat H(t_1)}\rangle\langle{\widehat H(t_2)}\rangle.$$ In the toggling frame of the driving field, the stochastic Hamiltonian is $\tilde {{\mathcal{H}}}(t)=\delta(t)N(t)\equiv\delta(t)\left[\cos(\Omega t) {\sigma_z}+ {\,\mathbb{SW}}\sin(\Omega t){\sigma_y}\right]$. Then the second cumulant for $n$ cycles is $K_2\!=\!n\triangle+\square\sum_{j=1}^n(n-j) G_j$ [@Cappellaro06], with $G_j=e^{-\frac{4k\pi j}{\Omega\tau_c}}$ and $$\triangle=\int_0^{4k\pi/\Omega}dt_1\,\int_0^{t_1}dt_2 {\widehat N}(t_1){\widehat N}(t_2) G(t_1-t_2),$$ $$\square=\int_0^{4k\pi/\Omega}dt_1\,\int_0^{2k\pi/\Omega}dt_2 {\widehat N}(t_1){\widehat N}(t_2) G(t_1-t_2).$$ The cumulant can be written as $$K_2=\frac{\alpha+\beta}2\hat S_z^2+\frac{\alpha-\beta}2\hat S_y^2+\frac{\sqrt{\gamma^2-\beta^2}}2(\hat S_y\hat S_z+\hat S_z\hat S_y) \label{eq:K2}$$ (see appendix for explicit expressions), yielding the signal $${\left\langle \mathcal{S}_{R_k}\right\rangle} ={\frac{1}{2}}[1+\mathcal D_R]={\frac{1}{2}}\left[1+e^{-\alpha} ( \cosh(\gamma)+\frac{\beta}{\gamma} \sinh(\gamma))\right].$$ Numerical simulations match well with these approximate analytical results. The longer coherence time under the RE sequence can be exploited either to reach a better sensitivity for a given frequency or to measure lower frequency fields at a given sensitivity, as shown in figure \[fig:SensitivityACFreq\]. The achievable improvement depends on the effective coherence time ratio, $\tau=T_{2R}/T_{2E}$, obtained from the two schemes. Because of the improved refocusing of RE with respect to PDD, the sensitivity can be improved for some noise regimes. In addition, RE-AC magnetometry provides the flexibility of using larger angles (larger $k$) to allow for longer interrogation times (Figure \[fig:SensitivityACtime\]) at lower frequencies, which could be beneficial in practical cases in combination with repeated readout schemes [@Neumann10b; @Aiello12]. We remark that besides the decay functions obtained above in the presence of dephasing noise, other sources of decay can arise from imperfect pulses or fluctuations in the driving power. To this effect, RE provides a good protection against slow fluctuation in the driving power [@Solomon57; @Aiello12] and it is thus expected to achieve much better overall sensitivities than a continuous driving. ![Sensitivity for AC magnetometry. We compare the achievable sensitivity for constant driving (green, dash-dotted), PDD of $n=50$ echoes (blue, dotted) and RE ($2\pi$-RE, red, achieving the same sensitivity of PDD at a lower frequency and $8\pi$-RE, black, achieving better sensitivity than PDD at the same frequency). We assumed $T_2=500\mu$s under OU noise, yielding a super-exponential decay $\propto e^{-T^3/(n^2 T_2^3)}$, and a single readout with $C=0.03$. The decay of the constant (Rabi) driving was calculated following Ref. [@Dobrovitski09] for long $\tau_c$. The dashed, thin lines correspond to the ideal limit with no driving or pulse errors.[]{data-label="fig:SensitivityACFreq"}](SensitivityACFreq1){width="40.00000%"} In conclusion, we analyzed a novel scheme for AC magnetometry based on continuous dynamical decoupling and compared its performance to pulsed DD schemes. While we focused on the simplest DD sequences, we note that more complex driving, such as composite pulses [@Levitt86; @Aiello12], could achieve even better refocusing of driving field instability and inhomogeneity while still providing comparable sensitivity. We further analyzed the response of AC magnetometry to fields of unknown frequencies, finding that some CoDD schemes (such as continuous driving or spin locking with alternating phases) are advantageous for spectroscopy. The sensitivity is ultimately limited not only by the theoretically achievable coherence time, but also by pulse errors or fluctuations in the driving field. While a full comparison of the limits due to imperfection in the control fields is beyond the scope of this work, the flexibility of CoDD schemes in modulating both phase and amplitude of the driving field can provide practical advantages, yielding a better compromise between the DD refocusing power and the frequencies of the field to be measured.\ Acknowledgments This work was supported in part by the ARO through grant No. W911NF-11-1-0400 and by DARPA. C. D. A. acknowledges support from the Schlumberger Foundation. Cumulant ======== We can calculate the time (ensemble) average of a time-ordered exponential operator by means of a cumulant expansion The first cumulant is zero since we assume a zero-average as zero. The second cumulant for the RE sequence is given by Eq. \[eq:K2\] with $$\begin{array}{ll} \alpha=&\frac{\sigma ^2 T^2 \tau _c e^{-\frac{n T}{\tau _c}}} {\left(e^{\frac{T}{2 \tau _c}}+1\right)^2 \left(16 \pi ^2 k^2 \tau _c^2+T^2\right)^2} \left[2 n \left(e^{\frac{T}{2 \tau _c}}+1\right){}^2 e^{\frac{n T}{\tau _c}} \left(16 \pi ^2 k^2 T \tau _c^2+64 \pi ^2 k^2 \tau _c^3 \tanh \left(\frac{T}{4 \tau _c}\right)+T^3\right)\right.\\ &\left.-8 \tau _c e^{\frac{(n+1) T}{2 \tau _c}} \left(\left(T^2-16 \pi ^2 k^2 \tau _c^2\right)+\left(16 \pi ^2 k^2 \tau _c^2+T^2\right) \cosh \left(\frac{T}{2 \tau _c}\right)\right) \sinh \left(\frac{n T}{2 \tau _c}\right)\right]\end{array}$$ $$\begin{array}{ll} \beta=&-\frac{2 \sigma ^2 T^2 \tau _c^2 e^{-\frac{n T}{\tau _c}}} {\left(e^{\frac{T}{2 \tau _c}}+1\right)^2 \left(16 \pi ^2 k^2 \tau _c^2+T^2\right)^2}\times \\&\left[16 \pi ^2 k^2 \tau _c^2 \left(e^{\frac{T}{2 \tau _c}}-1\right) \left(e^{\frac{n T}{\tau _c}} \left((4 n-1) e^{\frac{T}{2 \tau _c}}+4 n+1\right)+e^{\frac{T}{2 \tau _c}}-1\right)+T^2 \left(e^{\frac{T}{2 \tau _c}}+1\right){}^2 \left(e^{\frac{n T}{\tau _c}}-1\right)\right]\end{array}$$ $$\begin{array}{ll} \gamma=&-\frac{2 \sigma ^2 T^2 \tau _c^2 e^{-\frac{n T}{\tau _c}} } {\left(e^{\frac{T}{2 \tau _c}}+1\right)^2 \left(16 \pi ^2 k^2 \tau _c^2+T^2\right)^2} \left[64 \pi ^2 k^2 T^2 \tau _c^2 \left(e^{\frac{T}{2 \tau _c}}+1\right)^4 \left(e^{\frac{n T}{\tau _c}}-1\right)^2 \tanh\left(\frac{T}{4 \tau _c}\right)^2+\right.\\ &\left.\left(16 \pi ^2 k^2 \tau _c^2 \left(e^{\frac{T}{2 \tau _c}}-1\right) \left(e^{\frac{n T}{\tau _c}} \left((4 n-1) e^{\frac{T}{2 \tau _c}}+4 n+1\right)+e^{\frac{T}{2 \tau _c}}-1\right) + T^2 \left(e^{\frac{T}{2 \tau _c}}+1\right)^2 \left(e^{\frac{n T}{\tau _c}}-1\right)\right)^2\right]^{1/2}\end{array}$$ [32]{}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 [**, ()](\doibase 10.1063/1.1507838) [, ()](\doibase 10.1038/nphys1075) [, ()](\doibase 10.1038/nature07279) [, ()](\doibase 10.1038/nature07278) [, ()](\doibase 10.1038/nphys1969) [, ()](\doibase 10.1038/nnano.2012.50) [, ()](\doibase 10.1038/nnano.2011.64) [, ()](\doibase 10.1038/ncomms1856) [, ()](\doibase 10.1038/NPHYS1994) [, ()](\doibase 10.1103/PhysRev.94.630) [, ()](\doibase 10.1103/PhysRevLett.82.2417) [, ()](\doibase 10.1103/PhysRevLett.98.100504) [, ()](\doibase 10.1103/PhysRevA.75.062310) [, ()](\doibase 10.1088/0953-4075/44/15/154002) [, ()](\doibase 10.1103/PhysRevLett.106.080802) @noop [ ()]{}, [, ()](\doibase 10.1103/PhysRevLett.95.180501) [, ()](\doibase 10.1103/PhysRevB.85.085206) [, ()](\doibase 10.1016/0022-2364(81)90200-6) [, ()](\doibase 10.1016/S1090-7807(03)00010-7) [, ()](\doibase 10.1016/0022-2364(83)90035-5) [, ()](\doibase 10.1016/0079-6565(86)80005-X) [, ()](\doibase 10.1103/PhysRevA.80.032314) [ ()](http://arxiv.org/abs/1205.2402) [, ()](\doibase 10.1103/PhysRev.110.61) @noop [, ()](\doibase 10.1103/PhysRevB.84.161403) [, ()](\doibase 10.1103/PhysRev.98.1787) [, ()](\doibase 10.1143/JPSJ.17.1100) [, ()](\doibase 10.1063/1.2216702) [, ()](\doibase 10.1103/PhysRevLett.102.237601) [**, ()](\doibase 10.1126/science.1189075)	ArXiv
--- author: - Denija Crnojević bibliography: - 'biblio.bib' title: Resolved Stellar Populations as Tracers of Outskirts --- The Importance of Haloes {#intro} ======================== Our understanding of galaxy formation and evolution has dramatically evolved in the past fifty years. The first and simplest idea for the formation scenario of our own Milky Way (MW) Galaxy was put forward by [@eggen62], who proposed the bulk of a stellar halo to be formed in a rapid collapse of gas in the protogalaxy. This scenario, often referred to as “monolithic” collapse, is a dissipative process and takes place on dynamical timescales of the order of $\sim10^8$yr. This process gives birth to a metal-poor stellar component in the halo outer regions, while the inner regions ends up being more metal-rich due to the reprocessing of the gas as it collapses deeper into the protogalaxy potential well. This idea was later challenged by an alternative explanation, based on the observation that globular clusters (GCs) at different Galactocentric distances have a wide range of metallicities. In this scenario, the halo is formed on longer timescales ($\sim10^9$yr) and, instead of being a self-contained system, it comes together as the product of several protogalactic fragments (@searle78). These fragments can be pre-enriched before they are accreted. While both scenarios are capable of explaining many observed quantities of the Galactic halo, they cannot individually give a comprehensive picture (@norris91 [@chiba00]), which has led to the development of hybrid “two-phase” models. In the latter, the inner Galaxy regions are formed in a first phase as a result of a monolithic-like process, while the outer halo regions are built up over the Galaxy’s lifetime through dissipationless accretion events (@freeman02). In the past couple of decades, the most widely accepted paradigm of the hierarchical Lambda-Cold Dark Matter ($\rm \Lambda$CDM) structure formation model has prevailed, favouring the predominance of merger and accretion events in the build-up of galactic haloes (@white91 [@bullock05; @springel06; @johnston08]). These models predict the ubiquitous presence of haloes, which are characterized by old and metal-poor populations and often shows signs of recent interactions, in contrast with the smooth haloes predicted by dissipative models (@bullock05 [@abadi06; @font06]). The interaction events provide a mine of information on the assembly of haloes: dynamical timescales become relatively long (up to several Gyr) in the outer regions of a galaxy, and thus accretion/merger events that occurred a long time ago are often still visible as coherent structures like disrupting galaxies or streams, which readily testify the past assembly history of their host. The assembly itself depends on a variety of factors, such as number, mass, stellar content and structural properties of the accreted satellites, as well as orbital properties, timing and energy of the accretion event. Even when the progenitor is completely dissolved in the host’s halo (which is particularly true in the inner halo regions where dynamical timescales are relatively short), its stripped stellar content still retains a characteristic coherence in velocity space as well as in metallicity content, thus giving important clues about the progenitor’s properties. Observing the stellar “fossils” that populate galaxy haloes thus offers a unique opportunity to reconstruct the modes, timing, and statistics of the halo formation process. Besides being taletellers of their host system’s merger history, the shape and size of haloes also hold vital clues to the process of galaxy formation. In particular, they can teach us about the primordial power spectrum of density fluctuations at the smallest scales; about the reionization process, that shall lead to faint and concentrated haloes for an early suppression of star formation in low-mass dark matter (DM) subhaloes; or about the triaxiality of DM haloes, which are predicted to be more flattened for dissipationless formation scenarios (@abadi06). Despite only accounting for a mere $\sim1\%$ of a galaxy’s total mass (e.g., @morrison93), extended haloes are clearly extremely valuable to test and refine theoretical predictions on the halo assembly process. Due to their extreme faintness, however, haloes have not been as fully exploited as they should have been as key tests of galaxy formation models: they are not easily detected above the sky level, i.e., surface brightness values of $\mu_V\sim25$magarcsec$^{-2}$, posing a serious observing challenge to their investigation. Cosmological simulations predict the majority of past and ongoing accretion events to have surface brightness values well below this value (e.g., @bullock05). According to some models, reaching a surface brightness of $\mu_V\sim29$magarcsec$^{-2}$ should allow the detection of at least one stream per observed galaxy (@johnston08 [@cooper10]). How is it then possible to extract the information locked in the faint outskirts of galaxies? Resolved Stellar Populations {#rsp} ---------------------------- The best method to study faint haloes and their substructure in nearby galaxies is to resolve individual stars. Even when sparse and faint, resolved stars can be individually counted, and a stellar number density can easily be converted into a surface brightness value. When the Galactic extinction presents a high degree of spatial inhomogeneity (possibly mimicking faint irregular substructures), and the sky level is higher than the integrated light signal coming from extremely faint sources, resolved populations provide a very powerful means to trace them. This method is not free from complications: there will always be contamination coming both from foreground Galactic stars as well as from background unresolved galaxies. This can be accounted for statistically, by observing “field” regions away from the main target and quantifying the contaminants, while a direct confirmation of a star’s membership requires spectroscopy. At the same time, resolving individual stars poses constraints on the inherent nature and on the distance of the putative targets: for systems where the stellar density is so high that stars fall on top of each other on the sky, the “crowding” prevents the resolution of individual objects. This can of course occur also in the case of a relatively sparse galaxy which has a large line-of-sight distance, so that the stars are packed in a small region of the sky. Distance is also the principal enemy of depth: the larger the distance, the brighter the detection limit, i.e., the absolute magnitude/surface brightness that we can reach for a fixed apparent magnitude. Nonetheless, resolved stellar populations are able to deliver powerful information for galaxies located within $\sim10$Mpc, i.e., within the so-called Local Volume. The discovery of the Sagittarius dwarf galaxy by @ibata94 from the identification of a comoving group of stars opened the door to the era of halo studies and their substructure: a galaxy resembling the properties of classical dwarf spheroidals was clearly in the process of being disrupted by its giant host, our own MW. This evidence was the first to support theoretical predictions for the hierarchical assembly models and the existence of observable accretion events. Soon thereafter, stellar density maps allowed the discovery of a prominent low surface brightness stream around the MW’s closest giant spiral Andromeda (M31), the so-called Giant Stellar Stream (@ibata01). This feature, invisible to the naked eye, is a clear example of the elusive nature of haloes and their substructure: the surface brightness of the Giant Stellar Stream is $\mu_V\sim30$magarcsec$^{-2}$, which is prohibitive for integrated light images. As challenging as it is, the mere detection of haloes and their substructures is not enough to provide quantitative constraints on models of galaxy evolution. From the stars’ photometry and thus position in the colour-magnitude diagram (CMD), i.e., the observational counterpart of the Hertzsprung-Russel diagram, it is possible to characterize the properties of the considered stellar system. First and foremost, in contrast to integrated light, accurate distance measurements can be obtained from CMD features that act as standard candles, e.g., the luminosity of the tip of the red giant branch (TRGB) or of the horizontal branch (HB). Another key advantage of resolved populations is the possibility to constrain ages and metallicities more tightly than with integrated light alone. The CMD is used to quantify the star formation rate as a function of lookback time, and thus derive the star formation history (SFH) of a composite stellar population (e.g., @gallart05, and references therein). Spectroscopy of individual stars is the ultimate method to constrain their metallicity content and kinematical properties, such as radial velocity and proper motion, which allows for the full six-dimensional phase space to be investigated. The latter cannot, for the moment, be achieved beyond the LG limits, and still only occasionally for M31. Besides giving precious insights into galaxy haloes and their accretion histories, resolved stellar populations can help us characterizing the “surviving” low-mass galaxies that have not been accreted to date and reside in the outskirts of giant hosts. The Low-mass End of the Galaxy Luminosity Function {#satellites} -------------------------------------------------- The low-mass end of the galaxy luminosity function (LF) is of no less interest than haloes themselves. Besides the MW and M31, the Local Group (LG) contains tens of smaller galaxies which can be studied in detail due to their proximity (see @tolstoy09 for a review). While the $\rm \Lambda$CDM cosmological model has provided a convincing match to the large-scale structures observed in the high-redshift Universe, it falls short at the smallest, galactic scales, indicating an incomplete understanding of the physics involved in the evolution of galaxies: for example, the “missing-satellite problem” has been highlighted for the first time by @moore99 and @klypin99. Briefly, the number of DM subhaloes predicted in simulations exceeds the observed number of MW satellites by almost two orders of magnitude. The shape of the DM profile in the innermost regions of dwarf galaxies is also a matter of debate (the “cusp-core” problem; @walker11). In addition, the more massive among the MW satellites are less dense than what is expected from simulations, which is puzzling because they should be affected by fewer observational biases than their smaller, sparser siblings (the “too-big-to-fail” problem; @boylan11). In addition, the fact that many of the MW and M31 satellites are distributed along planes does not have a straightforward explanation in $\rm \Lambda$CDM models (e.g., @pawlowski14). From the theoretical point of view, the inclusion of baryonic physics in DM-only simulations is key to reconcile predictions with observations of the smallest galaxies. In particular, effects such as supernova feedback, stellar winds, cosmic reionisation, and tidal/ram pressure stripping all concur to reduce star formation efficiency in the least massive DM haloes. Tremendous progress is being made on this front, taking into account realistic physics as well as increasing the resolution of simulations (e.g., @stinson09 [@brooks13; @sawala16; @wetzel16]). At the same time, new observational discoveries keep offering intriguing challenges at the smallest galactic scales, as further described in Sect. \[mw\_sats\] and \[m31\_sats\]. Local Group {#sec:lg} =========== The galaxies closest to us give us the most detailed information because of the large number of stars that can be resolved. Here, I will summarize what we have learnt in the past two decades about our own Galaxy (even though an extensive picture of the MW outskirts goes beyond the scope of this contribution and can be found in Figueras, this volume), about its closest spiral neighbour M31 and about their lower-mass satellites. Milky Way {#mw} --------- The MW is traditionally divided into discrete components, i.e., the bulge, the disks (thin and thick) and the halo. The spheroidal portion of the MW is given by the central bulge, which consists mainly of metal-rich populations, and an extended diffuse component, which has a lower mean metallicity. Overall, stars and GCs in the halo have ages $\sim11-13$Gyr (@carollo07). The halo can be further deconstructed into an inner halo and an outer halo, even though the distinction could partly arise from observational biases (@schoenrich14). The inner and outer haloes also seem to have different chemical composition (\[Fe/H\]$\sim-1.6$ and \[Fe/H\]$\sim-2.2$, respectively; @ryan91 [@carollo07]). According to simulations, the two halo components should also have formed on different timescales: the inner halo ($<20$kpc) is partly constituted by early-formed in-situ stars, partly due to a violent relaxation process, and partly assembled from early, massive merging events that provide metal-rich populations (@abadi06 [@font11; @tissera13; @pillepich14; @cooper15]); the outer halo is assembled more recently, with its mass beyond $\sim30$kpc being mainly accreted in the past $\sim8$Gyr (@bullock05 [@cooper10]). These predictions are, however, still not sufficient at a quantitative level, and unconstrained as to the exact ratio of accreted stars versus in-situ populations. At the same time, observations of the MW halo with better statistics and precision are needed to inform them. Our position within the MW puts us at a clear disadvantage for global studies of its outskirts: the distant and sparse halo stars are observed from within the substantial disk component, which produces contamination both in terms of extinction and numerous disk stars along the line of sight, which completely “obscure” the sky at low Galactic latitudes. Nonetheless, thanks to the advent of wide-field imagers, the past two decades have revolutionized the large scale view of our Galaxy. Several stellar tracers can be used to dig into the MW halo at different distance ranges: old main sequence turnoff (MSTO) stars are identified mostly out to $\sim20$kpc, brighter RGB stars out to $\sim40-50$kpc, while RR Lyrae and blue horizontal branch (BHB) stars can be detected out to 100kpc. Spatial clustering of these stellar components indicate non-mixed substructure, which is often confirmed to be kinematically coherent. ### The Emergence of Streams {#mw_streams} ![Spatial density of SDSS stars around the Galactic cap, binned in $0.5\times0.5$deg$^2$; the colour scale is such that blue indicates the nearest stars while red is for the furthest ones. Labelled are the main halo substructures, which are in some cases streams associated with a GC or a dwarf galaxy; the circles show some newly discovered dwarf satellites of the MW. Plot adapted from @belokurov06a (http://www.ast.cam.ac.uk/$\sim$vasily/sdss/field\_of\_streams/dr6/) []{data-label="fig:fos"}](fos_dr6_marked.eps) ![Stellar density maps of the whole PanSTARRS footprint, obtained by selecting MSTO stars at a range of heliocentric distances (as indicated in each panel). The map is on a logarithmic scale, with darker areas indicating higher stellar densities. The many substructures are highlighted in each panel. Reproduced from [@bernard16], their Fig. 1, with permission of MNRAS[]{data-label="fig:panstarrs"}](f1LR.eps){width="12cm"} After the cornerstone discovery of the disrupting Sagittarius dwarf, it became clear that substructure is not only present in the MW halo, but it also might constitute a big portion of it. To put it in S. Majewski’s words (@majewski99a), > “There is good reason to believe that within a decade we will have a firm handle on the contribution of satellite mergers in the formation of the halo, as we move observationally from serendipitous discoveries of circumstantial evidence to more systematic surveys for the fossils left behind by the accretion process” . In the following decade, several stream-like features have indeed emerged from a variety of multi-band photometric and spectroscopic surveys indeed, and the Sloan Digital Sky Survey (SDSS) proved to be an especially prolific mine for such discoveries around the northern Galactic cap. The Sagittarius stream has been traced further, including in the Galactic anti-centre direction (e.g., @mateo98 [@majewski03]), and independent substructures have been uncovered (@ivezic00 [@yanny00; @newberg02; @grillmair06; @juric08]), most notably the Monoceros ring, the Virgo overdensity, the Orphan stream and the Hercules-Aquila cloud. Some of these have later been confirmed to be coherent with radial velocities (@duffau06). Note that most of these substructures are discovered at Galactocentric distances $>15$kpc, while the inner halo is smooth due to its shorter dynamical timescales. During the past decade, one of the most stunning vizualisations of the ongoing accretion events in the MW halo was provided by the Field of Streams (@belokurov06a), reproduced in Fig. \[fig:fos\]. The stunning stellar density map is derived from SDSS data of stars around the old MSTO at the distance of Sagittarius, with a range of magnitudes to account for a range in distances. This map not only shows the Sagittarius stream and its distance gradient, but also a plethora of less massive streams, as well as an abundance of previously unknown dwarf satellites (see Sect. \[mw\_sats\]). The Field of Streams has been now complemented with results from the latest state-of-the-art surveys, most notably the all-sky Panoramic Survey Telescope and Rapid Response System (PanSTARRS), which covers an area significantly larger than that of SDSS. In Fig. \[fig:panstarrs\] the first stellar density maps from PanSTARRS are shown, obtained in a similar way as the Field of Streams (@bernard16). The map highlights the fact that the deeper and wider we look at the Galaxy halo, the more substructures can be uncovered and used to constrain its past accretion history and the underlying DM halo properties. From this kind of maps, for example, the halo stellar mass that lies in substructures can be estimated, amounting to $\sim2-3 \times 10^8\,M_\odot$ (see @belokurov13). Using SDSS, [@bell08] also highlight the predominant role of accretion in the formation of the MW’s halo based on MSTO star counts, adding to up to $\sim40\%$ of the total halo stellar mass (note that, however, different tracers could indicate much smaller values; e.g., @deason11). Many of the known halo streams arise from tidally disrupting GCs, of which Palomar 5 is one of the most obvious examples (@odenkirchen01). This demonstrates the possible role of GCs, besides dwarf satellites, in building up the halo stellar population, and additionally implies that some of the halo GCs may be stripped remnants of nucleated accreted satellites (see @freeman02, and references therein). In order to discern between a dwarf or a cluster origin of halo stars, we need to perform chemical “tagging”, i.e., obtain spectroscopic abundances for tens of elements for these stars (e.g., @martell10): stars born within the same molecular cloud will retain the same chemical composition and allow us to trace the properties of their birthplace. A number of ambitious ongoing and upcoming spectroscopic surveys (SEGUE, APOGEE, Gaia-ESO, GALAH, WEAVE, 4MOST) is paving the path for this promising research line, even though theoretical models still struggle to provide robust predictions for the fraction of GC stars lost to the MW halo (e.g., @schiavon16, and references therein). ### The Smooth Halo Component {#mw_halo} Once the substructure in the halo is detected, it is important that it is “cut out” in order to gain insights into the smooth, in-situ stellar component (note that, however, the latter will inevitably suffer from residual contamination from accreted material that is now fully dissolved). The stellar profile of the Galactic halo is, in fact, not smooth at all: several studies have found a break at a radius $\sim25$kpc, with a marked steepening beyond this value (@watkins09 [@sesar13]), in qualitative agreement with halo formation models. Some of the explanations put forward suggest that a density break in the halo stellar profile is the likely consequence of a massive accretion event, corresponding to the apocentre of the involved stars (@deason13). The kinematics of halo stars, of GCs and of satellite galaxies, as well as the spatial distribution of streams and tidal features in satellites, can be further used as mass tracers for the DM halo. The total MW mass is to date still poorly constrained, given the difficulty of evaluating it with a broad range of different tracers. The general consensus is for a virial mass value of $\sim1.3\pm0.3 \times 10^{12}\,M_\odot$, even though values discrepant up to a factor of two have recently been suggested (see @bland16 for a compilation of estimates). Besides providing estimates for the total MW mass, studies of SDSS kinematical data, of the Sagittarius stream and of GCs tidal streams have provided discording conclusions on the shape of the MW DM halo: nearly spherical from the modelling of streams or strongly oblate from SDSS kinematics at Galactocentric distances $<20$kpc, while nearly spherical and oblate based on stream geometry or prolate from kinematical arguments for distances as large as $\sim100$kpc (see @bland16 for details). These constraints need a substantial improvement in the future to be able to inform cosmological models: the latter predict spherical/oblate shapes once baryons are included in DM-only flattened haloes (see @read14). ### Dwarf Satellites {#mw_sats} As mentioned above, the SDSS has revolutionized our notions of dwarf satellites of the MW. Bright enough to be easily recognized on photographic plates, a dozen “classical” MW dwarf satellites has been known for many decades before the advent of wide-field surveys (@mateo98 [@grebel00]). Starting with the SDSS, an entirely new class of objects has started to emerge with properties intermediate between the classical dwarfs and GCs (see @willman10, and references therein). The so-called ultra-faint satellites have magnitudes higher than $M_V\sim-8$ and surface brightness values so low that the only way to find them is to look for spatial overdensities of resolved main sequence/BHB stars. Their discovery ten years ago doubled the number of known MW satellites and revealed the most DM-dominated galaxies in the Universe, with mass-to-light ratios of up to several times $10^3\,M_\odot/L_\odot$ (@simon07). More recently, the interest in the low end of the galaxy LF has been revitalized once again with deep, wide-field surveys performed with CTIO/DECam, VST/Omegacam, and PanSTARRS: these have led to the discovery of more than 20 southern dwarfs in less than two years (@bechtol15 [@koposov15; @kim15a; @drlica16; @torrealba16], and references therein). Some of these discoveries represent extremes in the properties of MW satellites, with surface brightness values as low as $\sim30$magarcsec$^{-2}$, total luminosities of only a few hundred $L_\odot$ and surprisingly low stellar density regimes. One of the perhaps most intriguing properties of the newly discovered dwarfs is that many of them appear to be clustered around the Large Magellanic Cloud (LMC): this might be the smoking gun for the possible infall of a group of dwarfs onto the MW, which is predicted by simulations (@donghia08 [@sales16]). Low-mass galaxies are expected to have satellites on their own and to provide a large fraction of a giant galaxy’s dwarf companions (e.g., @wetzel15). The properties of the possible LMC satellites will give us a glimpse onto the conditions of galaxy formation and evolution in an environment much different from the LG as we know it today. These faintest galaxies, or their accreted and fully dispersed counterparts, are also excellent testbeds to look for the very most metal-poor stars and to investigate the star formation process in the early stages of the Universe (e.g., @frebel15). The study of the lowest mass galaxies holds the promise to challenge our knowledge of galaxy physics even further and pushes us to explore unexpected and exciting new limits. M31 (Andromeda) {#sec:m31} --------------- Our nearest giant neighbour has received growing attention in the past decade. Having a remarkable resemblance with the MW and a comparable mass (e.g., @veljanoski14), it is a natural ground of comparison for the study of spiral haloes. In terms of a global perspective, the M31 halo is arguably known better than that of the MW: our external point of view allows us to have a panoramic picture of the galaxy and its surrounding regions. The other side of the medal is that, at a distance of $\sim780$kpc, we can only resolve the brightest evolved stars in M31, and we are mostly limited to a two-dimensional view of its populations. Its proximity also implies a large angular size on the sky, underlining the need for wide field-of-view imagers to cover its entire area. At the distance of M31, ground-based observations are able to resolve at best the uppermost $\sim3-4$ magnitudes below the TRGB, which is found at a magnitude $i\sim21$. The RGB is an excellent tracer for old ($>1$Gyr) populations, but suffers from a degeneracy in age and metallicity: younger, metal-rich stars overlap in magnitude and colour with older, metal-poor stars (@koch06). Despite this, the RGB colour is often used as a photometric indicator for metallicity, once a fixed old age is assumed (@vandenberg06 [@crnojevic10]). This assumption is justified as long as a prominent young and intermediate-age population seems to be absent (i.e., as judged from the lack of luminous main sequence and asymptotic giant branch, AGB, stars), and it shows very good agreement with spectroscopic metallicity values where both methods have been applied. The very first resolved studies of M31’s halo introduced the puzzling evidence that the M31 halo stellar populations along the minor axis have a higher metallicity than that of the MW at similar galactocentric distances (e.g., @mould86). This was further confirmed by several studies targeting projected distances from 5 to 30kpc and returning an average value of \[Fe/H\]$\sim-0.8$: in particular, [@durrell01] study a halo region at a galactocentric distance of $\sim20$kpc and underline the difference between the properties of M31 and of the MW, suggesting that our own Galaxy might not represent the prototype of a typical spiral. In fact, it has later been suggested that the MW is instead fairly atypical based on its luminosity, structural parameters and the metallicity of its halo stars when compared to spirals of similar mass (@hammer07). This result was interpreted as the consequence of an abnormally quiet accretion history for the MW, which apparently lacked a major merger in its recent past. The wide-area studies of M31’s outskirts were pioneered $\sim15$ years ago with an Isaac Newton Telescope survey mapping $\sim40$deg$^2$ around M31, reaching significantly beyond its disk out to galactocentric distances of $\sim55$kpc (@ibata01 [@ferguson02]). As mentioned before, the southern Giant Stream was first uncovered with this survey, and the halo and its substructures could be studied with a dramatically increased detail. A metal-poor halo component (\[Fe/H\]$\sim-1.5$) was finally uncovered for regions beyond 30kpc and out to 160kpc (@irwin05 [@kalirai06; @chapman06]), similar to what had been observed for the MW both in terms of metallicity and for its stellar density profile. These studies do not detect a significant gradient in metallicity across the covered radial range. Nonetheless, the properties of the inner halo remained a matter of debate: while [@chapman06] found a metal-poor halo population within 30kpc above the disc, [@kalirai06] analysed a kinematically selected sample of stars within 20kpc along the minor axis and derived a significantly higher value of \[Fe/H\]$\sim-0.5$. At the same time, [@brown06] used deep, pencil beam [Hubble Space Telescope]{} ([HST]{}) pointings in M31’s inner halo to conclude that a significant fraction of its stellar populations have an intermediate age with an overall high metallicity. These results were later interpreted by [@ibata07] in light of their wider-field dataset: the samples from [@kalirai06] and [@brown06] are simply part of regions dominated by an extended disc component and with a high contamination from various accretion events, respectively. This underlines, once again, the importance of wide-field observations to reach a global understanding of halo properties. ![Stellar density maps of metal-poor RGB populations at the distance of M31, as derived from the PAndAS survey. The large circles lie at projected radii of 150kpc and 50kpc from M31 and M33, respectively. [Upper panel]{}: The Andromeda satellites are visible as clear overdensities and are marked with circles. The vast majority of them were uncovered by the PAndAS survey. Reproduced by permission of the AAS from [@richardson11], their Fig. 1. [Lower panel]{}: The main substructures around M31 are highlighted, showcasing a broad range of morphologies and likely progenitor type. Tidal debris is also present in the vicinities of the low-mass satellites M33 and NGC 147, indicating an ongoing interaction with M31. Reproduced by permission of the AAS from [@lewis13], their Fig. 1[]{data-label="pandas1"}](apj384661f1_hr.jpg "fig:"){width="7.5cm"} ![Stellar density maps of metal-poor RGB populations at the distance of M31, as derived from the PAndAS survey. The large circles lie at projected radii of 150kpc and 50kpc from M31 and M33, respectively. [Upper panel]{}: The Andromeda satellites are visible as clear overdensities and are marked with circles. The vast majority of them were uncovered by the PAndAS survey. Reproduced by permission of the AAS from [@richardson11], their Fig. 1. [Lower panel]{}: The main substructures around M31 are highlighted, showcasing a broad range of morphologies and likely progenitor type. Tidal debris is also present in the vicinities of the low-mass satellites M33 and NGC 147, indicating an ongoing interaction with M31. Reproduced by permission of the AAS from [@lewis13], their Fig. 1[]{data-label="pandas1"}](apj453276f1_hr.jpg "fig:"){width="6.8cm"} ![Stellar density map of M31 (akin to Fig. \[pandas1\]), this time subdivided into photometric metallicity bins (as indicated in each subpanel). The [upper]{} panels show high metallicity cuts, where the Giant Stream and Stream C are the most prominent features; note that the shape of the Giant Stream changes as a function of metallicity. The [lower]{} panels show lower metallicity cuts: the [lower left]{} panel is dominated by substructure at large radii, while the most metal-poor panel ([lower right]{}) is smoother and believed to mostly contain in-situ populations. Reproduced by permission of the AAS from [@ibata14], their Fig. 9.[]{data-label="pandas2"}](apj488754f9_hr.jpg){width="12cm"} The M31 INT survey was further extended out to $150$kpc (200kpc in the direction of the low-mass spiral M33) with the Canada-France-Hawaii Telescope/Megacam and dubbed Pan-Andromeda Archaeological Survey (PAndAS; @ibata07 [@mcconnachie09]). This survey contiguously covered an impressive 380deg$^2$ around M31, reaching 4mag below the TRGB. The PAndAS RGB stellar density map (see Fig. \[pandas1\]) is a striking example of an active accretion history, with a copious amount of tidal substructure at both small and large galactocentric radii. PAndAS also constituted a mine for the discovery of a number of very faint satellites and GCs (see below; @richardson11 [@huxor14; @martin16]). Fig. \[pandas2\] further shows the RGB stellar map broken into bins of photometric metallicity. The parallel Spectroscopic and Photometric Landscape of Andromeda’s Stellar Halo (SPLASH) survey (@guha06 [@kalirai06]) provides a comparison dataset with both photometric and spectroscopic information, the latter obtained with Keck/DEIMOS. The SPLASH pointings are significantly smaller than the PAndAS ones but strategically cover M31 halo regions out to $\sim225$kpc. Deeper, pencil-beam photometric follow-up studies have further made use of the [HST]{} to target some of the substructures uncovered in M31’s outskirts, resolving stars down to the oldest MSTO (e.g., @brown06 [@bernard15]). These observations reveal a high complexity in the stellar populations in M31, hinting at a high degree of mixing in its outskirts. Overall, M31 has evidently had a much richer recent accretion history than the MW (see also @ferguson16). ### Streams and Substructures {#m31_streams} As seen from the maps in Figs. \[pandas1\] and \[pandas2\], while the inner halo has a flattened shape and contains prominent, relatively metal-rich substructures (e.g., the Giant Stream), the outer halo ($>50$kpc) hosts significantly less extended, narrow, metal-poor tidal debris. The features in the innermost regions of M31 can be connected to its disk populations (e.g., the north-east structure or the G1 clump): kinematic studies show that a rotational component is present in fields as far out as 70kpc, and they retain a fairly high metallicity (@dorman13). This reinforces the possible interpretation as a vast structure, which can be explained as disk stars torn off or dynamically heated due to satellite accretion events. Deep [HST]{} pointings of these features indeed reveal relatively young populations, likely produced from pre-enriched gas in a continuous fashion, comparable to the outer disk (@ferguson05 [@brown06; @bernard15]). The most prominent feature in M31’s outer halo, the Giant Stream, was initially thought to originate from the disruption of either M32 or NGC 205, the two dwarf ellipticals located at only $\sim25-40$kpc from M31’s centre (@ibata01 [@ferguson02]). While both these dwarfs shows signs of tidal distortion, it was soon clear that none of them could produce the vast structure extending $\sim100$kpc into M31’s halo. Great effort has been spent into mapping this substructure both photometrically and spectroscopically, in order to trace its orbit and define its nature: a gradient in its line-of-sight distance was first highlighted by [@mcconnachie03], who found the outer stream regions to be located behind M31, the innermost regions at about the distance of M31, and an additional stream component on the opposite (northern) side of M31 to be actually in front of M31. The stream presents a metallicity gradient, with the core regions being more metal-rich and the envelope more metal-poor (see also Fig. \[pandas2\]), as well as a very narrow velocity dispersion, with the addition of a puzzling second kinematic component (@gilbert09); possible interpretations for the latter may be a wrap or bifurcation in the stream, as well as a component from M31’s populations. A number of increasingly sophisticated theoretical studies have tried to reproduce the appearance of the Giant Stream and picture its progenitor, which is undetected to date. The general consensus seems to be that a relatively massive ($\sim10^9\,M_\odot$) satellite, possibly with a rotating disk, impacted M31 from behind with a pericentric passage around $1-2$Gyr ago (most recently, @fardal13 [@sadoun14]). In particular, simulations can reproduce the current extension and shape of the stream and predict the progenitor to be located to the north-east of M31, just beyond its disk (@fardal13). This study also concludes that some of the substructures linked to M31’s inner regions are likely to have arisen from the same accretion event, i.e., the north-east structure and the G1 clump (Fig. \[pandas1\]): these shelf features would trace the second and third passage around M31, which is also supported by their radial velocities. CMDs of the Giant Stream populations are in agreement with these predictions: its stellar populations have mixed properties, consistent with both disk and stream-like halo features (@ferguson05 [@richardson08]). Detailed reconstruction of its SFH indicate that most star formation occurred at early ages, and was possibly quenched at the time of infall in M31’s potential (around 6Gyr ago) (@bernard15). Again, these studies deduce a likely origin of these populations as a dwarf elliptical or a spiral bulge. Besides the Giant Stream, the only other tidal feature with a relatively high metallicity is Stream C (see Fig. \[pandas1\] and \[pandas2\]), which appears in the metal-poor RGB maps as well. The origin of this feature is obscure, even though it is tempting to speculate that it could be part of the Giant Stream event. The lower left panel of Fig. \[pandas2\], showing metal-poor populations, encompasses all of the narrow streams and arcs beyond 100kpc, which extend for up to several tens of kpc in length. All these substructures are extremely faint ($\mu_V\sim31.5$magarcsec$^{-2}$), and their origin is mostly unknown because of the difficulty in following up such faint and sparse populations. As part of the [HST]{} imaging of these features, [@bernard15] find that their populations are mainly formed at early ages and undergo a more rapid chemical evolution with respect to the disk populations. Despite the metal-poor nature of these features, the hypothesis of a single accretion event producing most of the tidal features observed in the outer halo is not that unlikely, given the metallicity gradient present in the Giant Stream itself. An efficient alternative to investigate the nature of these streams is to study the halo GC population: the wide-field surveys of M31 have allowed to uncover a rich population of GCs beyond a radius of $\sim25$kpc (e.g., @huxor14, and references therein), significantly more numerous than that of the MW halo. [@mackey10] first highlighted a high spatial correlation between the streams in M31’s halo and the GC population, which would be extremely unlikely in a uniform distribution. Following the hypothesis that the disrupting satellites might be providing a high fraction of M31’s halo GCs, [@veljanoski14] obtained spectroscopic follow-up: they were able to confirm that streams and GCs often have correlated velocities and remarkably cold kinematics. This exciting result gives hope for studies of more distant galaxies, where halo populations cannot be resolved and GCs could be readily used to trace possible substructure. ### Smooth Halo {#m31_halo} One of the first spatially extended datasets to investigate the halo of M31 in detail is described in [@tanaka10]: their Subaru/SuprimeCam photometry along the minor axis in both directions are deeper, even though less extended, than PAndAS. The stellar density profile derived in this study extends out to 100kpc and shows a consistent power law for both directions. The authors also suggest that, given the inhomogeneities in the stellar populations, the M31 halo is likely not fully mixed. In the most metal-poor (lower right) panel of Fig. \[pandas2\], the substructures in the outer halo fade away, displaying a smoother component that can be identified with the in-situ M31 halo. Once the substructures are decoupled based on the lack of obvious spatial correlation and with an additional photometric metallicity cut, [@ibata14] derive a stellar density profile out to 150kpc. Again, the profile follows a power-law, which turns out to be steeper when increasingly more metal-rich populations are considered. [@ibata14] also conclude that only $5\%$ of M31’s total halo luminosity lies in its smooth halo, and the halo mass is as high as $\sim10^{10}\,M_\odot$, significantly larger than what estimated for the MW. The SPLASH survey extends further out than PAndAS, and benefits from kinematical information that is crucial to decontaminate the studied stellar samples from foreground stars and decreases the scatter in the radial profiles. Based on this dataset, [@gilbert12] find that the halo profile does not reveal any break out to 175kpc. This is somewhat surprising given the prediction from simulations that accreted M31-sized stellar haloes should exhibit a break beyond a radius of $\sim100$kpc (@bullock05 [@cooper10]). Beyond a radius of 90kpc, significant field-to-field variations are identified in their data, which suggests that the outer halo regions are mainly comprised of stars from accreted satellites, in agreement with previous studies. At the outermost radii probed by SPLASH ($\sim230$kpc), there is a tentative detection of M31 stars, but this is hard to confirm given the high contamination fraction. Finally, the [@gilbert12] stellar halo profile suggests a prolate DM distribution, which is also consistent with being spherical, in agreement with [@ibata14]. Both [@ibata14] and [@gilbert14] investigate the existence of a metallicity gradient in the smooth halo of M31: they found a steady decrease in metallicity of about 1dex from the very inner regions out to 100kpc. This might indicate the past accretion of (at least) one relatively massive satellite. At the same time, a large field-to-field metallicity variation could mean that the outer halo has been mainly built up by the accretion of several smaller progenitors. ### Andromeda Satellites {#m31_sats} Similarly to the boom of satellite discoveries around the MW, the vast majority of dwarfs in M31’s extended halo has been uncovered by the SDSS, PAndAS, and PanSTARRS surveys in the past decade (see @martin16, and references therein). The M31 satellites follow the same relations between luminosity, radius and metallicity defined by MW satellites, with the exception of systems that are likely undergoing tidal disruption (@collins14). Once more, the characterization of the lowest-mass galaxies raises new, unexpected questions: from the analysis of accurate distances and kinematics, [@ibata13] conclude that half of the M31 satellites lie in a vast ($\sim200$kpc) and thin ($\sim12$kpc) corotating plane, and share the same dynamical orbital properties. The extreme thinness of the plane is very hard to reconcile with $\rm \Lambda$CDM predictions, where such structures should not survive for a Hubble time. While several theoretical interpretations have been offered (e.g., @fernando16), none is conclusive, and this reinforces the allure of mystery surrounding low-mass satellites. Low-mass Galaxies In and Around the Local Group {#dwarfs_halo} ----------------------------------------------- Besides the detailed studies of the two LG spirals, increasing attention is being paid to lower-mass galaxies and their outskirts. Given the self-similar nature of DM, low-mass galaxies should naively be expected to possess haloes and satellites of their own; however, our difficulty in constraining star formation efficiency and physical processes affecting galaxy evolution at these scales blurs these expectations. In the last couple of years, the increasing resolution of cosmological simulations has allowed to make quantitative predictions about the halo and substructures in sub-MW-mass galaxies, and about the number of satellites around them (@wheeler15 [@dooley16]). Observations are thus much needed to test these predictions. Since the late 90s, numerous studies of star-forming dwarfs within or just beyond the LG have claimed the detection of an RGB component extending beyond the blue, young stars (see @stinson09, and references therein), hinting at a generic mode of galaxy formation independent on galaxy size. Such envelopes, however, were not characterized in detail, and in fact could not be identified uniquely as the product of hierarchical merging without, e.g., accurate age and metallicity estimates. The presence of extended haloes in the most luminous satellites of the MW and M31, i.e., the irregular LMC and the low-mass spiral M33, respectively, has not been confirmed to date despite the availability of exquisite datasets. [@gallart04] demonstrate how, out to a galactocentric distance of 7kpc, the stellar density profile of the LMC disk does not show a clear break, in contrast to previous tentative claims. Clearly, the question is complicated by the fact that the LMC is undergoing tidal disruption, and stripped stellar material could easily be misinterpreted as a halo component. Nonetheless, [@mcmonigal14] suggest to have found a sparse LMC halo population from a wide-field dataset around the nearby dwarf galaxy Carina, at galactocentric distances as large as $20$deg. The question might be settled in the near future with the help of wide-field surveys such as the Survey of MAgellanic Stellar History (@martin15). With regard to possible low-mass satellites, there is now tantalizing indication that the LMC might have fallen onto the MW with its own satellite system, as mentioned in Sect. \[mw\_sats\]. As part of the PAndAS survey, deep imaging of M33 has revealed prominent substructure in its outer disk reminiscent of a tidal disturbance, and a faint, diffuse substructure possibly identified as a halo component (@cockcroft13). This result was, however, carefully reconsidered by [@mcmonigal16], who claim that a definitive sign of a halo structure cannot be confirmed, and if present it must have a surface brightness below $\mu_V\sim35$magarcsec$^{-2}$. Besides the investigation of haloes and satellites, deep and wide-field views of low-mass galaxies are crucial to, e.g., assess the presence of tidal disturbances, which in turn are key to estimate mass values and constrain DM profiles (e.g., @sand12). As demonstrated by [@crnojevic14a], a striking similarity in the global properties (luminosity, average metallicity, size) of two low-mass galaxies, such as the M31 satellites NCG 185 and NGC 147, can be quite misleading: once deep imaging was obtained around these galaxies (within PAndAS), NCG 147 revealed extended, symmetric tidal tails, returning a much larger extent and luminosity for this dwarf than what was previously thought. This dataset further showed a flat metallicity gradient for NGC 147, in contrast with the marked gradient found in NGC 185. All these pieces of evidence point at an ongoing interaction of NGC 147 with M31. Large-scale studies of LG dwarfs also provide useful insights into their evolutionary history: by studying CMDs reaching below the MSTO, [@hidalgo13] trace significant age gradients that advocate an outside-in mode of star formation for dwarf galaxies. Clearly, systematic deep searches are needed to detect and characterize the outskirts of low-mass satellites. With this goal in mind, wide-field surveys of nearby ($<3$Mpc) dwarfs have started to be pursued. The first of these efforts targets NGC 3109, a sub-LMC-mass dwarf located just beyond the boundaries of the LG: several candidate satellites of NGC 3109 are identified from a CTIO/DECam survey targeting regions out to its virial radius (@sand15). One of them, confirmed to be at the distance of NGC 3109, is relatively bright ($M_V\sim-10$), and is already in excess of the predicted number by @dooley16 for this system. Other ongoing surveys are similarly looking for halo substructures and satellites in several relatively isolated dwarfs, e.g., the SOlitary LOcal dwarfs survey (@higgs16) and the Magellanic Analog Dwarf Companions And Stellar Halos survey (@carlin16), by using wide-field imagers on large telescopes such as CFHT/MegaCam, Magellan/Megacam, CTIO/DECam and Subaru/HyperSuprimeCam. These datasets will constitute a mine of information to constrain the role of baryonic processes at the smallest galactic scales. Beyond the Local Group {#sec:beyondlg} ====================== The ground-breaking photometric and kinematic surveys carried out in the past two decades have significantly advanced our knowledge of haloes and their substructures within LG galaxies. Nonetheless, the MW and M31 may not be representative of generic MW-sized haloes, given the stochasticity of the hierarchical assembly process: several marked differences in the stellar populations of their haloes underline the need for observations of a statistically significant sample of galaxy haloes with different morphologies, with surveys targeting large portions of their haloes. Cosmological simulations of MW-mass analogues show a wide variation in the properties of their haloes. As already mentioned, the relative contribution of in-situ star formation and disrupted satellites remains unclear: depending on the models (e.g., full hydrodynamical simulations, $N$-body models with particle tagging), they can vary from a negligible number of accretion events for a MW-sized halo, to making up for most of a stellar halo content (e.g., @lu14 [@tissera14]). Even within the same set of simulations, the number, mass ratio and morphology of accretion and merger events span a wide range of possible values (@bullock05 [@johnston08; @garrison14]). The chemical content of extended haloes can provide useful insights into their assembly history: mergers or accretion events of similar-mass satellites will generally tend to produce mild to flat gradients; in-situ populations will feature increasingly metal-poor populations as a function of increasing galactocentric radius, similarly to the accretion of one or two massive companions (e.g., @cooper10 [@font11]). More extended merger histories are also expected to return younger and relatively metal-rich populations with respect to those coming from a shorter assembly, and to produce more massive stellar haloes, with the final result that the mean halo metallicities of MW-mass spirals can range by up to 1dex (e.g., @renda05). Comprehensive observational constraints are key to guide future simulations of galaxy haloes: the past decade has seen a dramatic increase in the observational census of resolved galaxy haloes beyond the LG, thanks to deep imaging obtained with space facilities, as well as to the advent of wide-field imagers on large ground-based telescopes. While the increasing target distance means that it is easier to survey larger portions of their haloes, the drawback is that the depth of the images decreases dramatically, and thus we are only able to detect the brightest surface brightness features in the haloes, i.e., the uppermost $\sim2-3$mag below the TRGB in terms of resolved stars (see Fig. 6 in @radburn11 for a schematic visualization of the different stellar evolutionary phases recognizable in such shallow CMDs). A number of studies has surveyed relatively nearby and more distant galaxy haloes in integrated light despite the serious challenges posed by sky subtraction at such faint magnitudes, masking of bright stars, flat-fielding and scattered light effects, point spread function modelling, and/or spatially variable Galactic extinction. A few early studies have been able to uncover a halo component and tidal debris in the target galaxies (e.g., @malin83 [@morrison94; @sackett94]), without, however, settling the questions about their existence, nature or ubiquity. Different approaches have been adopted to detect haloes and their substructures, i.e., targeting either individual galaxies (e.g., @zheng99 [@pohlen04; @jablonka10; @janowiecki10; @martinez10; @adams12; @atkinson13]) or stacking the images of thousands of objects (e.g., @zibetti04 [@vandokkum05; @tal09]). A precise quantification of the occurrence of faint substructure in the outskirts of nearby galaxies seems as uncertain as it can be, ranging from a few percent to $\sim70\%$ (see, e.g., @atkinson13, and references therein). This is perhaps unsurprising given the heterogeneity of methods used, target galaxy samples, and surface brightness limits in such studies. Besides the identification of such features, the characterization of unresolved halo stellar populations constitutes an even harder challenge: integrated colours and spectra can at most reach a few effective radii, thus missing the outer haloes. Even for the available datasets, the degeneracies between age, metallicity and extinction are generally challenging to break (e.g., @dejong07); in addition, tidal features can rarely tell us about the mass ratio of a merger event or its orbit (with the exception of tails). Here, we do not intend to discuss the detection of haloes and the variety of fractions and morphologies for tidal features observed in integrated light studies; Knapen & Trujillo (this volume) treat this topic in detail, while this contribution focusses on resolved populations. Obtaining resolved photometry beyond the LG is a daunting task as well, due to the very faint luminosities involved—the brightest RGB stars for galaxies at have magnitudes of $I\sim24-28.5$, and thus this approach is so far really limited to the Local Volume. Early attempts to perform photometry of individual stars in the outskirts of nearby galaxies have been made using large photographic plates and the first CCDs (e.g., @humphreys86 [@davidge89; @georgiev92]). The brightest populations (i.e., the youngest) could often be reconciled with being members of the parent galaxy, but the critical information on the faint, old stars was still out of reach. With the advent of wide-format CCDs in the mid 90s, photometry finally became robust enough to open up new perspectives on the resolved stellar content of our closest neighbours. The first studies of this kind date back to twenty years ago and mainly focus on the inner regions of the target galaxies, most commonly their disks or inner haloes, with the goal of studying their recent star formation and of deriving TRGB distances (see, e.g., @soria96 for CenA, @sakai99 for M81 and M82). [@elson97], in particular, resolved individual stars in the halo of the S0 galaxy NGC 3115 with [HST]{}. By analysing the uppermost 1.5mag of the RGB at a galactocentric distance of 30kpc, they derived a distance of $\sim11$Mpc, and additionally discovered for the first time a bimodality in the photometric metallicity distribution function of this early-type galaxy. [@tikhonov03] studied for the first time the resolved content of the nearest ($\sim3.5$Mpc) S0 galaxy NGC 404 with combined ground-based and [HST]{} imaging. Their furthermost [HST]{} pointings ($\sim20$kpc in projection) contain RGB stars that are clearly older than the main disk population, with similar colour (metallicity). The authors conclude that the disk of NGC 404 extends out to this galactocentric distance, but they do not mention a halo component. Beyond these early studies of individual galaxies, the need for systematic investigations of resolved stellar haloes was soon recognized. Next we describe the design and results of some systematic surveys targeting samples of galaxies in the Local Volume. Systematic Studies {#systematic} ------------------ A decade ago, [@mouhcine05a; @mouhcine05b; @mouhcine05c] started an effort to systematically observe the haloes of eight nearby ($<7$Mpc) spiral galaxies with the resolution of [HST]{}. In particular, they utilized WFPC2 to target fields off of the galaxies’ disks (2 to 13kpc in projection along the minor axis) with the goal of investigating their stellar populations, and obtaining accurate distance estimates as well as photometric metallicity distribution functions, to gain insights into the halo formation process. [@mouhcine05c] find the haloes to predominantly contain old populations, with no younger components and little to no intermediate-age populations. Interestingly, [@mouhcine05b] find a correlation between luminosity and metallicity for the target galaxies, where the metallicity is derived from the mean colour of the resolved RGB. Both the spiral galaxies from their sample (NGC 253, NGC 4244, NGC 4945, NGC 4258, NGC 55, NGC 247, NGC 300, and NGC 3031 or M81) and the two ellipticals (NGC 3115 and NGC 5128 or Centaurus A, included in their comparison from previous literature data) fall on the same relation, indicating that haloes might have a common origin regardless of the galaxy morphological type. Interestingly enough, the MW halo turns out to be substantially more metal-poor than those of the other galaxies of comparable luminosity, based on kinematically selected pressure-supported halo stars within $\sim10$kpc above the disk (see also Sect. \[sec:m31\]). This relation is consistent with a scenario where halo field stars form in the potential well of the parent galaxy in a gradual way from pre-enriched gas. Moreover, the relatively high metallicities of the target haloes seem to suggest that they likely originate from the disruption of intermediate-mass galaxies, rather than smaller metal-poor dwarf galaxies (@mouhcine05c). Interestingly, the dataset presented and studied in [@mouhcine05a; @mouhcine05b; @mouhcine05c] is further analyzed by [@mouhcine06] to find that each spiral of the sample presents a bimodal metallicity distribution. In particular, both a metal-poor and a metal-rich component are present in the outskirts of the target galaxies, and both components correlate with the host’s luminosity. This is taken as a hint that these populations are born in subgalactic fragments that were already embedded in the dark haloes of the host galaxy; the metal-poor component additionally has a broader dispersion than that of the metal-rich population. These properties show similarities with GC subpopulations in the haloes of early-type galaxies (e.g., @peng06). [@mouhcine06] argues that the metal-poor component may arise from the accretion of low-mass satellites, while the metal-rich one could be linked to the formation of the bulge or the disk. The shortcoming of this ambitious study is, however, twofold: first, the limited field of view (FoV) of [HST]{} hampers global conclusions on the galaxies’ haloes, and the stellar populations at even larger radii may have different properties than those in the observed fields; second, perhaps most importantly, it is not obvious what structure of the galaxy is really targeted, i.e., the halo, the outer bulge or disk, or a mixture of these. Along the same lines of these studies, [@radburn11] present an even more ambitious [HST]{} survey of 14 nearby disk galaxies within 17Mpc, with a range of luminosities, inclinations and morphological types. The Galaxy Halos, Outer disks, Substructure, Thick disks, and Star clusters (GHOSTS) survey aims at investigating radial light profiles, axis ratios, metallicity distribution functions (MDFs), SFHs, possible tidal streams and GC populations, all to be considered as a function of galaxy type and position within the galaxies. The 76 ACS pointings of the survey are located along both major and minor axes for most of the targets, and reach $\sim2-3$mag below the TRGB, down to surface brightness values of $V\sim30$magarcsec$^{-2}$. This dataset thus represents a very valuable resource for testing hierarchical halo formation models. [@monachesi16] investigate six of the galaxies in this sample (NGC 253, NGC 891, M81, NGC 4565, NGC 4945, and NGC 7814) and conclude that all of them contain a halo component out to 50kpc, and two of them out to 70kpc along their minor axis. The colour (i.e., photometric metallicity) distribution of RGB stars in the target haloes is analysed and reveals a non-homogeneity which likely indicates the presence of non-mixed populations from accreted objects. The average metallicity out to the largest radii probed remains relatively high when compared to the values of the MW halo; metallicity gradients are also detected in half of the considered galaxies. Surprisingly, and in contrast to the results presented by [@mouhcine05b], the spiral galaxies in this sample do not show a strong correlation between the halo metallicity and the total mass of the galaxies, highlighting instead the stochasticity inherent to the halo formation process through accretion events (e.g., @cooper10). The advantage of the GHOSTS dataset over the one from [@mouhcine05b] is that the GHOSTS fields are deeper, there are several pointings per galaxy and they reach significantly larger galactocentric distances, thus offering a more global view of the haloes of the targets. In an effort to increase the sample of nearby galaxies for which stellar haloes are resolved and characterized, several groups have individually targeted Local Volume objects with either ground-based or space-borne facilities: the low-mass spirals NGC 2403 (@barker11, with Subaru/SuprimeCam), NGC 300 (@vlajic09, with Gemini/GMOS), and NGC 55 (@tanaka11, with Subaru/SuprimeCam), the ellipticals NGC 3379 (@harris07a, with [HST]{}) and NGC 3377 (@harris07b, with [HST]{}), and the lenticular NGC 3115 (@peacock15, with [HST]{}). In most of these galaxies, a resolved faint halo (or at least an extended, faint and diffuse component) has been detected and is characterized by populations more metal-poor than the central/disk regions. Most of these haloes also show signs of substructure, pointing at past accretion/merger events as predicted by a hierarchical galaxy formation model. Even galaxies as far as the central elliptical of the Virgo cluster, M87, ($\sim16$Mpc) are starting to be targeted with [HST]{}, although pushing its resolution capabilities to the technical limits (@bird10). While spectroscopically targeting individual RGB stars to obtain radial velocity and metallicity information is still prohibitive beyond the LG (see Sect. \[dwarfs\_halo\]), some cutting-edge studies have pushed the limits of spectroscopy for dwarf galaxies within $\sim1.5$Mpc (e.g., @kirby12, and references therein). At the same time, novel spectroscopic techniques are being developed to take full advantage of the information locked into galaxy haloes. One example is the use of co-added spectra of individual stars, or stellar blends, to obtain radial velocities, metallicities and possibly gradients in galaxies within $\sim4$Mpc, as robustly demonstrated by [@toloba16]. The development of new analysis methods and the advent of high-resolution spectrographs will soon allow for systematic spectroscopic investigations of nearby galaxy haloes which will importantly complement the available photometric studies, similarly to the studies of LG galaxies. Besides the systematic studies presented here, which mostly involve deep space observations, an increasing effort is being invested in producing spatial density maps of outer haloes in some of the closest galaxies with ground-based observations, akin to the panoramic view of M31 offered by PAndAS. In the following Section we describe some of these efforts. Panoramic Views of Individual Galaxies {#panoramic} -------------------------------------- Panoramic views of nearby galaxies can be obtained with the use of remarkable ground-based wide-field imagers such as Subaru/SuprimeCam and HyperSuprimeCam and CFTH/MegaCam in the northern hemisphere, and Magellan/Megacam, CTIO/DECam and VISTA/VIRCAM in the southern hemisphere. Clearly, such CMDs cannot reach the depth of those obtained for M31; these studies nevertheless represent cornerstones for our investigation of global halo properties, and serve as precursor science cases for the next generation of telescopes that will open new perspectives for this kind of studies to be performed on a significantly larger sample of galaxies. As mentioned in Sect. \[dwarfs\_halo\], the haloes of low-mass galaxies are also starting to be systematically investigated, to gain a more complete picture of galaxy formation at all mass scales. Here we further describe the few examples of spatially extended imaging obtained to date for some of the closest spiral and elliptical galaxies. ### NGC 891 ![Surface density map of RGB stars in the halo of NGC 891, obtained with Subaru/SuprimeCam. The overdensities of old RGB stars reveal a large complex of arcing streams that loops around the galaxy, tracing the remnants of an ancient accretion. The second spectacular morphological feature is the dark cocoon-like structure enveloping the high surface brightness disk and bulge. Fig. 1 from [@mouhcine10], reproduced by permission of the AAS[]{data-label="fig:mouhcine10"}](ngc891_v2.eps){width="7.cm"} Despite its relatively large distance ($\sim9$Mpc, @radburn11), the “MW-twin” NCG 891 (@vanderkruit84) is one of the first spirals to be individually investigated in resolved light. Its high inclination and absence of a prominent bulge make it an appealing target for halo studies. @mouhcine07 exploit three [HST]{} pointings located approximately 10kpc above the disk of NGC 891 to investigate the properties of this galaxy’s halo. The broad observed RGB indicates a wide range of metallicities in this population, with metal-rich peaks and extended metal-poor tails. The three fields also show a decreasing mean metallicity trend as a function of increasing distance along the major axis. The mean metallicity of this sample of RGB stars (\[Fe/H\]$\sim-1$) falls on the halo metallicity-galaxy luminosity relation pointed out by [@mouhcine05b]: this, together with the gradient mentioned before, is in contrast with the lower metallicities and absence of a gradient for non-rotating stars in the inner haloes of the MW and M31 (@chapman06 [@kalirai06]). @mouhcine07 thus suggest that not all massive galaxies’ outskirts are dominated by metal-poor, pressure-supported stellar populations (because of the inclination and absence of a bulge, the studied RGB sample is thought to be representative of the true halo population). A possible explanation is suggested with the presence of two separate populations: a metal-rich one that is present in the most massive galaxies’ outskirts, and one constituting the metal-poor, pressure-supported halo, coming from the accretion of moderate-mass satellites. For smaller-mass galaxies, the halo would instead be dominated by debris of small satellites with lower metallicities. Follow-up analysis on the same [HST]{} dataset has been carried out by [@ibata09] and [@rejkuba09]. After careful accounting for the internal reddening of the galaxy, a mild metallicity gradient is confirmed in NGC 891’s spheroidal component, which is surveyed out to $\sim20$kpc (assuming elliptical radii), and suggested to arise from the presence of a distinct outer halo, similarly to the MW (@ibata09). Most importantly, and for the first time, this refined analysis reveals a substantial amount of substructure not only in the RGB spatial distribution but also as metallicity fluctuations in the halo of NGC 891. This evidence points at multiple small accretion events that have not fully blended into the smooth halo. Motivated by these studies, @mouhcine10 provide the first attempt to derive a PAndAS-like map of a MW-analogue beyond the LG: their wide-field map of NGC 891’s halo is shown in Fig. \[fig:mouhcine10\]. The panoramic survey, performed contiguously with Subaru/SuprimeCam, covers an impressive $\sim90\times90$kpc$^2$ in the halo of NGC 891 with the $V$ and $i$ filters, reaching $\sim2$mag below the TRGB. Among the abundant substructures uncovered by the RGB map around NGC 891, a system of arcs/streams reaches out some $\sim50$kpc into the halo, including the first giant stream detected beyond the LG with ground-based imaging. The latter’s shape does not rule out a single accretion event origin, but a possible progenitor cannot be identified as a surviving stellar overdensity. These structures appear to be old, given the absence of corresponding overdensities in the luminous AGB (i.e., intermediate-age populations) maps. Another surprising feature highlighted by the RGB map is a flattened, super-thick envelope surrounding the disk and bulge of NGC 891, which does not seem to constitute a simple extension of its thick disk but is instead believed to generate from the tidal disruption of satellites given its non-smooth nature (@ibata09). ### M81 ![Isodensity contour map of red RGB stars in the M81 group, as observed by Subaru/HyperSuprimeCam. Structures up to $20\sigma$ above the background level are visible; the cross marks represent the centres of known M81 group members, while solid lines are ${R}_{25}$ of galaxies. The high degree of substructure underlines the ongoing tidal interactions in this group; note in particular the S-shape of the outer regions in NGC 3077 and M82. Fig. 4 from [@okamoto15], reproduced by permission of the AAS[]{data-label="fig:okamoto"}](okamoto_m81.eps){width="7.cm"} Located at a distance of 3.6Mpc (@radburn11) and with a dynamical mass inside 20kpc of $\sim10^{11}\,M_\odot$, M81 is one of the closest MW-analogues, and has thus been among the first targets for extended halo studies beyond the LG. The earliest H[i]{} imaging of the galaxy group dominated by this spiral unambiguously shows a spectacular amount of substructure, most prominently a bridge of gas between M81 and its brightest companions NGC 3077 and M82, located at a projected distance of $\sim60$kpc (@vanderhulst79 [@yun94]). Given the high level of interaction and H[i]{} substructure in a group that can be considered as a LG-analogue, it is natural to pursue the investigation of this complex environment even further. The intergalactic gas clouds embedding this environment are traced by young stellar systems identified in resolved stellar studies (@durrell04 [@davidge08; @demello08]). Some of them are classified as tidal dwarf galaxies, such as Holmberg IX and the Garland (@makarova02 [@kara04; @sabbi08; @weisz08]), characterized by a predominance of young stellar populations. This type of galaxy has no counterpart in our own LG, and it is believed to be DM-free (see, e.g., @duc00). The first detailed look into the resolved populations in the outskirts of M81 is through the eye of [HST]{}: the predominantly old halo RGB stars show a broad range of metallicities and a radial gradient (@tikhonov05 [@mouhcine05c]). The radial stellar counts (along several different directions) also reveal a break at a radius of $\sim25$kpc, which is interpreted as the transition point between thick disk and halo (@tikhonov05). In a similar fashion, the ground-based wide-field imager Subaru/SuprimeCam has been used to uncover a faint and extended component beyond M81’s disk with a flat surface brightness profile extending out to $\sim0.5$deg (or $\sim30$kpc) to the north of M81 (@barker09). This low surface brightness feature ($\sim28$magarcsec$^{-2}$) traced by the brightest RGB star counts appears bluer than the disk, suggesting a metallicity lower than that of M81’s main body, but its true nature remains unclear. The authors suggest this component to have intermediate properties between the MW’s halo and its thick disk, but the limited surveyed area ($0.3$deg$^2$) precludes any robust conclusions. As part of a campaign to obtain panoramic views of nearby galaxy haloes, [@mouhcine09] present a $0.9\times 0.9$deg$^2$ view of M81’s surroundings obtained with the CFHT/MegaCam imager. The images resolve individual RGB stars down to $\sim2$mag below the TRGB, but this study focusses on the younger, bright populations such as massive main sequence stars and red supergiants, which reveal further young systems tracing the H[i]{} tidal distribution between M81 and its companions. These systems are younger than the estimated dynamical age of the large-scale interaction and do not have an old population counterpart, suggesting that they are not simply being detached from the main body of the primary galaxies but are instead formed within the H[i]{} clouds. [@durrell10] recently conducted a deeper, albeit spatially limited, [HST]{} study of a field at a galactocentric distance of $\sim20$kpc. This field reveals an \[M/H\]$\sim-1.15$ population with an approximate old age of $\sim9$Gyr. This field thus contains the most metal-poor stars found in M81’s halo to that date, which led the authors to the conclusion that they were dealing with an authentic halo component. This study is extended by [@monachesi13] with the [HST]{} GHOSTS dataset (see Sect. \[systematic\]): they construct a colour profile out to a radius of $\sim50$kpc, and this dataset does not show a significant gradient. The mean photometric metallicity derived is \[Fe/H\]$\sim-1.2$, similarly to @durrell10. This result is found to be in good agreement with simulations and the authors suggest that the halo of M81 could have been assembled through an early accretion of satellites with comparable mass (e.g., @cooper10 [@font06]). As a further step in the investigation of M81’s halo, the [@barker09] and [@mouhcine09] ground-based imaging of M81 is being improved by means of the Subaru/HyperSuprimeCam. The first $\sim2\times2$deg$^2$ ($\sim100\times115$kpc$^2$) resolved stellar maps from different subpopulations (upper main sequence, red supergiants, RGB and AGB stars) are presented in [@okamoto15] and constitute a preview of an even wider-field effort to map the extended halo of this group. These first maps (see Fig. \[fig:okamoto\]) confirm a high degree of substructure, most interestingly: the youngest populations nicely trace the H[i]{} gas content, confirming previous small FoV studies; the RGB distributions are smoother and significantly more extended than the young component, and show stream-like overlaps between the dominant group galaxies, e.g., M82’s stars clearly being stripped by M81; a redder RGB distribution is detected for M81 and NGC 3077 compared to M82, indicating a lower metallicity in the latter; in addition, M82 and NGC 3077’s outer regions present S-shaped morphologies, a smoking gun of the tidal interaction with M81 and typical of interacting dwarf galaxies with larger companions (e.g., @penarrubia09). Not less importantly, the widest-field survey to date ($\sim65$deg$^2$) of the M81 group has been performed by [@chiboucas09] with CFHT/MegaCam, although with only one filter. The main goal of this survey was to identify new, faint dwarf galaxies and investigate the satellite LF in a highly interacting group environment as compared to the LG. This is the first survey to systematically search for faint dwarfs beyond the LG. Resolved spatial overdensities consistent with candidate dwarfs have been followed up with two-band [HST]{}/ACS and [HST]{}/WFPC2 observations. Fourteen of the 22 candidates turned out to be real satellites of M81 based on their CMDs and TRGB distances, extending the previously known galaxy LF in this group by three orders of magnitude down to $M_r\sim-9.0$ (@chiboucas13), with an additional possibly ultra-faint member at $M_r\sim-7.0$. The measured slope of the LF in the M81 group appears to be flatter than cosmological predictions ($\alpha\sim-1.27$, in contrast to the theoretical value of $\alpha\sim-1.8$), similar to what has been found for the MW and M31 satellites. ### NGC 253 Another obvious MW-mass spiral target for halo studies is NCG 253 ($\sim3.5$Mpc, @radburn11). Its role of brightest object within the loose Sculptor filament of galaxies makes it ideally suited to investigate the effects of external environment on the assembly of haloes. As already apparent from old photographic plates, NGC 253’s outskirts show faint perturbation signs, such as an extended shelf to the south of its disk (@malin97), pointing at a possible accretion event. This spiral galaxy, despite its relative isolation, is experiencing a recent starburst and a pronounced nuclear outflow: the latter is believed to host local star formation extending as high as $\sim15$kpc above the disk in the minor axis direction (see @comeron01, and references therein). The resolved near-infrared study of [@davidge10] allowed them to detect bright AGB stars, but not RGB stars, extending out to $\sim13$kpc from the disk plane in the south direction: these are interpreted as being expelled from the disk into the halo as consequence of a recent interaction. Subsequently, [@bailin11] exploited a combination of [HST]{} data from the GHOSTS survey and ground-based Magellan/IMACS imaging, the former being deeper while the latter have a more extended FoV (out to $\sim30$kpc in the halo NGC 253 in the south direction). The authors are able to estimate NGC 253’s halo mass as $\sim2\times10^9\,M_\odot$, or 6% of the galaxy’s total stellar mass: this value is broadly consistent with those derived from the MW and M31 but higher, reminiscent of the halo-to-halo scatter seen in simulations. A power law is fit to the RGB radial profile which is found to be slightly steeper than that of the two LG spirals, and appears to be flattened in the same direction as the disk component. This is the one of the few studies to date to quantitatively measure such parameters for a halo beyond the LG, and it sets the stage for the possibilities opened by similar studies of other nearby galaxies. The RGB density maps derived in [@bailin11] from IMACS imaging confirm the early detection of a shelf structure, and uncover several additional kpc-scale substructures in the halo of this spiral. A more recent wide-field study of NGC 253 is presented by [@greggio14], who exploit the near-infrared VISTA/VIRCAM imager to study the RGB and AGB stellar content of this galaxy out to $\sim40-50$kpc, covering also the northern portion which was not included in previous studies. This portion, in particular, reveals an RGB substructure symmetric (and likely connected) to the one in the south. A prominent arc ($\sim20$kpc in length) to the north-west of the disk is detected and estimated to arise from a progenitor with a stellar mass of roughly $\sim7\times10^6\,M_\odot$. The RGB radial density profile shows a break at a radius of $\sim25$kpc, indicative of the transition from disk to halo. The elongated halo component already discussed in [@bailin11] is confirmed here, but is considered to be an inner halo: an outer, more spherical and homogeneous component extends at least out to the galactocentric distances covered by this survey. Intriguingly, the AGB density map reveals that 25% of this intermediate-age (i.e., up to a few Gyr old) population is spread out to $\sim30$kpc above the disk: this component cannot easily be explained with either an in-situ or an accreted origin. NGC 253 is also one of the two targets of the Panoramic Imaging Survey of Centaurus and Sculptor (PISCeS), recently initiated with the wide-field imager Magellan/Megacam. This ambitious survey aims at obtaining RGB stellar maps of this galaxy and of the elliptical Centaurus A (Cen A; see next Section) out to galactocentric radii of $\sim150$kpc, similarly to the PAndAS survey of M31. Early results from this survey include the discovery of two new faint satellites of NGC 253, one of which is clearly elongated and in the process of being disrupted by its host (@sand14 [@toloba16b]). ### NGC 5128 (Centaurus A) {#cena} ![Surface density map of RGB stars in the halo of Cen A, obtained with Magellan/Megacam as part of the PISCeS survey. The map extends out to a radius of 150kpc in the north and east directions (physical and density scales are reported). Several tidal features are easily recognized, including a stunning disrupting dwarf with tails 2deg long in the outer halo, an extended sparse cloud to the south of the galaxy, as well as arcs and plumes around the inner regions, tracing both ongoing and past accretion events. Fig. 3 from [@crnojevic16], reproduced by permission of the AAS[]{data-label="fig:crnojevic16"}](fig2_1.eps){width="11cm"} It is important to target galaxies of different morphologies and environments to thoroughly investigate the assembly of haloes. The closest ($\sim3.8$Mpc; @harrisg09) elliptical galaxy is Centaurus A (Cen A; technically speaking, Maffei 1 is slightly closer but it lies behind the Galactic disk and is thus heavily reddened, see @wu14). Cen A is the dominant galaxy of a rich and dense group, which also has a second subgroup component centred on the spiral M83 (e.g., @kara07). Despite having often been referred to as a peculiar galaxy, due to its pronounced radio activity, its central dust lanes, and a perturbed morphology, the luminosity of Cen A is quite typical of field elliptical galaxies: a recent ($<1$Gyr) merger event is believed to be the culprit for its peculiar features (see @israel98, and references therein). Besides this main merger event, [@peng02] uncover a system of faint shells and an arc within $\sim25$kpc of Cen A’s centre from integrated light observations; the arc is believed to have been produced by the infall of a low-mass, star forming galaxy around $\sim300$Myr ago. This elliptical galaxy has been the subject of a systematic study conducted with [HST]{}/ACS and [HST]{}/WFPC2 throughout the past couple of decades: a number of pointings at increasingly large galactocentric radii (from a few out to $\sim150$kpc) have been used to investigate the properties and gradients of Cen A’s halo populations (@rejkuba14, and references therein). The considered pointings out to 40kpc reveal metal-rich populations (\[Fe/H\]$>-1.0$), not dissimilar to what has been observed for the haloes of spiral galaxies. The deepest CMD to date of this elliptical is presented by [@rejkuba11] for the [HST]{} field at 40kpc: this study concludes that the vast majority of Cen A’s halo population is old ($\sim12$Gyr), with a younger ($\sim2-4$Gyr) component accounting for $\sim20\%$ of the total population. The first wide-field study of Cen A was performed with the ground-based VLT/VIMOS imager, reaching out to $\sim85$kpc along both minor and major axes (@crnojevic13). Cen A’s halo population seems to extend all the way out to this large radius. This study confirms the relatively high metallicity for halo populations found by the [HST]{} studies, although with a considerable presence of metal-poor stars at all radii; the authors also highlight the absence of a strong metallicity gradient from a $\sim30$kpc radius out to the most distant regions probed. This study suggests that the outer regions of Cen A’s halo show an increase in ellipticity as a function of radius, which could, however, be interpreted as the presence of substructure contaminating the observed fields. A subsequent study exploits additional [HST]{} pointings out to a remarkably large radius of $\sim150$kpc: the edge of Cen A’s halo is not reached even by this study (@rejkuba14). This dataset, analysed together with the previous [HST]{} pointings, confirms that a very mild metallicity gradient is present, with median metallicities remaining high out to the largest distances probed. [@rejkuba14], however, also detect a significant pointing-to-pointing variation in both the RGB star counts and the median metallicity, which is likely indicative of non-mixed accreted populations. Recently, the PISCeS survey (see previous Section) has sketched a PAndAS-like picture of Cen A’s halo out to $\sim150$kpc: the RGB stellar density map derived from a mosaic of Magellan/Megacam images is presented in Fig. \[fig:crnojevic16\]. This map, very much like the ones obtained for M31 and NGC 891, uncovers a plethora of faint substructures, both in the inner regions of the target galaxy and in its outskirts. The morphological variety of these features is reminiscent of that observed in PAndAS, with shells, plumes, an extended cloud and long tidal streams. In particular, one of the newly discovered dwarf satellites of Cen A is clearly in the process of being disrupted, with $\sim2$deg long tails: taking into account the stellar content of these tails, this galaxy’s pre-disruption luminosity could have been similar to that of Sagittarius in the LG. This survey also led to the discovery of nine (confirmed) dwarf satellites down to $M_V\sim-7$. Their properties are consistent with those of faint LG satellites, but some of them lie at the faint/diffuse end of the LG luminosity/surface brightness/radius distribution: this indicates that we might be looking at previously unexplored physical regimes for these faintest satellites, which opens new exciting perspectives for future studies. Summary and Future Prospects {#conclusions} ============================ In a $\rm \Lambda$CDM hierarchical model, all galaxies are predicted to have experienced mergers, of which many should be recognizable as debris/streams that make up for a large fraction of their haloes. Haloes and their substructures thus provide a unique glimpse into the assembly history of galaxies, and can inform the models at the smallest galactic scales, where they still fall short in reproducing observations. The time is now ripe for in-depth systematic studies of the resolved stellar populations in galaxy haloes, which will dramatically increase our understanding of galaxy evolution over the next decade. The challenges for this type of studies are of a different nature: for our own Galaxy, state-of-the-art results on its halo shape, profile and mass inevitably suffer from assumptions on underlying density models and extrapolations of the available data to radii larger than observed. The major current limitation of MW halo studies lies in observational biases due to small field-of-view samples, which preclude the identification of possible substructure contamination. Future surveys hold the promise to advance the knowledge of our Galaxy by obtaining significantly larger samples of tracers, especially in areas so far not covered. Most notably, the astrometric [Gaia]{} mission (which will provide unprecedented six-dimensional phase space information for two billion stars out to the inner MW halo) and the Large Synoptic Survey Telescope (LSST; designed to provide a southern sky counterpart to SDSS, and reaching $\sim4$ magnitudes fainter than its predecessor for a total sample of tens of billions of stars), are going to revolutionize our view of the MW. At the same time, the current and future generation of high-resolution spectrographs will follow up these surveys from the ground, providing comprehensive kinematic and chemical information to assess the origin of halo stars and characterize their birthplaces (see also Figueras, this volume). The pioneering studies of an increasing number of haloes beyond the LG, and across a range of masses, will soon be extended by the next generation of ground-based extremely large telescopes (E-ELT, GMT, TMT), as well as space-borne missions ([JWST]{}, [Euclid]{}, [WFIRST]{}). The PAndAS survey of M31 has extensively demonstrated that only the synergy of wide-field ground-based observations, deep (but spatially limited) observations from space, and spectroscopy can return a truly global understanding of haloes made up of a complex mixture of in-situ and accreted populations. The aforementioned facilities will open new perspectives with wide-field optical and infrared imagers in concert with high-resolution spectrographs, which will allow us to systematically survey hundreds of galaxies within tens of Mpc in the next decade or two. For example, with the E-ELT/MICADO and [JWST]{}/NIRcam imagers (the former having higher resolving power and the latter a wider field-of-view), we should resolve stars down to the HB within $\sim10$Mpc, thus identifying and characterizing the SFHs of streams and faint satellites; derive radial profiles, MDFs and stellar population gradients in haloes within 20Mpc from the uppermost few magnitudes of the RGB; and trace the halo shape and possible overdensities down to $\mu_V\sim33$magarcsec$^{-2}$ from the uppermost $\sim0.5$mag of the RGB out to 50Mpc (@greggio16). These observational constraints will be crucial to inform increasingly sophisticated theoretical models, and ultimately answer intriguing open questions (as well as possibly unexpected ones that will likely be raised by these observations themselves), such as: - [Do all galaxies have haloes?]{} - [What are the relative fractions of in-situ versus accreted populations in galaxy haloes, and how does this depend on galactocentric distance, galaxy morphology, and environment?]{} - [What are the properties of the objects currently being accreted, i.e., mass, chemical content, SFH, orbital properties, and how do they relate to those of the present day low-mass satellites?]{} - [Do low-mass galaxies possess haloes/satellites of their own, and what is their fate and contribution upon infall onto a massive galaxy?]{} - [How extended really are the haloes of massive galaxies?]{} - [What is the shape and mass of the DM haloes underlying galaxies?]{} - [What is the relation between the outer halo and the bulge/disk of a galaxy?]{} - [What is the role of internal versus external processes in shaping a galaxy’s properties, especially at the low-mass end of the galaxy LF?]{} - [What is the relation between the present-day haloes/satellites and their unresolved, high-redshift counterparts?]{} The era of resolved populations in galaxy haloes has just begun, and it holds the promise to be a golden one. I would like to thank the organizers for a lively and stimulating conference. I am indebted to S. Pasetto for his advice and support throughout the preparation of this contribution. I acknowledge the hospitality of the Carnegie Observatories during the completion of this work.	ArXiv
--- abstract: 'Using a sample of $1.31\times10^{9} ~J/\psi$ events collected with the BESIII detector, we perform a study of $J/\psi\to\gamma K\bar{K}\eta''$. The $X(2370)$ is observed in the $K\bar{K}\eta''$ invariant-mass distribution with a statistical significance of 8.3$\sigma$. Its resonance parameters are measured to be $M=2341.6\pm 6.5\text{(stat.)}\pm5.7\text{(syst.)}$ MeV/$c^{2}$ and $\Gamma = 117\pm10\text{(stat.)}\pm8\text{(syst.)}$ MeV. The product branching fractions for ${J/\psi}\to \gamma X(2370),X(2370)\to {K^{+}K^{-}}\eta''$ and ${J/\psi}\to \gamma X(2370),X(2370)\to {K_{S}^{0}K_{S}^{0}}\eta''$ are determined to be $(1.79\pm0.23\text{(stat.)}\pm0.65\text{(syst.)})\times10^{-5}$ and $(1.18\pm0.32\text{(stat.)}\pm0.39\text{(syst.)})\times10^{-5}$, respectively. No evident signal for the $X(2120)$ is observed in the $K\bar{K}\eta''$ invariant-mass distribution. The upper limits for the product branching fractions of $\mathcal{B}({J/\psi}\to\gamma X(2120)\to\gamma K^{+} K^{-} \eta'')$ and $\mathcal{B}({J/\psi}\to\gamma X(2120)\to\gamma K_{S}^{0} K_{S}^{0} \eta'')$ are determined to be $1.49\times10^{-5}$ and $6.38\times10^{-6}$ at the 90% confidence level, respectively.' author: - \| M. Ablikim$^{1}$, M. N. Achasov$^{10,e}$, P. Adlarson$^{59}$, S. Ahmed$^{15}$, M. Albrecht$^{4}$, M. Alekseev$^{58A,58C}$, A. Amoroso$^{58A,58C}$, Q. An$^{55,43}$, Anita$^{21}$, Y. Bai$^{42}$, O. Bakina$^{27}$, R. Baldini Ferroli$^{23A}$, I. Balossino$^{24A}$, Y. Ban$^{35,l}$, K. Begzsuren$^{25}$, J. V. Bennett$^{5}$, N. Berger$^{26}$, M. Bertani$^{23A}$, D. Bettoni$^{24A}$, F. Bianchi$^{58A,58C}$, J Biernat$^{59}$, J. Bloms$^{52}$, I. Boyko$^{27}$, R. A. Briere$^{5}$, H. Cai$^{60}$, X. Cai$^{1,43}$, A. Calcaterra$^{23A}$, G. F. Cao$^{1,47}$, N. Cao$^{1,47}$, S. A. Cetin$^{46B}$, J. Chai$^{58C}$, J. F. Chang$^{1,43}$, W. L. Chang$^{1,47}$, G. Chelkov$^{27,c,d}$, D. Y. Chen$^{6}$, G. Chen$^{1}$, H. S. Chen$^{1,47}$, J. C. Chen$^{1}$, M. L. Chen$^{1,43}$, S. J. Chen$^{33}$, Y. B. Chen$^{1,43}$, W. Cheng$^{58C}$, G. Cibinetto$^{24A}$, F. Cossio$^{58C}$, X. F. Cui$^{34}$, H. L. Dai$^{1,43}$, J. P. Dai$^{38,i}$, X. C. Dai$^{1,47}$, A. Dbeyssi$^{15}$, D. Dedovich$^{27}$, Z. Y. Deng$^{1}$, A. Denig$^{26}$, I. Denysenko$^{27}$, M. Destefanis$^{58A,58C}$, F. De Mori$^{58A,58C}$, Y. Ding$^{31}$, C. Dong$^{34}$, J. Dong$^{1,43}$, L. Y. Dong$^{1,47}$, M. Y. Dong$^{1,43,47}$, Z. L. Dou$^{33}$, S. X. Du$^{63}$, J. Z. Fan$^{45}$, J. Fang$^{1,43}$, S. S. Fang$^{1,47}$, Y. Fang$^{1}$, R. Farinelli$^{24A,24B}$, L. Fava$^{58B,58C}$, F. Feldbauer$^{4}$, G. Felici$^{23A}$, C. Q. Feng$^{55,43}$, M. Fritsch$^{4}$, C. D. Fu$^{1}$, Y. Fu$^{1}$, X. L. Gao$^{55,43}$, Y. Gao$^{56}$, Y. Gao$^{35,l}$, Y. G. Gao$^{6}$, Z. Gao$^{55,43}$, I. Garzia$^{24A,24B}$, E. M. Gersabeck$^{50}$, A. Gilman$^{51}$, K. Goetzen$^{11}$, L. Gong$^{34}$, W. X. Gong$^{1,43}$, W. Gradl$^{26}$, M. Greco$^{58A,58C}$, L. M. Gu$^{33}$, M. H. Gu$^{1,43}$, S. Gu$^{2}$, Y. T. Gu$^{13}$, A. Q. Guo$^{22}$, L. B. Guo$^{32}$, R. P. Guo$^{36}$, Y. P. Guo$^{26}$, Y. P. Guo$^{9,j}$, A. Guskov$^{27}$, S. Han$^{60}$, X. Q. Hao$^{16}$, F. A. Harris$^{48}$, K. L. He$^{1,47}$, F. H. Heinsius$^{4}$, T. Held$^{4}$, Y. K. Heng$^{1,43,47}$, M. Himmelreich$^{11,h}$, T. Holtmann$^{4}$, Y. R. Hou$^{47}$, Z. L. Hou$^{1}$, H. M. Hu$^{1,47}$, J. F. Hu$^{38,i}$, T. Hu$^{1,43,47}$, Y. Hu$^{1}$, G. S. Huang$^{55,43}$, J. S. Huang$^{16}$, X. T. Huang$^{37}$, X. Z. Huang$^{33}$, N. Huesken$^{52}$, T. Hussain$^{57}$, W. Ikegami Andersson$^{59}$, W. Imoehl$^{22}$, M. Irshad$^{55,43}$, S. Jaeger$^{4}$, Q. Ji$^{1}$, Q. P. Ji$^{16}$, X. B. Ji$^{1,47}$, X. L. Ji$^{1,43}$, H. B. Jiang$^{37}$, X. S. Jiang$^{1,43,47}$, X. Y. Jiang$^{34}$, J. B. Jiao$^{37}$, Z. Jiao$^{18}$, D. P. Jin$^{1,43,47}$, S. Jin$^{33}$, Y. Jin$^{49}$, T. Johansson$^{59}$, N. Kalantar-Nayestanaki$^{29}$, X. S. Kang$^{31}$, R. Kappert$^{29}$, M. Kavatsyuk$^{29}$, B. C. Ke$^{1}$, I. K. Keshk$^{4}$, A. Khoukaz$^{52}$, P. Kiese$^{26}$, R. Kiuchi$^{1}$, R. Kliemt$^{11}$, L. Koch$^{28}$, O. B. Kolcu$^{46B,g}$, B. Kopf$^{4}$, M. Kuemmel$^{4}$, M. Kuessner$^{4}$, A. Kupsc$^{59}$, M. G. Kurth$^{1,47}$, W. Kühn$^{28}$, J. S. Lange$^{28}$, P. Larin$^{15}$, L. Lavezzi$^{58C}$, H. Leithoff$^{26}$, T. Lenz$^{26}$, C. Li$^{59}$, Cheng Li$^{55,43}$, D. M. Li$^{63}$, F. Li$^{1,43}$, G. Li$^{1}$, H. B. Li$^{1,47}$, H. J. Li$^{9,j}$, J. C. Li$^{1}$, J. W. Li$^{41}$, Ke Li$^{1}$, L. K. Li$^{1}$, Lei Li$^{3}$, P. L. Li$^{55,43}$, P. R. Li$^{30}$, Q. Y. Li$^{37}$, S. Y. Li$^{45}$, W. D. Li$^{1,47}$, W. G. Li$^{1}$, X. H. Li$^{55,43}$, X. L. Li$^{37}$, X. N. Li$^{1,43}$, Z. B. Li$^{44}$, Z. Y. Li$^{44}$, H. Liang$^{55,43}$, H. Liang$^{1,47}$, Y. F. Liang$^{40}$, Y. T. Liang$^{28}$, G. R. Liao$^{12}$, L. Z. Liao$^{1,47}$, J. Libby$^{21}$, C. X. Lin$^{44}$, D. X. Lin$^{15}$, Y. J. Lin$^{13}$, B. Liu$^{38,i}$, B. J. Liu$^{1}$, C. X. Liu$^{1}$, D. Liu$^{55,43}$, D. Y. Liu$^{38,i}$, F. H. Liu$^{39}$, Fang Liu$^{1}$, Feng Liu$^{6}$, H. B. Liu$^{13}$, H. M. Liu$^{1,47}$, Huanhuan Liu$^{1}$, Huihui Liu$^{17}$, J. B. Liu$^{55,43}$, J. Y. Liu$^{1,47}$, K. Liu$^{1}$, K. Y. Liu$^{31}$, Ke Liu$^{6}$, L. Liu$^{55,43}$, L. Y. Liu$^{13}$, Q. Liu$^{47}$, S. B. Liu$^{55,43}$, T. Liu$^{1,47}$, X. Liu$^{30}$, X. Y. Liu$^{1,47}$, Y. B. Liu$^{34}$, Z. A. Liu$^{1,43,47}$, Z. Q. Liu$^{37}$, Y. F. Long$^{35,l}$, X. C. Lou$^{1,43,47}$, H. J. Lu$^{18}$, J. D. Lu$^{1,47}$, J. G. Lu$^{1,43}$, Y. Lu$^{1}$, Y. P. Lu$^{1,43}$, C. L. Luo$^{32}$, M. X. Luo$^{62}$, P. W. Luo$^{44}$, T. Luo$^{9,j}$, X. L. Luo$^{1,43}$, S. Lusso$^{58C}$, X. R. Lyu$^{47}$, F. C. Ma$^{31}$, H. L. Ma$^{1}$, L. L. Ma$^{37}$, M. M. Ma$^{1,47}$, Q. M. Ma$^{1}$, X. N. Ma$^{34}$, X. X. Ma$^{1,47}$, X. Y. Ma$^{1,43}$, Y. M. Ma$^{37}$, F. E. Maas$^{15}$, M. Maggiora$^{58A,58C}$, S. Maldaner$^{26}$, S. Malde$^{53}$, Q. A. Malik$^{57}$, A. Mangoni$^{23B}$, Y. J. Mao$^{35,l}$, Z. P. Mao$^{1}$, S. Marcello$^{58A,58C}$, Z. X. Meng$^{49}$, J. G. Messchendorp$^{29}$, G. Mezzadri$^{24A}$, J. Min$^{1,43}$, T. J. Min$^{33}$, R. E. Mitchell$^{22}$, X. H. Mo$^{1,43,47}$, Y. J. Mo$^{6}$, C. Morales Morales$^{15}$, N. Yu. Muchnoi$^{10,e}$, H. Muramatsu$^{51}$, A. Mustafa$^{4}$, S. Nakhoul$^{11,h}$, Y. Nefedov$^{27}$, F. Nerling$^{11,h}$, I. B. Nikolaev$^{10,e}$, Z. Ning$^{1,43}$, S. Nisar$^{8,k}$, S. L. Olsen$^{47}$, Q. Ouyang$^{1,43,47}$, S. Pacetti$^{23B}$, Y. Pan$^{55,43}$, M. Papenbrock$^{59}$, P. Patteri$^{23A}$, M. Pelizaeus$^{4}$, H. P. Peng$^{55,43}$, K. Peters$^{11,h}$, J. Pettersson$^{59}$, J. L. Ping$^{32}$, R. G. Ping$^{1,47}$, A. Pitka$^{4}$, R. Poling$^{51}$, V. Prasad$^{55,43}$, H. R. Qi$^{2}$, H. R. Qi$^{45}$, M. Qi$^{33}$, T. Y. Qi$^{2}$, S. Qian$^{1,43}$, C. F. Qiao$^{47}$, N. Qin$^{60}$, X. P. Qin$^{13}$, X. S. Qin$^{4}$, Z. H. Qin$^{1,43}$, J. F. Qiu$^{1}$, S. Q. Qu$^{34}$, K. H. Rashid$^{57}$, K. Ravindran$^{21}$, C. F. Redmer$^{26}$, M. Richter$^{4}$, A. Rivetti$^{58C}$, V. Rodin$^{29}$, M. Rolo$^{58C}$, G. Rong$^{1,47}$, Ch. Rosner$^{15}$, M. Rump$^{52}$, A. Sarantsev$^{27,f}$, M. Savrié$^{24B}$, Y. Schelhaas$^{26}$, C. Schnier$^{4}$, K. Schoenning$^{59}$, W. Shan$^{19}$, X. Y. Shan$^{55,43}$, M. Shao$^{55,43}$, C. P. Shen$^{2}$, P. X. Shen$^{34}$, X. Y. Shen$^{1,47}$, H. Y. Sheng$^{1}$, X. Shi$^{1,43}$, X. D Shi$^{55,43}$, J. J. Song$^{37}$, Q. Q. Song$^{55,43}$, X. Y. Song$^{1}$, S. Sosio$^{58A,58C}$, C. Sowa$^{4}$, S. Spataro$^{58A,58C}$, F. F. Sui$^{37}$, G. X. Sun$^{1}$, J. F. Sun$^{16}$, L. Sun$^{60}$, S. S. Sun$^{1,47}$, Y. J. Sun$^{55,43}$, Y. K Sun$^{55,43}$, Y. Z. Sun$^{1}$, Z. J. Sun$^{1,43}$, Z. T. Sun$^{1}$, Y. X. Tan$^{55,43}$, C. J. Tang$^{40}$, G. Y. Tang$^{1}$, X. Tang$^{1}$, V. Thoren$^{59}$, B. Tsednee$^{25}$, I. Uman$^{46D}$, B. Wang$^{1}$, B. L. Wang$^{47}$, C. W. Wang$^{33}$, D. Y. Wang$^{35,l}$, K. Wang$^{1,43}$, L. L. Wang$^{1}$, L. S. Wang$^{1}$, M. Wang$^{37}$, M. Z. Wang$^{35,l}$, Meng Wang$^{1,47}$, P. L. Wang$^{1}$, W. P. Wang$^{55,43}$, X. Wang$^{35,l}$, X. F. Wang$^{1}$, X. L. Wang$^{9,j}$, Y. Wang$^{55,43}$, Y. Wang$^{44}$, Y. D. Wang$^{15}$, Y. F. Wang$^{1,43,47}$, Y. Q. Wang$^{1}$, Z. Wang$^{1,43}$, Z. G. Wang$^{1,43}$, Z. Y. Wang$^{1}$, Zongyuan Wang$^{1,47}$, T. Weber$^{4}$, D. H. Wei$^{12}$, P. Weidenkaff$^{26}$, F. Weidner$^{52}$, H. W. Wen$^{32,a}$, S. P. Wen$^{1}$, U. Wiedner$^{4}$, G. Wilkinson$^{53}$, M. Wolke$^{59}$, L. H. Wu$^{1}$, L. J. Wu$^{1,47}$, Z. Wu$^{1,43}$, L. Xia$^{55,43}$, S. Y. Xiao$^{1}$, Y. J. Xiao$^{1,47}$, Z. J. Xiao$^{32}$, Y. G. Xie$^{1,43}$, Y. H. Xie$^{6}$, T. Y. Xing$^{1,47}$, X. A. Xiong$^{1,47}$, G. F. Xu$^{1}$, J. J. Xu$^{33}$, Q. J. Xu$^{14}$, W. Xu$^{1,47}$, X. P. Xu$^{41}$, F. Yan$^{56}$, L. Yan$^{58A,58C}$, L. Yan$^{9,j}$, W. B. Yan$^{55,43}$, W. C. Yan$^{2}$, W. C. Yan$^{63}$, H. J. Yang$^{38,i}$, H. X. Yang$^{1}$, L. Yang$^{60}$, R. X. Yang$^{55,43}$, S. L. Yang$^{1,47}$, Y. H. Yang$^{33}$, Y. X. Yang$^{12}$, Yifan Yang$^{1,47}$, M. Ye$^{1,43}$, M. H. Ye$^{7}$, J. H. Yin$^{1}$, Z. Y. You$^{44}$, B. X. Yu$^{1,43,47}$, C. X. Yu$^{34}$, J. S. Yu$^{20}$, T. Yu$^{56}$, C. Z. Yuan$^{1,47}$, X. Q. Yuan$^{35,l}$, Y. Yuan$^{1}$, A. Yuncu$^{46B,b}$, A. A. Zafar$^{57}$, Y. Zeng$^{20}$, B. X. Zhang$^{1}$, B. Y. Zhang$^{1,43}$, C. C. Zhang$^{1}$, D. H. Zhang$^{1}$, H. H. Zhang$^{44}$, H. Y. Zhang$^{1,43}$, J. Zhang$^{1,47}$, J. L. Zhang$^{61}$, J. Q. Zhang$^{4}$, J. W. Zhang$^{1,43,47}$, J. Y. Zhang$^{1}$, J. Z. Zhang$^{1,47}$, K. Zhang$^{1,47}$, L. Zhang$^{1}$, Lei Zhang$^{33}$, S. F. Zhang$^{33}$, T. J. Zhang$^{38,i}$, X. Y. Zhang$^{37}$, Y. H. Zhang$^{1,43}$, Y. T. Zhang$^{55,43}$, Yan Zhang$^{55,43}$, Yao Zhang$^{1}$, Yi Zhang$^{9,j}$, Yu Zhang$^{47}$, Z. H. Zhang$^{6}$, Z. P. Zhang$^{55}$, Z. Y. Zhang$^{60}$, G. Zhao$^{1}$, J. W. Zhao$^{1,43}$, J. Y. Zhao$^{1,47}$, J. Z. Zhao$^{1,43}$, Lei Zhao$^{55,43}$, Ling Zhao$^{1}$, M. G. Zhao$^{34}$, Q. Zhao$^{1}$, S. J. Zhao$^{63}$, T. C. Zhao$^{1}$, Y. B. Zhao$^{1,43}$, Z. G. Zhao$^{55,43}$, A. Zhemchugov$^{27,c}$, B. Zheng$^{56}$, J. P. Zheng$^{1,43}$, Y. Zheng$^{35,l}$, Y. H. Zheng$^{47}$, B. Zhong$^{32}$, L. Zhou$^{1,43}$, L. P. Zhou$^{1,47}$, Q. Zhou$^{1,47}$, X. Zhou$^{60}$, X. K. Zhou$^{47}$, X. R. Zhou$^{55,43}$, A. N. Zhu$^{1,47}$, J. Zhu$^{34}$, K. Zhu$^{1}$, K. J. Zhu$^{1,43,47}$, S. H. Zhu$^{54}$, W. J. Zhu$^{34}$, X. L. Zhu$^{45}$, Y. C. Zhu$^{55,43}$, Y. S. Zhu$^{1,47}$, Z. A. Zhu$^{1,47}$, J. Zhuang$^{1,43}$, B. S. Zou$^{1}$, J. H. Zou$^{1}$\ (BESIII Collaboration)\ title: 'Observation of the $X(2370)$ and search for the $X(2120)$ in $J/\psi\to\gamma K\bar{K} \eta''$' --- INTRODUCTION ============ Quantum chromodynamics (QCD), a non-Abelian gauge field theory, predicts the existence of new types of hadrons with explicit gluonic degrees of freedom (e.g., glueballs, hybrids) [@bibg1; @bibg2; @bibg3]. The search for glueballs is an important field of research in hadron physics. It is, however, challenging since possible mixing of pure glueball states with nearby $q\bar{q}$ nonet mesons makes the identification of glueballs difficult in both experiment and theory. Lattice QCD (LQCD) predicts the lowest-lying glueballs which are scalar (mass 1.5$-$1.7 GeV/$c^2$), tensor (mass 2.3$-$2.4 GeV/$c^2$), and pseudoscalar (mass 2.3$-$2.6 GeV/$c^2$) [@bib3]. Radiative $J/\psi$ decay is a gluon-rich process and it is therefore regarded as one of the most promising hunting grounds for glueballs [@bibjpsi1; @bibjpsi2]. Recently, three states, the $X(1835)$, $X(2120)$ and $X(2370)$, are observed in the BESIII experiment in the $\pi^{+}\pi^{-}\eta'$ invariant-mass distribution through the decay of $J/\psi\to\gamma\pi^{+}\pi^{-}\eta'$ with statistical significances larger than 20$\sigma$, 7.2$\sigma$ and 6.4$\sigma$, respectively [@PRL1]. The measured mass of the $X(2370)$ is consistent with the pseudoscalar glueball candidate predicted by LQCD calculations [@bib3]. In the case of a pseudoscalar glueball, the branching fractions of the $X(2370)$ decaying into $KK\eta'$ and $\pi\pi\eta'$ are predicted to be 0.011 and 0.090 [@PRD1], respectively, in accordance with calculations that are based upon the chiral effective Lagrangian. Study on the decays to $K\bar{K}\eta'$ of the glue-ball candidate X states is helpful to identify their natures. In this paper, the $X(2370)$ as well as the $X(2120)$ are studied via the decays of $J/\psi\to\gamma K^{+}K^{-}\eta'$ and $J/\psi\to\gamma K_{S}^{0}K_{S}^{0}\eta'$($K_{S}^{0}\to\pi^{+}\pi^{-}$) using (1310.6$\pm$7.0)$\times$10$^6$ $J/\psi$ decays [@jpsinumber] collected with the BESIII detector in 2009 and 2012. Two $\eta'$ decay modes are used, namely $\eta'\to\gamma\rho^{0}(\rho^{0}\to\pi^{+}\pi^{-})$ and $\eta'\to\pi^{+}\pi^{-}\eta(\eta\to\gamma\gamma)$. DETECTOR AND MONTE CARLO SIMULATIONS ==================================== The BESIII detector is a magnetic spectrometer [@Ablikim:2009aa] located at the Beijing Electron Positron Collider II(BEPCII) [@Yu:IPAC2016-TUYA01]. The cylindrical core of the BESIII detector consists of a helium-based multilayer drift chamber (MDC), a plastic scintillator time-of-flight system (TOF), and a CsI(Tl) electromagnetic calorimeter (EMC), which are all enclosed in a superconducting solenoidal magnet providing a 1.0 T (0.9 T in 2012) magnetic field. The solenoid is supported by an octagonal flux-return yoke with resistive plate counter muon identifier modules interleaved with steel. The acceptance of charged particles and photons is 93% over $4\pi$ solid angle. The charged-particle momentum resolution at $1~{\rm GeV}/c$ is $0.5\%$, and the $dE/dx$ resolution is $6\%$ for the electrons from Bhabha scattering. The EMC measures photon energies with a resolution of $2.5\%$ ($5\%$) at $1$ GeV in the barrel (end cap) region. The time resolution of the TOF barrel part is 68 ps, while that of the end cap part is 110 ps. Simulated samples produced with the [geant4]{}-based [@geant4] Monte Carlo (MC) package which includes the geometric description of the BESIII detector and the detector response, are used to determine the detection efficiency and to estimate the backgrounds. The simulation includes the beam energy spread and initial-state radiation (ISR) in the $e^+e^-$ annihilations modeled with the generator [kkmc]{} [@ref:kkmc]. The inclusive MC sample consists of the production of the $J/\psi$ resonance, and the continuum processes incorporated in [kkmc]{} [@ref:kkmc]. The known decay modes are modeled with [evtgen]{} [@ref:evtgen] using branching fractions taken from the Particle Data Group [@pdg], and the remaining unknown decays from the charmonium states are generated with [lundcharm]{} [@ref:lundcharm]. The final-state radiations (FSR) from charged final-state particles are incorporated with the [photos]{} package [@photos]. Background is studied using a sample of $1.2 $$\times$$ 10^{9}$ simulated ${J/\psi}$ events. Phase-space (PHSP) MC samples of ${J/\psi}\to\gamma {K^{+}K^{-}}\eta'$ and ${J/\psi}\to\gamma {K_{S}^{0}K_{S}^{0}}\eta'$ are generated to describe the nonresonant contribution. To estimate the selection efficiency and to optimize the selection criteria, signal MC events are generated for ${J/\psi}\to\gamma X(2120)/X(2370) \to\gamma{K^{+}K^{-}}\eta'$ and ${J/\psi}\to\gamma X(2120)/X(2370)\to\gamma{K_{S}^{0}K_{S}^{0}}\eta'$ channel, respectively. The polar angle of the photon in the ${J/\psi}$ center-of-mass system, $\theta_{\gamma}$, follows a $1+ \mathrm{cos}^{2}\theta_{\gamma}$ function. For the process of $\eta'\to\gamma\rho^{0}, \rho^{0}\to\pi^{+}\pi^{-}$, a generator taking into account both the $\rho - \omega$ interference and the box anomaly is used [@gammapipiDIY]. The analysis is performed in the framework of the BESIII offline software system (BOSS) [@ref:boss] incorporating the detector calibration, event reconstruction and data storage. EVENT SELECTION =============== Charged-particle tracks in the polar angle range $\|\cos\theta\| < 0.93$ are reconstructed from hits in the MDC. Tracks (excluding those from $K_{S}^{0}$ decays) are selected that extrapolated to be within 10 cm from the interaction point in the beam direction and 1 cm in the plane perpendicular to the beam. The combined information from energy-loss ($dE/dx$) measurements in the MDC and time in the TOF is used to obtain confidence levels for particle identification (PID) for $\pi$, $K$ and $p$ hypotheses. For ${J/\psi}\to\gamma{K^{+}K^{-}}\eta'$ decay, each track is assigned to the particle type corresponding to the highest confidence level; candidate events are required to have four charged tracks with zero net charge and with two opposite charged tracks identified as kaons and the other two identified as pions. For the ${J/\psi}\to\gamma{K_{S}^{0}K_{S}^{0}}\eta'$ decay, each track is assumed to be a pion and no PID restrictions are applied; candidate events are required to have six charged tracks with zero net charge. ${K_{S}^{0}}$ candidates are reconstructed from a secondary vertex fit to all ${\pi^{+}\pi^{-}}$ pairs, and each ${K_{S}^{0}}$ candidate is required to satisfy $\|M_{{\pi^{+}}{\pi^{-}}}-m_{{K_{S}^{0}}}\|<$9 MeV/$c^{2}$, where $m_{{K_{S}^{0}}}$ is the nominal mass of the ${K_{S}^{0}}$ [@pdg]. The reconstructed ${K_{S}^{0}}$ candidates are used as an input for the subsequent kinematic fit. Photon candidates are required to have an energy deposition above 25 MeV in the barrel region ($\|\cos\theta\|<0.80$) and 50 MeV in the end cap ($0.86<\|\cos\theta\|<0.92$). To exclude showers from charged tracks, the angle between the shower position and the charged tracks extrapolated to the EMC must be greater than $5^{\circ}$. A timing requirement in the EMC is used to suppress electronic noise and energy deposits unrelated to the event. At least two (three) photons are required for the $\eta'\to\gamma\rho^{0}$ ($\eta'\to{\pi^{+}\pi^{-}\eta}$) mode. For the ${J/\psi}\to\gamma{K^{+}K^{-}}\eta' (\eta'\to\gamma\rho^{0})$ channel, a four-constraint (4C) kinematic fit is performed by requiring the total energy and each momentum component to be conserved to the hypothesis of ${J/\psi}\to\gamma\gamma{K^{+}K^{-}}{\pi^{+}}{\pi^{-}}$. For events with more than two photon candidates, the combination with the minimum $\chi_{4C}^{2}$ is selected, and $\chi_{4C}^{2}<$ 25 is required. Events with $\|M_{\gamma\gamma} - m_{\pi^{0}}\| <$ 30 MeV/$c^{2}$ or $\|M_{\gamma\gamma} - m_{\eta}\| <$ 30 MeV/$c^{2}$ are rejected to suppress background containing $\pi^{0}$ or $\eta$, where the $m_{\pi^{0}}$ and $m_{\eta}$ are the nominal masses of $\pi^{0}$ and $\eta$ [@pdg]. A clear $\eta'$ signal is observed in the invariant-mass distribution of ${\gamma\pi^{+}\pi^{-}}$ ($M_{\gamma\pi^{+}\pi^{-}}$), as shown in Fig. \[selectkketap\](a). Candidates of $\rho$ and $\eta'$ are reconstructed from the ${\pi^{+}\pi^{-}}$ and ${\gamma\pi^{+}\pi^{-}}$ combinations with 0.55 GeV/$c^{2} < M_{\pi^{+}\pi^{-}} < $ 0.85 GeV/$c^{2}$ and $\|M_{\gamma\pi^{+}\pi^{-}} - m_{\eta'}\| <$ 20 MeV/$c^{2}$, where $m_{\eta'}$ is the nominal mass of $\eta'$ [@pdg], respectively. If there are more than one combination satisfing the selection criteria, the combination with $M_{\gamma\pi^{+}\pi^{-}}$ closest to $m_{\eta'}$ is selected. After applying the above requirements, we obtain the invariant-mass distribution of ${K^{+}K^{-}}\eta'$ ($M_{{K^{+}K^{-}}\eta'}$) as shown in Fig. \[selectkketap\](b). ![Invariant-mass distributions for the selected candidates of ${J/\psi}\to\gamma{K^{+}K^{-}}\eta'$. Plots (a) and (b) are invariant-mass distributions of ${\gamma\pi^{+}\pi^{-}}$ and ${K^{+}K^{-}}\eta'$ for $\eta'\to\gamma\rho^{0}$, $\rho^{0}\to\pi^{+}\pi^{-}$, respectively; plots (c) and (d) are the invariant-mass distributions of ${\pi^{+}\pi^{-}\eta}$ and ${K^{+}K^{-}}\eta'$ for $\eta'\to{\pi^{+}\pi^{-}\eta}$, $\eta\to\gamma\gamma$, respectively. The dots with error bars correspond to data and the histograms are the results of PHSP MC simulations (arbitrary normalization).[]{data-label="selectkketap"}](Fig01a.eps "fig:"){width="24.00000%"}(-30,85)[[(a)]{}]{} ![Invariant-mass distributions for the selected candidates of ${J/\psi}\to\gamma{K^{+}K^{-}}\eta'$. Plots (a) and (b) are invariant-mass distributions of ${\gamma\pi^{+}\pi^{-}}$ and ${K^{+}K^{-}}\eta'$ for $\eta'\to\gamma\rho^{0}$, $\rho^{0}\to\pi^{+}\pi^{-}$, respectively; plots (c) and (d) are the invariant-mass distributions of ${\pi^{+}\pi^{-}\eta}$ and ${K^{+}K^{-}}\eta'$ for $\eta'\to{\pi^{+}\pi^{-}\eta}$, $\eta\to\gamma\gamma$, respectively. The dots with error bars correspond to data and the histograms are the results of PHSP MC simulations (arbitrary normalization).[]{data-label="selectkketap"}](Fig01b.eps "fig:"){width="24.00000%"}(-30,85)[[(b)]{}]{} -0.03cm ![Invariant-mass distributions for the selected candidates of ${J/\psi}\to\gamma{K^{+}K^{-}}\eta'$. Plots (a) and (b) are invariant-mass distributions of ${\gamma\pi^{+}\pi^{-}}$ and ${K^{+}K^{-}}\eta'$ for $\eta'\to\gamma\rho^{0}$, $\rho^{0}\to\pi^{+}\pi^{-}$, respectively; plots (c) and (d) are the invariant-mass distributions of ${\pi^{+}\pi^{-}\eta}$ and ${K^{+}K^{-}}\eta'$ for $\eta'\to{\pi^{+}\pi^{-}\eta}$, $\eta\to\gamma\gamma$, respectively. The dots with error bars correspond to data and the histograms are the results of PHSP MC simulations (arbitrary normalization).[]{data-label="selectkketap"}](Fig01c.eps "fig:"){width="24.00000%"}(-30,85)[[(c)]{}]{} ![Invariant-mass distributions for the selected candidates of ${J/\psi}\to\gamma{K^{+}K^{-}}\eta'$. Plots (a) and (b) are invariant-mass distributions of ${\gamma\pi^{+}\pi^{-}}$ and ${K^{+}K^{-}}\eta'$ for $\eta'\to\gamma\rho^{0}$, $\rho^{0}\to\pi^{+}\pi^{-}$, respectively; plots (c) and (d) are the invariant-mass distributions of ${\pi^{+}\pi^{-}\eta}$ and ${K^{+}K^{-}}\eta'$ for $\eta'\to{\pi^{+}\pi^{-}\eta}$, $\eta\to\gamma\gamma$, respectively. The dots with error bars correspond to data and the histograms are the results of PHSP MC simulations (arbitrary normalization).[]{data-label="selectkketap"}](Fig01d.eps "fig:"){width="24.00000%"}(-30,85)[[(d)]{}]{} To reduce background and to improve the mass resolution of the ${J/\psi}\to\gamma{K^{+}K^{-}}\eta' (\eta'\to{\pi^{+}\pi^{-}\eta})$ channel, a five-constraint (5C) kinematic fit is performed whereby the total four momentum of the final-state particles are constrained to the total initial four momentum of the colliding beams and the invariant mass of the two photons from the decay of the $\eta$ is constrained to its nominal mass. If there are more than three photon candidates, the combination with the minimum $\chi_{5C}^{2}$ is retained, and $\chi_{5C}^{2} < $ 45 is required. To suppress background from $\pi^{0}\to\gamma\gamma$, $\|M_{\gamma\gamma} - m_{\pi^{0}}\| >$ 30 MeV/$c^{2}$ is required for all photon pairs. The $\eta'$ candidates are formed from the ${\pi^{+}\pi^{-}\eta}$ combination satisfying $\|M_{\pi^{+}\pi^{-}\eta} - m_{\eta'}\| <$ 15 MeV/$c^{2}$, where $M_{\pi^{+}\pi^{-}\eta}$ is the invariant mass of $\pi^{+}\pi^{-}\eta$, as shown in Fig. \[selectkketap\](c). After applying the mass restrictions, we obtain the invariant-mass distribution of ${K^{+}K^{-}}\eta'$($\eta'\to{\pi^{+}\pi^{-}\eta}$) as shown in Fig. \[selectkketap\](d). ![Invariant-mass distributions for the selected ${J/\psi}\to\gamma{K_{S}^{0}K_{S}^{0}}\eta'$ candidate events. (a) and (b) are the invariant-mass distributions of ${\gamma\pi^{+}\pi^{-}}$ and ${K_{S}^{0}K_{S}^{0}}\eta'$ for $\eta'\to\gamma\rho^{0}$, $\rho^{0}\to\pi^{+}\pi^{-}$, respectively; (c) and (d) are the invariant-mass distribution of ${\pi^{+}\pi^{-}\eta}$ and ${K_{S}^{0}K_{S}^{0}}\eta'$ for $\eta'\to{\pi^{+}\pi^{-}\eta}$, $\eta\to\gamma\gamma$, respectively. The dots with error bars represent the data and the histograms are the results of PHSP MC simulations (arbitrary normalization).[]{data-label="selectksksetap"}](Fig02a.eps "fig:"){width="24.00000%"}(-30,85)[[(a)]{}]{} ![Invariant-mass distributions for the selected ${J/\psi}\to\gamma{K_{S}^{0}K_{S}^{0}}\eta'$ candidate events. (a) and (b) are the invariant-mass distributions of ${\gamma\pi^{+}\pi^{-}}$ and ${K_{S}^{0}K_{S}^{0}}\eta'$ for $\eta'\to\gamma\rho^{0}$, $\rho^{0}\to\pi^{+}\pi^{-}$, respectively; (c) and (d) are the invariant-mass distribution of ${\pi^{+}\pi^{-}\eta}$ and ${K_{S}^{0}K_{S}^{0}}\eta'$ for $\eta'\to{\pi^{+}\pi^{-}\eta}$, $\eta\to\gamma\gamma$, respectively. The dots with error bars represent the data and the histograms are the results of PHSP MC simulations (arbitrary normalization).[]{data-label="selectksksetap"}](Fig02b.eps "fig:"){width="24.00000%"}(-30,85)[[(b)]{}]{} -0.03cm ![Invariant-mass distributions for the selected ${J/\psi}\to\gamma{K_{S}^{0}K_{S}^{0}}\eta'$ candidate events. (a) and (b) are the invariant-mass distributions of ${\gamma\pi^{+}\pi^{-}}$ and ${K_{S}^{0}K_{S}^{0}}\eta'$ for $\eta'\to\gamma\rho^{0}$, $\rho^{0}\to\pi^{+}\pi^{-}$, respectively; (c) and (d) are the invariant-mass distribution of ${\pi^{+}\pi^{-}\eta}$ and ${K_{S}^{0}K_{S}^{0}}\eta'$ for $\eta'\to{\pi^{+}\pi^{-}\eta}$, $\eta\to\gamma\gamma$, respectively. The dots with error bars represent the data and the histograms are the results of PHSP MC simulations (arbitrary normalization).[]{data-label="selectksksetap"}](Fig02c.eps "fig:"){width="24.00000%"}(-30,85)[[(c)]{}]{} ![Invariant-mass distributions for the selected ${J/\psi}\to\gamma{K_{S}^{0}K_{S}^{0}}\eta'$ candidate events. (a) and (b) are the invariant-mass distributions of ${\gamma\pi^{+}\pi^{-}}$ and ${K_{S}^{0}K_{S}^{0}}\eta'$ for $\eta'\to\gamma\rho^{0}$, $\rho^{0}\to\pi^{+}\pi^{-}$, respectively; (c) and (d) are the invariant-mass distribution of ${\pi^{+}\pi^{-}\eta}$ and ${K_{S}^{0}K_{S}^{0}}\eta'$ for $\eta'\to{\pi^{+}\pi^{-}\eta}$, $\eta\to\gamma\gamma$, respectively. The dots with error bars represent the data and the histograms are the results of PHSP MC simulations (arbitrary normalization).[]{data-label="selectksksetap"}](Fig02d.eps "fig:"){width="24.00000%"}(-30,85)[[(d)]{}]{} ![image](Fig03a.eps){width="45.00000%"}(-30,145)[[(a)]{}]{} ![image](Fig03b.eps){width="45.00000%"}(-30,145)[[(b)]{}]{} -0.03cm ![image](Fig03c.eps){width="45.00000%"}(-30,145)[[(c)]{}]{} ![image](Fig03d.eps){width="45.00000%"}(-30,145)[[(d)]{}]{} For the ${J/\psi}\to\gamma{K_{S}^{0}K_{S}^{0}}\eta' (\eta'\to\gamma\rho^{0})$ channel, the $\gamma \gamma {K_{S}^{0}K_{S}^{0}}\pi^+\pi^-$ candidates are subjected to a 4C kinematic fit. For events with more than two photons or two $K^{0}_{S}$ candidates, the combination with the smallest $\chi^{2}_{4C}$ is retained, and $\chi^{2}_{4C} < 45$ is required. To suppress background events containing a $\pi^{0}$ or $\eta$, events with $\|M_{\gamma\gamma} - m_{\pi^{0}}\| <$ 30 MeV/$c^{2}$ or $\|M_{\gamma\gamma} - m_{\eta}\| <$ 30 GeV/$c^{2}$ are rejected. The ${\pi^{+}\pi^{-}}$ invariant mass is required to be in the $\rho$ mass region, 0.55 GeV/$c^{2} < M_{\pi^{+}\pi^{-}} < $ 0.85 GeV/$c^{2}$, and $\|M_{\gamma\pi^{+}\pi^{-}} - m_{\eta'}\| <$ 20 MeV/$c^{2}$ is applied to select $\eta'$ signal. If more than one combination of $\gamma\pi^{+}\pi^{-}$ are obtained, the combination with $M_{\gamma\pi^{+}\pi^{-}}$ closest to $m_{\eta'}$ is selected as shown in Fig. \[selectksksetap\](a). After applying the above requirements, we obtain the ${K_{S}^{0}K_{S}^{0}}\eta'$($\eta'\to\gamma\rho^{0}$) invariant-mass spectrum as illustrated in Fig. \[selectksksetap\](b). Candidate events of the ${J/\psi}\to\gamma{K_{S}^{0}K_{S}^{0}}\eta'$ $(\eta'\to{\pi^{+}\pi^{-}\eta})$ channel are subjected to a 5C kinematic fit, which is similar to that for the ${J/\psi}\to\gamma{K^{+}K^{-}}\eta'$ $(\eta'\to{\pi^{+}\pi^{-}\eta})$ mode. If there are more than three photons or more than two $K^{0}_{S}$ candidates, only the combination with the minimum $\chi_{5C}^{2}$ is selected and $\chi_{5C}^{2} < $50 is required. To reduce the combinatorial background from $\pi^{0}\to\gamma\gamma$ events, $\|M_{\gamma\gamma} - m_{\pi^{0}}\| >$ 30 MeV/$c^{2}$ is required for all photon pairs. For selecting the $\eta'$ signal, the ${\pi^{+}\pi^{-}\eta}$ combination satisfying $\|M_{\pi^{+}\pi^{-}\eta} - m_{\eta'}\| <$ 15 MeV/$c^{2}$ is required, as shown in Fig. \[selectksksetap\](c). After applying the above selection criteria, we obtain the invariant-mass distribution of ${K_{S}^{0}K_{S}^{0}}\eta'$$(\eta'\to{\pi^{+}\pi^{-}\eta})$ events as shown in Fig. \[selectksksetap\](d). SIGNAL EXTRACTION ================= Potential backgrounds are studied using an inclusive MC sample of $1.2\times10^{9}$ ${J/\psi}$ decays. No significant peaking background is identified in the invariant-mass distributions of ${K^{+}K^{-}}\eta'$ and ${K_{S}^{0}K_{S}^{0}}\eta'$. Non-$\eta'$ processes are studied using the $\eta'$ mass sidebands. The major background in the decay ${J/\psi}\to\gamma{K^{+}K^{-}}\eta'$ stem from ${J/\psi}\to K^{+} K^{-}\eta' (K^{+}\to K^{+}\pi^{0}) + c.c.$. The contribution of ${J/\psi}\to K^{+} K^{-}\eta' (K^{+}\to K^{+}\pi^{0}) + c.c.$ is estimated by the background-subtracted ${K^{+}K^{-}}\eta'$ spectrum of ${J/\psi}\to K^{+} K^{-}\eta' (K^{+}\to K^{+}\pi^{0}) + c.c.$ events selected from data. The spectrum is reweighted according to the ratio of efficiency of ${J/\psi}\to\gamma{K^{+}K^{-}}\eta'$ and ${J/\psi}\to K^{+} K^{-}\eta' (K^{+}\to K^{+}\pi^{0}) + c.c.$. For the ${J/\psi}\to\gamma{K_{S}^{0}K_{S}^{0}}\eta'$ case, backgrounds from the process ${J/\psi}\to{\pi^{0}}{K_{S}^{0}K_{S}^{0}}\eta'$ are negligible, as it is forbidden due to charge conjugation invariance. A structure near 2.34 GeV/$c^{2}$ is observed in the invariant-mass distribution of ${K^{+}K^{-}}\eta'$ and ${K_{S}^{0}K_{S}^{0}}\eta'$. We performed a simultaneous unbinned maximum-likelihood fit to the ${K^{+}K^{-}}\eta'$ and ${K_{S}^{0}K_{S}^{0}}\eta'$ invariant-mass distributions between 2.0 and 2.7 GeV/$c^{2}$, as shown in Fig \[fit2370\]. The signal is represented by an efficiency-weighted non-relativistic Breit-Wigner (BW) function convolved with a double Gaussian function to account for the mass resolution. The mass and width of BW function are left free in the fit while the parameters of the double Gaussian function are fixed on the results obtained from the fit of signal MC samples generated with zero width. The non-$\eta'$ background events are described with $\eta'$ sideband data and the yields from these sources are fixed; the ${J/\psi}\to K^{+}K^{-}\eta' + c.c.$ contributions in ${J/\psi}\to\gamma{K^{+}K^{-}}\eta'$ decay channel are studied as discussed above and its shape as well as the yields are fixed in the fit; the contribution from the nonresonant $\gamma K\bar{K}\eta'$ production is described by the shape from the PHSP MC sample of $J/\psi\to\gamma K\bar{K} \eta'$ and its absolute yield is set as a free parameter in the fit; the remaining background is described by a second order Chebychev polynomial function and its parameters are left to be free. In the simultaneous fit, the resonance parameters are free parameters and constrained to be the same for all four channels. The signal ratio for the two $\eta'$ decay modes is fixed with a factor calculated by their branching fractions and efficiencies. The signal ratio between ${J/\psi}\to\gamma X(2370)\to\gamma{K^{+}K^{-}}\eta'$ and ${J/\psi}\to\gamma X(2370)\to\gamma{K_{S}^{0}K_{S}^{0}}\eta'$ is a free parameter in the fit. The obtained mass, width and the number of signal events for the $X(2370)$ are listed in Table \[fitresult2370\]. A variety of fits with different fit ranges, $\eta'$ sideband regions and background shapes are performed, and the smallest statistical significance among these fits is found to be 8.3$\sigma$. With the detection efficiencies listed in Table \[effi\_all\], the product branching fractions for ${J/\psi}\to \gamma X(2370),X(2370)\to {K^{+}K^{-}}\eta'$ and ${J/\psi}\to \gamma X(2370),X(2370)\to {K_{S}^{0}K_{S}^{0}}\eta'$ are determined to be $(1.79\pm0.23)\times10^{-5}$ and $(1.18\pm0.32)\times10^{-5}$, respectively, where the uncertainties are statistical only. $\eta'\to\gamma\rho^{0}$ $\eta'\to\pi^{+}\pi^{-}\eta$ ------------------------------------ -------------------------- ------------------------------ $M_{X(2370)}$ (MeV/$c^{2}$) $\Gamma_{X(2370)}$ (MeV) $N(J/\psi \to \gamma X(2370)^{a})$ $882\pm112$ $320\pm40$ $N(J/\psi \to \gamma X(2370)^{b})$ $174\pm47\phantom{2}$ $55\pm15$ $N(J/\psi \to \gamma X(2120)^{a})$ $<553.5$ $<187.3$ $N(J/\psi \to \gamma X(2120)^{b})$ $<88.7$ $<30.0$ : \[fitresult\]Fit results for the structure around 2.34 GeV/$c^{2}$ and 2.12 GeV/$c^{2}$. The superscripts $a$ and $b$ represent the decay modes of $ X\to K^{+}K^{-}\eta'$ and $X\to K_{S}^{0}K_{S}^{0}\eta'$, respectively. The uncertainties are statistical only.[]{data-label="fitresult2370"} Decay modes $\varepsilon_{\eta'\to\gamma\rho^{0}}$ $\varepsilon_{\eta'\to\pi^{+}\pi^{-}\eta}$ --------------------------------- ---------------------------------------- -------------------------------------------- $J/\psi \to \gamma X(2370)^{a}$ 12.9 % 8.0 % $J/\psi \to \gamma X(2370)^{b}$ 8.1 % 4.4 % $J/\psi \to \gamma X(2120)^{a}$ 10.3 % 6.0 % $J/\psi \to \gamma X(2120)^{b}$ 7.9 % 4.6 % : Summary of the MC detection efficiencies of the signal yields for the two $\eta'$ modes where the $K\bar{K} \eta'$ invariant-mass is constrained to the applied fitting range between 2.0 and 2.7 GeV/$c^{2}$. The superscripts $a$ and $b$ represent the decay modes of $ X\to K^{+}K^{-}\eta'$ and $X\to K_{S}^{0}K_{S}^{0}\eta'$, respectively.[]{data-label="effi_all"} There is no obvious signal of the $X(2120)$ found in the $K\bar{K}\eta'$ invariant-mass distribution. We performed a simultaneous unbinned maximum-likelihood fit to the $K\bar{K}\eta'$ invariant-mass distribution in the range of \[2.0, 2.7\] GeV/$c^2$. The signal, $X(2120)$, is described with an efficiency-weighted BW function convolved with a double Gaussian function. The mass and width of the BW function are fixed to previously published BESIII results [@PRL1]. The backgrounds are modeled with the same components as used in the fit of the $X(2370)$ as mentioned above. The contribution from the $X(2370)$ is included in the fit and its mass, width and the number of events are set free. The distribution of normalized likelihood values for a series of input signal event yields is taken as the probability density function (PDF) for the expected number of events. The number of events at 90$\%$ of the integral of the PDF from zero to the given number of events is defined as the upper limit, $N^{UL}$, at the 90$\%$ confidence level (C.L.). We repeated this procedure with different signal shape parameters of $X(2120)$ (by varying the values of mass and width with 1$\sigma$ of the uncertainties cited from [@PRL1]), fit ranges, $\eta'$ sideband regions and background shapes, and the maximum upper limit among these cases is selected. The statistical significance of the $X(2120)$ is determined to be 2.2$\sigma$. To calculate $N^{UL}$ for the ${J/\psi}\to\gamma X(2120)\to\gamma{K^{+}K^{-}}\eta'$ (${J/\psi}\to\gamma X(2120)\to\gamma{K_{S}^{0}K_{S}^{0}}\eta'$) channel, the number of signal events for ${J/\psi}\to\gamma X(2120)\to\gamma{K_{S}^{0}K_{S}^{0}}\eta'$ (${J/\psi}\to\gamma X(2120)\to\gamma{K^{+}K^{-}}\eta'$) channel is left free. The obtained upper limits of the signal yields are listed in Table \[fitresult2370\], and the upper limit for the product branching fractions are calculated to be $\mathcal{B}({J/\psi}\to\gamma X(2120)\to\gamma K^{+} K^{-} \eta') < 1.41\times10^{-5}$ and $\mathcal{B}({J/\psi}\to\gamma X(2120)\to\gamma K_{S}^{0} K_{S}^{0} \eta') < 6.15\times10^{-6}$, respectively. ----------------------------------- ----- ---------- ----- ---------- ----- ---------- ----- ---------- $M$ $\Gamma$ $M$ $\Gamma$ $M$ $\Gamma$ $M$ $\Gamma$ Veto $\pi^{0}$ 0.0 1 0.3 1 0.2 1 0.2 1 Veto $\eta$ 0.2 1 – – 0.2 1 – – Fit range 0.1 3 0.1 3 0.1 3 0.1 3 Sideband region 0.1 2 0.1 2 0.2 1 0.1 1 Chebychev function 0.2 3 0.1 3 0.2 1 0.1 3 $J/\psi\to K^{+}K^{-}\eta'+c.c.$ 0.2 5 0.2 5 0.2 5 0.2 5 X(2120)\* 5.7 7 5.7 7 5.7 7 5.7 7 Total 5.7 10 5.7 10 5.7 9 5.7 10 ----------------------------------- ----- ---------- ----- ---------- ----- ---------- ----- ---------- SYSTEMATIC UNCERTAINTIES {#sec::sys} ======================== Several sources of systematic uncertainties are considered for the determination of the mass and width of the $X(2370)$ and the product branching fractions. These include the efficiency differences between data and MC simulation in the MDC tracking, PID, the photon detection, ${K_{S}^{0}}$ reconstruction, the kinematic fitting, and the mass window requirements of $\pi^{0}$, $\eta$, $\rho$ and $\eta'$. Furthermore, uncertainties associated with the fit ranges, the background shapes, the sideband regions, the signal shape parameters of $X(2120)$, intermediate resonance decay branching fractions and the total number of ${J/\psi}$ events are considered. ----------------------------------------------------------------- ------ ------ ------ ------ I II I II MDC tracking\* 4.0 4.0 2.0 2.0 Photon detection\* 2.0 3.0 2.0 3.0 $K_{S}^{0}$ reconstruction\* – – 3.0 3.0 PID\* 4.0 4.0 – – Kinematic fit 1.7 1.0 3.8 2.2 $\rho$ mass window 0.2 – 0.3 – $\eta'$ mass window 0.1 0.4 0.1 0.3 Veto $\pi^{0}$ 1.2 1.6 1.7 0.6 Veto $\eta$ 1.0 – 0.6 – Fit range 2.4 2.4 1.7 1.7 Sideband region 5.4 2.8 2.8 1.2 Chebychev function 4.9 5.5 1.7 1.7 $J/\psi\to K^{+}K^{-}\eta'+c.c.$ 4.0 4.0 2.2 2.2 $\mathcal{B}(\eta'\to \gamma\rho^{0}\to\gamma{\pi^{+}\pi^{-}})$ 1.7 – 1.7 – $\mathcal{B}(\eta'\to\eta{\pi^{+}\pi^{-}})$ – 1.6 – 1.6 $\mathcal{B}(\eta\to\gamma\gamma)$ – 0.5 – 0.5 $\mathcal{B}(K_{S}^{0}\to{\pi^{+}\pi^{-}})$\ – – 0.1 0.1 Number of ${J/\psi}$ events\* 0.5 0.5 0.5 0.5 Quantum number of $X$ 16.7 13.6 16.0 19.0 X(2120)\* 33.7 33.7 30.5 30.5 Total 39.2 37.7 35.3 36.5 ----------------------------------------------------------------- ------ ------ ------ ------ : Systematic uncertainties for determination of the branching fraction of ${J/\psi}\to\gamma X(2370)\to\gamma K \bar{K} \eta'$ (in %). The items with \* are common uncertainties of both $\eta'$ decay modes. I and II represent the decay modes of $ \eta'\to\gamma\rho^{0}$, $\rho^{0}\to\pi^{+}\pi^{-}$ and $\eta'\to{\pi^{+}\pi^{-}\eta}$, $\eta\to\gamma\gamma$, respectively. []{data-label="sumsysBR2370"} ----------------------------------------------------------------- ------ ------ ------ ------ I II I II MDC tracking\* 4.0 4.0 2.0 2.0 Photon detection\* 2.0 3.0 2.0 3.0 $K_{S}^{0}$ reconstruction\* – – 3.0 3.0 PID\* 4.0 4.0 – – Kinematic fit 1.7 0.8 4.0 3.5 $\rho$ mass window 0.2 – 0.3 – $\eta'$ mass window 0.1 0.1 0.2 0.2 Veto $\pi^{0}$ 0.8 1.0 1.4 1.5 Veto $\eta$ 0.8 – 1.4 – $\mathcal{B}(\eta'\to \gamma\rho^{0}\to\gamma{\pi^{+}\pi^{-}})$ 1.7 – 1.7 – $\mathcal{B}(\eta'\to\eta{\pi^{+}\pi^{-}})$ – 1.6 – 1.6 $\mathcal{B}(\eta\to\gamma\gamma)$ – 0.5 – 0.5 $\mathcal{B}(K_{S}^{0}\to{\pi^{+}\pi^{-}})$\* – – 0.1 0.1 Number of ${J/\psi}$ events\* 0.5 0.5 0.5 0.5 Quantum number of $X$ 18.2 16.4 20.9 19.3 Total 19.3 17.6 21.8 20.2 ----------------------------------------------------------------- ------ ------ ------ ------ : Systematic uncertainties for the determination of the upper limit of the branching fraction of ${J/\psi}\to\gamma X(2120)\to\gamma K \bar{K} \eta'$(in %). The items with \* are common uncertainties of both $\eta'$ decay modes. I and II represent the decay modes of $ \eta'\to\gamma\rho^{0}$, $\rho^{0}\to\pi^{+}\pi^{-}$ and $\eta'\to{\pi^{+}\pi^{-}\eta}$, $\eta\to\gamma\gamma$, respectively.[]{data-label="sumsysBR2120"} Efficiency estimation --------------------- The MDC tracking efficiencies of charged pions and kaons are investigated using nearly background-free (clean) control samples of ${J/\psi}\to p\bar{p}{\pi^{+}\pi^{-}}$ and ${J/\psi}\to{K_{S}^{0}}K^{\pm}\pi^{\mp}$ [@MDCpi; @MDC], respectively. The difference in tracking efficiencies between data and MC is 1.0% for each charged pion and kaon. The photon detection efficiency is studied with a clean sample of $J/\psi\to\rho^{0}\pi^{0}$ [@Photon], and the result shows that the difference of photon detection efficiencies between data and MC simulation is 1.0% for each photon. The systematic uncertainty from $K_{S}^{0}$ reconstruction is determined from the control samples of ${J/\psi}\to K^{\pm}K^{\mp}$ and ${J/\psi}\to\phi{K_{S}^{0}}K^{\pm}\pi^{\mp}$, which indicate that the efficiency difference between data and MC is less than 1.5$\%$ for each ${K_{S}^{0}}$. Therefore, 3.0$\%$ is taken as the systematic uncertainty for the two ${K_{S}^{0}}$ in ${J/\psi}\to\gamma{K_{S}^{0}K_{S}^{0}}\eta'$ channel. For the decay channel of $J/\psi\to\gamma K^{+}K^{-}\eta'$, the PID has been used to identify the kaons and pions. Using a clean sample of $J/\psi\to p\bar{p}\pi^{+}\pi^{-}$, the PID efficiency of $\pi^{+}/\pi^{-}$ has been studied, which indicates that the $\pi^{+}/\pi^{-}$ PID efficiency for data agrees with MC simulation within 1%. The PID efficiency for the kaon is measured with a clean sample of $J/\psi\rightarrow K^+ K^-\eta$. The difference of the PID efficiency between data and MC is less than 1% for each kaon. Hence, In this analysis, four charged tracks are required to be identified as two pions and two kaons, 4% is taken as the systematic uncertainty associated with the PID. The systematic uncertainties associated with the kinematic fit are studied with the track helix parameter correction method, as described in Ref. [@4cError]. The differences with respect to those without corrections are taken as systematic uncertainties. Due to the difference in the mass resolution between data and MC, uncertainties related to the $\rho^{0}$ and $\eta'$ mass window requirements are investigated by smearing the MC simulation to improve the consistency between data and MC simulation. The differences in the detection efficiency before and after smearing are assigned as systematic uncertainties for the $\rho^{0}$ and $\eta'$ mass window requirement. The uncertainties from the $\pi^{0}$ and $\eta$ mass-window requirements are estimated by varying the mass windows of $\pi^{0}$ and $\eta$, and differences in the resulting branching fractions are assigned as the systematic uncertainties of this item. Furthermore, we considered the effects arising from different quantum numbers of the $X(2120)$ and $X(2370)$. We generated ${J/\psi}\rightarrow\gamma X(2120)$ and ${J/\psi}\rightarrow\gamma X(2370)$ decays following a $\mathrm{sin}^2\theta_{\gamma}$ angular distribution. The resulting differences in efficiency with respect to the nominal value are taken as systematic uncertainties. Fit to the signal ----------------- To study the uncertainties from the fit range and $\eta'$ sideband region, the fits are repeated with different fit ranges and sideband regions, the largest differences among these signal yields are taken as systematic uncertainties, respectively. To estimate the uncertainties in the description of various background contributions, we performed alternative fits with third-order Chebychev polynomials modeling the background of the ${K^{+}K^{-}}\eta'$ and ${K_{S}^{0}K_{S}^{0}}\eta'$ channels. The maximum differences in signal yields with respect to the nominal fit are taken as systematic uncertainties. The uncertainties from the background of $J/\psi\to K^{+}K^{-}\eta' + c.c.$ are estimated by absorbing this component into a Chebychev polynomial function, and the differences obtained by using the description with or without the background component of $J/\psi\to K^{+}K^{-}\eta' + c.c.$ are taken as systematic uncertainties. The impact of the $X(2120)$ is also considered as a systematic uncertainty in the study of the $X(2370)$. The difference between a fit with and without a $X(2120)$ contribution is taken as a systematic uncertainty associated to this item. Others ------ Since no evident structures are observed in the invariant-mass distributions of $M(K\eta')$, $M(\bar{K}\eta')$ and $M(K\bar{K})$ for the events with a $K\bar{K}\eta'$ invariant mass within the $X(2370)$ mass region (2.2 GeV/$c^{2} < M_{K\bar{K}\eta'} <$ 2.5 GeV/$c^{2}$), the systematic uncertainties of the reconstruction efficiency due to the possible intermediate states on the $K\eta'$, $\bar{K}\eta'$ and $K\bar{K}$ mass spectra are ignored. The uncertainties on the intermediate decay branching fractions of $\eta'\to\gamma\rho^{0}\to\gamma\pi^{+}\pi^{-}$, $\eta'\to\pi^{+}\pi^{-}\eta$, $\eta\to\gamma\gamma$ and $K_{S}^{0}\to\pi^{+}\pi^{-}$ are taken from the world average values [@pdg], which are $1.7\%$, $1.6\%$, $0.5\%$ and $0.1\%$, respectively. The systematic uncertainty due to the number of ${J/\psi}$ events is determined as 0.5$\%$ according to Ref. [@jpsinumber]. A summary of all the uncertainties is shown in Table \[sumsys2370\], \[sumsysBR2370\] and \[sumsysBR2120\]. The total systematic uncertainties are obtained by adding all individual uncertainties in quadrature, assuming all sources to be independent. The $X(2120)$ and $X(2370)$ are studied via ${J/\psi}\to\gamma {K^{+}K^{-}}\eta'$ and $J/\psi\to\gamma {K_{S}^{0}K_{S}^{0}}\eta'$ with two $\eta'$ decay modes, respectively. The measurements from the two $\eta'$ decay modes are, therefore, combined by considering the difference in uncertainties of these two measurements. The combined systematic uncertainties are calculated with weighted least squares method [@combinepaper] and the results are shown in Table \[combinesystable\]. ---------------------------------------------------------------------------- -------------------------------------------------------------------- $M_{X(2370)}$ (MeV/$c^{2}$) $2341.6\pm 6.5\text{(stat.)}\pm 5.7\text{(syst.)}$ $\Gamma_{X(2370)}$ (MeV) $117\pm10\text{(stat.)}\pm 8\text{(syst.)}$ $\mathcal{B}(J/\psi \to \gamma X(2370)\to \gamma K^{+}K^{-}\eta')$ $(1.79\pm0.23~\text{(stat.)}\pm 0.65~\text{(syst.)})\times10^{-5}$ $\mathcal{B}(J/\psi \to \gamma X(2370)\to \gamma K_{S}^{0}K_{S}^{0}\eta')$ $(1.18\pm0.32~\text{(stat.)}\pm 0.39~\text{(syst.)})\times10^{-5}$ $\mathcal{B}(J/\psi \to \gamma X(2120)\to \gamma K^{+}K^{-}\eta')$ $<1.49\times10^{-5}$ $\mathcal{B}(J/\psi \to \gamma X(2120)\to \gamma K_{S}^{0}K_{S}^{0}\eta')$ $<6.38\times10^{-6}$ ---------------------------------------------------------------------------- -------------------------------------------------------------------- RESULTS AND SUMMARY =================== Using a sample of $1.31\times10^{9} ~J/\psi$ events collected with the BESIII detector, the decays of $J/\psi\to\gamma {K^{+}K^{-}}\eta'$ and $J/\psi\to\gamma {K_{S}^{0}K_{S}^{0}}\eta'$ are investigated using the two $\eta'$ decay modes, $\eta'\to\gamma\rho^{0}(\rho^{0}\to\pi^{+}\pi^{-})$ and $\eta'\to\pi^{+}\pi^{-}\eta( \eta\to\gamma\gamma)$. The $X(2370)$ is observed in the $K\bar{K}\eta'$ invariant-mass distribution with a statistical significance of 8.3$\sigma$. The mass and width are determined to be\ ,\ ,\ which are found to be consistent with those of the $X(2370)$ observed in the previous BESIII results [@PRL1]. The product branching fractions of $\mathcal{B}({J/\psi}\to\gamma X(2370)\to\gamma K^{+} K^{-} \eta')$ and $\mathcal{B}({J/\psi}\to\gamma X(2370)\to\gamma K_{S}^{0} K_{S}^{0} \eta')$ are measured to be $(1.79\pm0.23~\text{(stat.)}\pm0.65~\text{(syst.)})\times10^{-5}$ and $(1.18\pm0.32~\text{(stat.)}\pm0.39~\text{(syst.)})\times10^{-5}$, respectively. No evident signal for the $X(2120)$ is observed in the $K\bar{K}\eta'$ invariant-mass distribution. For a conservative estimate of the upper limits of the product branching fractions of $J/\psi\to\gamma X(2120)\to{K^{+}K^{-}}\eta'$ and $J/\psi\to\gamma X(2120)\to{K_{S}^{0}K_{S}^{0}}\eta'$, the multiplicative uncertainties are considered by convolving the normalized likelihood function with a Gaussian function. Upper limits for product branching fractions at 90% C. L. are determined to be $\mathcal{B}({J/\psi}\to\gamma X(2120)\to\gamma K^{+} K^{-} \eta') < 1.49\times10^{-5}$ and $\mathcal{B}({J/\psi}\to\gamma X(2120)\to\gamma K_{S}^{0} K_{S}^{0} \eta') < 6.38\times10^{-6}$. To understand the nature of the $X(2120)$ and $X(2370)$, it is critical to measure their spin and parity and to search for them in more decay modes. A partial-wave analysis is needed to measure their masses and widths more precisely, and to determine their spin and parity. This might become possible in the future with the foreseen higher statistics of ${J/\psi}$ data samples. The BESIII collaboration thanks the staff of BEPCII and the IHEP computing center for their strong support. This work is supported in part by National Key Basic Research Program of China under Contract No. 2015CB856700; National Natural Science Foundation of China (NSFC) under Contracts Nos. 11625523, 11635010, 1332201, 11735014, 11565006; National Natural Science Foundation of China (NSFC) under Contract No. 11835012; the Chinese Academy of Sciences (CAS) Large-Scale Scientific Facility Program; Joint Large-Scale Scientific Facility Funds of the NSFC and CAS under Contracts Nos. U1532257, U1532258, U1732263, U1832207; CAS Key Research Program of Frontier Sciences under Contracts Nos. 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--- abstract: 'In this study interest centers on regional differences in the response of housing prices to monetary policy shocks in the US. We address this issue by analyzing monthly home price data for metropolitan regions using a factor-augmented vector autoregression (FAVAR) model. Bayesian model estimation is based on Gibbs sampling with Normal-Gamma shrinkage priors for the autoregressive coefficients and factor loadings, while monetary policy shocks are identified using high-frequency surprises around policy announcements as external instruments. The empirical results indicate that monetary policy actions typically have sizeable and significant positive effects on regional housing prices, revealing differences in magnitude and duration. The largest effects are observed in regions located in states on both the East and West Coasts, notably California, Arizona and Florida.' author: - \| Manfred M. Fischer, Florian Huber, Michael Pfarrhofer and Petra Staufer-Steinnocher\ Vienna University of Economics and Business bibliography: - 'favar.bib' - 'mpShocks.bib' - 'additional.bib' date: - - title: 'The dynamic impact of monetary policy on regional housing prices in the US: Evidence based on factor-augmented vector autoregressions' --- --------------- --------------------------------------------------------------------------------------------------- Keywords: Regional housing prices, metropolitan regions, Bayesian estimation, high-frequency identification --------------- --------------------------------------------------------------------------------------------------- ---------------- -------------------- JEL Codes: C11, C32, E52, R31 ---------------- -------------------- Introduction ============ This paper examines the impact of monetary policy on housing prices in the US.[^1] The literature on this relationship is fairly limited. Previous studies generally rely on two competing approaches. The first uses a structural model to analyze the relationship between monetary policy and housing prices (see, for example, [@MANC:MANC332; @ungerer2015monetary]). Such models impose a priori restrictions on the coefficients. The major strength of this model-based approach is to provide a theoretically grounded answer to the question of interest. Its potential shortcoming, however, is that the answer is only as good as the model is adequately representing the relationships in the real world. The second approach – labeled evidence-based – focuses more on the empirical evidence and relies less directly on economic theory. Researchers have commonly used vector autoregressive (VAR) models to measure the impact of monetary policy [see @Baffoe-Bonnie1998; @fratatoni2003monetary; @delnegro2007luftballons; @jarocinski2008house; @vargassilva2008monetary; @beltratti2010international; @ECTJ:ECTJ319]. Such models allow the data rather than the researcher to specify the dynamic structure of the model, and provide a plausible assessment of the response of macroeconomic variables to monetary policy shocks without the need of a complete structural model of the economy. In the tradition of the latter approach, this paper differs from previous literature both in terms of focus and methodology. With [@fratatoni2003monetary], we share the focus on regional differences in the response of housing prices, using metropolitan-level rather than state-level data.[^2] In terms of methodology, similar to [@vargassilva2008monetary] and in contrast to [@fratatoni2003monetary], we use a factor-augmented vector autoregressive (FAVAR) model to explore regional housing price responses to a national monetary policy shock.[^3] The effects are measured by considering idiosyncratic impulse responses of regions to the shock that is normalized to yield a 25 basis-points decline in the one-year government bond rate. Differently from [@vargassilva2008monetary] and [@ECTJ:ECTJ319], we employ a full Bayesian approach that is based on shrinkage priors for several parts of the parameter space. In particular, we make use of Markov Chain Monte Carlo (MCMC) methods to estimate the model parameters and the latent factors simultaneously. A full Bayesian approach has the advantage of directly controlling for uncertainty surrounding the latent factors and the model parameters. We follow [@gertler2015monetary] to identify monetary policy shocks by using high-frequency surprises around policy announcements as external instruments. The paper provides a rich picture on how an expansionary monetary policy shock affects housing prices in 417 US metropolitan regions over a time horizon of 72 months after impact. The findings reveal regional housing price effects to vary substantially over space, with size and modest sign differences among the regions. Some few regions in Utah, New Mexico, Kansas, Oklahoma, Mississippi and West Virginia show no significant impact or even slightly negative cumulative responses. In most of the regions, however, the cumulative responses of housing prices are positive, in line with theory. This regional heterogeneity may have different reasons, with heterogeneous regional housing markets playing a major role. The largest positive effects are observed in states on both the East and West Coasts, notably in Miami-Fort Lauderdale in Florida and Riverside-Sun Bernardino-Ontario in California, but also in Las Vegas in Nevada. In general, housing impulse responses tend to be similar within states and adjacent regions in neighboring states, evidenced by a high degree of spatial autocorrelation. The remainder of the paper is structured as follows. The next section presents the FAVAR model along with the Bayesian approach for estimation. describes the data and the sample of regions, and outlines the model specification. The empirical results are discussed in , while the final section concludes. Econometric framework {#sec:framework} ===================== A factor-augmented vector autoregressive model ---------------------------------------------- The econometric approach employed in this study is a FAVAR model, as introduced in [@bernanke2005measuring]. In our implementation, we let $\bm{H}_t=(H_{1t},...,H_{Rt})'$ denote an $R \times 1$ vector of housing prices at time $t$ ($t = 1, \hdots, T$) across $R=417$ US regions. The model postulates that regional housing prices depend on a number of latent factors, monetary and macroeconomic national aggregates and region-specific shocks. Specifically, the measurement equation can be written as $$\begin{bmatrix} \bm{H}_t \\ \bm{M}_t \end{bmatrix} = \begin{bmatrix}[l] \bm{\Lambda}^F & \bm{\Lambda}^M \\ \bm{0}_{K \times S} & \bm{I}_K \end{bmatrix} \begin{bmatrix} \bm{F}_t \\ \bm{M}_t \end{bmatrix} + \begin{bmatrix}[l] \bm{\epsilon} _t \\ \bm{0}_{K \times 1} \end{bmatrix}, \label{eq:main-struct}$$ where $\bm{F}_t=(F_{1t},...,F_{St})'$ is an $S \times 1$ vector of latent (unobservable) factors which capture co-movement at the regional level ($F_{rt}$, $r = 1, \hdots, R$, $t = 1, \hdots, T$). $\bm{M}_t=(M_{1t},...,M_{Kt})'$ is a $K \times 1$ vector of economic and monetary national aggregates that are treated as observable factors, and $\bm{\epsilon}_t$ ($t = 1, \hdots, T$) an $R \times 1 $ vector of normally distributed zero mean disturbances with an $R \times R$ variance-covariance matrix $\bm{\Sigma}_{\epsilon}=\text{diag}(\sigma_1^2, \dots, \sigma_R^2)$. These disturbances arise from measurement errors and special features that are specific to individual regional time series. $\bm{\Lambda}^F = (\lambda^{F}_{rs}:$ $r = 1, \hdots, R$; $s = 1, \hdots, S)$ is an $R \times S$ matrix of factor loadings with typical elements $\lambda^{F}_{rs}$, while $\bm{\Lambda}^M = (\lambda^{M}_{rk}:$ $r = 1, \hdots, R$; $k = 1, \hdots, K)$ a coefficient matrix of dimension $R \times K$ with typical elements $\lambda^{M}_{rk}$. The number of latent factors is much smaller than the number of regions, i.e. $S \ll R$. Note that the diagonal structure of $\bm{\Sigma}_\epsilon$ implies that any co-movement between the elements in $\bm{H}_t$ and $\bm{M}_t$ stems exclusively from the presence of the factors. The evolution of the factors $\bm{y}_t=(\bm{F}'_t, \bm{M}'_t)'$ is given by the state equation, governed by a VAR process of order $Q$, $$\bm{y}_t = \bm{A} \bm{x}_t + \bm{u}_t \label{eq: stateEQ},$$ with $\bm{x}_t=(\bm{y}'_{t-1}, \dots, \bm{y}'_{t-Q})'$ and the associated $(S+K)\times Q(S+K)$-dimensional coefficient matrix $\bm{A}$. Moreover, $\bm{u}_t$ is an $(S+K)$-dimensional vector of normally distributed shocks, with zero mean and variance covariance matrix $\bm{\Sigma}_u$. The parameters $\bm{\Lambda}^F$, $\bm{\Lambda}^{M}$ and $\bm{A}$ as well as the latent dynamic factors $\bm{F}_{t}$ are unkown and have to be estimated. To identify the model, we follow @bernanke2005measuring and assume that the upper $(S\times S)$-dimensional submatrix of $\bm{\Lambda}^F$ equals an identity matrix $\bm{I}_S$ while the first $S$ rows of $\bm{\Lambda}^{M}$ are set equal to zero. This identification strategy implies that the first $S$ elements in $\bm{H}_t$ are effectively the factors plus noise. A Bayesian approach to estimation --------------------------------- The model described above is highly parameterized, containing more parameters that can be reasonably estimated with the data at hand. In this study, we use a Bayesian estimation approach to incorporate knowledge about parameter values via prior distributions. Before proceeding with the prior setting employed it is convenient to stack the free elements of the factor loadings in an $L$-dimensional vector $\bm{\lambda}=\text{vec}[\bm{\Lambda}^F, \bm{\Lambda}^M]$ with $L=R (S+K)$, and the VAR coefficients in a $J$-dimensional vector $\bm{a}=\text{vec}(\bm{A})$ with $J=(S+K)^2 Q$. Prior distributions for the state equation {#prior-distributions-for-the-state-equation .unnumbered} ------------------------------------------ For the VAR coefficients $a_j$ ($j = 1,\hdots,J$) we impose the Normal-Gamma shrinkage prior proposed in @griffin2010inference [@griffin2016hierarchical], and subsequently applied in a VAR framework in @Huber2017, $$a_j\| \xi_a, \tau_{a j}^2 \sim \mathcal{N}\left(0, 2\ \xi_a^{-1} \tau^2_{a j}\right),$$ that is controlled by Gamma priors on $\tau^2_{a j}~(j = 1, \hdots, J)$ and $\xi_a$, $$\begin{aligned} \tau^2_{a j} \sim \mathcal{G}(\vartheta_a, \vartheta_a), \label{eq:tausq}\\ \xi_a \sim \mathcal{G}(d_0, d_1), \label{eq:xi_a}\end{aligned}$$ with hyperparameters $\vartheta_a$ and $d_0$, $d_1$ respectively. $\tau^2_{a j}$ operates as a local scaling and $\xi_a$ as a global shrinkage parameter. This hierarchical prior shows two convenient features. First, $\xi_a$ applies to all $J$ elements in $\bm{a}$. Higher values of $\xi_a$ yield stronger global shrinkage towards the origin whereas smaller values induce only little shrinkage. Second, the local scaling parameters $\tau^2_{a j}$ place sufficient prior mass of $a_j$ away from zero in the presence of strong overall shrinkage involved by large values for $\xi_a$. The hyperparameter $\vartheta_a$ in controls the excess kurtosis of the marginal prior, $$p(a_j \| \xi_a) = \int p(a_j\| \xi_a, \tau_{a j}^2 ) d\tau^2_{a j},$$ obtained after integrating over the local scales. Lower values of $\vartheta_a$ generally place increasing mass on zero, but at the same time lead to heavy tails, allowing for large deviations of $a_j$ from zero, if necessary. The hyperparameters $d_0$ and $d_1$ in are usually set to rather small values to induce heavy overall shrinkage [see @griffin2010inference for more details]. For the variance-covariance matrix $\bm{\Sigma}_u$ we use an inverted Wishart prior, $$\bm{\Sigma}_u \sim \mathcal{IW}(v, \bm{\ubar{\Sigma}}),$$ with $v$ denoting prior degrees of freedom, while $\bm{\ubar{\Sigma}}$ is a prior scaling matrix of dimension $(S+K)\times(S+K)$. Prior distributions for the observation equation {#prior-distributions-for-the-observation-equation .unnumbered} ------------------------------------------------ For the factor loadings $\lambda_{\ell}$ ($\ell = 1, \hdots, L$) we employ a Normal-Gamma prior similar to the one used for the VAR coefficients $a_j$ ($j = 1, \hdots, J$). The set-up follows @kastner with a single global shrinkage parameter $\xi_\lambda$ that applies to all free elements $\lambda_\ell$ in the factor loadings matrix. Specifically, we impose a hierarchical Gaussian prior on $\lambda_\ell$, $$\lambda_\ell\|\xi_\lambda, \tau^2_{\lambda \ell} \sim \mathcal{N}\left(0, 2\ \xi_\lambda^{-1} \tau^2_{\lambda \ell}\right)$$ that depends on Gamma priors for $\tau^2_{\lambda \ell}~(\ell = 1, \hdots, L)$ and $\xi_\lambda$, $$\begin{aligned} \tau^2_{\lambda\ell} \sim \mathcal{G}(\vartheta_\lambda, \vartheta_\lambda),\\ \xi_\lambda \sim \mathcal{G}(c_0, c_1).\end{aligned}$$ The hyperparameters $\vartheta_\lambda, c_0$ and $c_1$ control the tail behavior and overall degree of shrinkage of the prior. For the measurement error variances $\sigma^2_r~(r = 1,\hdots,R)$ we rely on a sequence of independent inverted Gamma priors, $$\sigma_r^2 \sim \mathcal{G}^{-1}(e_0, e_1),$$ where the hyperparameters $e_0$ and $e_1$ are typically set to small values to reduce prior influence on $\sigma^2_r$. Estimation of the model parameters and the latent factors is based on the MCMC algorithm described in . More specifically, we use Gibbs sampling to simulate a chain consisting of 20,000 draws, where we discard the first 10,000 draws as burn-in. It is worth noting that this algorithm shows fast mixing and satisfactory convergence properties. Data and model implementation {#sec:dataimplementation} ============================= Regions and Data ---------------- To explore regional differences in the impact of monetary policy on housing prices in the US, we need to define our notion of regions. Throughout the paper, we use $R = 417$ regions, a subsample of the 917 core-based statistical areas.[^4] These 417 regions include 264 metropolitan and 153 micropolitan statistical areas, briefly termed metropolitan regions in this paper. They have been selected based on the availability of the data over time. For the list of regions used, see . Our dataset consists of a panel of monthly time series ranging from 1997:04 to 2012:06. The $R \times 1$ vector of housing prices at time $t$ is constructed using the Zillow Home Value Index.[^5] In contrast to the FHFA (Federal Housing Finance Agency) Index and the Standard & Poor’s Case-Shiller Index, the Zillow Home Value Index does not use a repeat sales methodology, but statistical models along with information from sales assessments to generate valuations for all homes (single family residences, condominiums and co-operative homes) in any given region.[^6] These valuations are aggregated to determine the Zillow Home Value Index, measured in US dollars. An estimate for any given property is meant to indicate the fair value of a home sold as a conventional non-foreclosure, arms-length sale [@winkler2013]. We include $K = 7$ variables in the $K \times 1$ vector of observable national aggregates: three economic variables, namely housing investment (measured as the quantity of housing starts), the industrial production index and the consumer price index. The one-year government bond rate serves as policy indicator in line with @gertler2015monetary. In addition, three credit-spreads are included: the ten-year treasury minus the federal funds rate, the prime mortgage spread calculated over the ten-year government bonds and the @gilchrist2012credit excess bond premium.[^7] The economic variables capture housing, price and output movements. The mortgage spread is relevant to the cost of housing finance and the excess bond premium to the cost of long term credit in the business sector, while the term spread measures expectations on short-term interest rates [@gertler2015monetary]. All observable national aggregates are taken from the FRED database [@McCracken2016], with the exception of the excess bond premium and the mortgage spread that we obtained from the dataset provided in @gertler2015monetary. All data series are seasonally adjusted, if applicable, and transformed to be approximately stationary. Model implementation -------------------- For implementation of the FAVAR, we have to specify the order $Q$ of the VAR process and the number of latent factors, $S$. As is standard in the literature, we pick $Q = 2$ lags of the endogenous variables. To decide on the number of factors we use the deviance information criterion [@spiegelhalter2002bayesian] where the full data likelihood is obtained by running the Kalman filter and integrating out the latent states. This procedure yields $S=1$, a choice that is also consistent with traditional criteria (Bayesian information criterion or Kaiser criterion) to select the number of factors. Next and finally, a brief word on hyperparameter selection for the prior set-up. We specify $\vartheta_a = \vartheta_\lambda=0.1$, a choice that yields strong shrinkage but, at the same time, leads to heavy tails in the underlying marginal prior. Recent literature [@Huber2017] integrates out $\vartheta_a, \vartheta_\lambda$ and finds that, for US data, the posterior is centered on values between $0.10$ and $0.15$. The hyperparameters on the global shrinkage parameters are set equal to $c_0=c_1= d_0 =d_1 = 0.01$, a choice that is consistent with heavy shrinkage towards the origin representing a standard in the literature [@griffin2010inference]. The prior on $\bm{\Sigma}_u$ is specified to be weakly informative, i.e. $\nu= S+K+1$ and $\bm{\ubar{\Sigma}}= 10^{-2} \bm{I}_{S+K}$. Likewise, for the inverted Gamma prior on $\sigma_r^2~(r = 1, \hdots, R)$ we set $e_0=e_1=0.01$ to render the prior only weakly influential. Impulse response analysis {#sec:results} ========================= Structural identification of the model -------------------------------------- The high-frequency variant of the external instruments identification approach [@kuttner2001monetary; @gurkaynak2005sensitivity] employed in this paper is based on the surprises in the three-months-ahead futures rate that reflect expectations on interest rate movements further into the future, measured within a 30 minutes time window surrounding Federal Reserve announcements [@gertler2015monetary]. Note that in contrast to the Cholesky identification strategy, there is no need to impose zero restrictions. To implement the approach we follow @paul20017 and use high-frequency surprises as a proxy for the structural monetary policy shock. This is achived by integrating the surprises into as an exogenous variable ${z}_t$, $$\bm{y}_t = \bm{A} \bm{x}_t + \bm{\zeta} {z}_t+ \bm{u}_t.$$ Hereby $\bm{\zeta}$ is a $Q(S+K)$-dimensional vector of regression coefficients that collects the impulses of the shocks. @paul20017 shows that under mild conditions, the contemporaneous relative impulse responses can be estimated consistently.[^8] Note that the contemporaneous response of $\bm{y}_t$ to changes in $z_t$ is given by $\bm{\zeta}$. Higher order responses are defined recursively by exploiting the state space representation of the VAR model in . Impulse responses of macroeconomic quantities --------------------------------------------- We first consider the dynamic responses of the endogenous variables included in the $K \times 1$ vector $\bm{M}_t~(t = 1, \hdots, T)$ to illustrate that the results of the model are consistent with established findings in the literature. An expansionary monetary policy shock is modeled by taking the one-year government bond rate as the relevant policy indicator, rather than the federal funds rate that is commonly used in the literature based on arguments presented in @gertler2015monetary.[^9] Normalization is achieved by assuming that a monetary policy shock yields a 25 basis-points decrease in the policy indicator. [.329]{} \[fig:ip\] ![[]{data-label="fig:IRF_macro"}](logip.pdf "fig:"){width="\textwidth"} [.329]{} \[fig:houst\] ![[]{data-label="fig:IRF_macro"}](HOUST.pdf "fig:"){width="\textwidth"} [.329]{} \[fig:cpi\] ![[]{data-label="fig:IRF_macro"}](logcpi.pdf "fig:"){width="\textwidth"} \ [.329]{} \[fig:t10yty\] ![[]{data-label="fig:IRF_macro"}](gs1.pdf "fig:"){width="\textwidth"} [.329]{} \[fig:termspread\] ![[]{data-label="fig:IRF_macro"}](T10YFFM.pdf "fig:"){width="\textwidth"} [.329]{} \[fig:mortgage\_spread\] ![[]{data-label="fig:IRF_macro"}](mortg_spread_m.pdf "fig:"){width="\textwidth"} \ [.329]{} \[fig:ebp\] ![[]{data-label="fig:IRF_macro"}](ebp.pdf "fig:"){width="\textwidth"} \ \[fig:IRF\_macro\] depicts the impulse response functions of the endogenous variables. All plots include the median response (in blue) for 72 months after impact along with 68 percent posterior coverage intervals reflecting posterior uncertainty. An unanticipated decrease in the government bond rate by 25 basis-points causes a significant increase in real activity, with industrial production, housing investment and consumer prices all increasing over the next months after the impact. From a quantitative standpoint, the effects of the monetary shock on industrial production and consumer price index are considerably larger than the impact on housing investment, although uncertainty surrounding the size of impacts is large, and posterior coverage intervals include zero during the first months after impact. Housing investment shows a reaction similar in shape to real activity measured in terms of the industrial production index, suggesting a positive relationship between expansionary monetary policy and housing investment at the national level. Turning to the responses of financial market indicators, it should be noted that the one-year government bond rate falls by 25 basis-points on impact by construction, then increases significantly before it turns non-significant after about nine months. The term spread reacts adversely on impact, and we find significant deviations from zero that die out after about 16 months. This result points towards an imperfect pass-through of monetary policy on long-term rates, implying that long-term yields display a weaker decline as compared to short-term rates. The prime mortgage spread does not show a significant effect on impact, while responses between ten to 20 months ahead indicate a slightly negative overall reaction to expansionary monetary policy. Consistent with @gilchrist2012credit, one implication of this finding is that movements in key short-term interest rates tend to impact credit markets, with mortage spreads showing a tendency to decline. The responses of the excess bond premium almost perfectly mirror the reaction of the mortgage spread. The effects, however, are much larger from a quantitative point of view. To sum up, the results obtained by the impulse response analysis provide empirical support that monetary policy shocks, identified by using high-frequency surprises around policy announcements as external instruments, generate impulse responses of the endogenous variables that are consistent with the findings by @gertler2015monetary. The dynamic factor and its loadings ----------------------------------- Before moving to the impulse responses of regional housing markets to a monetary policy shock, we briefly consider the estimated latent factor as well as its loadings, with two aims in mind: first, to provide a rough intuition on how the latent factor captures co-movement in regional house price variations, and second, to give indication of the relative importance of individual regions shaping the evolution of the common factor. \[fig:dynamic-factor\] shows the evolution of the negative latent factor (in solid red) and provides evidence that the common factor co-moves with the average growth rate of housing prices (in solid blue, calculated using the arithmetic mean of the individual regional housing prices) nearly perfectly. The figure illustrates that during the 2001 recession, housing price declines have been mild, while being substantial during the Great Recession, with large variations across space. It is worth noting that home prices fell the most during the late 2000s in regions with the largest declines in economic activity [@beraja2017]. ![](Figuresfactor_ts_2.pdf) While provides intuition on the shape of the latent housing factor, the question on how individual regions are linked to it still needs to be addressed. For this purpose, reports the posterior mean of the region-specific factor loadings in form of a geographic map in which thinner lines denote the boundaries of the regions, while thicker lines signify US state boundaries. Visualization is based on a classification scheme with equal-interval breaks. We see that the great majority of regions exhibit negative loadings, and only 22 regions show positive values. Eighty regions have zero loadings or loadings where the 16th and 84th credible sets (68 percent posterior coverage) of the respective posterior distributions include zero. The pattern of factor loadings, evidenced by the map, indicates that the latent factor is largely driven by regions located in California, Arizona and Florida. Regions in the rest of the country, with loadings being either small in absolute terms or not significantly different from zero, tend to play only a minor role in shaping national housing prices. Impulse responses of housing prices ----------------------------------- \[fig:irf-factor\] displays the impulse response function of the latent factor over 72 months after impact to an expansionary monetary policy shock. The latent factor reacts positively after the shock; however, the posterior coverage interval includes zero for the first few months. This is consistent with economic theory which suggests that as the costs of financing a home purchase decrease, the demand for housing increases and as a result, real housing prices increase. ![](factor2.pdf){width=".329\textwidth"} Housing price responses, cumulated over the time horizon of six years, are displayed in .[^10] The results are presented in form of a geographic map with a classification scheme that generates class breaks in standard deviation measures (SD $= 2.98$) above and below the mean of $3.43$. Again thinner lines denote the boundaries of the metropolitan regions and thicker lines those of US states. Five points are worth noting here. First, cumulative regional housing price effects vary substantially over space, with size and modest sign differences among the regions. Some few regions in Utah, New Mexico, Kansas, Oklahoma, Mississippi and West Virginia, but also in Louisiana and North Carolina show no significant impact or even negative cumulative responses. In more than 97 percent of the regions, however, the cumulative response of housing prices is positive. Second, this heterogeneity may be due to varying sensitivity of housing to interest rates across space, and regional differences in housing markets, such as supply and demand elasticities [@fratatoni2003monetary]. For example, supply elasticities are relatively low on the East and West Coasts, but higher in the South and Southwest parts of the US. Third, the largest cumulative effects can be observed in states on both the East and West Coasts, notably Riverside, Madera, Merced and Bakersfield in California, Miami-Fort Lauderdale and Key West in Florida, but also Las Vegas and Fernley in Nevada. These regions, expecially those in California, seem to play an important role in shaping the movement of the US housing price following a monetary shock. Fourth, the regions in the East North Central states (as defined by the Census Bureau), but also in Georgia and Massachusetts have cumulative home price responses that resemble the mean response of the US regions within a 0.5 standard deviation band from the mean. Prominent examples include Atlanta ($2.97$), Boston ($3.76$) and Chicago ($3.88$). Fifth and finally, cumulative responses tend to be similar within states and adjacent regions in neighboring states. Looking at the map, this spatial autocorrelation phenomenon becomes particularly evident in the case of the Californian regions. This is most likely due to the importance of new house construction industries in California, along with the spatial influence the Californian housing market has on regions in neighboring states, especially Nevada and Arizona. For reasons of space limits we cannot present the 417 impulse response functions of the individual metropolitan regions, but report those of six regions in . We pick these regions to show examples for metropolitan regions with larger positive cumulative responses (Riverside and Miami-Fort Lauderdale) and those with negative cumulative responses (Salt Lake City and Hickory). Recall that there are only eleven regions belonging to this latter category. For comparison, we also display the impulse response function of two regions that closely resemble the mean response of US regions (Chicago and Boston). Again, the solid blue line denotes the median response and the shaded areas (in light blue) the 68th posterior coverage intervals. [.329]{} \[fig:irf\_select5\] ![](Riverside.pdf "fig:"){width="\textwidth"} [.329]{} \[fig:irf\_select7\] ![](SaltLakeCity.pdf "fig:"){width="\textwidth"} [.329]{} \[fig:irf\_select2\] ![](Chicago.pdf "fig:"){width="\textwidth"} \ [.329]{} \[fig:irf\_select6\] ![](Miami-FortLauderdale.pdf "fig:"){width="\textwidth"} [.329]{} \[fig:irf\_select8\] ![](Hickory.pdf "fig:"){width="\textwidth"} [.329]{} \[fig:irf\_select1\] ![](Boston.pdf "fig:"){width="\textwidth"} \[fig:IRF\_selected\] reveals profound differences in dynamic responses between the three regional categories, especially in shape and duration of effects. The differences within the categories tend to be rather small. In case of the first category, represented by the metropolitan regions Riverside and Miami-Fort Lauderdale, an expansionary monetary policy shock generates a significant increase of housing prices. This level remains stable and significant in the short-run, before fading away after approximately three years. The charts of Salt Lake City and Hickory, examples of the second regional category, show the housing price responses to fall strongly immediately, and these effects remain significantly negative for less than one year after impact. The response pattern of housing prices in Chicago and Boston is different. The effects are small in size, and hardly different from zero, with the exception of weakly significant effects between the third and fourth year after impact. Closing remarks =============== This paper has examined the relationship between monetary policy and the US housing market, focusing on monetary policy shocks. The analysis is based on a Bayesian FAVAR model where monetary policy shocks are identified using high-frequency surprises around policy announcements as external instruments. Bayesian model estimation uses Gibbs sampling with Normal-Gamma shrinkage priors for both the autoregressive coefficients and factor loadings, relying on a panel of monthly time series for a set of 417 regions that range from 1997:04 to 2012:06. The main findings of our analysis can be summarized as follows. The results provide empirical evidence that metropolitan regions react differently to an expansionary monetary policy shock, revealing magnitude and duration differences, and pointing to some modest sign differences, in additon. The extent and nature of regional heterogeneity are consistent with @fratatoni2003monetary who report impulse responses for house prices in 27 US metropolitan regions. Since our sample of regions covers the whole US (except Alaska, Maine, South Dakota and Wyoming) rather than only 16 US states, we find considerably greater regional heterogeneity in the results. In line with theory, the great majority of regions exhibit positive home price responses. The largest positive effects, cumulated over the time horizon of six years, can be observed in regions located in states on both the East and West Coasts, notably California, Arizona and Florida, and in Nevada. Impulse responses of regions tend to be similar within states and adjacent regions in neighboring states, evidenced by a high degree of spatial dependence among the impulse responses, as measured in terms of Moran’s I statistic. Finally, it is worth noting that our analysis is confined to a linear setting, implying the underlying transmission mechanism to be constant over time. This assumption simplifies the analysis, but may be overly simplistic in turbulent economic times such as the collapse of the housing market around the Great Recesssion. Hence, an extension of the linear setting to allow for non-linearities – in the spirit of @huber2018 – might be a promising avenue for future research. The MCMC algorithm {#app:mcmc} ================== We estimate the model by running an MCMC algorithm. The full conditional posterior distributions are available in closed form implying that we can apply Gibbs sampling to obtain draws from the joint posterior distribution. More specifically, our MCMC algorithm involves the following steps: 1. Simulate the VAR coefficients $a_j~(j = 1,\hdots,J)$ conditional on the factors and remaining model parameters from a multivariate Gaussian distribution that takes a standard form [see, for instance, @george2008bayesian for further information]. 2. Simulate the latent factors $\bm{F}_t~(t = 1, \hdots, T)$ by using forward filtering backward sampling [@carter1994gibbs; @fruhwirth1994data]. 3. The error variance-covariance matrix $\bm{\Sigma}_u$ is simulated from an inverted Wishart posterior distribution with degrees of freedom equal to $\nu = v + T$ and scaling matrix equal to $\bm{P}=\sum_{t=1}^T (\bm{y}_t-\bm{A} \bm{x}_t)'(\bm{y}_t-\bm{A} \bm{x}_t) + \bm{\ubar{\Sigma}}$. 4. Simulate the factor loadings $\lambda_\ell~(\ell = 1, \hdots, L)$ from Gaussian posteriors (conditioned on the remaining parameters and the latent factors) by running a sequence of $(R-S)$ unrelated regression models. 5. The measurement error variances $\sigma_r^2$ for $r = S+1, \dots, R$ are simulated independently from an inverse Gamma distribution $\sigma_r^2\| \Xi \sim \mathcal{G}^{-1}(\alpha_r, \beta_r)$ with $\alpha_r = \frac{1}{2} T + e_0$ and $\beta_r = \frac{1}{2} \sum_{t=1}^T (H_{rt} - \bm{\Lambda}^F_{r \bullet} \bm{F}_t - \bm{\Lambda}^M_{r \bullet} \bm{M}_t)^2+e_1$. The notation $\bm{\Lambda}^F_{r \bullet}$ indicates that the $r$th row of the matrix concerned is selected and $\Xi$ stands for conditioning on the remaining parameters and the data. 6. Simulate $\tau^2_{a j}~(j=1,\dots,J)$ from a generalized inverted Gaussian distributed posterior distribution with $$\tau^2_{a j}\|\Xi \sim \mathcal{GIG}\left(\vartheta_a - \frac{1}{2}, a_j^2, \vartheta_a \xi_a\right).$$ 7. Draw $\xi_a$ from a Gamma distributed posterior given by $$\xi_a\|\Xi \sim \mathcal{G}\left(c_0 + \vartheta_a J, c_1 + \frac{1}{2} \vartheta_a \sum_{\ell=1}^L \tau^2_{a \ell}\right).$$ 8. Simulate the posterior of $\tau^2_{\lambda \ell}~(\ell=1,\dots,L)$ from a generalized inverted Gaussian distribution, $$\tau^2_{\lambda \ell}\|\Xi \sim \mathcal{GIG}\left(\vartheta_\lambda-\frac{1}{2}, \lambda_\ell^2, \vartheta_\lambda \xi_\lambda\right).$$ 9. Finally, the global shrinkage parameter $\xi_\lambda$ associated with the prior on the factor loadings is simulated from a Gamma distribution, $$\xi_\lambda\|\Xi \sim \mathcal{G}\left(d_0 +\vartheta_\lambda L, d_1 + \frac{1}{2}\vartheta_\lambda \sum_{\ell=1}^L \tau^2_{\lambda \ell}\right).$$ Steps described above are iterated for 20,000 cycles, where we discard the first 10,000 draws as burn-in. Regions used in the study {#app:regions} ========================= Regions in this study are defined as core-based statistical areas (CBSA) that – by definition of the United States Office of Management and Budget – are based on the concept of a core area of at least 10,000 population, plus adjacent counties having at least 25 percent of employed residents of the county who work in the core area. Core-based statistical areas may be categorized as being either metropolitan or micropolitan. The 917 core-based statistical areas include 381 metropolitan statistical areas which have an urban core population of at least 50,000, and 536 micropolitan statistical areas which have an urban core population of at least 10,000 but less than 50,000. In this study we use 264 metropolitan and 153 micropolitan statistical areas, due to limited availability of data. These 417 regions, briefly termed metropolitan regions in this paper, represent all US states except Alaska, Maine, South Dakota and Wyoming. [p[.2]{} p[.75]{}]{} \ State & Region\ &\ Alabama & Birmingham, Daphne, Mobile, Montgomery, Tuscaloosa\ Arizona & Flagstaff, Lake Havasu City, Phoenix, Prescott, Sierra Vista, Tucson, Yuma\ Arkansas & Fayetteville, Fort Smith, Hot Springs, Jonesboro, Little Rock\ California & Bakersfield, Chico, El Centro, Fresno, Hanford, Los Angeles-Long Beach-Anaheim, Madera, Merced, Modesto, Napa, Redding, Riverside, Sacramento, Salinas, San Diego, San Francisco, San Jose, San Luis Obispo, Santa Cruz, Santa Maria-Santa Barbara, Santa Rosa, Stockton, Vallejo, Ventura, Visalia, Yuba City\ Colorado & Boulder, Colorado Springs, Denver, Fort Collins, Grand Junction, Greeley, Pueblo\ Connecticut & Hartford, New Haven, New London, Stamford\ Delaware & Dover\ District of Columbia & Washington\ Florida & Crestview-Fort Walton Beach-Destin, Daytona Beach, Fort Myers, Gainesville, Homosassa Springs, Jacksonville, Lakeland, Melbourne, Miami-Fort Lauderdale, Naples, North Port-Sarasota-Bradenton, Ocala, Orlando, Panama City, Pensacola, Port St. Lucie, Punta Gorda, Sebring, Tallahassee, Tampa, The Villages, Vero Beach\ Georgia & Albany, Athens, Atlanta, Augusta, Columbus, Dalton, Gainesville, Hinesville, Macon, Savannah, Valdosta, Warner Robins\ Hawaii & Kahului, Urban Honolulu\ Idaho & Boise City, Idaho Falls, Lewiston\ Illinois & Bloomington, Chicago, Davenport, Kankakee, Springfield\ Indiana & Bloomington, Elkhart, Evansville, Fort Wayne, Lafayette-West Lafayette, Muncie, South Bend, Terre Haute\ Iowa & Des Moines\ Kansas & Lawrence\ Kentucky & Lexington, Louisville-Jefferson County\ Louisiana & Alexandria, Baton Rouge, Houma, Lafayette, Lake Charles\ Nebraska & Grand Island, Lincoln, Omaha\ Nevada & Las Vegas, Reno\ New Hampshire & Manchester\ New Jersey & Ocean City, Trenton, Vineland\ New Mexico & Albuquerque, Las Cruces, Santa Fe\ New York & Albany, Binghamton, Elmira, Glens Falls, Ithaca, Kingston, New York, Rochester, Syracuse, Watertown\ North Carolina & Asheville, Burlington, Charlotte, Durham, Fayetteville, Greensboro, Hickory, Raleigh, Rocky Mount, Wilmington, Winston-Salem\ North Dakota & Fargo\ Ohio & Akron, Canton, Cincinnati, Cleveland, Columbus, Dayton, Lima, Springfield, Toledo, Youngstown\ Oklahoma & Oklahoma City, Tulsa\ Oregon & Albany, Bend, Corvallis, Eugene, Grants Pass, Medford, Portland, Salem\ Maryland & Baltimore, California-Lexington Park, Cumberland, Hagerstown, Salisbury\ Massachusetts & Boston, Cape Cod, Pittsfield, Springfield, Worcester\ Michigan & Ann Arbor, Battle Creek, Bay City, Grand Rapids, Jackson, Lansing, Midland, Monroe, Muskegon, Saginaw\ Minnesota & Mankato, Minneapolis-St Paul, Rochester\ Mississippi & Hattiesburg, Jackson\ Missouri & Columbia, Joplin, Springfield, St. Louis\ Pennsylvania & Allentown, Altoona, Erie, Harrisburg, Lancaster, Philadelphia, Pittsburgh, Reading, Scranton, State College, York\ Rhode Island & Providence\ South Carolina & Columbia, Florence, Greenville, Hilton Head Island, Myrtle Beach, Spartanburg\ Tennessee & Chattanooga, Clarksville, Cleveland, Jackson, Johnson City, Kingsport, Knoxville, Nashville\ Texas & Amarillo, Brownsville, College Station, Dallas-Fort Worth, El Paso, Killeen, Laredo, Midland, Texarkana\ Utah & Ogden, Provo, Salt Lake City, St. George\ Virginia & Charlottesville, Harrisonburg, Richmond, Roanoke, Staunton, Virginia Beach, Winchester\ Washington & Bellingham, Kennewick, Longview, Olympia, Seattle, Spokane, Walla Walla, Yakima\ West Virginia & Charleston\ Wisconsin & Appleton, Eau Claire, Fond du Lac, Janesville, La Crosse, Madison, Oshkosh, Racine\ [p[.2]{} p[.75]{}]{} \ State & Region\ &\ Arizona & Nogales, Payson, Safford\ Arkansas & Batesville, Harrison, Paragould, Russellville, Searcy\ California & Clearlake, Eureka, Red Bluff, Susanville, Truckee\ Colorado & Durango, Glenwood Springs, Montrose, Sterling\ Connecticut & Torrington\ Florida & Clewiston, Key West, Lake City, Okeechobee, Palatka\ Georgia & Bainbridge, Calhoun, Cedartown, Dublin, Jesup, Moultrie, St. Marys, Thomaston, Tifton, Vidalia, Waycross\ Hawaii & Hilo\ Idaho & Burley\ Illinois & Effingham, Jacksonville\ Indiana & Angola, Auburn, Bedford, Connersville, Crawfordsville, Decatur, Frankfort, Greensburg, Huntington, Jasper, Kendallville, Logansport, Madison, Marion, New Castle, North Vernon, Peru, Plymouth, Richmond, Seymour, Vincennes, Wabash, Warsaw, Washington\ Kansas & Garden City\ Kentucky & Danville, Murray\ Louisiana & Opelousas\ Nebraska & North Platte\ Nevada & Elko, Fernley, Gardnerville Ranchos\ New Hampshire & Concord, Keene, Laconia\ New York & Amsterdam, Batavia, Corning, Cortland, Gloversville, Hudson, Olean, Oneonta, Plattsburgh, Seneca Falls\ North Carolina & Albemarle, Morehead City, Sanford, Wilson\ Ohio & Ashtabula, Coshocton, Defiance, Findlay, Jackson, New Philadelphia, Portsmouth, Sandusky, Urbana, Wooster\ Oklahoma & Ardmore, Bartlesville, Duncan, Durant, Enid, McAlester, Tahlequah\ Oregon & Coos Bay, Hermiston-Pendleton, Klamath Falls, Ontario, Roseburg, The Dalles\ Maryland & Cambridge, Easton\ Massachusetts & Greenfield Town, Vineyard Haven\ Michigan & Adrian, Hillsdale, Holland, Ionia, Ludington, Owosso\ Minnesota & Owatonna, Willmar, Winona\ Mississippi & Cleveland, Columbus, Corinth, Grenada, Laurel, Oxford, Picayune, Tupelo, Vicksburg\ Missouri & Mexico\ Pennsylvania & Indiana, Lock Haven, Oil City, Pottsville\ South Carolina & Orangeburg\ Tennessee & Cookeville, Lawrenceburg, Lewisburg, Martin, Paris, Sevierville, Shelbyville, Tullahoma\ Virginia & Danville, Martinsville\ Washington & Oak Harbor, Port Angeles, Shelton\ Wisconsin & Baraboo, Marinette, Whitewater\ Robustness check {#app:robustness} ================ To assess the sensitivity of our results with respect to identification of the monetary policy shock, we use an alternative strategy based on contemporaneous sign restrictions [see @uhlig2005effects; @DEDOLA2007512]. Technical implementation is achieved by using the algorithm proposed in @arias2014inference that collapses to the procedure outlined in @rubio2010structural in the absence of zero restrictions. For each iteration of the MCMC algorithm we draw a rotation matrix and assess whether the following set of sign restrictions is satisfied. Consistent with economic common sense, output (measured in terms of the industrial production index), housing investment (measured in terms of housing starts) and consumer prices (measured in terms of the consumer price index) are bound to increase on impact. Moreover, we assume that the term-spread also widens on impact. Finally, consistent with the normalization adopted when using external instruments, we assume that the one-year yield declines. If this is the case, we keep the rotation matrix and store the associated structural coefficients, while if the sign restrictions are not met, we reject the draw and repeat the procedure. The results are displayed in form of a geographic map with a classification scheme that generates class breaks in standard deviation measures above and below the mean, see . A comparison with provides evidence of the robustness of our results. [^1]: Housing is defined here to include family residences, condominiums and co-operative homes. [^2]: Their empirical analysis uses a small set of 27 US regions to analyze the effects of monetary policy, based on quarterly data from 1986 to 1996. Aside from this study, metropolitan-level housing data have not been explored very much. [^3]: For the definition of our notion of region and the list of regions used, see . [^4]: A core-based statistical area is a US geographic area – defined by the Office of Management and Budget – that consists of one or more counties anchored by an urban center of at least 10,000 people plus adjacent counties that are socioeconomically tied to the urban center. The term core-based statistical area refers collectively to both metropolitan and micropolitan statistical areas. [^5]: The Zillow Home Value Index has the benefit of a broad coverage of the large set of core-based statistical areas. The set of data we use in our study is available for download at <https://www.zillow.com/research/data/>. [^6]: For more information on the proprietary valuation model used by Zillow to estimate the market value of a home, see @bruce2014zillow. [^7]: The excess bond premium may roughly be seen as the component of the spread between an index of yields on corporate fixed income securities and a similar maturity government bond rate that is left after removing the component due to default risk [@gertler2015monetary]. @gilchrist2012credit show that this variable provides a convenient summary of additional information not included in the FAVAR that may be relevant to economic activity. [^8]: Relative impulse responses are obtained by normalizing the absolute impulse responses, i.e. the change in $\bm{y}_{t+h}$ to a change in $z_t$, by the contemporaneous response of some element in $\bm{y}_{t}$. [^9]: @gertler2015monetary have shown that the one-year bond rate has a much stronger impact on market interests than the funds rate does. [^10]: The quantitative and qualitative nature of the results is robust to an alternative identification scheme, in which sign restrictions have been employed (for the results see \[app:robustness\]).	ArXiv
--- abstract: 'A detailed analytical inspection of light scattering by a particle with high refractive index $m+i\kappa$ and small dissipative constant $\kappa$ is presented. We have shown that there is a dramatic difference in the behavior of the electromagnetic field within the particle (inner problem) and the scattered field outside it (outer problem). With an increase in $m$ at fix values of the other parameters, the field within the particle asymptotically converges to a periodic function of $m$. The electric and magnetic type Mie resonances of different orders overlap substantially. It may lead to a giant concentration of the electromagnetic energy within the particle. At the same time, we demonstrate that identical transformations of the solution for the outer problem allow to present each partial scattered wave as a sum of two partitions. One of them corresponds to the $m$-independent wave, scattered by a perfectly reflecting particle and plays the role of a background, while the other is associated with the excitation of a sharply-$m$-dependent resonant Mie mode. The interference of the partitions brings about a typical asymmetric Fano profile. The explicit expressions for the parameters of the Fano profile have been obtained “from the first principles” without any additional assumptions and/or fitting. In contrast to the inner problem, at an increase in m the resonant modes of the outer problem die out, and the scattered field converges to the universal, $m$-independent profile of the perfectly reflecting sphere. Numerical estimates of the discussed effects for a gallium phosphide particle are presented.' author: - - 'Andrey E. Miroshnichenko' bibliography: - 'Mie\_Fano.bib' title: 'Giant In-Particle Field Concentration and Fano Resonances at Light Scattering by High-Refractive Index Particles' --- Introduction ============ Presently the resonant light scattering by particles related to excitation of different eigenmodes attracts a great deal of attention of researchers all around the world [@Zhao:MT:2009; @Rybin:PRL:2009; @Evlyukhin:PRB:2011; @Staude:ACSN:2013; @Hancu:NL:2014; @Kuznetsov:NC:2014]. In addition to purely academic interest there is a broad spectrum of applications of the phenomenon in physics, chemistry, biology, medicine, data storage and processing, telecommunications, micro- and nanotechnologies, etc., see, e.g., [@Novotny:Book:2006; @Rybin:PRB:2013]. In particular, plenty of hopes were pinned on the resonant excitation of localized and/or bulk plasmons in metal nanoparticles [@Klimov:Book:2014]. Unfortunately, plasmonic resonances in such nanoparticles are usually accompanied with rather large dissipative losses, which in many cases diminish the advantages of the resonances. For this reason recently the frontier of the corresponding study has been shifted to light scattering by dielectric particles with low losses and high refractive index (HRI) $m$. In contrast to the plasmonic resonances, they exhibit the high $Q$-factor Mie resonances of both electric and magnetic types which bring more opportunities for wider applications in sensing, spontaneous emission enhancement, and unidirectional scattering. Despite the fact that the exact Mie solution, describing light scattering by a sphere with an arbitrary size and material properties, is known for more than a hundred years, and the case of a sphere with HRI has been repeatedly discussed in textbooks and monographs, see, e.g. [@Landau:T8:1984; @Hulst::1981], some important peculiarities of this problem have not been disclosed yet. Meanwhile, the quantitative feature of HRI brings about qualitatively new effects and paradoxes, which merely do not exist at moderate values of the refractive index, see, e.g. [@Evlyukhin:PRB:2010; @Tribelsky:EPL:2012; @Geffrin:NC:2012]. In the present paper we produce a detailed, systematic study of light scattering by a HRI particle. Specifically, we show that the scattering may be accompanied with a giant concentration of the electromagnetic field within the particle and reveal the nature of the Fano resonances exhibited by partial scattered waves. It is well known that at the limit $m \rightarrow \infty$ a small (relative to the incident light wavelength) dielectric sphere scatters light as a perfectly reflecting one (PRS) “into which neither the electric, nor the magnetic field penetrates" [@Landau:T8:1984]. Then, it may be concluded that the electromagnetic field within the scattering particle should vanish at $m \rightarrow \infty$. Seemingly, the conclusion is supported by the argument that a HRI implies a high polarizability of the sphere. Then, at $m \rightarrow \infty$ from the point of view of the polarizability by an external electric field a dielectric sphere becomes equivalent to a perfectly conducting one [@Landau:T8:1984], and the electric field induced within the particle owing to its polarization by the incident light should compensate the field inducing the polarization. That is to say, the field within the particle should vanish. In fact, the question is much more subtle, and the actual situation is far from this simple picture. The point is that the wavelength inside the particle vanishes at $m \rightarrow \infty$. Then, no matter how small the particle is, at large enough $m$ the wavelength within the particle becomes smaller than the particle size. In this case the incident wave may resonantly excite the Mie electromagnetic eigenmodes in the particle. Moreover, an unlimited growth in $m$ results in infinite cascades of these resonances. The interference of the resonant eigenmodes with the incident wave and, what is the most important, with each other gives rise to dramatic changes in the aforementioned simple scattering process. To reveal these changes is the goal of our study. To this end the full Mie theory is employed. We show that while at $m \rightarrow \infty$ the scattered field for the outer problem does converge to the one for the PRS (no matter, whether the sphere is small, or large), the field within the particle, though it sounds paradoxical, does not have any limit at all. Such a difference between the outer and inner problems is related to the different lineshapes of the Mie resonances in the two problems. For the former an increase in $m$ makes the resonances less pronounced. For the latter in the non-dissipative limit the amplitude of the resonances increases with an increase in $m$. In the case of a finite dissipation rate (regardless how small it is), the growth of the amplitudes eventually saturates and the resonance lines become periodic functions of $m$. In both the cases the field within the particle does not tend to any fixed limit at $m \rightarrow \infty$. It is important to stress, that the resonance lines of different orders and different origin (i.e., electric and magnetic) may overlap substantially. The mentioned peculiarities of the inner field in the vicinity of the resonances may result in a giant concentration of the electromagnetic energy inside the particle. At realistic values of the refractive index and the proper selection of the particle radius the field inside the particle may exceed the one in the incident wave in several orders of magnitude. Such a huge field may give rise to numerous nonlinear effects. For this reason the discussed results may appear extremely important in the design and fabrication of highly-nonlinear nanostructures. Regarding the outer problem, it is known that the scattering coefficients in this problem have the well-pronounced asymmetric Fano resonance lines. Recent publications of Rybin et al, Ref. [@Rybin:OE:2013; @Rybin:FTT:2014; @Rybin:SciRep:2014] should be mentioned in this context. Based on the analysis of the exact Lorenz-Mie solution for a cylinder the authors of these publications have revealed that the resonant Mie scattering can be presented through infinite cascades of the Fano resonances between the narrow-line resonant Mie scattering and the non-resonant (background) scattering from the object. The analytical expressions for both the partitions have been obtained through the Maxwell boundary conditions. The numerical fit of the lineshape, resulting from the exact solution in the vicinity of the resonances, to the conventional Fano profile [@Fano:PR:1961] has allowed the authors to obtain the dependence of the Fano asymmetry parameter $q$ [@Fano:PR:1961] on the ratio of the radius of the cylinder R to the wavelength of the incident light $x =2\pi R/\lambda$ (size parameter) in rather a broad range of its variations. They also have shown that in the inspected cases $q(x) \sim −\cot x$. This dependence agrees with their previous results for disordered photonic crystals [@Rybin:NatCom:2012], as well as with the general expression for $q$ in terms of the phase shift of the background partition [@Connerade:RPP:1988]. Despite the study of these authors is a big step forward to understanding the essence of the Fano resonances at light scattering by a particle, they have not disclosed the physical nature of the background partition. Regarding the results obtained by the numerical fit, the great advantage of this procedure is the possibility to fit any curve with any set of the basic functions. However, precisely because of that, based on the fitting solely, one never can answer the question whether the studied profile is the Fano profile indeed, or it is just fitted to that profile. It also remains unclear how far beyond the inspected numerical domain the obtained results could be extended, e.g., what happens with the modes with the multipolarity higher than that, examined by the authors, etc. For this reason, a self-consistent analytical examination of the problem, connecting the parameters of the Fano profiles with the fundamental parameters of the light scattering $x$ and $m$ would be highly desirable. Such a study is produced in the present paper. Specifically, we show that the Fano profile could be obtained by identical transformations of the exact Mie solution. As a result, the exact analytical expressions for the parameters of the profile are obtained automatically. Regarding the background partition, we reveal that it is just the corresponding partial wave scattered by a PRS. We should emphasize that the resonances discussed in the present paper have nothing to do with the well known whispering gallery modes. These modes are associated with waves propagating along the surface of the particle, confined there by “continuous total internal reflection" [@Vahala:Nature:2003] and cannot be excited by a plane incident wave. The closest to the topic of our study is another well known phenomenon: the ripple structure in the spectrum at light scattering by droplets with $m \approx 1.5$ and related problems, see, e.g., [@Chylek:JOSA:1976; @Hulst::1981; @Chylek:ApplPhys:1985]. The resonances we discuss and the ripple structure, both have the same nature. However, the large value of the refractive index makes the manifestation of our resonances quite different from what is known for the ripple structure. In addition, the specific characteristics of the resonances examining in our paper, usually are not discussed in connection with the ripple structure. Note also the very close effects discussed in Ref. [@Arruda:JOSA:2010] for a magnetic particle. However, up to now materials with large magnetic permeability at optical frequencies remain hypothetical objects, while the ones with HRI may be easily found among the common semiconductors, see below. It is necessary to stress in this context that all the effects discussed in the present paper occur to optically thick particles, i.e., $mx$ should be of the order of unity, or larger than that. However, an important peculiarity of the HRI particles is that this condition may hold for geometrically small particles with $x \ll 1$, cf. [@Tribelsky:EPL:2012]. The paper is arranged as follows. In Section II we consider general properties of the Mie solution at the limit of large, purely real $m$. In Section III the lineshapes and linewidths of partial resonances for the scattered field and the field, concentrated within the particle are inspected. In Section IV the generic nature of the Fano resonances for partial modes of the scattered field is disclosed, and explicit expressions for the parameters of the Fano profile are obtained from the first principles by means of identical transformations of the exact Mie solution. In we show that in the non-dissipative limit the entire set of the infinite number of the cascades of the resonances possesses a certain scaling and may be reduced to a universal set of lines by simple scale transformations. In Section VI, effects of finite dissipation are inspected. In Section VII the resonances at a fixed $m$ and varying size parameter are discussed and the manifestation of the resonances in a particle made of gallium phosphide is presented as an example. In Conclusion a brief summary of the obtained results is presented. In Appendix certain cumbersome but important calculations are performed. Last, but not least. The only thing we do below is a detailed analysis of the well known Mie solution at the range of high refractive index of the scattering sphere. It would have been nothing but a mathematical exercise, if it did not reveal new unusual features of the phenomenon. It did. Let us proceed discussing these features in detail. Large refractive index limit {#sec:2} ============================ The subsequent analysis is based upon the exact Mie solution describing light scattering by a spatially homogeneous sphere with an arbitrary radius $R$ and a given permittivity $\varepsilon$. The case of a cylinder and core-shell structures may be inspected in the same manner. Since we are interested in the optical properties of a particle with low dissipation, we first focus on a discussion of the non-dissipative limit, i.e., a purely real positive refractive index $m \equiv \sqrt{\varepsilon}$. The general case of a complex refractive index along with the applicability condition for the non-dissipative limit will be produced later on. According to the Mie solution, the scattered field is presented as an infinite series of partial multipolar contributions (dipolar, quadrupolar, etc.) of the two types: TE and TM — the so-called electric and magnetic modes. For the sake of briefness the electric responses only are discussed here in detail. The behavior of the magnetic submodes is alike. Therefore, the corresponding discussion of these modes will be rather brief. The key quantities of the Mie solution are the properly normalized complex amplitudes of the scattered $a_n,\;b_n$ and internal $c_n,\;d_n$ field components (scattering coefficients). These coefficients should satisfy the boundary conditions, following from the continuity of the tangential components of the electric and magnetic fields at the surface of the particle. The conditions are split into two independent pairs for the electric and magnetic modes, respectively [@Bohren::1998] $$\begin{aligned} & & \xi_n(x)a_n + \psi_n(mx)d_n= \psi_n(x), \label{Eq:BC1electr} \\ & & m\xi'_n(x)a_n +\psi'_n(mx)d_n = m \psi'_n(x), \label{Eq:BC2electr}\\ & & m\xi_n(x)b_n + \psi_n(mx)c_n = m\psi_n(x), \label{Eq:BC1magn}\\ & & \xi'_n(x)b_n + \psi'_n(mx)c_n = \psi'_n(x) \label{Eq:BC2magn}.\end{aligned}$$ Here $x=2\pi R/\lambda$ is the size parameter; $\psi_n(z)$, , $\psi_n(z)=zj_n(z)$ and $\chi_n(z)=-zy_n(z)$ are the Riccati-Bessel functions; $j_n(z),\;y_n(x)$ stand for the spherical Bessel functions [@Bohren::1998]; ${}'=\partial/\partial z$ designates derivative with respect to the entire argument. Integer $n$ indicates the multipolarity of the mode, so that $n=1,\;2...$ correspond to the dipole mode, quadrupole mode, etc. It should be stressed that functions $\psi_n(z)$ and $\chi_n(z)$ are real for Solving Eqs. , with respect to $a_n$, $d_n$, we obtain [@Bohren::1998] $$\begin{aligned} a_n &=& \frac{m\psi_n(mx)\psi_n'(x) - \psi_n(x)\psi_n'(mx)}{m\psi_n(mx)\xi_n'(x) - \xi_n(x)\psi_n'(mx)} \label{Eq:a_n} \\ d_n &=& \frac{im}{m\psi_n(mx)\xi_n'(x) - \xi_n(x)\psi_n'(mx)},\label{Eq:d_n}\end{aligned}$$ where the identity $$\label{Eq:identity_1} \psi_n(x)\xi'_n(x) - \psi'_n(x)\xi_n(x) \equiv i,$$ following from the expression for the Wronskian of the Riccati-Bessel functions [@Abramowitz::1965], has been employed. The corresponding treatment of the magnetic modes yields $$\begin{aligned} b_n &=& \frac{m\psi_n(x)\psi'_n(mx)-\psi_n(mx)\psi'_n(x)}{m\xi_n(x)\psi'_n(mx)-\psi_n(mx)\xi'_n(x)} \label{Eq:b_n} \\ c_n &=& -\frac{im}{m\xi_n(x)\psi'_n(mx)-\psi_n(mx)\xi'_n(x)} \label{Eq:c_n}\end{aligned}$$ Note, that according to Eqs. , , $d_n \equiv c_n \equiv 1$ at $m=1$, i.e., when the optical properties of the particle are identical to those of the embedding medium. Thus, $\|d_n\|,\;\|c_n\|$ may be regarded as the enhancement parameters for the field within the particle. Eqs. , , , may be written in the equivalent, identical form: $$\begin{aligned} a_n & = & \frac{F_n^{(a)}}{F_n^{(a)}+i G_n^{(a)}}\;,\;\;d_n=\frac{i\;m}{F_n^{(a)}+i G_n^{(a)}},\label{Eq:an_dn_FG}\\ b_n & = & \frac{F_n^{(b)}}{F_n^{(b)}+i G_n^{(b)}}\;,\;\;c_n=-\frac{i\;m}{F_n^{(b)}+i G_n^{(b)}},\end{aligned}$$ with $$\begin{aligned} F_n^{(a)}&=&m\psi'_n(x)\psi_n(m x)-\psi_n(x)\psi'_n(m x)\;,\label{Eq:Fna}\\ G_n^{(a)}&=&-m\chi'_n(x)\psi_n(m x)+\chi_n(x)\psi'_n(m x)\;,\label{Eq:Gna}\\ F_n^{(b)}&=&m\psi_n(x)\psi'_n(mx)-\psi_n(mx)\psi'_n(x)\;,\label{Eq:Fnb}\\ G_n^{(b)}&=&-m\chi_n(x)\psi'_n(mx) + \chi'_n(x)\psi_n(mx). \label{Eq:Gnb}\end{aligned}$$ Let us focus on the electric modes. At the limit of the large refractive index we can keep just the first terms in Eqs.–: $$\begin{aligned} F_n^{(a)}\xrightarrow[{m \gg 1}]{}m\psi'_n(x)\psi_n(m x)\;,\label{Eq:Fna_m->infty}\\ G_n^{(a)}\xrightarrow[{m \gg 1}]{}-m\chi'_n(x)\psi_n(m x)\;. \label{Eq:Gna_m->infty}\end{aligned}$$ Eqs. – lead to two important conclusions. First, the partial scattering coefficients $a_n$ converge to the $m$-[independent]{} form valid for the PRS: $$a_n\xrightarrow[{m \gg 1}]{}a_n^{\rm (PRS)}\!=\!\frac{F_n^{(a,{\rm PRS})}}{F_n^{(a,{\rm PRS})}+i G_n^{(a,{\rm PRS})}}\!=\!\frac{\psi'_n(x)}{\xi'_n(x)}, \label{Eq:a_n-PRS}$$ with $$\label{Eq:Fna_PRS-Gna_PRS} F_n^{(a,{\rm PRS})}=\psi'_n(x),\;\;G_n^{(a,{\rm PRS})}=-\chi'_n(x).$$ Note that expression, Eq. can be also obtained from the boundary condition, Eq. , if we suppose there $d_n = 0$, as it should be for a PRS, or by direct solution of the light scattering problem for a PRS [^1]. Second, the internal coefficients $d_n$ in this limit still keep the dependence on $m$, converging to $$\begin{aligned} \label{Eq:d_n_at_m->infty} d_n\xrightarrow[m \gg 1]{}d_n^{(\lim)}=\frac{i}{\psi_n(mx)\xi'_n(x)}.\end{aligned}$$ It is relevant to mention that while expressions, Eqs. , satisfy boundary condition, Eq. identically, boundary condition, Eq. for these expressions is satisfied only asymptotically at It is interesting to note also, that the limit, Eqs. , corresponds to the scattering coefficient for a sphere made of a perfect electric conductor [@Balanis:Book:2012]. However, in the problem in question a sphere made of a perfect insulator with zero conductivity (Im$\,\varepsilon = 0$) is considered. The coincidence of the two limits occurs owing to the fact, that despite the conductivity current in our case is zero, the displacement current plays its role. If the field [E]{} inside the particle does not vanish, the displacement current diverges at Re$\,\varepsilon \rightarrow \infty$. It brings about exactly the same result as that, following from the divergence of the conductivity current in a perfect electric conductor. It should be stressed that limit, Eq. (\[Eq:a\_n-PRS\]) is not valid in the vicinities of the points, where $\psi_n(mx)=0$. Moreover, exactly at these points the internal coefficient diverges $d_n^{(\lim)}\rightarrow\infty$, see Eq. (\[Eq:d\_n\_at\_m->infty\]). It allows us to conclude that the condition $$\label{Eq:psi=0} \psi_n(mx)=0$$ corresponds to the resonances excited in a HRI dielectric sphere at large enough $m$. This resonance condition can be further simplified taking into account that in the range $m x>n^2$ (known as the Fraunhofer regime) the Riccati-Bessel functions are reduced to simple trigonometrical ones, so that the condition $\psi_n(mx)=0$ reads $$\begin{aligned} \psi_n(m x)\cong\sin\left(m x-\frac{n\pi}{2}\right)=0 \label{Eq:sin=0}\;.\end{aligned}$$ An analogous treatment of the magnetic coefficients $b_n$ (for the outer problem) and $c_n$ (for the inner) brings about the following expressions: $$\begin{aligned} F_n^{(b)}\xrightarrow[{m \gg 1}]{}m\psi_n(x)\psi'_n(m x)\;,\label{Eq:Fnb_m->infty}\\ G_n^{(b)}\xrightarrow[{m \gg 1}]{}-m\chi_n(x)\psi'_n(m x)\;. \label{Eq:Gnb_m->infty}\end{aligned}$$ $$b_n\xrightarrow[{m \gg 1}]{}b_n^{\rm (PRS)}\!=\!\frac{F_n^{(b,{\rm PRS})}}{F_n^{(b,{\rm PRS})}+i G_n^{(b,{\rm PRS})}}\!=\!\frac{\psi_n(x)}{\xi_n(x)}, \label{Eq:b_n-PRS}$$ $$\label{Eq:Fnb_PRS-Gnb_PRS} F_n^{(b,{\rm PRS})}=\psi_n(x),\;\;G_n^{(b,{\rm PRS})}=-\chi_n(x),$$ $$\begin{aligned} \label{Eq:c_n_at_m->infty} c_n\xrightarrow[m \gg 1]{}c_n^{(\lim)}=-\frac{i}{\psi'_n(mx)\xi_n(x)},\end{aligned}$$ with the following condition for the magnetic modes resonances: $$\begin{aligned} \psi'_n(m x)=0\;, \label{Eq:psi_prime=0}\end{aligned}$$ leading in the Fraunhofer regime to the equation $$\label{Eq:cos=0} \psi'_n(m x)\cong \cos\left(m x-\frac{n\pi}{2}\right)=0.$$ The two resonance conditions may be rewritten in a unified form: $$\begin{aligned} & & m^{({\rm res},E)}_{n,p}\cong(n+2p)\frac{\pi}{2x}, \label{Eq:m_res_E}\\ & & m^{({\rm res},H)}_{n,p}\cong(n+2p+1)\frac{\pi}{2x} \label{Eq:m_res_H}.\end{aligned}$$ where $p$ is a non-negative integer number. The meaning of resonance conditions, Eqs. , becomes absolutely clear, if we recall that $x = 2\pi R/\lambda$ and $\lambda/m$ is the wavelength inside the scattering particle. Then, Eqs. , may be rewritten as follows: $$\begin{aligned} & & \frac{\lambda}{m^{({\rm res},E)}_{n,p}}\left(\frac{n}{2}+p\right) \cong 2R, \\ & & \frac{\lambda}{m^{({\rm res},H)}_{n,p}}\left(\frac{n+1}{2}+p\right) \cong 2R,\end{aligned}$$ i.e., the resonances occur when an integer number of the half-waves equals the diameter of the sphere, cf. [@Kuznetsov:SR:2012]. These resonance conditions lead to a number of interesting conclusions, cf. [@Hulst::1981]. First, each multipole has, in general, infinite number of resonances, associated with $p=0,1,2,\ldots$. Second, owing to the additional degree of freedom related to variations of $p$, there is the multiple degeneracy of the resonances. Specifically, the resonances with different multipolarity $n$ occur at one and the same value of $m$, provided for these resonances the variation of $n$ is compensated by the corresponding variation of $p$, i.e., $n+2p$ for the electric resonant modes remains the same and equals $n+1+2p$ for magnetic. Next, at a give $n$ the points of the $n$th electric resonances correspond to those of the $(n+1)th$ magnetic and vice versa. Note also that at a fixed $n$ the points of resonances of one type (i.e., either electric, or magnetic) are situated in the $m$-axis just in the middle of the spacing between the points of the other type, so that the maxima for the electric modes correspond to the minima of magnetic and the other way around, see Eqs. , . In Fig. \[fig:fig2\] we plot the dependence of the first two electric $d_{1,2}$ and magnetic $c_{1,2}$ internal coefficients vs. size parameter $x$ and refractive index $m$, calculated according to the exact Mie solution. In this figure we also plot asymptotic resonance conditions, Eqs. (\[Eq:m\_res\_E\]), (\[Eq:m\_res\_H\]). The agreement between the position of the resonances according to the exact solution and approximate conditions, Eqs. (\[Eq:m\_res\_E\]), (\[Eq:m\_res\_H\]) is surprisingly good. It should be stressed, however, that the results obtained do not mean a strong overlap of the resonances yet. The point is that the right-hand-sides of Eqs. , are just the first terms of the corresponding asymptotic expansions in powers of small $1/(mx)$. Higher order terms, dropped in these equations bring about the mismatches between the points of the resonances for different modes. To claim the strong overlap of the resonances we must make sure that the mismatches are smaller than the corresponding linewidths. Let us proceed to a discussion of these issues. ![(color online) Dependence of electric $d_{1,2}$ (a,c) and magnetic $c_{1,2}$ (b,d) internal coefficients vs size parameter $x$ and refractive index $m$, calculated according to the exact Mie solution. Dashed lines indicate the asymptotic resonance conditions, obtained in the limit $m\rightarrow\infty$ , see Eqs. (\[Eq:m\_res\_E\]), (\[Eq:m\_res\_H\]) at $p = 0,\,1,\,2,\,3.$ Despite the employed values of $m$ are not so large, the agreement between the dashed lines and the positions of the actual resonances is very good. []{data-label="fig:fig2"}](Fig2 "fig:"){width="\columnwidth"}\ Linewidth and lineshape {#sec:3} ======================= To understand the meaning of the existence of the limit at $m \rightarrow \infty$ for scattering coefficients $a_n$, and its absence for internal ones, $d_n$, we have to inspect Eqs. – more carefully. First of all note, direct calculations of $a_n$ based upon Eqs. , , , , brings about the uncertainty of the type 0/0. To resolve the lineshape, let us consider a small departure of the refractive index from a certain resonant value: $\delta m = m - m^{\rm (res)}$, where $m^{\rm (res)}$ satisfies Eq. . Here and in what follows for simplicity of notations we put $m^{\rm (res)}$ for $m^{({\rm res}, E)}_{n,p}$. Then, bearing in mind that in the vicinity of the resonance $\psi_n\left((m^{{\rm (res)}}+\delta m)x\right) \cong \psi'_n(m^{{\rm (res)}}x)x\delta m$, we readily obtain the following formulas for $F^{(a)}_n\!\!,\; G^{(a)}_n$: $$\begin{aligned} \!\!\!\!\!\!\!\!F^{(a)}_n &\cong& \psi_n'(m^{\rm (res)}x)\left\{m^{\rm (res)}\psi_n'(x)x\delta m - \psi_n(x) \right\}\!, \label{Eq:Fn_res_a} \\ \!\!\!\!\!\!\!\!G^{(a)}_n &\cong& -\psi_n'(m^{\rm (res)}x)\left\{m^{\rm (res)}\chi_n'(x)x\delta m - \chi_n(x) \right\}\!. \label{Eq:Gn_res_a}\end{aligned}$$ It is seen from Eqs. , , that scattering coefficient reaches its maximum (the points of the constructive interference), $a_n =1$ at $$\label{Eq:delta_m_G_a} \delta m^{(a)}_G \cong \frac{1}{x m^{\rm (res)}}\frac{\chi_n(x)}{\chi_n'(x)},$$ and minimal value, $a_n=0$ (the points of the destructive interference) at $$\label{Eq:delta_m_F_a} \delta m^{(a)}_F \cong \frac{1}{x m^{\rm (res)}}\frac{\psi_n(x)}{\psi_n'(x)}.$$ The corresponding expressions for $b_n$ read $$\begin{aligned} F^{(b)}_n &\cong& m^{\rm (res)}\psi_n(x)\psi''_n(m^{\rm (res)}x)x\delta m \label{Eq:Fn_res_b}\\ & & - \psi'_n(x)\psi_n(m^{\rm (res)}x), \nonumber\\ G^{(b)}_n &\cong& -m^{\rm (res)}\chi_n(x)\psi''_n(m^{\rm (res)}x)x\delta m \label{Eq:Gn_res_b}\\ & & + \chi'_n(x)\psi_n(m^{\rm (res)}x) \nonumber.\end{aligned}$$ $$\label{Eq:delta_m_G_b} \delta m^{(b)}_G \cong \frac{1}{x m^{\rm (res)}}\frac{\chi'_n(x)\psi_n(m^{\rm (res)}x)}{\chi_n(x)\psi''_n(m^{\rm (res)}x)},$$ $$\label{Eq:delta_m_F_b} \delta m^{(b)}_F \cong \frac{1}{x m^{\rm (res)}}\frac{\psi'_n(x)\psi_n(m^{\rm (res)}x)}{\psi_n(x)\psi''_n(m^{\rm (res)}x)}.$$ where now $m^{{\rm (res)}}$ stands for $m^{({\rm res},H)}_{n,p}\!\!\!,$ satisfying Eq. . Note, $\psi_n(m^{\rm (res)}x)/\psi''_n(m^{\rm (res)}x) \cong -1$ at $m^{\rm (res)}x > n^2$, see Eqs. . The obtained results explain the mentioned convergence of $a_n$ to the $m$-independent form at $m \rightarrow \infty$: at an increase in $m$ the width of the resonance line $\|\delta m_G - \delta m_F\|$ contracts as $1/xm^{\rm (res)}$, while the amplitude of the resonances and spacing between two adjacent maxima (minima), both remain $m$-independent. Asymptotically, at $m \rightarrow \infty$ the maxima and minima merge with each other, and the resonance profile vanishes. An example of such a process for $\|a_1\|^2$ at $x=1$ is presented in Fig. \[fig:a1\]. ![(color online) Contraction of the Fano resonance lines for $\|a_1\|^2$ with an increase in $m;\;x=1;\;\zeta = m - m^{{\rm (res)}}.$ Here (solid line) and 111.5 (dashed line). A yellow horizontal line corresponds to $a_1^{\rm (PRS)}$. []{data-label="fig:a1"}](a1_contraction){width="\linewidth"} Now let us inspect coefficients $d_n$. As it follows from $$\label{Eq:a_n_trough_aPRS_and_d_n} a_n = a_n^{\rm (PRS)} -\frac{\psi_n'(mx)}{m\xi_n'(x)}d_n.$$ Then, it is convenient to introduce the rescaled coefficient $$\label{Eq:d_n_tilde} \tilde{d_n} = \frac{\psi_n'(mx)}{m\xi_n'(x)}d_n \equiv \frac{i}{m\frac{\psi_n(mx)}{\psi_n'(mx)} - \frac{\xi_n(x)}{\xi_n'(x)}}\frac{1}{\xi'^2_n(x)}.$$ Next, utilizing the identity $$\label{Eq:identity_2} \left\|\xi'^2_n(x){\rm Im}\,\left(\frac{\xi_n(x)}{\xi_n'(x)}\right)\right\| \equiv 1,$$ see Appendix A, its modulus may be presented as follows: $$\label{Eq:d_n_tilde_AB} \|\tilde{d_n}\|^2 = \frac{1}{\left(m\frac{\psi_n(mx)}{\psi'_n(mx)}-A_n^{(d)}(x)\right)^2B_n^{(d)2}(x)+1},$$ where $$\label{Eq:A_n^{(d)}_B_n^{(d)}} A_n^{(d)}(x) = {\rm Re}\,\left(\frac{\xi_n(x)}{\xi_n(x)'}\right);\;\;\; B_n^{(d)}(x) = \left\|\xi_n'^2(x)\right\| \equiv \left\|\xi_n'(x)\right\|^2.$$ The conclusion, which immediately follows from Eq. , is that the resonant values of $\|\tilde{d_n}\|^2$ equals unity and the resonances are achieved at the values of $m$, satisfying the equation: $$\label{Eq:resonant_cond_\|d_n\|} m\frac{\psi_n(mx)}{\psi'_n(mx)}=A_n^{(d)}(x).$$ We should emphasize that the resonance condition stipulated by Eq. is an exact result, valid at any $m$ and $x$. To obtain the lineshape in the proximity of $m=m^{\rm (res)}$, with $m^{\rm (res)}$ defined by Eq. , as usual, we have to expand the argument of $\psi_n(mx)$ in powers of a small $\delta m = m-m^{\rm (res)}$, so that $\psi_n(mx) \cong \psi'_n(m^{\rm (res)}x)x\delta m$. Then, the general expression, Eq. is reduced to the following simple form: $$\label{Eq:lineshape_\|d_n\|^2} \|\tilde{d_n}\|^2 = \frac{1}{\left(m^{\rm (res)}x\delta m -A_n^{(d)}(x)\right)^2B_n^{(d)2}(x)+1}.$$ It is a typical Lorentzian profile with a maximum at $$\label{Eq:delta_m_\|d\|_max} \delta m_{\|d_n\|^2}^{\rm (res)} = \frac{A_n^{(d)}(x)}{m^{\rm (res)}x}$$ and the half-maximum linewidth (FWHM) $$\label{Eq:linewidth_\|d_n\|^2} \gamma_{\|d_n\|^2} = \frac{2}{m^{({\rm res})}xB_n^{(d)}(x)}.$$ Note, that Eq. gives the mismatch between the positions of the resonance points defined according to Eq. and the ones corresponding to the maxima of $\|d_n\|^2$. It seems we have encountered a paradox. On one side, we have obtained that in contrast to the outer problem, the inner one does not have a definite limit at $m \rightarrow \infty$, see Eq. . On the other side, $\tilde{d_n} = a_n^{\rm (PRS)} - a_n$, see Eqs. , . This equality means that since $a_n \xrightarrow[m\rightarrow\infty]{}a_n^{\rm (PRS)}$, coefficient $\tilde{d_n}$ should vanish in this limit. The latter reasoning agrees well with the just obtained for $\tilde{d_n}$ lineshape. Indeed, being equal in its maximum to unity, the linewidth for $\|\tilde{d_n}\|^2$ vanishes at $m \rightarrow \infty$ as $1/m^{\rm (res)}$, while the value of $\|\tilde{d_n}\|^2$ at off-resonance regions vanishes as $1/m^2$, see Eq. . Then, at $m \rightarrow \infty$ the resonance lines become infinitesimally narrow and the entire profile $\|\tilde{d_n}(m)\|^2$ vanishes. Finally, because the difference between $\tilde{d_n}$ and ${d_n}$ is just in the multiplicative scaling factor the same conclusion, seemingly, may be applied to the profile $\|{d_n(m)}\|^2$. However, one should be careful here, because the scaling factor itself depends on $m$. Since , the maximal value of $\|d_n\|^2$ increases as $m^{(\rm{res})\,2}$, see Eq. . It means that the total area under a given resonance line increases with an increase in $m^{\rm (res)}$ linearly in $m^{\rm (res)}$, see Eq. , making (in contrast to $a_n$) the resonances more and more pronounced. Note, however, that it does not mean nonexistence of any universality in the profile $\|d_n(m)\|^2$ at $m \rightarrow \infty$. Let us consider a part of the profile from its bottom to any fixed value $D_n^2$. According to Eq. $$\label{Eq:d_n_through_d_n_tilde} d_n =\frac{m\xi_n'(x)}{\psi_n'(mx)}\tilde{d}_n \equiv \frac{m(\psi_n'(x) -i\chi_n'(x))}{\psi_n'(mx)}\tilde{d}_n .$$ Then, bearing in mind that in the Fraunhofer regime for the discussed resonances $\|\psi_n'(m^{({\rm res})}x)\| = 1$, see Eqs. , , the width of the line at this distance from the bottom ($\gamma_D$) is given by the difference , where $\delta m_{1,2}$ are the two roots of the equation $$\label{Eq:eq_for Gamma_D} \frac{m^{(\rm{res})\,2}\left(\psi_n'^2(x)+\chi_n'^2(x)\right)}{\left(m^{\rm (res)}x\delta m -A_n^{(d)}(x)\right)^2B_n^{(d)2}(x)+1} = D^2,$$ ![(color online) Independence of $m^{{\rm (res)}}$ of the bottom parts of the resonance lines for profiles $\|c_1\|^2$ and $\|d_1\|^2$ at $x =1$ and large values of $m^{{\rm (res)}};\; \zeta = m - m^{{\rm (res)}}$. Here profiles $\|c_1(\zeta)\|^2$ and $\|d_1(\zeta)\|^2$ are presented at $m^{{\rm (res)}} = 14.1019...$ (thick blue solid line), $m^{{\rm (res)}} = 1013.1631...$ (thick yellow dashed line) and $m^{{\rm (res)}} = 15.5792$ (red solid line), $m^{{\rm (res)}} = 1014.7325...$ (black dashed line), respectively. Increase in $m^{{\rm (res)}}$ in the three orders of magnitudes, practically, does not affect the shape of the lines. Note the perfect coincidence of the corresponding lines, even in the off-resonance regions, despite Eq. is valid only in the vicinity of the points of the resonances and, generally speaking, cannot be applied to describe the lineshape far from them.[]{data-label="fig:d1"}](c1d1__bottom){width="\linewidth"} Trivial calculations show that at the quantity $\gamma_D$ converges to the following $m$-independent expression: $$\label{Eq:Gamma_D} \gamma_D \cong \frac{2\sqrt{\psi_n'^2(x) + \chi_n'^2(x)}}{DB(x)x}.$$ This asymptotics is valid for any $D$ satisfying the aforementioned constraint. It means that at large $m$ a part of the resonance lineshape close to the bottom of the resonance line becomes universal. An example of this universality for $n=1$ and $x=1$ is shown in Fig. \[fig:d1\]. Thus, instead of vanishing [(as it should be for a PRS)]{} at $m \rightarrow \infty$ the resonance lines for coefficient $\|d_n\|^2$ at the bottom converge to a certain universal form, while the maximal value of $\|d_n\|^2$ at the peak of the resonance increases as $m^{(\rm{res})\,2}$ and the half-maximum linewidth contracts as $1/m^{\rm (res)}$. The behavior of $c_n$ is quite analogous to that of $d_n$. The expressions describing this behavior are presented below for references: $$\begin{aligned} & & b_n = b_n^{\rm (PRS)} -\tilde{c}_n,\;\;\tilde{c}_n \equiv \frac{\psi_n(mx)}{m\xi_n(x)}c_n, \label{Eq:b_n_trough_bPRS_and_c_n--c_n^tilde} \\ & & \|\tilde{c_n}\|^2 = \frac{1}{\left(m\frac{\psi'_n(mx)}{\psi_n(mx)}-A_n^{(c)}(x)\right)^2B_n^{(c)2}(x)+1} \label{Eq:c_n_tilde_AB} \\ & & \approx \frac{1}{\left(\frac{\psi''_n(m^{\rm (res)}x)}{\psi_n(m^{\rm (res)}x)}m^{\rm (res)}x\delta m -A_n^{(c)}(x)\right)^2B_n^{(c)2}(x)+1},\nonumber \\ & & \delta m_{\|c_n\|^2}^{\rm (res)} = \frac{A_n^{(c)}(x)\psi_n(m^{\rm (res)}x)}{m^{\rm (res)}x\psi''_n(m^{\rm (res)}x)},\; \label{Eq:delta_m_\|c\|_max}\\ & & \gamma_{\|c_n\|^2} = \frac{2}{m^{({\rm res})}xB_n^{(c)}(x)}\left\|\frac{\psi_n(m^{\rm (res)}x)}{\psi''_n(m^{\rm (res)}x)}\right\|. \label{Eq:linewidth_\|c_n\|^2}\end{aligned}$$ Here $m^{\rm (res)}$ is defined according to Eq. , $$\label{Eq:A_n^{(c)}_B_n^{(c)}} A_n^{(c)}(x) = {\rm Re}\,\left(\frac{\xi'_n(x)}{\xi_n(x)}\right);\;\;\; B_n^{(c)}(x) = \left\|\xi_n^2(x)\right\| \equiv \left\|\xi_n(x)\right\|^2,$$ and to obtain Eq. the identity $$\left\|\xi^2_n(x){\rm Im}\,\left(\frac{\xi'_n(x)}{\xi_n(x)}\right)\right\| \equiv 1,$$ whose proof is completely analogous to that for Eq. , has been employed. Let us make numerical estimates. At and (typical values for a number of semiconductors in the visible and IR diapason) the resonant ; ; ; . It gives an estimate for the range of the growth of the electric field inside the particle with respect to the incident wave, while the corresponding values for the volume density of the electromagnetic energy will be of the orders of squires of these values. To conclude this section a certain important remark should be made. Though it is convenient for the theoretical study adopted in the present paper to inspect the resonances at varying $m$ and fixed values of the other problem parameters, such an approach seems completely irrelevant from the physical viewpoint. Really, while in an actual experiment, it is rather easy to change the value of the size parameter, $x$ just varying the wavelength of the incident light, it is very difficult to change $m$. The refractive index is a given property of the material the particle made of. To change it either materials with strong dispersion should be employed, or one has to have a set of particles with the same size but different refractive indices varying in small steps. Both the options look unrealistic. Thus, it seems our study is quite meaningless. Fortunately, this is not the case. The point is that the actual parameter of the theory is the product, $\rho = mx$. The problem in question corresponds to $\rho \gg 1$, while the spacing between two sequential resonances $\delta \rho = \pi$, see Eqs. , . Now, if instead of variations of $m$ at a fixed $x$ we consider variations of $x$ at a fixed $m$, to cover the distance between the two sequential resonances we have to consider departures of $x$ from a resonant value $x^{{\rm(res)}}$ (defined by the same conditions, Eqs. , ) of the order $\delta x \sim 1/m \ll 1$. It means that in slowly-varying functions of $x$ solely, namely in $\xi_n(x),\;\psi_n(x)$ and $\chi_n(x)$ we may neglect these small variations of $x$, replacing these functions by their values at $x = x^{{\rm(res)}}$. The only remaining step is to replace $\delta \rho = x\delta m$ by $\delta \rho = m\delta x$. Then, all the expressions obtained in this section and in what follows are readily recalculated for the case of varying $x$ and fixed $m$. Fano resonances in partial wave scattering =========================================== Up to now we have inspected the modula of the scattering coefficients. Let us focus on their phases. Following the approach, described in monograph [@Hulst::1981], it is convenient to introduce a real angle $\Delta_n^{(a)}$ according to the expression $$\begin{aligned} \label{Eq:Delta} \tan\Delta_n^{(a)} \equiv \frac{F_n^{(a)}}{G_n^{(a)}}\;.\end{aligned}$$\ Then, $a_n$ can be written in the following form $$\begin{aligned} \label{Eq:Sca2} a_n=\frac{\tan\Delta_n^{(a)}}{\tan\Delta_n^{(a)}+i}=\frac{i}{2}(1-e^{2i\Delta_n^{(a)}})\;.\end{aligned}$$ Let us recall now that $a_n$ may be presented as a sum of two terms: $a_n^{\rm (PRS)}$ and $\tilde{d}_n=d_n\psi_n'(mx)/(m\xi_n'(x))$, see Eq. . Here the first term does not depend on $m$, while the second is a sharp function of $m$. Then, it makes sense to split $\Delta_n^{(a)}$ into two corresponding parts, namely to present it as $$\label{Eq:Delta_as_a_sum} \Delta_n^{(a)} \equiv \Delta_n^{(a,{\rm PRS})} + \Delta_n^{(a,{\rm res})},$$ where $\Delta_n^{(a,{\rm PRS})}$ is defined for a PRS in the same manner as that for the just introduced $\Delta_n^{(a)}$ for a particle with arbitrary $m$, i.e., $$\label{Eq:tan_delta_PRS} \tan \Delta_n^{(a,{\rm PRS})} \equiv \frac{F_n^{(a,{\rm PRS})}}{G_n^{(a,{\rm PRS})}} = - \frac{\psi_n'(x)}{\chi_n'(x)},$$ see Eq. . Then, the quantity $\Delta_n^{(a,{\rm res)}}$ is defined by the identity $$\begin{aligned} \tan\Delta_n^{(a)} &\equiv& \tan(\Delta_n^{(a,{\rm res)}} + \Delta_n^{(a,{\rm PRS})})=\nonumber \\ & & \frac{\tan\Delta_n^{(a,{\rm res)}} +\tan\Delta_n^{(a,{\rm PRS})}}{1-\tan\Delta_n^{(a,{\rm res)}}\tan\Delta_n^{(a,{\rm PRS)}}}\;.\label{Eq:Delta_res}\end{aligned}$$ Taking into account Eqs. , and , after some algebra it is possible to show that to satisfy Eq. identically, tangent of $\Delta_n^{(a,{\rm res)}}$ must be equal $$\label{Eq:tan_Delta_res} \tan \Delta_n^{(a,{\rm res)}} =-\frac{\psi'_n(mx)}{G^{(a,{\rm PRS})}_n G_n^{(a)}+F^{(a,{\rm PRS})}_n F_n^{(a)}}.$$ If now we introduce the notations $$\label{Eq:q_and_epsilon_a} q_n^{(a)} \equiv -\cot \Delta_n^{(a,{\rm PRS})}= \frac{\chi_n'(x)}{\psi_n'(x)},\;\; \epsilon_n^{(a)} \equiv -\cot \Delta_n^{(a,{\rm res)}},$$ the expression for $\|a_n\|^2$ transforms into the following: $$\begin{aligned} \|a_n\|^2=\frac{\left(\epsilon_n^{(a)}+q_n^{(a)}\right)^2}{\left(1+q^{(a)2}\right)\left(1+\epsilon^{(a)2}\right)}\;.\label{Eq:\|a_n\|_Fano}\end{aligned}$$ It is the conventional Fano profile, normalized to its maximal value, [@Fano:PR:1961]. A similar treatment of $b_n$ brings about analogous results with $$\label{Eq:qb_n} q_n^{(b)} = \frac{\chi_n(x)}{\psi_n(x)}.$$ ![(color online) Comparison of profile $\|a_1(m)\|^2$, given by the exact Mie solution (blue solid line), with that, produced by Eqs. , ; (black dashed line); $x=1$. A yellow horizontal line corresponds to $a_1^{\rm (PRS)}$.[]{data-label="fig:Fano_exact_vs_approx"}](Fano_exact_vs_approx "fig:"){width="\linewidth"}\ ![(color online) $q_1^{(a)}(x)$ – solid and $q_1^{(b)}(x)$ – dashed according to Eqs. , . []{data-label="fig:qab"}](qab "fig:"){width="\linewidth"}\ Once again we encounter a discrepancy, this time with our own results. It seems that by means of just identical transformations we have proven that the profiles of $\|a_n\|^2,\;\|b_n\|^2$ are of the Fano type always. On the other hand, it has been shown in our previous publication [@Tribelsky:PRL:2008] that at least in the case of small particles the Fano resonances in partial scattered coefficients cannot happen. Naturally, the discrepancy, as usual, is an illusion. The fact is that the condition $mx \ll 1$ does not allow $m$ to be large enough to reach the point of the first Fano resonance. Therefore, to prove that the profile, Eq. does correspond to the Fano lineshape, we have to show that (i) in the specified range of variations of $m\;\;\epsilon_n^{(a)}(m)$ reaches the values corresponding to the constructive ($\epsilon_n^{(a)} = 1/q_n^{(a)}$) and destructive ($\epsilon_n^{(a)} = -q_n^{(a)}$) conditions, i.e., the problem in question may exhibit the Fano resonances, indeed, and (ii) for a given Fano profile in the vicinity of both the constructive and destructive interference $\epsilon$ is one and the same linear function of $m$. Regarding the former, the manifestation of the constructive and destructive resonances in the problem has been already shown in the previous sections of this paper and in other publications [@Tribelsky:EPL:2012; @Rybin:OE:2013; @Rybin:SciRep:2014]. As for the latter, employing Eqs. , , after simple calculations, it is possible to show that: ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ![(color online) An example of a reduction of the Fano profile to Lorentzian at $x=0.3$. While a logarithmic plot of $\|a_1(m)\|^2$ (a) clearly exhibits the Fano line with the point of the constructive resonance ($\|a_1(m)\|^2=1$) at $m \approx 14.905$ and the destructive resonance ($\|a_1(m)\|^2=0$) at $m \approx 15.012$, a linear plot of the same dependence (b) is, practically, indistinguishable from the Lorentzian profile, centered about the point of the constructive resonance. The high-resolution-lineshape in the vicinity of the constructive and destructive resonances is shown in the inset. The background level $\|a_1^{{\rm (PRS)}}\|^2 \approx 0.0003404$. It corresponds to $q^{(a)2} \approx 2937$.[]{data-label="fig:Fano-Lorentz"}](Fano-Lorentz_a "fig:"){width="50.00000%"} ![(color online) An example of a reduction of the Fano profile to Lorentzian at $x=0.3$. While a logarithmic plot of $\|a_1(m)\|^2$ (a) clearly exhibits the Fano line with the point of the constructive resonance ($\|a_1(m)\|^2=1$) at $m \approx 14.905$ and the destructive resonance ($\|a_1(m)\|^2=0$) at $m \approx 15.012$, a linear plot of the same dependence (b) is, practically, indistinguishable from the Lorentzian profile, centered about the point of the constructive resonance. The high-resolution-lineshape in the vicinity of the constructive and destructive resonances is shown in the inset. The background level $\|a_1^{{\rm (PRS)}}\|^2 \approx 0.0003404$. It corresponds to $q^{(a)2} \approx 2937$.[]{data-label="fig:Fano-Lorentz"}](Fano-Lorentz_b "fig:"){width="50.00000%"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ $$\begin{aligned} \epsilon_n^{(a)} &=& \left(\psi_n'^2(x) + \chi_n'^2(x)\right)m^{{\rm(res})}x\delta m\nonumber\\ & & - \psi_n'(x)\psi_n(x) - \chi_n'(x)\chi_n(x).\label{Eq:epsilon_though_delta_m}\end{aligned}$$ It means that in the vicinity of the resonances $\epsilon_n^{(a)}$ is a linear function of $\delta m$, indeed. Regarding the validity of this approximation up to the points of the extrema of the profile, it follows from Eqs. , . Note that Eq yields that the characteristic scale for the Fano profiles in $\delta m$ is $1/m$, which agrees with the previous consideration of the linewidths, see Section \[sec:3\]. It is worthwhile mentioning that the accuracy of approximation, is surprisingly good. As an example a comparison of presentation, Eqs. , , with the exact Mie solution for $\|a_1\|^2$ at $x=1$ in the vicinity of $m=4.5$ is presented in Fig. \[fig:Fano\_exact\_vs\_approx\]. It is important that the asymmetry parameter $q_n^{(a,b)}$ in this case is expressed by simple formulas, Eqs. , in terms of $\psi_n(x)$ and $\chi_n(x)$, i.e., it depends just on the multipolarity of the scattered partial wave $n$, radius of the scattering sphere and wavelength of the incident light (we remind that $x \equiv 2\pi R/\lambda$) and does not depend on the reflective index of the particle $m$. Note also that the expressions for $q_n^{(a,b)}$ may be obtained in a less formal way too. According to the preceding analysis in the off-resonance regions In terms of the Fano resonances these regions correspond to the limit $\epsilon \rightarrow \infty$, when the Fano profile tends to $1/(1+q^2)$, see Eq. . Equalizing $a^{({\rm PRS})}$ to $1/(1+q^{(a)2})$ and $b^{({\rm PRS})}$ to $1/(1+q^{(b)2})$ we again arrive at Eqs. , for $q_n^{(a,b)}$. Let us discuss the dependence $q_n^{(a,b)}(x)$ in detail. At $x > n^2$ we just can take the first term of the asymptotical expansions of the Riccati-Bessel functions. For $\psi_n(x)$ it is given by Eq. . The corresponding asymptotics for $\chi_n(x)$ reads [@DLMF::2015] $$\label{Eq:chi_n_at_large_x} \chi_n(x) \cong \cos\left(x-\frac{n\pi}{2}\right).$$ Then, at $x > n^2$ $$\label{Eq:qab_at_large_x} q_n^{(a)} \cong - \tan\left(x-\frac{n\pi}{2}\right),\;\; q_n^{(b)} \cong \cot\left(x-\frac{n\pi}{2}\right).$$ Note, Eq. yields $q_{n}^{(a)}= q_{n \pm 1}^{(b)}=q_{n+2}^{(a)}$. In the opposite limit of small $x$, utilizing the known asymptotic expressions for the Bessel functions at a small value of the argument, we arrive at the following expressions for $q_n^{(a,b)}$ at $x \ll 1$: $$\begin{aligned} q_n^{(a)} & \cong & -\frac{n}{n+1} \frac{2^{1 + 2 n}\Gamma(n +\frac{1}{2}) \Gamma(n +\frac{3}{2})}{\pi x^{2n+1}}, \label{Eq:qa_at_small_x} \\ & \mbox{}&\nonumber\\ q_n^{(b)} & \cong & \frac{2^{1 + 2 n}\Gamma(n +\frac{1}{2}) \Gamma(n +\frac{3}{2})}{\pi x^{2n+1}}, \label{Eq:qb_at_small_x}\end{aligned}$$ where $\Gamma(z)$ stands for the gamma function. As an example, the dependences $q_1^{(a,b)}(x)$ in domain $0\leq x \leq 10$, are shown in Fig. \[fig:qab\]. Both expressions, Eqs. and Eq. diverge as $1/x^{2n+1}$ at $x \rightarrow \infty$. On the other hand, at large $q^2$ the Fano profile, Eq. is reduced to the Lorentzian one: $$\label{Fano_to_Lorentz} \frac{\left(\epsilon+q\right)^2}{\left(1+q^{2}\right)\left(1+\epsilon^{2}\right)} \xrightarrow[q \rightarrow \infty]{}\frac{1}{1+\epsilon^{2}}.$$ ------------------------------------ ------------------------------------ ![image](a_bcd){width="50.00000%"} ![image](ab_cd){width="50.00000%"} ![image](abc_d){width="50.00000%"} ![image](abcd_){width="50.00000%"} ------------------------------------ ------------------------------------ To understand the physical grounds for this reduction, we have to highlight the following. As it has been pointed out already, in the off-resonance regions the Fano profile tends to a non-zero constant background value, $1/(1+q^2)$, while the Lorentzian profile tends to zero, cf. the limits of the right- and left-hand-side of Eqs. . At the point of the constructive resonance ($\epsilon = 1/q$) the Fano profile exhibits its maximal value, equal to unity. So does the Lorentzian profile. However, if for the Lorentzian profile its increase from the off-resonance region also equals unity, for the Fano profile it equals the difference between unity and the background level, i.e., $1-[1/(1+q^2)] = q^2/(1+q^2)$. In the point of the destructive resonance the Fano profile drops from the background level to zero, that is to say, the amplitude of the corresponding modulation of the profile equals the background level $1/(1+q^2)$. The ratio of the amplitudes of the two modulations of the profile (constructive to destructive) is just $q^2$. The larger $q^2$, the less pronounced the destructive resonance relative to constructive and the closer the Fano profile to Lorentzian. If now we remember, that the case $x \ll 1$ corresponds to a small particle and a small particle in the off-resonance regions is a weak scatterer, we immediately understand why the background level at $x \ll 1$ is small and the Fano profile is close to Lorentzian. Thus, we have arrived at an important conclusion: though, formally, the Mie resonances for the partial scattered waves of the outer problem for small particles still belong to the Fano type, actually, the corresponding lineshape is very close to the Lorentzian, cf. [@Tribelsky:EPL:2012]. An example of such a profile is presented in Fig. \[fig:Fano-Lorentz\]. Scale invariance ================ The results discussed in the previous sections give rise to a simple but very important conclusion. Namely, we have obtained that at any fixed $x$ and $n$ the profiles of $\|a_n(m)\|^2,\;\|b_n(m)\|^2$ exhibit infinite sequences of the Fano resonances. All these resonances have one and the same, $m$-independent value of $q$, see Eq. , and the characteristic scale, decreasing as $1/m^{{(\rm res})}$, see Eqs. , . Since the shape of the Fano profile is completely defined by the value of $q$, the latter means that at a given $x$ all the Fano resonances are identical and may be reduced to a single universal form by the scale transformation $$\label{Eq:scale_m} \delta m \rightarrow m^{({\rm res})}\delta m.$$ Regarding $\|c_n\|^2,\;\|d_n\|^2$, at a fixed $m$ these profiles are Lorentzian, and, therefore, also universal. The corresponding scale transformations are Eq. , supplemented with the rescaling of the coefficients $$\label{Eq:scale_cd} c_n \rightarrow c_n/m^{({\rm res})},\;\; d_n \rightarrow d_n/m^{({\rm res})}.$$ Finally, bearing in mind that the mismatches in the positions of the points of the resonances for modes with different $n$ are also scaled as $1/m^{({\rm res})}$, we obtain that the stipulated scale transformations should reduce the entire variety of the resonances to a single universal set, including the mutual position of the resonances with different $n$. An example of such a collapse is shown in Fig. \[fig:Fig4\]. It is also seen from Fig. \[fig:Fig4\] that an increase in $n$ results in shifts of the points of resonances for $\|c_n\|^2$ and $\|d_n\|^2$ to the left. For the Fano resonances (Fig. \[fig:Fig4\]a,b) the increase in $n$ is also accompanied by the change of the asymmetry parameter $q$, see Eq. , and hence, by the change of the corresponding lineshape. Thus, to identify the shift one should look at the position of the local extrema. Note also, that for the lines $\|c_n\|^2,\;\|d_n\|^2$ the increase in $n$ results in an increase in the $Q$-factor of the resonances, see Fig. \[fig:Fig4\]c,d. These peculiarities are generic for the problem in question and valid for any $n$. Dissipative effects {#sec:6} =================== Remember now, that the non-dissipative limit discussed above is a nonexistent in nature abstraction. In fact, dissipative losses always remain finite, as long as a real material is a concern. Then, a natural question “How the obtained results are affected by the dissipative losses?" arises. In this section we are going to answer the question. To this end, we have to introduce a complex refractive index $$\label{Eq:mhat} \hat{m} = m + i\kappa$$ Now note that, actually, there are two cases: strong dissipation ($m$ and $\kappa$ are of the same order of magnitude) and weak dissipation ($\kappa \ll m$). The former case is trivial — the dissipation just suppresses the resonances. Thus, the most interesting is the weak-dissipation-case, especially its limit of the extremely weak dissipation ($\kappa \ll 1$), when the dissipative damping may compete with the small radiative one, and effects similar to the anomalous scattering [@Tribelsky:PRL:2006] may be observed. Examples of the extremely weak dissipation may be readily found among widely used semiconductors. For instance, at the wavelength of 532 nm (the second harmonic of a Nd:YAG laser) the complex refractive indices for gallium phosphide, silicon and gallium arsenide are respectively [@refractiveindex.info]. In what follows all numerical examples, illustrating the developed theory, will be given for a particle made of gallium phosphate embedded into a transparent matrix with a refractive index close to unity. The dispersion of will be neglected, since in the proximity of the specified wavelength it is rather weak. Let us discuss the weak dissipation case in detail. For this purpose, we have to consider small complex departures of $\delta\hat{m} = \delta m + i\kappa$ from a purely real $m^{{(\rm res)}}$, defined by the condition . Expansion of $\psi(\hat{m}x)=0$ in powers of small $x\delta\hat{m}$ about the point $m^{{(\rm res)}}x$ results in the following trivial generalization of expression, Eq. for $\|\tilde{d}_n\|^2$: $$\label{Eq:lineshape_\|tilde_d_n\|^2_with_kappa} \|\tilde{d_n}\|^2 = \frac{1}{\left(m^{\rm (res)}x\delta m -A_n^{(d)}(x)\right)^2B_n^{(d)2}(x) +\left(1+\kappa B_n^{(d)}(x)m^{\rm (res)}x\right)^2},$$ with the connection between $\|\tilde{d}_n\|^2$ and $\|{d}_n\|^2$ $$\label{Eq:\|dn\|^2_through_\|tilde_dn\|^2} \|d_n\|^2 = \frac{m^2B_n^{(d)}(x)}{\|\psi'_n(mx)\|^2} \|\tilde{d}_n\|^2,$$ following from Eqs. , . Eq. yields the linewidth $$\label{Eq:Gamma_\|dn\|^2_kappa} \gamma_{\|d_n\|^2}^{(\kappa)} = 2\frac{1+\kappa B_n^{(d)}(x)m^{\rm (res)}x}{B_n^{(d)}(x)m^{\rm (res)}x}.$$ The profile $\|d_n\|^2$ is maximized by the same $\delta m_{\|d_n\|^2}^{\rm (res)}$, given by Eq. , but the maximal value now is different: $$\label{Eq:Max_\|dn\|_kappa} \text{Max}\left\{ \|d_n\|^2\right\} \cong \frac{m^{{\rm (res)}2}B_n^{(d)}(x) }{\|\psi'_n(m^{\rm(res)}x)\|^2\left(1+\kappa B_n^{(d)}(x)m^{\rm (res)}x\right)^2}.$$ The obtained results give rise to important conclusions. Namely, as it is clearly seen from Eq. , there is a point of crossover, $m_{\rm cr}$ from the non-dissipative regime (at $m \ll m_{\rm cr}$) to dissipative (at $m \gg m_{\rm cr}$), where $m_{\rm cr}$ is a solution of the equation: $$\label{Eq:m_cr} \kappa m x B_n^{(d)}(x) = 1.$$ If at $m \ll m_{\rm cr}$ the linewidth is determined by Eq. , at $m \gg m_{\rm cr}$ it converges to the universal $m$-$x$-$n$-independent value $2\kappa$. However, the most dramatic changes happen with the amplitude of the resonances. While in the non-dissipative limit the amplitude of the profile $\|d_n\|^2$ at the resonance points at large $m$ increases as $m^2$, see Eq. , now the entire profile converges to the universal form $$\label{Eq:\|dn\|_kappa_large_m} \|d_n\|^2 \xrightarrow[{m \gg {m_{_{cr}}}}]{}\frac{1}{\|\psi'_n(m^{\rm(res)}x)\|^2x^2B_n^{(d)}(x)}\frac{1}{\delta m^2 + \kappa^2},$$ which becomes completely $m^{\rm(res)}$-independent in the Fraunhofer regime, $m^{\rm(res)}x > n^2$ (we remind that in this regime $\|\psi'_n(m^{\rm(res)}x)\| \cong 1$). Coefficients $c_n$ may be inspected exactly in the same manner. The corresponding expressions this inspection yields are quite similar to those, obtained above for $d_n$. For example, $$\begin{aligned} & & \text{Max}\left\{ \|c_n\|^2\right\} \label{Eq:Max_\|cn\|_kappa}\\ & & \cong \frac{m^{{\rm (res)}2}B_n^{(c)}(x) }{\|\psi_n(m^{\rm(res)}x)\|^2\left(1-\kappa\frac{\psi''(m^{\rm (res)}x)}{\psi(m^{\rm (res)}x)} B_n^{(c)}(x)m^{\rm (res)}x\right)^2}.\nonumber\end{aligned}$$ The reader should not be confused with sing minus in front of the term with the dissipative constant $\kappa$. The point it that this term includes also the factor ${\psi''(m^{\rm (res)}x)}/{\psi(m^{\rm (res)}x)}$, which is always negative and tends to minus unity in the Fraunhofer regime, see Eq. . Thus, actually, the sign in front of the dissipative constant is plus. The other expressions for $c_n$, analogous to those discussed above for $d_n$, are not presented here owing to the triviality of the corresponding calculations. They result in the behavior of $c_n$ quite similar to that for $d_n$. These peculiarities of $c_n$ and $d_n$ brings about a completely different scenario for vanishing of the Fano resonances for coefficients $a_n$ and $b_n$. For definiteness, let us focus on $a_n$. This coefficient is expressed in terms of $m$-independent $a^{{\rm (PRS)}}_n$ and $m$-dependent $d_n$ according to Eq. . Thus, the entire $m$-dependence of $a_n$ is given by the second term in the right-hand-side of Eq. . We remind the reader that in the non-dissipative limit $\|a_n\|^2$ always vanishes at the points of the destructive Fano resonances and reaches unity at the points of the constructive resonances, see Fig. \[fig:a1\] and Eq. . Then, the vanishing of the Fano resonances with an increase in $m$ occurs owing to the contraction of the resonance lines. Asymptotically the points of the constructive and destructive resonances merge, while the amplitude of modulations of $\|a_n\|^2$ caused by the Fano resonances all the time remains the same: each resonance forces $\|a_n\|^2$ to vary from zero to unity. In contrast, now at large enough $m$ the width of the resonances for $\|a_n\|^2$ becomes $m$-independent and equal to $2\kappa$ (according to Eq. the characteristic widths of the resonances for $a_n$ and $d_n$ have the same order of magnitude), while the amplitude of the modulations of $\|a_n - a^{{\rm (PRS)}}_n\|^2$ decreases as $1/m^2$, see Eqs. , . Thus, the vanishing of the Fano resonances (convergence of $a_n$ to $a^{{\rm (PRS)}}_n$) occurs owing to the vanishing of the amplitude of the modulations of $a_n$. A crossover from the non-dissipative scenario to dissipative is again determined by $m_{{\rm cr}}$, see Eq. . The behavior of $b_n$ is analogous. Resonances at varying size parameter and fixed refractive index =============================================================== Amplitudes of resonances ------------------------ Up to now the resonances have been studied at a fixed $x$ (and $\kappa$) at varying $m$. On the other hand, as it has been mentioned above, the most interesting from the experimental viewpoint is the dependence of the resonances on the size of the particle at a fixed value of the refractive index. In this case at any fixed $n$ an increase in $x$ again results in a cascade of resonances, whose position is determined by the same conditions Eqs. , , regarded now as equations At $mx \gg 1$ these two problem formulations (fixed $x$ at varying $m$ and fixed $m$ at varying $x$) are easily reduced to each other by the set of transformations as long as the shape of a single resonance line is a concern, see the corresponding remark in the end of Sec. \[sec:3\]. However, this is not the case anymore, if we are interested in rather a large range of variations of the size parameter. The corresponding behavior of the scattering coefficients is discussed in the present section. ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![(color online) The behavior of $B_n^{(d)}(x) \equiv \|\xi_n'(x)\|^2$ and $B_n^{(c)}(x) \equiv \|\xi_n(x)\|^2$ for the first five multipoles. The curves from left to right correspond to $n=1,\;2,\;3,\;4$ and 5, respectively.[]{data-label="fig:Bn(x)"}](Bdn "fig:"){width="50.00000%"} ![(color online) The behavior of $B_n^{(d)}(x) \equiv \|\xi_n'(x)\|^2$ and $B_n^{(c)}(x) \equiv \|\xi_n(x)\|^2$ for the first five multipoles. The curves from left to right correspond to $n=1,\;2,\;3,\;4$ and 5, respectively.[]{data-label="fig:Bn(x)"}](Bcn "fig:"){width="50.00000%"} ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- The results obtained in the previous Sec. \[sec:6\] show that the key function determining the behavior of the resonances at large variations of $x$, is $B_n^{(d)}(x) = \|\xi_n'(x)\|^2$. The known expansions of the Bessel functions in powers of their small argument yield the following behavior of $B_n^{(d)}(x)$ at $x \rightarrow 0$: $$\label{Eq:Bn_at_x->0} B_n^{(d)}(x) \cong \frac{n^2 4^n \Gamma^2(n+\frac{1}{2})}{\pi x^{2(n+1)}}.$$ An increase in $x$ is supplemented with a monotonic decrease in $B_n^{(d)}$ until this function reaches its minimal value at $x \approx n$. A further increase in $x$ results in a slow monotonic growth of $B_n^{(d)}$, asymptotically approaching unity at $x \rightarrow \infty$, see Eqs. , . Accordingly, the entire domain $0 \leq x < \infty$ of variations of $x$ is partitioned into two subdomains: the subdomain $0 < x \leq n$ of a sharp fall of $B_n^{(d)}(x)$ from infinity to its minimal value below unity and the one of a slow asymptotical growth $B_n^{(d)}(x)$ to unity $(n < x <\infty)$. As an example, functions $B_n^{(d)}(x)$ for the first five multipoles are presented in Fig. \[fig:Bn(x)\]a. Thus, at any $n$ both $B_n^{(d)}(x)$ and $xB_n^{(d)}(x)$ are singular functions at $x\rightarrow 0$. It brings about a dramatic enhancement of the dissipative effects at small $x$, see Eq. . On the other hand, an increase in $B_n^{(d)}(x)$ increases the numerator in the right-hand-side of Eq. , defining the amplitude of the resonance. That is to say, at small $x$ there is a competition between the growth of the amplitude of the resonance owing to the increase in $B_n^{(d)}(x)$ in the numerator of Eq. and its suppression because of the growth of the same quantity in the denominator of the same equation. In this case we have to distinguish to limits: \(i) Despite the large value of $B_n^{(d)}$ the first (smallest) resonant $x^{({\rm res})}_{n,1}<n$ still corresponds to weak dissipation . Then, the amplitude of this resonance is the largest for the given $n$. An increase in $x$ gives rise to the fall of the amplitudes of the sequential resonances, first due to the decrease of $B_n^{(d)}(x)$ in the numerator of Eq. \[Eq:Max\_\|dn\|\_kappa\], and then due to the transition to the dissipation-controlled region due to increase of $\kappa B_n^{(d)}(x)m x$ in the denominator of this expression. \(ii) $x^{({\rm res})}_{n,1}<n$ is so small (i.e., $B_n^{(d)}(x^{\rm (res)}_{n,1})$ is so large), that $\kappa B_n^{(d)}(x^{\rm (res)}_{n,1})m x^{\rm (res)}_{n,1} \gg 1$, despite smallness of $\kappa$. Then, the amplitude of the corresponding resonance approximately equals $$\text{Max}\{\|d_n\|^2\} \cong \frac{1}{(\psi'_n(mx^{\rm (res)})\kappa x^{\rm (res)})^2B_n^{(d)}(x^{\rm (res)})},$$ see, Eq. . Since, by definition, in a given cascade $x^{\rm (res)}_{n,1}<x^{\rm (res)}_{n,2}<x^{\rm (res)}_{n,3}<\ldots,$ the next resonances in the same cascade result in a decrease of $B_n^{(d)}(x^{\rm (res)}_{n,p})$ and, hence, in an increase of their amplitude. It goes on in this manner until the decease of $B_n^{(d)}$, eventually, drives the particle out of the dissipation-controlled regime. Then, a further increase in $x^{\rm (res)}$ results in the effects, described above in Thus, now we have two dissipation-controlled domains: the first at small $x$ and the second at large, separated by a non-dissipative domain. The maximal amplitude of the resonance mode is achieved at $x^{\rm (res)}$ situated at the boundary between the first dissipation-control domain the non-dissipative domain. It is important to stress also the quite different asymptotic behavior of profiles $\|d_n(m)\|^2$ at a fixed $x$, and $\|d_n(x)\|^2$ at a fixed $m$, respectively. If $\|d_n(m)\|^2$ at $m \rightarrow \infty$ converges to a certain universal periodic function, see Eq. , $\|d_n(m)\|^2$ at $x \rightarrow \infty$ vanishes as $1/(\kappa x)^2$, owing to Eq. and limits $$\|\psi'_n(mx^{\rm (res)})\| \xrightarrow[{x^{\rm (res)} \to \infty }]{}1,\;\; B_n^{(d)}(x) \xrightarrow[{x \to \infty }]{}1.$$ The behavior of $\|c_n\|^2$ in general is analogous to the discussed above for $\|d_n\|$. However, in contrast to $B_n^{(d)}(x)$, function $B_n^{(c)}(x)$ is monotonically decreasing, see Fig. \[fig:Bn(x)\]b. This difference in $B_n$ results in a certain difference in the shape of the envelopes of the resonances for $\|d_n(x)\|$ and $\|c_n(x)\|$, see Fig. \[fig:GaP\_d20\]. -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![(color online) A typical behavior of $\|d_n(x)\|$ and $\|c_n\|$ on the example of $n=20$ for a particle made of GaP. Blue solid lines corresponds to the exact solution, Eqs. ; the envelopes, calculated according to the approximate Eq. are shown as red dashed lines. Note strongly non-monotonic dependence of the amplitude of the resonant oscillations on $x$ due to the enhancement of the dissipative effects at certain domains of variations of $x$ and a “bottle neck" in the dependence $\|d_n(x)\|$ at the proximity of $x=n=20$. The bottle neck is originated in the corresponding minimum in $B_n^{(d)}(x)$, which does not have $B_n^{(c)}(x)$. For more details, see the text. []{data-label="fig:GaP_d20"}](GaP_d20 "fig:"){width="50.00000%"} ![(color online) A typical behavior of $\|d_n(x)\|$ and $\|c_n\|$ on the example of $n=20$ for a particle made of GaP. Blue solid lines corresponds to the exact solution, Eqs. ; the envelopes, calculated according to the approximate Eq. are shown as red dashed lines. Note strongly non-monotonic dependence of the amplitude of the resonant oscillations on $x$ due to the enhancement of the dissipative effects at certain domains of variations of $x$ and a “bottle neck" in the dependence $\|d_n(x)\|$ at the proximity of $x=n=20$. The bottle neck is originated in the corresponding minimum in $B_n^{(d)}(x)$, which does not have $B_n^{(c)}(x)$. For more details, see the text. []{data-label="fig:GaP_d20"}](GaP_c20 "fig:"){width="50.00000%"} -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Overlap of resonances --------------------- One of the key features of the problem in question is an overlap of a large number of resonances, which may result in a creation within the particle “hot spots" with a giant concentration of the electromagnetic field. The corresponding field structure is determined by the coordinate dependence of the resonant modes. Its detailed inspection requires a separate consideration and will be reported elsewhere. In the present paper we discuss just the necessary conditions for such hot spots to come into being, namely the overlap of profiles of the modula of the scattering coefficients, describing the field within the particle. As usual, for the sake of briefness we restrict the discussion with the properties of $\|d_n\|^2$ solely. The behavior of $\|c_n\|^2$ may be studied in the same manner and exhibits similar peculiarities. Overlap of two resonances means that the mismatch between the positions of the maxima of their lines is smaller than the largest linewidth. Therefore, to begin with, we have to discuss the linewidths of the resonances at a fixed $m$ and varying $x$. To this end, according to the stipulated above general rules, we have to replace in Eq. $x\delta m \rightarrow m\delta x,\; m^{\rm (res)} \rightarrow m$ and $x \rightarrow x^{\rm (res)}$. This brings about the following expression for the linewidth: $$\label{Eq:gamma_x} \gamma_{\|d_n\|^2}^{(\kappa,x)} = 2\frac{1+\kappa B_n^{(d)}(x^{\rm (res)})mx^{\rm (res)}}{B_n^{(d)}(x^{\rm (res)})m^2}.$$ Thus, in the non-dissipative region ($\kappa B_n^{(d)}(x)mx \ll 1$) the linewidth is $2/[B_n^{(d)}(x)m^2]$, while in the dissipation-controlled region ($\kappa B_n^{(d)}(x)mx \gg 1$) it is $2\kappa x/m$. On the other hand, the resonant values of $x$ are defined by the condition $\psi_n(mx^{{\rm (res,}E)}) = 0$ for the electric modes and for magnetic, see Eqs. , . Below the Fraunhofer regime, at $mx < n^2$ different solutions of these equations are situated at distances of the order of $1/m$. It is always much larger than the linewidth in the non-dissipative region and still larger than that in the dissipation-controlled case, provided $x$ is of the order of unity, or smaller. Therefore, the overlap of the resonances in these regions may occur just accidentally. However, the case is changed drastically in the Fraunhofer regime, when $mx$ becomes larger than $n^2$. In this regime points of different resonances are going to merge. In the leading (in $1/mx$) approximation they are just coincide, see Sec. \[sec:2\]. To resolve the mismatch between the points of different resonances we have to go beyond the leading approximation. Employing the asymptotic expansion of the Riccati-Bessel functions at large values of their argument [@DLMF::2015] and taking into account the first subleading term, instead of Eq. we arrive at the following equation, determining the points of the electric mode resonances: $$\label{Eq:sin=0_subleading} \sin\left(\rho_{_E}-\frac{n\pi}{2}\right) + \frac{n(n+1)}{2\rho_{_E}}\cos\left(\rho_{_E}-\frac{n\pi}{2}\right) \cong 0,$$ where $\rho_{_E} > n^2$ stands for $mx^{{\rm (res,}E)}$. Looking for a solution of this equation in the form $\rho_{_E} = \rho_{_E}^{(0)} + \delta \rho_{_E};$ $\|\delta\rho_{_E}\| \ll 1$, where $\rho_{_E}^{(0)}=[(2p+n)\pi/2]$ it is easy to obtain that $$\label{Eq:delta_rho_E} \delta\rho_{_E} \cong - \frac{n(n+1)}{\pi(n+2p)}.$$ The solution is valid at $n(n+1) \ll \pi(n+2p)$. The corresponding mismatch between different resonances is $\delta \rho_{_E}/m$. Strictly speaking, we have to add to this mismatch another one, caused by the departure of the position of the maxima of $\|d_n\|^2$ from the resonance points determined according to the employed condition $\psi_n(mx) =0$. However, according to Eq. this mismatch equals $A_n^{(d)}(x)/m^2$ (we recall, that to apply Eq. to our case we have to replace $x\delta m \to m\delta x$). At large $m$ this quantity is small relative to $\delta\rho_{_E}/m$ and may be neglected. ![(color online) Cascades of resonances for $\|c_n\|$ and $\|d_n\|$ at a fixed refractive index and varying size parameter, $x$ for a spherical particle made of GaP, embedded in a transparent medium with the refractive index close to unity. Calculations according to Eqs. , . Gradual overlap of resonances for and at an increase in $x$ is seen clearly. Note broadening of the resonance lines with an increase in $x$ and an increase of the amplitudes of the first resonances in each cascade with an increase in $n$. The first resonances in each cascades have the largest amplitudes. The large mismatches between the first resonances in the cascades for $\|c_1\|$ – $\|d_2\|$ and $\|c_2\|$ – $\|d_3\|$ are generic, while the overlaps of the first resonances in the cascades for $\|d_1\|$ – $\|c_2\|$ and $\|d_2\|$ – $\|c_3\|$ are accidental. For more details, see the text.[]{data-label="fig:GaP_ovlp"}](GaP_ovlp){width="50.00000%"} Similar inspection of $\|c_n\|^2$ gives rise to $\rho_{_H} = \rho_{_H}^{(0)} + \delta \rho_{_H}$ with $$\label{Eq:delta_rho_H} \rho_{_H}^{(0)} = \frac{(n+2p+1)\pi}{2},\;\; \delta\rho_{_H} \cong \frac{n(n+1)}{\pi(n+2p+1)}.$$ Let us try to collect all together. According to the results discussed in Sec. \[sec:2\], there are two types of possible overlaps. First, corresponds to the overlap of the modes of the same type (e.g., electric – electric, or magnetic – magnetic) with different $n$ and $p$, changed in such a manner that sum $n+2p$ remains fixed. The adjacent candidates for this overlap have the difference in $n$ equals 2 and the difference in $p$ equals 1. According to Eqs. , the mismatches between the resonance points for these two modes are $$\label{Delta_x_n_n+2} (\Delta x_{_E})_{n,n+2} \cong (\Delta x_{_H})_{n,n+2} \cong \frac{2(3+2n)}{m\pi N},$$ where integer $N$ satisfies the condition $\pi N/2 = \rho_{_H}^{(0)}=\rho_{_H}^{(0)} = mx^{\rm (res)}$. The candidates for the second type of the overlap are modes with the different nature: electric – magnetic. In this case the adjacent modes have both $n$ and $p$ differed in unity, however, the signs of these mismatches are opposite, see Eqs. , . Then, the corresponding mismatch is $$\label{Delta_x_EH_n_n+1} (\Delta x_{_{E,H}})_{n,n+1} \cong \frac{2(n+1)^2}{m\pi N}.$$ In all the cases the mismatches monotonically increase with an increase in $n$. A rough estimate of the number of modes, which may overlap yields from the applicability condition of the Fraunhofer regime: $\rho > n^2$. For a given $x^{\rm (res)}$ it results in the inequality $n < \sqrt{N} \sim \sqrt{mx^{\rm (res)}}$. Thus, at $n \gg 1$ the mismatches between the modes of the same type are scaled as while the ones between the modes of different types are scaled as $n^2/(mN)$. On the other hand, the linewidth in the dissipation-controlled region is scaled as $\kappa x/m$, see Eq. . An increase in $x$ decreases $\Delta x$ and increases the linewidth. The overlap begins, when the later becomes larger than the former. First, it happens with the modes of the same type at $x = x_{\rm ovlp}$, where $$\label{Eq:x_ovlp} x_{\rm ovlp} \sim \sqrt{\frac{n}{m\kappa}}.$$ A further growth of $x$ increases the number of the overlapping modes, until all modes lying in the Fraunhofer range overlap. It happens at $x>X_{\rm ovlp}$, where $X_{\rm ovlp}$ is such a value of $x$, when $(\Delta x)_{1,\sqrt{N}}$ becomes equal, or smaller than the linewidth. Here $(\Delta x)_{1,\sqrt{N}}$ stands for the mismatch between the mode with $n=1$ and the mode with the maximal $n$ still belonging to the Fraunhofer range, i.e., with $n \sim \sqrt{N}$. According to Eqs. , , $(\Delta x)_{1,\sqrt{N}} \sim (n^2/mN)_{_{n=\sqrt{N}}} = 1/m$. It yields a simple estimate $X_{\rm ovlp} \sim 1/\kappa$. As it follows from Eq. Max$\{\|d_n(X_{\rm ovlp})\|\} \sim 1$. Then, in the point, where all the overlapping modes are summarized coherently with the weight factor of the order of unity, the resulting amplitude of the electric (magnetic) field is $\sqrt{N}$ times greater (the density of energy N times greater) the one in the incident wave. To give a certain impression about a manifestation of the discussed resonances in a possible real experiment we produce some results for a particle made of gallium phosphide. The corresponding plots are presented in Figs. \[fig:GaP\_d20\], \[fig:GaP\_ovlp\]. Our calculations for GaP show also that the absolute maximum for a single resonant mode is achieved for the first resonance of mode $c_5$ at $x \approx 2.318$. The corresponding value of $\|c_5\|$ equals 34.641. The first resonant of $\|d_5\|$ gives rise to the close value: 33.161 at $x \approx 2.626$. The density of the electromagnetic energy in the hot spots corresponding to these solitary resonances is about $10^3$ greater than that for the incident wave. Meanwhile, multiple overlap of not so sharp resonances for a particle with $x \approx 25$ may produce hot spots with the density of energy $10^4$ relative to that in the incident wave. Conclusions =========== Summarizing the results obtained, we may say that the detailed study of light scattering by a high refractive index particle with low dissipation discussed in the present paper, has revealed a number of new important features of the problem. In particular, we have shown that while at increasing $m$ the partial scattered waves outside the particle at any fixed $x$ and $n$ tend to the limits, corresponding to the light scattering by a perfectly reflective sphere, the field within the particle does not have any limit at all. The reason for this difference in the behavior between the solutions of the outer and inner problems is related to the different manifestation of the infinite sequences of cascades of the Mie resonances in the two problems. For the scattered field outside the particle each Mie resonance in a cascade has the asymmetric Fano lineshape. As $m$ increases the resonances are suppressed, and the expressions for the scattering coefficients converge to the corresponding $m$-independent quantities for the perfectly reflecting sphere. For the field within the particle, each Mie resonance in a given cascade has the Lorentzian lineshape. In this case, while at large $m$ the profile of the cascade at a fixed distance from its bottom converges to a certain universal form, the peak value of the modulus of the electric (magnetic) field amplitude increases with an increase in $m$. Thus, the increase in $m$ makes the resonances more pronounced. At finite dissipation the growth of the amplitude of the resonances eventually saturates, and the lineshape becomes a periodic function It is important that at large enough $m$ the positions of the electric resonances for a partial mode with multipolarity $n$ correspond to those for magnetic resonances with the multipolarity $n+1$, the electric mode with the multipolarity $n+2$, etc. The same is true for the magnetic modes. For a given multipolarity the points of both the types of the resonances together (i.e., electric and magnetic) are situated in the $m$-axis periodically with the period equal to the half of the period for each type of the resonances separately. It means the points of a magnetic resonance are situated just in the middle of the two adjacent points of the electric resonances and vice versa. We have shown that the cascade of the Fano resonances, exhibited with an increase in the refractive index of a scatterer, is a general, intrinsic feature of the light scattering problem. In this problem the field scattering by the same scatterer, but with the perfectly reflecting properties plays the role of a background partition, while the resonantly excited in the scatterer Mie modes correspond to the resonant ones. The characteristics of the Fano profiles, including a simple expression for the asymmetry parameter $q$, see Eqs. , , have been obtained from “the first principles" based upon the identical transformations of the exact Mie solution. We have demonstrated that in the non-dissipative limit the discussed resonances (for both the inner and outer problems) possess a scale invariance, so that at any fixed value of the size parameter $x$ any resonance line in the infinite cascades of the resonances may be reduced to the universal, $m^{({\rm res})}$-independent form by the scale transformations, Eqs. , . It should be stressed that the universality extends to both the shape of the lines and the mutual positions of lines with different $n$ with respect to each other. The quantitative applicability conditions for the non-dissipative and dissipation-controlled regimes as well as the corresponding crossover points have been obtained in the explicit form. The peculiarities of the resonances at a fixed refractive index and a varying size parameter, $x$ have been revealed. It has been shown that, generally speaking, the entire domain $0 \leq x < \infty$ is partitioned into three subdomains: two dissipation-controlled subdomains (at small and very large $x$) are separated by a non-dissipative one. The explicit expressions, determining the boundaries between the subdomains as well as the formulas for the lineshape and linewidth have been derived. The linewidth is minimal in the non-dissipative domain and increases in the dissipation-controlled subdomains with an increase in the departure of $x$ from the corresponding boundary. For the size parameter lying in the non-dissipative subdomain a high concentration of the electromagnetic field within the particle may be achieved owing to individual narrow-line partial resonances of modes with a high $Q$-factor. For the realistic values of the complex refractive index, typical for a number of common semiconductors in the visible and near IR range of the spectrum, the peak value of the density of electromagnetic energy in this case may exceed the one in the incident plane wave in the three orders of magnitude. In contrast, for the size parameter, lying in the dissipation-controlled subdomains even larger concentration may be realized due to multiple overlap of rather broad resonance lines of modes with of different types (magnetic and electric) and different multipolarity. The discussed effect of the giant concentration of the electromagnetic field within a particle with a high refractive index may have great importance in medical applications (such as, e.g., cancer therapy [@El-Sayed:LMS:2008]), in the design and fabrication of high-nonlinear nanostructures, etc. A detailed study of this important issue is a separate problem and will be reported elsewhere. The authors are grateful to B.S. Luk’yanchuk for the critical reading of the manuscript and valuable comments. The work of AEM was supported by the Australian Research Council via Future Fellowship\ Proof of identity, Eq. ======================= The proof is based upon Eq. . According to it and presentation of $\xi_n$ in the form with real functions $\psi_n,\;\chi_n$, we may write the following chain of identities: $$\begin{aligned} \frac{\xi_n(x)}{\xi_n'(x)} \equiv \frac{\psi_n'(x)\xi_n(x)}{\psi_n'(x)\xi_n'(x)} \equiv \frac{\psi_n(x)\xi_n'(x) -i}{\psi_n'(x)\xi_n'(x)} \tag{A.1}\label{Eq:xsi_n/xsi_n_prime}\\ \equiv \frac{\psi_n(x)}{\psi_n'(x)} - \frac{i\xi_n'^\ast(x)}{\psi_n'(x)\|\xi_n'(x)\|^2}, \nonumber\end{aligned}$$ where asterisk means complex conjugation. Then, bearing in mind the same presentation $\xi_n = \psi_n - i\chi_n$, and taking the imaginary part of Eq. we obtain $${\rm Im}\frac{\xi_n(x)}{\xi_n'(x)} = -\frac{1}{\|\xi_n'(x)\|^2}.$$\ Finally, multiplying it by $\xi_n'^2(x)$ and taking modulus, we arrive at identity, Eq. . [^1]: The problem of light scattering by a PRS is a bit tricky. The point is that the field inside the PRS is zero identically. Then, instead of the four independent scattering coefficients: two for the scattered field outside the particle and two for the field within the particle, only the former two remain. The reduction of the independent constants of integration of the Maxwell equations, requires the corresponding reduction of the boundary conditions. It could be shown, that for the electric modes discussed here only boundary condition, Eq. remains, while the boundary condition, Eq. should be removed at this limit.	ArXiv
--- title: Jet Physics with Identified Particles at RHIC and LHC --- Introduction ============ The measurement of particle identified high momentum spectra and two-particle correlations can be considered the next step in the characterization of the dense partonic phase created in relativistic heavy ion collisions at RHIC. Much emphasis is presently given to the similarities in the light and heavy quark behavior in the plasma, but these measurements are detailed in other contributions to this workshop [@dasilva; @harris]. Here I will mostly focus on the flavor dependence in light hadron production, i.e. hadrons consisting of u,d,s quarks. I will address in particular the question of baryon versus meson production through fragmentation and recombination as competing mechanisms. In order to determine the baryon production mechanism in AA collisions we need to understand the production in pp first. The main question is whether there is a difference between the light baryon production in the medium and in the vacuum. In the following section I will briefly review the latest particle identified pp results from STAR and then move on to results from AuAu collisions. The ambiguity in the production mechanism as deduced from single particle properties such as the suppression at high pt and elliptic flow v2 can be partially resolved by analyzing two particle correlations. I will show that early results of intermediate p$_{T}$ associated particle production are surprisingly featureless regarding flavor dependencies that were expected from perturbative QCD calculations in vacuum. In contrast, the single particle spectra suppression at high p$_{T}$ in AA seems very much affected by flavor suppression in the proton-proton system, in particular for strange particles. I will try to argue that this effect should be unique to the strange quark, only if the strange quark generation is gluon dominated and the heavy quark production is quark dominated. Many of the studies at RHIC lack the statistics to map out the transition from medium effects to pure vacuum fragmentation. I will show that these measurements are readily accessible at the LHC, in particular with the ALICE detector. Why is pp so important ? ======================== The latest strange and non-strange hadron production results obtained in proton-proton collisions by STAR, were presented at this workshop [@heinz; @ruan]. They show that simple leading order fragmentation codes are not sufficient to describe the baryon production at RHIC energies. Next to leading order calculations do well as soon as quark separated fragmentation functions are used [@akk], in other words there are contributions in the baryon spectra from non-valence quark fragmentation. In a leading order calculation this fact can apparently be approximated by increasing the K-factor and the multiple parton scattering contribution to the underlying event [@bellwied1]. These effects are expected to be significantly less pronounced at higher incident energies [@eskola]. In other words, next to leading order contributions are more important at RHIC energies than at LHC energies. One important new pp result is the break-down of the so called m$_{T}$-scaling [@gatoff; @schaffner] at sufficiently high transverse mass at RHIC. Fig.1a shows the measured spectra of all particle identified species obtained by STAR. After appropriate scaling of each spectrum the m$_{T}$-scaling plot (Fig.1b) can be devised. Clearly the m$_{T}$-scaling at lower m$_{T}$, which had previously been established through measurements at lower energies [@isr1; @isr2], can be confirmed. The noticeable deviation from simple scaling at sufficiently high m$_{T}$ (m$_{T}$ $>$ 3 GeV/c) is a new feature, though. The spectra seem to group into common baryon and meson curves at these momenta. This is a surprising effect in elementary pp collisions which is nevertheless described quantitatively in PYTHIA, as shown in Fig.2. ![(a) Identified transverse mass spectra as measured by STAR in pp collisions. (b) Same spectra as in Fig.1a but scaled via multiplicative factors to the measured pion spectrum.](fig1a-mt-unscaled.eps "fig:"){width="2.in"} \[fig:1a\] ![(a) Identified transverse mass spectra as measured by STAR in pp collisions. (b) Same spectra as in Fig.1a but scaled via multiplicative factors to the measured pion spectrum.](fig1b-mt-scaled.eps "fig:"){width="2.in"} \[fig:1b\] Figs.2a and 2b break down the effect, as modeled in PYTHIA, for gluon and quark jets. The combined spectrum (Fig.2c) is dominated by the gluon jets. According to PYTHIA the ratio of gluon to quark jets is about 2:1 at RHIC energies. ![PYTHIA simulation of the scaled identified mt spectra in Fig.1b from (a) gluon jet fragmentation (g-g or q-g) only, (b) quark jet fragmentation (q-q) only, and (c) sum of all parton fragmentation processes based on the relative ratio of quark to gluon jets at 200 GeV according to PYTHIA.](fig2a-pythia-gluon-mt.ps "fig:"){width="1.5in"} \[fig:2a\] ![PYTHIA simulation of the scaled identified mt spectra in Fig.1b from (a) gluon jet fragmentation (g-g or q-g) only, (b) quark jet fragmentation (q-q) only, and (c) sum of all parton fragmentation processes based on the relative ratio of quark to gluon jets at 200 GeV according to PYTHIA.](fig2b-pythia-quark-mt.ps "fig:"){width="1.5in"} \[fig:2b\] ![PYTHIA simulation of the scaled identified mt spectra in Fig.1b from (a) gluon jet fragmentation (g-g or q-g) only, (b) quark jet fragmentation (q-q) only, and (c) sum of all parton fragmentation processes based on the relative ratio of quark to gluon jets at 200 GeV according to PYTHIA.](fig2c-pythia-sum-mt.ps "fig:"){width="1.5in"} \[fig:2c\] One very interesting feature in the parton separated spectra (Figs.2a and 2b) is that apparently the gluon jet fragmentation leads to a baryon/meson difference at high m$_{T}$, whereas the quark jets show a mass ordering of the high m$_{T}$ spectra. Gluon fragmentation dominates at RHIC (and even more at LHC) energies, and based on PYTHIA the baryon/meson splitting effect in the kinematic spectra is due to the di-quark formation process which is a pre-requisite for baryon formation in LUND type string fragmentation [@lund-baryon]. The initial di-quark formation leads to a lowering of the $<$p$_{T}$$>$ for the baryons (the so called di-quark suppression factor), whereas for mesons the simple quark-antiquark hadronization does not affect the $<$pT$>$. This feature in the fragmentation of elementary gluon jets could be considered the seed of the baryon meson differences in AA collisions. The effect by itself is not big enough, though, to quantitatively describe the strong peak in the baryon/meson ratio at intermediate pt in AA, but the overall feature (a bump at intermediate p$_{T}$) can already be seen in pp collisions and might be simply enhanced by additional effects in AA. In AA one needs the combined effects of radial flow and quenching to ’pile-up’ the baryons over mesons in the intermediate p$_{T}$ range. Still it is important to note that the baryon/meson differences already have their origin in the basic fragmentation process in pp collisions. This measurement also hints at the validity of the fragmentation process as modeled by PYTHIA. It is my opinion that in its nature the diquark-quark formation process is similar to a three quark coalescence process and thus exhibits equivalent features. Another conclusion from the pp modeling is that, although early pQCD calculations predicted a stronger drop in the anti-baryon over baryon ratio, our results are in accordance with PYTHIA calculations if one takes into account the strong gluon dominance at RHIC energies as shown in Fig.3. ![Measured Anti-$\Lambda$ over $\Lambda$ ratio compared to PYTHIA ratio from gluon and quark jets.[]{data-label="fig:3"}](fig3-lambdaratio.eps){width="2.5in"} In order to unambiguously determine the predicted drop, which is due to the difference between quark and gluon distribution functions and the relative contribution of these partons to the baryon and anti-baryon production, one needs measurements at higher p$_{T}$. The momentum range of the spectra will be greatly enhanced at the LHC. Fig.4 shows a projection of p$_{T}$-ranges for identified spectra based on a scaled LHC pQCD calculation. Generally the light quark spectra will reach out to at least 20 GeV/c in the first year of running, which allows for a more unambiguous study of the flavor effects at high p$_{T}$. ![Annual hard process yields expected in the ALICE Calorimeter acceptance for minimum bias Pb-Pb collisions at 5.5 TeV.[]{data-label="fig:6"}](fig4-leading.ps){width="2.5in"} ![Annual hard process yields expected in the ALICE Calorimeter acceptance for minimum bias Pb-Pb collisions at 5.5 TeV.[]{data-label="fig:6"}](fig5.eps){width="2.5in"} One interesting feature in the gluon dominated regime at RHIC and the LHC is that the $\gamma$-jet process is dominated by Compton scattering, which leads to a $\gamma$-quark-jet combination in the outgoing channel. This means not only is the tagged $\gamma$ the ’standard candle’ for the jet energy measurements, but it is also a good trigger for quark-jet events in the gluon dominated regime. The rate, based on early simulations using the ALICE calorimeter, is significantly limited compared to di-jet events, but one can still expect on the order 10,000 events with a $\gamma$ energy above 50 GeV. A similar quark-jet selection can be achieved by triggering on heavy mesons as leading particles. Projected rates for all jet measurements with the proposed ALICE EMCal are shown in Fig.5. Flavor dependencies in identified high pt AA spectra ==================================================== One of the most surprising results of the past year was the apparent difference between R$_{AA}$ and R$_{CP}$ measurements, in particular for strange baryons. Figs.6 and 7 show a direct comparison of strange and non-strange baryons and mesons. ![Nuclear suppression factor (R$_{CP}$) as measured by STAR.[]{data-label="fig:6"}](fig6-raa.eps){width="2.in"} ![Nuclear suppression factor (R$_{CP}$) as measured by STAR.[]{data-label="fig:6"}](fig7-rcp.eps){width="2.in"} The strong jet quenching effect at high p$_{T}$ and the slight baryon/meson splitting due to recombination at moderate p$_{T}$ as shown on the R$_{CP}$ plot is not visible in the R$_{AA}$ plot. The R$_{CP}$ suppression has been established for all particle species from pions to charmed mesons, and besides the already mentioned baryon/meson difference at intermediate p$_{T}$ the suppression is surprisingly flavor independent. The R$_{AA}$ pattern though exhibits a very strong flavor dependence, in particular for baryons. An enhancement of the high p$_{T}$ yield, rather than a suppression compared to the pp spectrum, actually increases as a function of the strangeness content in baryons. A similar enhancement pattern was previously measured for integrated particle yields, and is generally attributed to canonical suppression of strange quarks in small systems [@tounsi], but it is unexpected that apparently this effect of a small correlation volume in an equilibrated system should also affect the high p$_{T}$ particle production. Not only does it lead to an enhancement of the intermediate p$_{T}$ yield from pp to AA but there is also no evidence of quenching in the strange baryon R$_{AA}$ plot at high p$_{T}$. This effect is actually not seen in preliminary results of charmed meson suppression, so it could indeed be unique to the strange quarks. A measurement of a charmed baryon ($\Lambda_c$) is needed to unambiguously determine the difference between the quark flavors. Because of the difference between R$_{CP}$ and R$_{AA}$ the strange quark flavor effects need to have their origin in the particle production in the pp system. This is a good indicator that even intermediate p$_{T}$ strange baryons are predominantly produced through coalescence from a thermalized partonic system. In other words the initial gluon dominated scattering processes leads to thermalized strange quarks which coalesce into strange baryons. The effect of the correlation volume during hadronization is still dominant even at rather large (up to 3 GeV/c) transverse momentum. Beyond the intermediate p$_{T}$ range the spectrum gets quenched, but it is still enhanced in AA collisions compared to the pp system. The question is whether this strange particle production mechanism drives the pp to AA comparison even in the pure fragmentation regime above 7 GeV/c. The RHIC experiments do not have a big enough reach to establish an answer. Fig.4 shows that these measurements can be achieved at the LHC, though. We know from e$^{+}$e$^{-}$ experiments that the strangeness suppression factor in the quark condensate is about 0.4 [@e+e-], but this is just the relative quark production probability in the hadronization sea, which should also exist in the medium. In addition, the strangeness saturation factor increases from pp to AA by a factor two at RHIC energies based on measurements of integrated strange particle yields [@cleymans]. The factor in AA can be described quantitatively in lattic QCD [@gavai]. The difference between baryons and mesons in R$_{CP}$ (Fig.7), which is generally attributed to either recombination or an interplay between radial flow and jet quenching [@lamont], can also be described by the so-called Corona effect, which was shown elsewhere at this conference [@werner; @pantuev]. The principle here is that the formed medium consists of a dense core, which follows hydrodynamics and a corona of pp interactions dominated by multiple scattering. The main reason for such a distinction and the strong contribution from the corona is the relative diffuseness of the nuclear surface, which is not well described by hard spheres. The pp interactions can be modeled by codes such as EPOS which take into account the increased parton cascade activity in the low momentum sector. In these models effects such as baryon/meson splitting in v$_{2}$ occur because the corona, which carries very little v$_{2}$, has a much stronger contribution to the light particle spectrum than the heavy particle spectrum, so it pulls down the hydro v$_{2}$ for mesons to lower values than for baryons. Finally one can measure identified two particle correlations at intermediate p$_{T}$ in order to detect flavor dependencies that are expected from simple fragmentation arguments. The correlations shown in Fig.8 show surprisingly little trigger particle flavor dependence. Again it seems that non-fragmentation processes, such as recombination, dominate in this p$_{T}$ range. There is evidence for long-range correlations in $\Delta\eta$ which could be due production mechanisms that do not exhibit the flavor dependencies of simple fragmentation. These correlations lead to a significant enhancement of the associated yield for any trigger species over the associated yields measured in pp, which are in agreement with the PYTHIA simulations shown in Fig.8b. ![High p$_T$ two particle correlations using identified trigger particles and charged hadron as associated particles. a.) measurements in 0-5% centrality Au-Au collisions in STAR, b.) PYTHIA simulations of the same correlations in pp collisions.[]{data-label="fig:7"}](fig8-complete.eps){width="3.in"} Summary ======= The interpretation of our pp collision results reveal that the baryon production yields require either multiple scattering through a soft particle production model such as EPOS or NLO corrections in pQCD models such as PYTHIA. It is interesting to note that the basic string fragmentation differences in baryon and meson production lead to a breakdown of the universal m$_{T}$-scaling of identified particle spectra. Apparently this breakdown is driven by the baryon production mechanism in gluon jets and manifests itself as a slope difference at high m$_{T}$ when comparing baryon to meson spectra. This basic effect in pp is not sufficient though to describe the large baryon over meson yield enhancement at intermediate p$_{T}$ in AA collisions. Besides the baryon/meson difference there is a surprising absence of strong flavor effects in the particle to anti-particle ratios in AA, the identified two particle correlations in AA, and even the jet quenching in the medium. Thus, it is still an open question whether the partonic energy loss in AA shows the expected Casimir factor when comparing hadrons from a fragmenting gluon jet to a quark jet, i.e. is the energy loss really non-Abelian ? The only strong flavor effect is in the strangeness sector. High p$_{T}$ strange baryon production in AA is enhanced instead of suppressed compared to pp . This could be due to simple canonical suppression in pp. This thermodynamic effect, which is due to a limited strangeness phase space occupancy, has been measured for the first time as a function of transverse momentum, and it is obvious that the effect is not limited to low momentum or simply bulk properties. Surprisingly the intermediate p$_{T}$ part scales well with the canonical suppression factors, which indicates that the hadronization mechanism of strange baryons, even at higher p$_{T}$, is driven by a correlation volume, which is distinctly different from charmed meson production. The D-meson yield and high p$_{T}$ suppression factors in AA are consistent with scaled hard scattering cross section, i.e. production from string fragmentation. In identified two particle correlations in AA collisions we see a strongly enhanced associated particle yield compared to pp, independent of the trigger particle species. Long range $\Delta\eta$ correlations might account for that and they might be due to recombination [@hwa1]. A small baryon/meson trend can be found in those correlations but the effect is not very significant. Larger predicted effects for $\phi$ and $\Omega$ triggered correlations [@hwa2] are under investigation. In summary, I believe that studies of the hadronization mechanism at RHIC and LHC energies in vacuum and in medium hold the key to the puzzle of baryonic matter formation in the universe. We need to first understand the basic baryon production mechanism(s) in pp (string fragmentation vs. recombination, di-quark formation ?). Then we need to determine whether the baryon production mechanism in AA collisions is modified from the vacuum production. For a more detailed study the high pt reach and the particle identification properties of the LHC detectors are crucial. Acknowledgements {#acknowledgements .unnumbered} ================ I thank Helen Caines, Mark Heinz, and Klaus Werner for useful discussions. [99]{} C.L. DaSilva, contribution to these proceedings J. Harris, contribution to these proceedings M. Heinz, contribution to these proceedings L. Ruan, contribution to these proceedings S. Albino et al., hep-ph/0502188 R. Bellwied for the STAR collaboration, QM05 proceedings, nucl-ex/0511006 K. Eskola et al., Nucl. Phys. A713 (2003) G. Gatoff and C. Y. Wong, Phys. Rev. D 46, 997 (1992) J. Schaffner-Bielich et al., arXiv:nucl-th/0202054 P.V. Chliapnikov and V.A. Uvarov, Phys.Lett. B345, 313, 1995 M. Szczekowski, Phys.Lett. B359, 387, 1995 P. Eden, G. Gustafson, Z. Phys. C75, 41 (1997) and hep-ph/9606454 K. Redlich and A. Tounsi, Eur. Phys. J. [C24]{} (2002) 589 G. Abbiendi et al. (OPAL), Eur. Phys. J. C16, 407 (2000) and hep-ex/0001054 J. Cleymans, J. Phys. G28, 1575 (2002) R.V. Gavai and S. Gupta, Phys. REv. D73, 014004 (2006) J. Adams et al. (STAR), nucl-ex/0601042 K. Werner, contribution to these proceedings V. Pantuev, contribution to these proceedings C.B. Chiu, R. Hwa, Phys. Rev. C72, 034903 (2005) R. Hwa, C.B. Yang, nucl-th/0602024	ArXiv
	ArXiv
--- abstract: 'This work deals with partial MDS (PMDS) codes, a special class of locally repairable codes, used for distributed storage system. We first show that a known construction of these codes, using Gabidulin codes, can be extended to use any maximum rank distance code. Then we define a standard form for the generator matrices of PMDS codes and use this form to give an algebraic description of PMDS generator matrices. This implies that over a sufficiently large finite field a randomly chosen generator matrix in PMDS standard form generates a PMDS code with high probability. This also provides sufficient conditions on the field size for the existence of PMDS codes.' author: - 'Alessandro Neri and Anna-Lena Horlemann-Trautmann' bibliography: - 'PMDS\_stuff.bib' title: Random Construction of Partial MDS Codes --- Introduction ============ In a distributed storage system a file $x\in {{\mathbb{F}}}_q^k$ (where ${{\mathbb{F}}}_q$ denotes the finite field of cardinality $q$), is encoded and stored as some codeword $c\in {{\mathbb{F}}}_q^n$, over several storage nodes. Each of these nodes is assumed, for simplicity, to store exactly one coordinate of $c$. In case some of the nodes fail, we want to be able to recover the lost information using as little effort as possible. The locality of a code plays an important role in this context, and it denotes the number of nodes one has to contact for repairing a lost node. We call the set of nodes one has to contact if a given node fails, the locality group of that node. For this work we assume that the locality groups are distinct. Partial MDS codes are maximally recoverable codes in this setting, i.e., any erasure pattern that is information theoretically correctable is correctable with such a code. It is known that maximally recoverable codes in general, and PMDS codes in particular, exist for any locality configuration if the field size is large enough [@ch07]. Furthermore, some constructions of PMDS codes are known, e.g. [@bl13; @bl14; @bl16; @ch15; @go14]. In this paper we describe a random construction of PMDS codes, by prescribing a generator matrix of the respective code in a specific form, which we will call PMDS standard form. If we fill the non-prescribed coordinates of this generator matrix with random values, by high probability, the resulting code is PMDS, if the underlying field size is large enough. We derive a lower bound on this probability (depending on the field size). This gives rise to a lower bound on the necessary field size for PMDS codes to exist. With some final adjustments on the random construction we get a lower bound that improves the one of [@ch07]. Preliminaries ============= PMDS codes ---------- Consider a distributed storage system with $m$ disjoint locality groups, where the $i$-th group is of size $n_i$ ($i=1,\dots,m$) and can correct any $r_i$ erasures. First we set the locality for the code to be $\ell \in \mathbb N$. We can divide the coordinates of the code into blocks of length $n_1, \dots, n_m$, where $n_i=\ell+r_i$, such that each block represents a locality group. We denote an MDS code of length $n$ and dimension $k$ by $[n,k]$-MDS code. We use the definition of PMDS codes given in [@ht17], which generalizes the definition of [@bl13]. Let $\ell,m, r_1,\dots,r_m \in \mathbb N$. Define $n:=\sum_{i=1}^m (r_i+\ell)$ and let $C\subseteq {{\mathbb{F}}}_q^n$ be a linear code of dimension $k<n$ with generator matrix $$\label{eq:genmatrix} G= \left( B_1 \mid \dots \mid B_m \right) \in {{\mathbb{F}}}_q^{k\times n}$$ such that $B_i\in {{\mathbb{F}}}_q^{k\times ( r_i + \ell)}$. Then $C$ is a $[n,k, \ell ; r_1,\dots,r_m]$-partial-MDS (PMDS) code (with locality $\ell$) if - for $i\in \{1,\dots,m\}$ the row space of $B_i$ is a $[r_i+\ell, \ell ]$-MDS code, and - for any $r_i$ erasures in the $i$-th block ($i=1,\dots, m$), the remaining code (after puncturing the coordinates of the erasures) is a $[m\ell, k]$-MDS code. The erasure correction capability of PMDS codes is as follows: \[lem:PMDScap\][@ht17 Lemma 3] A $[n,k, \ell ; r_1,\dots,r_m]$-PMDS code can correct any $r_i$ erasures in the $i$-th block (simultaneously) plus $s:= m\ell -k$ additional erasures anywhere in the code. We can see that the definition of PMDS codes given makes sense only for $k\geq \ell$. In case of equality, or in the case that $m=1$ there exist only trivial PMDS codes, i.e. the only PMDS codes are MDS codes. It was shown in [@ht17] that a code is a $[n,k, \ell; r_1,\dots,r_m]$-PMDS code if and only if it is maximally recoverable (for the respective locality group configuration). The same results had previously been shown in [@go14 Lemma 4] for the case $r_1=r_2=\dots = r_m$. Now we give a summary on known results about PMDS codes. [@ch07]\[prop4\] Maximally recoverable (MR) codes of length $n$ and dimension $k$ exist for any locality configuration over any finite field of size $q> \binom{n-1}{k-1}$. MR codes are PMDS codes for disjoint locality blocks. Therefore, Proposition \[prop4\] implies that PMDS codes exist for any set of parameters when the field size is large enough. A construction of PMDS codes based on rank-metric and MDS codes was given in [@ca17], when $r_1=r_2=\dots=r_m$. This gives the following existence result: [@ca17] $[n,k,\ell; r,\dots,r]$-PMDS codes with $m$ locality blocks of the same length exist over a finite field of size $q^{n-mr}$. Furthermore, some specific constructions of PMDS codes, for particular values of $s$ or of the $r_i$, are given in [@bl13; @bl14; @bl16; @go14]. In particular, a general construction for PMDS codes with $s=1$ was given in [@ht17]. This construction is based on the concatenation of several MDS codes as building blocks. [@ht17 Corollary 14] 1. For any integers $m\geq 2 $ and $\ell,r_1,\dots,r_m \geq 1$ there exists a $[n,k=m\ell-1, \ell; r_1,\dots,r_m]$-PMDS code over any field ${{\mathbb{F}}}_q$ with $q\geq \max_i\{r_i\}+\ell$. 2. If there exists $h\in \mathbb N$ such that $\ell\in \{3,2^h-1\}$ and $\max_i \{r_i\} +\ell = 2^h+1$, then there exists a $[n,k=m\ell-1, \ell; r_1,\dots,r_m]$-PMDS code over ${{\mathbb{F}}}_q$ with $q=2^h= \max_i\{r_i\}+\ell -1$. In [@ht17] the authors also show that this construction is basically the only one possible, i.e., every PMDS with $s=1$ is of this form, giving thus a characterization for this set of parameters. However, for $s\geq 2$ there is no characterization yet for PMDS codes. Zarisky topology over finite fields ----------------------------------- Let ${{\mathbb{F}}}$ be a field, and ${{\mathbb{F}}}[x_1,\ldots,x_N]$ be the polynomial ring over ${{\mathbb{F}}}$. Denote by $\bar{{{\mathbb{F}}}}$ the algebraic closure of ${{\mathbb{F}}}$. For a subset $S\subseteq {{\mathbb{F}}}[x_1,\ldots,x_N]$ we define the algebraic set $$V(S): = \{{\boldsymbol}\alpha \in \bar{{\mathbb{F}}}_q^r \mid f({\boldsymbol}\alpha) = 0, \forall f \in S\} .$$ The Zariski topology on $\bar{{{\mathbb{F}}}}^N$ is defined as the topology whose closed sets are the algebraic sets, while the complements of the Zariski-closed sets are the Zariski-open sets [@ha13b Ch. I, Sec. 1]. A subset $A\subset\bar{{{\mathbb{F}}}}^N$ is called a generic set if $A$ contains a non-empty Zariski-open set. In classical geometry one studies the Zariski topology over the complex numbers. In this framework, a generic set inside $\mathbb C^N$ is dense and its complement is contained in an algebraic set of dimension at most $N-1$. If one wants to consider generic sets restricted to a finite field ${{\mathbb{F}}}_q$, the situation is slightly different. Here, for a subset $T\subseteq{{\mathbb{F}}}_q^N$ one can always find a set of polynomials $S\subseteq {{\mathbb{F}}}_q[x_1,\dots,x_N]$ such that $$T=\{{\boldsymbol}\alpha \in {{\mathbb{F}}}_q^N \mid f({\boldsymbol}\alpha) = 0, \forall f \in S\}.$$ and therefore the Zariski topology restricted to ${{\mathbb{F}}}_q^N$ is the discrete topology. This means that it is not useful to extend the notion of generic sets to finite fields since it would not give any information. However, given a set of polynomials $S\subseteq{{\mathbb{F}}}_q[x_1,\ldots,x_N]$, we can define the set of ${{\mathbb{F}}}_q$-rational points as $$V(S;{{\mathbb{F}}}_{q}): = \{{\boldsymbol}\alpha \in {{\mathbb{F}}}_{q}^N \mid f({\boldsymbol}\alpha) = 0, \forall f \in S\} .$$ In this setting the Schwartz-Zippel Lemma implies an analog result to the one of generic sets, as explained in the following. [@le98p Lemma 1.1]\[lem:SZ\] Let $f\in {{\mathbb{F}}}_q[x_1,\dots,x_N]$ be a non-zero polynomial of total degree $d \geq 0$. Let $T\subseteq \bar{{{\mathbb{F}}}}$ be a finite set and let $\alpha_1, \dots, \alpha_N$ be selected at random independently and uniformly from $T$. Then $$\Pr\big(f(\alpha_1,\ldots,\alpha_N)=0\big)\leq\frac{d}{\|T\|}.$$ As a consequence of this result we have that, in case the size of $S$ and the total degrees of the polynomials in $S$ do not depend on the finite field, the proportion between the cardinality of $ V(S;{{\mathbb{F}}}_{q})$ and the cardinality of the whole space ${{\mathbb{F}}}_q^N$ goes to $0$ as $q$ grows. Vice versa, for growing $q$ the probability that a random point is in the complement of $ V(S;{{\mathbb{F}}}_{q})$ tends to $1$. This result will be crucial in Section \[sec:topprob\] for our random construction of PMDS codes. Rank-metric codes ----------------- We now give some known facts about rank-metric codes. Recall that ${\mathbb{F}_{q^N}}$ is isomorphic to ${\mathbb{F}_q}^N$ as an ${\mathbb{F}_q}$- vector space. From this it easily follows that ${\mathbb{F}_{q^N}}^n\cong {\mathbb{F}_q}^{N\times n}$. Then we can give the following definition. The rank distance $d_R$ on ${{\mathbb{F}}}_q^{N\times n}$ is defined by $$d_R(U,V):= {\mathrm{rank}}(U-V) , \quad U,V \in {{\mathbb{F}}}_q^{N\times n}.$$ Analogously, if $\boldsymbol u,\boldsymbol v \in {\mathbb{F}_{q^N}}^n$, then $d_R(\boldsymbol u,\boldsymbol v)$ is the rank of the difference of the respective matrix representations in ${{\mathbb{F}}}_q^{N\times n}$. Observe that the definition of rank distance in the case of vectors in ${\mathbb{F}_{q^N}}^n$ does not depend on the choice of the basis. Moreover it can be shown that the function $d_R: {\mathbb{F}_{q^N}}^n \times {\mathbb{F}_{q^N}}^n \rightarrow {{\mathbb{R}}}_{\geq 0}$ is a metric. An ${\mathbb{F}_{q^N}}$-linear rank-metric code ${\mathcal{C}}$ of length $n$ and dimension $k$ is a $k$-dimensional subspace of ${\mathbb{F}_{q^N}}^n$ equipped with the rank distance. The minimum distance of ${\mathcal{C}}$ is defined as $$d_R({\mathcal{C}}):=\min\left\{d_R({\boldsymbol}u,{\boldsymbol}v) \mid {\boldsymbol}u,{\boldsymbol}v\in {\mathcal{C}}, {\boldsymbol}u\neq {\boldsymbol}v\right\}.$$ \[th:SB\][@ro91 Theorem 1] Let ${\mathcal{C}}\subseteq {\mathbb{F}_{q^N}}^n$ be an ${\mathbb{F}_{q^N}}$-linear rank-metric code of dimension $k$. Then $$d_R({\mathcal{C}})\leq n-k+1.$$ Codes attaining the Singleton-like bound are called Maximum Rank Distance (MRD) Codes. A necessary and sufficient condition for the existence of MRD codes is that $n\leq N$. In this framework, a characterization for ${\mathbb{F}_{q^N}}$-linear MRD codes in terms of their generator matrices was given in [@ho16 Corollary 2.12], which in turn is based on a result given in [@ga85]. For this we define the set $${\mathcal E}_q(k,n):=\left\{E \in {{\mathbb{F}}}_q^{k\times n} \mid {\mathrm{rank}}(E)=k\right\}.$$ \[prop:MRDCrit\] Let $G\in {{\mathbb{F}}}_{q^m}^{k\times n}$ be a generator matrix of a rank-metric code $\mathcal{C}\subseteq {{\mathbb{F}}}_{q^m}^n$. Then $\mathcal{C}$ is an MRD code if and only if $${\mathrm{rank}}(GE^T) =k$$ for all $E\in {\mathcal E}_q(k,n)$. General construction using rank metric codes ============================================ In this section we generalize the construction given in [@ca17]. In that work the authors use Gabidulin codes in order to build $[n,k, \ell, r,\ldots,r]$-PMDS codes. We will show that this construction also works for different $r_i$, and that Gabidulin codes can be replaced by any linear MRD codes. Fix $n, k,\ell, r_1,\ldots,r_m$, and let $\widetilde{G} \in {\mathbb{F}_{q^N}}^{k\times m\ell}$ be the generator matrix of a MRD code. For the existence of an MRD code we need $N\geq m\ell$. Moreover, for every $i=1,\ldots, m$, we consider a $[\ell+r_i,\ell]$-MDS code over ${\mathbb{F}_q}$ with generator matrix $M_i$, and define $$\label{eq:MDSblock} M:=\left(\begin{array}{cccc} M_1 & 0 & \ldots & 0 \\ 0 & M_2 &\ldots & 0 \\ \vdots & & \ddots & \vdots \\ 0 & \ldots & 0 & M_m \end{array}\right) \in {\mathbb{F}_q}^{m\ell \times n}.$$ We can now formulate our PMDS construction. \[thm16\] Let $\widetilde{G} \in {\mathbb{F}_{q^N}}^{k\times m\ell}$ be the generator matrix of a MRD code and let $M$ be the matrix defined in (\[eq:MDSblock\]). Then the matrix $\widetilde{G}M$ is a generator matrix for a $[n,k,\ell,r_1,\ldots,r_m]$-PMDS code over ${\mathbb{F}_{q^N}}$. Let $G:=\widetilde{G}M$ and let $S\in \mathcal T_{k,\ell}(G)$ be the submatrix obtained by selecting columns $h_1,\ldots,h_{k_j}$ from the $j$th block for $j=1,\ldots,m$, where $k_i\leq \ell$ and $k_1+\ldots+k_m=k$. $S$ is equal to $\widetilde{G}\widetilde{M}$, where $$\widetilde{M}=\left(\begin{array}{cccc} N_1 & 0 & \ldots & 0 \\ 0 & N_2 &\ldots & 0 \\ \vdots & & \ddots & \vdots \\ 0 & \ldots & 0 & N_m \end{array}\right),$$ and $N_j$ is the $\ell \times k_j$ submatrix of $M_j$ obtained by the respective selected columns. Since $M_i$ generates an $[\ell+r_i, \ell]$-MDS code, any $\ell$ columns of $M_i$ are linearly independent. Thus, ${\mathrm{rank}}(N_i)=k_i$ and ${\mathrm{rank}}(\widetilde{M})=k_1+\ldots+k_m=k$. By Proposition \[prop:MRDCrit\] we have that $\det(\widetilde{G}\widetilde{M})\neq 0$, and we conclude the proof using Proposition \[prop:PMDS\]. \[cor:exPMDS\] Let $m\geq 2 $ and $\ell,r_1,\dots,r_m \geq 1$, $k\geq \ell$ be positive integers. Then, for every prime $p$ and every positive integer $L\geq n_0m\ell$ there exists a $[n,k,\ell,r_1,\ldots,r_m]$-PMDS code over ${{\mathbb{F}}}_{p^L}$, where $$n_0=\min\{j \in {{\mathbb{N}}}\mid p^j \geq \ell + r_i-1, \mbox{ for } i=1,\ldots,m\}.$$ A MRD code in ${\mathbb{F}_{q^N}}^{ m\ell}$ exists if $N\geq m\ell$. Suitable MDS codes over ${{\mathbb{F}}}_q$ for the matrix in exist if $q\geq \max\{\ell + r_i - 1\}$. The statement follows from Theorem \[thm16\] with $q=p^{n_0}$ and $L=n_0N$. Algebraic description of PMDS codes =================================== We will now define a standard form for generator matrices of PMDS codes. This standard form is the main tool for the random construction of PMDS codes. \[thm:stform\] Let $m\geq 2 $ and $s, \ell,r_1,\dots,r_m \geq 1$ and let ${\mathcal{C}}$ be a $[n,k=m\ell-s, \ell; r_1,\dots,r_m]$-PMDS code over a field ${{\mathbb{F}}}_q$. Then ${\mathcal{C}}$ has a generator matrix of the form $$\label{eq:stform} {G}= \left({B}_1 \mid \dots \mid {B}_m \right),$$ where - ${B}_i=({C}_i \mid {D}_i)$, ${C}_i \in {{\mathbb{F}}}_q^{k \times \ell}$ and ${D}_i \in {{\mathbb{F}}}_q^{k\times r_i}$ for $i=1,\ldots, m$, and - the submatrix ${G}_{C}=\left({C}_1 \mid \dots \mid {C}_m \right)$ is of the form $${G}_C=\left[ I_{k} \mid A \right],$$ with $A$ being superregular. Let $\widetilde{G}$ be a generator matrix for ${\mathcal{C}}$ of the form (\[eq:genmatrix\]), i.e. $$\widetilde{G}= \left(\widetilde{B}_1 \mid \dots \mid \widetilde{B}_m \right).$$ Puncturing every block $\widetilde{B}_i$ in the last $r_i$ columns, we get that the submatrix $\widetilde{G}_C$ is the generator matrix of a $[m\ell,k]$-MDS code. Operating on the rows of such a submatrix we can transform it to a matrix $G_C=\left[ I_{k} \mid A \right]$, with $A$ superregular. I.e., there exists an invertible matrix $P \in \mathrm{GL}_k({\mathbb{F}_q})$ such that $P\widetilde{G}_C=\left[ I_{k} \mid A \right]$, and therefore the matrix $G:=P\widetilde{G}$ is a generator matrix of ${\mathcal{C}}$ of the required form. We now consider the entries $a_{w,z}$ of $A$ as variables $x_{w,z}$ for $w=1,\ldots,k$ amd $z=1,\ldots,s$. We know that the column space of ${D}_i$ is inside the column space of ${C}_i$, by the parameters of the block MDS codes. This means that every column in ${{D}_i}$ is a linear combination of the columns of ${{C}_i}$. If we denote by ${{D}_i}^{(j)}$ the $j$th column of ${{D}_i}$, then $$\label{eq:columns} {{D}_i}^{(j)}=\sum_{t=1}^{\ell}y_{t,i,j}{{C}_i}^{(t)}$$ for some $y_{t,i,j}$, which we also consider variable. This way we can consider a $k\times n$ generator matrix as a matrix in ${{\mathbb{F}}}_q[x_{w,z}, y_{t,i,j}]^{k\times n}$ (where ${{\mathbb{F}}}_q[x_{w,z}, y_{t,i,j}]$ denotes the polynomial ring in all $x_{w,z}, y_{t,i,j}$). Let $R=\sum_{i=1}^m r_i$. We denote $\boldsymbol{\alpha} := (\alpha_{w,z})_{w,z}\in {{\mathbb{F}}}_q^{sk}$ and $\boldsymbol{\beta} := (\beta_{t,i,j})_{t,i,j} \in {{\mathbb{F}}}_q^{\ell R}$. If we replace the variables $x_{w,z}, y_{t,i,j}$ described above in a matrix in PMDS standard form by the values $\alpha_{w,z}, \beta_{t,i,j}$, we denote the corresponding generator matrix by $$G(\boldsymbol{\alpha},\boldsymbol{\beta}).$$ Analogously we will denote the variable form by $ G(\boldsymbol{x},\boldsymbol{y})$. However, a general matrix of this form is not necessarily a generator matrix of a PMDS code for any values $\boldsymbol{\alpha},\boldsymbol{\beta}$. The following proposition shows what needs to be fulfilled to generate a PMDS code: \[prop:PMDS\] A matrix $G\in {{\mathbb{F}}}_q^{k\times n}$ generates a $[n,k=m\ell-s, \ell; r_1,\dots,r_m]$-PMDS code if and only if, every submatrix in the set $$\mathcal T_{k,\ell}(G):=\left\{S\in {{\mathbb{F}}}^{k \times k} \mid \begin{array}{l}S \mbox{ is a submatrix of } G \mbox{ with }\\ \mbox{ at most } \ell \mbox{ columns per block } B_i \end{array}\right\}$$ has non-zero determinant. This follows from the definition of PMDS, cf. also [@ht17]. The above results give an algebraic description of the generator matrix of a $[n,k=m\ell-s, \ell; r_1,\dots,r_m]$-PMDS code over ${{\mathbb{F}}}_q$, as follows. If we consider the variable form of a generator matrix $G$ as above, and the polynomial $$\label{eq:poly} p(\boldsymbol{x},\boldsymbol{y}):= \mathrm{lcm} \{ \det S \mid S\in\mathcal T_{k,\ell}(G)\} \in {{\mathbb{F}}}_q[x_{w,z}, y_{t,i,j}],$$ then, we have that $G(\boldsymbol{\alpha},\boldsymbol{\beta})$ generates a $[n,k=m\ell-s, \ell; r_1,\dots,r_m]$-PMDS code over ${{\mathbb{F}}}_q$ if and only if $p(\boldsymbol{\alpha},\boldsymbol{\beta})$ is non-zero. Topological and probability results {#sec:topprob} =================================== In this section we first deal with the algebraic description of the generator matrix of a PMDS code in the algebraic closure of the finite field where we want our code to be built. After that, we analyze the probability that a code whose generator matrix is of the form $G(\boldsymbol{\alpha},\boldsymbol{\beta})$ is PMDS. Moreover, we also study the existence of PMDS codes for given parameters $n,k,\ell, s, r_1,\ldots,r_m$ and $R=\sum_{i=1}^m r_i$, giving sufficient conditions on the field size. Although for $s=1$ this problem was completely solved in [@ht17], for $s\geq 2$ this is still an open problem. We denote the set of valid entries for PMDS generator matrices over the algebraic closure of the finite field ${\mathbb{F}_q}$ by $$\small{\mathcal A_{\mathrm{PMDS}}:=\left\{(\boldsymbol{\alpha},\boldsymbol{\beta}) \in \bar{{{\mathbb{F}}}}_q^{sk}\times \bar{{{\mathbb{F}}}}_q^{\ell R} \mid \mathrm{row space}(G(\boldsymbol{\alpha},\boldsymbol{\beta})) \mbox{ is PMDS}\right\},}$$ Then the following result holds. \[th:gen\] $\mathcal A_{\mathrm{PMDS}}$ is a generic set. By Proposition \[prop:PMDS\] we have that $$\mathcal A_{\mathrm{PMDS}}:=\left\{(\boldsymbol{\alpha},\boldsymbol{\beta}) \in \bar{{{\mathbb{F}}}}_q^{sk}\times \bar{{{\mathbb{F}}}}_q^{\ell R} \mid p(\boldsymbol{\alpha},\boldsymbol{\beta})\neq 0 \right\},$$ and therefore $\mathcal A_{\mathrm{PMDS}}$ is a Zariski open set. Concerning the non-emptiness, let $q=p^{t_0}$. From Corollary \[cor:exPMDS\] there exists a $[n,k,\ell,r_1,\ldots,r_m]$-PMDS code ${\mathcal{C}}$ over ${{\mathbb{F}}}_{p^L}$, for some $L$ multiple of $t_0$. By Theorem \[thm:stform\], ${\mathcal{C}}$ has a generator matrix of the form $G(\boldsymbol{\alpha},\boldsymbol{\beta})$, therefore $(\boldsymbol{\alpha},\boldsymbol{\beta})\in \mathcal A_{\mathrm{PMDS}}$. This means that over the algebraic closure, by probability $1$, for randomly chosen $\boldsymbol{\alpha},\boldsymbol{\beta}$ the matrix $G(\boldsymbol{\alpha},\boldsymbol{\beta})$ generates a PMDS code. For underlying finite fields, this implies that for growing field size this probability will tend to $1$. We now derive a probability formula depending on the field size. We can easily observe that the entries of $G(\boldsymbol{x},\boldsymbol{y})$ are polynomials of total degree $0,1$ or $2$. In particular, if $t:=\lceil\frac{s}{\ell}\rceil$ and we write $G(\boldsymbol{x},\boldsymbol{y})$ as in , then the entries of the blocks ${{D}_i}$ are polynomials of degree at most $1$ for $i=1,\ldots, m-t$, and of degree at most $2$ for the last $t$ blocks. To estimate the degree of $p(\boldsymbol{x},\boldsymbol{y}) $ we need the following lemma. \[lem:Vid\] $$\sum_{j=0}^r (r-j)\binom{m}{j}\binom{n}{r-j}=n\binom{m+n-1}{r-1}.$$ \[lem:deg\] The total degree of the polynomial $p(\boldsymbol{x},\boldsymbol{y})$, defined as in (\[eq:poly\]), satisfies the inequality $$\deg p(\boldsymbol{x},\boldsymbol{y}) \leq 2(n-k)\binom{n-1}{k-1}.$$ It holds that $$\mathcal T_{k,\ell}(G) \subset \mathcal M_k(G):=\{ S \in {\mathbb{F}_q}^{k\times k} \mid S \mbox{ is a submatrix of } G\},$$ hence the polynomial $p(\boldsymbol{x},\boldsymbol{y})$ divides the polynomial $$q(\boldsymbol{x},\boldsymbol{y}):=\mathrm{lcm}\{\det S \mid S\in \mathcal M_k(G)\}.$$ Observe that the entries of the first $k$ columns of the submatrix ${G}_C$ have degree 0. Let $t:=\lceil\frac{s}{\ell}\rceil$. Then the entries of the columns corresponding to the blocks ${{D}_i}$ for $i=1,\ldots, m-t$ have degree at most $1$, as well as the last $m\ell -k$ columns of ${G}_C$. Finally, the columns of the blocks ${{D}_i}$ for $i=m-t+1,\ldots, m$, have degree at most $2$. In particular, all the entries of the blocks ${{D}_i}$ and the last $m\ell -k$ columns of ${G}_C$ have degree at most $2$. Therefore, $$\begin{aligned} \deg q(\boldsymbol{x},\boldsymbol{y}) &\leq \sum_{S \in \mathcal M_k(G)} \deg \det S \\ &\leq \sum_{j_0=0}^k2(k-j_0) \binom{k}{j_0}\binom{n-k}{k-j_0}\\ &=2(n-k)\binom{n-1}{k-1}\end{aligned}$$ where the last equality follows from Lemma \[lem:Vid\]. Since $\deg p(\boldsymbol{x},\boldsymbol{y}) \leq \deg q(\boldsymbol{x},\boldsymbol{y})$ we conclude the proof. We can now formulate a lower bound for the probability that a randomly chosen generator matrix in PMDS standard form generates a PMDS code over a finite field ${{\mathbb{F}}}_q$: \[thm:Prob\] Let the entries of $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ be uniformly and independently chosen at random in ${\mathbb{F}_q}$. Then $$\mathrm{Pr}\{\mathrm{row space}(G(\boldsymbol{\alpha},\boldsymbol{\beta})) \mbox{ is PMDS }\}\geq 1- \frac{2(n-k)\binom{n-1}{k-1}}{q}.$$ We have $$\begin{aligned} &\mathrm{Pr}\{\mathrm{row space}(G(\boldsymbol{\alpha},\boldsymbol{\beta})) \mbox{ is PMDS }\} \\ =&\mathrm{Pr}\{(\boldsymbol{\alpha},\boldsymbol{\beta}) \notin V(p(\boldsymbol{x},\boldsymbol{y});{\mathbb{F}_q})\} \\ =&1-\mathrm{Pr}\{p(\boldsymbol{\alpha},\boldsymbol{\beta})=0\} \\ \geq& 1-\frac{\deg p(\boldsymbol{x},\boldsymbol{y})}{q}\geq 1 - \frac{2(n-k)\binom{n-1}{k-1}}{q},\end{aligned}$$ where the last two inequalities follow from Lemmas \[lem:SZ\] and \[lem:deg\], respectively. From this we can deduce an existence result for PMDS codes over finite fields of a given minimal size. If $q>2(n-k)\binom{n-1}{k-1}$ then there exists a $[n,k,\ell,r_1,\ldots,r_m]$-PMDS code over the finite field ${\mathbb{F}_q}$. One notices that this is not an improvement over the known existence result from Proposition \[prop4\]. However, we can improve the above result, considering a step-by-step construction. We will again consider a generator matrix in PMDS standard form as in . We start with an $[m\ell,k]$-MDS code over a finite field ${\mathbb{F}_q}$ and write its generator matrix as $(C_1 \mid \dots \mid C_m)$. For this purpose it is sufficient that $q\geq m\ell -1$. Then we construct the first column $D_1^{(1)}$ of the block $D_1$ as in (\[eq:columns\]). Every entry will be a degree $1$ polynomial in the variables $y_{t,1,1}$ for $t=1,\ldots,\ell$. Imposing that every $k\times k$ minor of $$G':=\left( {C}_1\mid {D}_1^{(1)} \mid {C}_2\mid \ldots \mid {C}_m \right)$$ is non-zero, we get the condition $p'(y_{t,1,1})\neq 0$, where $$p'(y_{t,1,1})=\mathrm{lcm}\{ \det S \mid S \in \mathcal T_{k,\ell}(G')\}.$$ Using Lemma \[lem:SZ\], we obtain $$\mathrm{Pr}\{p'(\beta_{t,1,1})= 0\}\leq \frac{\deg p'}{q}.$$ In this situation $\deg p' \leq \binom{m\ell}{k-1}$, therefore for $q> \binom{m\ell}{k-1}$ we have that there exists at least one evaluation of $p'$ that is non-zero and such that $G'(\beta_{t,1,1})$ generates a $[n,k,\ell, 1,0,\ldots,0]$-PMDS code. Repeating this construction step by step, we get a $[n,k,\ell,r_1,\ldots,r_{m-1},r_{m}-1]$-PMDS code. From that code we build the last column ${D}_m^{(r_m)}$ of the block ${D}_m$ again as in (\[eq:columns\]): $${D}_m^{(r_m)}=\sum_{t=1}^{\ell}y_{t,m,r_m}{C}_m^{(t)}.$$ In the end we get the matrix $$\label{eq:G} {G}(y_{t,m,r_m})=({C}_1\mid {D}_1 \mid \ldots \mid {C}_m \mid {D}_m ),$$ where the matrix ${G}(y_{t,m,r_m})$ without the last column generates a $[n,k,\ell,r_1,\ldots,r_{m-1},r_{m}-1]$-PMDS code, and the entries of the last column are polynomials of total degree at most $1$ in the variables $y_{t,m,r_m}$, for $t=1,\ldots,\ell$. Let $m, n, k, n_1,\ldots,n_m, f_1,\ldots,f_m$ be positive integers such that $n=\sum_i n_i$. Let $N_0:=0$, $N_i:=\sum_{j=1}^i n_j$ and $J_i=\{N_{i-1}+1,\ldots,N_i\}$ for $i=1,\ldots,m-1$. We define the set $$\mathcal M(k;n_1\ldots,n_m;f_1,\ldots,f_m) =\> \left\{I \subset \{1,\ldots,n\}\mid \|I\|=k, \|I\cap J_i\|\leq f_i\right\}$$ and $M(k;n_1\ldots,n_m;f_1,\ldots,f_m)$ as its cardinality. Let ${G}(y_{t,m,r_m})$ be as in (\[eq:G\]). The total degree of the polynomial $$\tilde{p}(y_{t,m,r_m}):=\mathrm{lcm}\{\det S \mid S\in \mathcal T_{k,\ell}({G}(y_{t,m,r_m}))\}$$ is less or equal to $$M(k-1;\ell+r_1\ldots,\ell+r_{m-1},\ell+r_m-1;\ell,\ldots,\ell,\ell-1) =: M^.$$ The polynomial $\tilde{p}(y_{t,m,r_m})$ has degree less or equal to $\sum \deg \det S$. By assumption all the determinants $\det S$ for $S$ not containing the last column are non zero elements in ${\mathbb{F}_q}$. The only polynomials with degree $1$ are the determinants of $k\times k$ submatrices involving the last column, and they are exactly $M^$ many. \[cor25\] If $q>M^$ then there exists a $[n,k,\ell,r_1,\ldots,r_m]$-PMDS code over the finite field ${\mathbb{F}_q}$. To our knowledge there is no closed formula for $M^$. However, it is easy to see that $M^* \leq \binom{n-1}{k-1}$ and that the inequality is strict if any of the conditions $\|I\cap J_i\|\leq f_i$ is non-empty. Hence, Corollary \[cor25\] improves upon Proposition \[prop4\]. Conclusion ========== We gave a generalization of a known PMDS code construction based on rank-metric codes. Furthermore, we investigated a random construction of PMDS codes by prescribing a PMDS standard form. We derived a lower bound on the probability that a randomly filled matrix in PMDS standard form generates a PMDS code. This probability implies a lower bound on the field size needed for such codes to exist. In the end we gave a step-by-step construction of such a generator matrix to improve this lower bound on the necessary field size.	ArXiv
--- author: - \| Bahman Dehnadi ${}^a$\ E-mail: - \| Andre H. Hoang ${}^{b,c}$\ E-mail: - \| Vicent Mateu ${}^{c,d,e}$\ E-mail: - \| S. Mohammad Zebarjad ${}^a$\ E-mail:\ ${}^a$ Shiraz University, Physics Department, Shiraz 71454, Iran.\ ${}^b$ University of Vienna, Faculty of Physics, Boltzmanngasse 5, A-1090 Vienna, Austria.\ ${}^c$ Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6, D-80805 München, Germany.\ ${}^d$ Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139.\ ${}^e$ Instituto de Física Corpuscular, UVEG - Consejo Superior de Investigaciones Científicas, Apartado de Correos 22085, E-46071, Valencia, Spain. bibliography: - 'charm2.bib' title: 'Charm Mass Determination from QCD Charmonium Sum Rules at Order $\alpha_s^3$' --- Introduction {#sectionintroduction} ============ Accurate determinations of the charm quark mass are an important ingredient in the prediction of inclusive and radiative $B$ decays or exclusive kaon decays such as $K\to\pi\nu\bar{\nu}$. Since these decays are instruments to either measure CKM matrix elements or to search for new physics effects, appropriate and realistic estimates of the uncertainties are also an important element of these analyses [@Antonelli:2009ws]. One of the most powerful methods to determine the charm quark mass is based on sum rules for the charm-anticharm production rate in $e^{+}e^{-}$ annihilation [@Novikov:1977dq]. Here, moments of the correlation function of two charm vector currents at zero momentum transfer $$\begin{aligned} \label{momentdef1} M_{n}^{{\rm th}} & = & \dfrac{12\pi^2 Q_c^2}{n!}\,\dfrac{{\rm d}}{{\rm d}q^{2n}}\left.\Pi(q^{2})\right\|_{q^{2}=0}\,,\\ \left(g_{\mu\nu}q^{2}-q_{\mu}q_{\nu}\right)\,\Pi(q^{2}) & = & -\, i\int\mathrm{d}x\, e^{iqx}\left\langle \,0\left\|T\, j_{\mu}(x)j_{\nu}(0)\right\|0\,\right\rangle \,,\nonumber \\[2mm] j^{\mu}(x) & = & \bar{\psi}(x)\gamma^{\mu}\psi(x) \,,\nonumber \end{aligned}$$ $Q_c$ being the charm quark electric charge, can be related to weighted integrals of the normalized charm cross section $$\begin{aligned} \label{momentdef2} M_{n} & = & \int\dfrac{{\rm d}s}{s^{n+1}}R_{e^{+}e^{-}\to\, c\bar{c}\,+X}(s)\,,\\ R_{e^{+}e^{-}\to\, c\bar{c}\,+X}(s) & = & \dfrac{\sigma_{e^{+}e^{-}\to\, c\bar{c}\,+X}(s)}{\sigma_{e^{+}e^{-}\to\,\mu^{+}\mu^{-}}(s)}\,,\nonumber \end{aligned}$$ which can be obtained from experiments. For small values of $n$ such that $m_{c}/n\gtrsim\Lambda_{{\rm QCD}}$ the theoretical moments $M_{n}^{{\rm th}}$ can be computed in an operator product expansion (OPE) where the dominant part is provided by perturbative QCD supplemented by small vacuum condensates that parametrize nonperturbative effects [@Shifman:1978bx; @Shifman:1978by]. The leading gluon condensate power correction term has a surprisingly small numerical effect and is essentially negligible for the numerical analysis as long as $n$ is small. This allows to determine the charm mass in a short distance scheme such as $\overline{{\rm MS}}$ to high precision. This method to determine the $\overline{{\rm MS}}$ charm mass is frequently called charmonium sum rules. For the theoretical moments the perturbative part of the OPE is known at ${\mathcal O}(\alpha_{s}^{0})$ and ${\mathcal O}(\alpha_{s})$ for any value of $n$ [@Kallen:1955fb]. At ${\mathcal O}(\alpha_{s}^{2})$ the first 30 moments are known [@Boughezal:2006uu; @Maier:2007yn], and to ${\mathcal O}(\alpha_{s}^{3})$ for $n=1$ [@Chetyrkin:2006xg; @Boughezal:2006px], $n=2$ [@Maier:2008he], and $n=3$ [@Maier:2009fz]. Higher moments at ${\cal O}(\alpha_s^3)$ have been determined by a semianalytical procedure [@Hoang:2008qy; @Kiyo:2009gb] (see also [@Greynat:2010kx]). The Wilson coefficient of the gluon condensate contribution is known to ${\mathcal O}(\alpha_{s})$ [@Broadhurst:1994qj]. On the experimental side the total hadronic cross section in $e^{+}e^{-}$ annihilation is known from various experimental measurements for c.m. energies up to $10.538\,$GeV. None of the experimental analyses actually ranges over the entire energy region between the charmonium region and $10.538$ GeV, but different analyses overlapping in energy exist such that energies up to $10.538\,$GeV are completely covered [@Bai:1999pk; @Bai:2001ct; @Ablikim:2004ck; @Ablikim:2006aj; @Ablikim:2006mb; @:2009jsa; @Osterheld:1986hw; @Edwards:1990pc; @Ammar:1997sk; @Besson:1984bd; @:2007qwa; @CroninHennessy:2008yi; @Blinov:1993fw; @Criegee:1981qx; @Siegrist:1976br; @Rapidis:1977cv; @Abrams:1979cx; @Siegrist:1981zp].[^1] Interestingly, to the best of our knowledge, the complete set of all available experimental data on the hadronic cross section has never been used in previous charmonium sum rule analyses to determine the experimental moments. Rather, sum rule analyses have relied heavily on theoretical input using different approaches to determine the corresponding “experimental error” and intrinsically leading to a sizable modeling uncertainty for energy regions below $10.538\,$GeV for low values of $n$ [@Hoang:2004xm]. The most recent charmonium sum rule analysis based on Eqs. (\[momentdef1\]) and (\[momentdef2\]), carried out by Kühn et al. [@Chetyrkin:2009fv; @Kuhn:2007vp] using input from perturbative QCD (pQCD) at ${\mathcal O}(\alpha_{s}^{3})$ for the perturbative contribution, obtained $\overline{m}_{c}(\overline{m}_{c})= 1279\pm(2)_{{\rm pert}}\pm(9)_{{\rm exp}} \pm(9)_{\alpha_{s}}\pm(1)_{{\rm \left\langle GG\right\rangle}}\,$ MeV where the first error is the perturbative uncertainty and the second is the experimental one. The third and the fourth uncertainties come from $\alpha_s$ and the gluon condensate correction, respectively. To our knowledge this result, the outcome of similar analyses in Ref. [@Chetyrkin:2006xg] and by Boughezal, Czakon and Schutzmeier [@Boughezal:2006px][^2], and a closely related analysis based on lattice results instead of data for pseudoscalar moments [@Allison:2008xk; @McNeile:2010ji] represent the analyses with the highest precision achieved so far in the literature. If confirmed, any further investigations and attempts concerning a more precise charm quark $\overline{{\rm MS}}$ mass would likely be irrelevant for any foreseeable future. We therefore find it warranted to reexamine the charmonium sum rule analysis with special attention on the way how perturbative and experimental uncertainties have been treated in Refs. [@Chetyrkin:2009fv; @Kuhn:2007vp]. A closer look into their analysis reveals that the quoted perturbative uncertainty results from a specific way to arrange the $\alpha_s$ expansion for the charm mass extractions and, in addition, by setting the $\overline{{\rm MS}}$ renormalization scales in $\alpha_{s}$ and in the charm mass (which we call $\mu_{\alpha}$ and $\mu_{m}$, respectively) equal to each other (i.e., they use $\mu_{\alpha}=\mu_{m}$). Moreover, concerning the experimental moments, only data up to $\sqrt{s}=4.8\,$GeV from the BES experiments [@Bai:2001ct; @Ablikim:2006mb] were used, while for $\sqrt{s}>4.8$ GeV perturbative QCD predictions were employed. Conceptually this approach is somewhat related to the method of finite energy sum rules (see e.g. Ref. [@Penarrocha:2001ig]), which we, however, do not discuss in this work. While this approach might be justified to estimate the overall nominal contribution for the experimental moments from $\sqrt{s}>4.8$ GeV, since perturbative QCD predictions describe quite well the measured total hadronic cross section outside the resonance regions, it is certainly not a suitable method to determine the experimental uncertainty. Since the region $\sqrt{s}>4.8$ GeV constitutes about $30\%$ of the first moment $M_{1}$, which is theoretically most reliable, this approach contains a significant intrinsic model dependence that cannot be quantified unambiguously. In this work we reexamine the charmonium sum rules analysis for low values of $n$ using the latest ${\mathcal O}(\alpha_{s}^{3})$ perturbative results, and we implement improvements which concern the two issues just mentioned: 1. We analyze several different types of perturbative expansions and examine in detail how the result for the $\overline{{\rm MS}}$ charm mass depends on independent choices of $\mu_{\alpha}$ and $\mu_{m}$. We show in particular that the interplay of the perturbative expansion and the scale setting $\mu_{\alpha}=\mu_{m}$ used in previous ${\mathcal O}(\alpha_s^3)$ analyses leads to sizable cancellations of the dependence on $\mu_{\alpha}$ and $\mu_{m}$ that in the light of our new analysis has to be considered as accidental. As the outcome of our analysis we estimate the current ${\mathcal O}(\alpha_{s}^{3})$ perturbative error as around $20$ MeV, which is an order of magnitude larger than that of Refs. [@Chetyrkin:2009fv; @Kuhn:2007vp; @Chetyrkin:2006xg; @Boughezal:2006px]. 2. Using a clustering method [@D'Agostini:1993uj; @Takeuchi:1995xe; @Hagiwara:2003da] to combine correlated data from many different experimental measurements we show that the $e^{+}e^{-}$ total hadronic cross section relevant for the charmonium sum rules can be determined with a complete coverage of center of mass energies above the $J/\psi$ and $\psi'$ resonances up to $10.538\,$GeV. Conservatively estimated modeling uncertainties coming from the energy range above $10.538\,$GeV then only lead to an insignificant contribution to the total uncertainty of the experimental moments. This paper is organized as follows: In Sec. \[sectiontheory\] we introduce the theoretical framework and review the current status of perturbative computations. We also show various equivalent ways of arranging the perturbative series in $\alpha_s$ for the charm mass. Finally we discuss how to properly estimate theoretical uncertainties due to the truncation of the perturbative series. In Sec. \[sectiondata\] we present all the experimental information that goes into our analysis. We discuss a clustering fit procedure that allows to combine data from different experiments accounting for their correlation and show the results. In Sec. \[sectionanalysis\] we carry out the numerical charm mass analysis concentrating on the first moment using arbitrary values of $\alpha_s$, and we present our final charm mass result. In Appendix A we present more details on the outcome of our clustering fit procedure for the charm , and in Appendix B we prove the equivalence of different versions of $\chi^2$ functions when auxiliary fit parameters are employed. Theoretical Input {#sectiontheory} ================= Perturbative Contribution {#subsectionperturbative} ------------------------- The moments of the vector current correlator are defined in Eq. (\[momentdef1\]). Their perturbative contribution in the framework of the OPE has a non-linear dependence on the charm quark mass. Thus in principle no conceptual preference can be imposed on any of the possible perturbative series that arises when solving for the charm mass. As a consequence, different versions of the expansion should be considered to obtain reliable estimates of the perturbative uncertainty. As indicated in Sec. \[sectionintroduction\] we use in the following $\mu_\alpha$ as the renormalization scale in $\alpha_s$ and $\mu_m$ as the renormalization scale in the $\overline{\rm MS}$ charm quark mass $\overline m_c$. [(a) Standard fixed-order expansion]{}\ Writing the perturbative vacuum polarization function as $$\begin{aligned} \label{Mnpertfixedorder1} \Pi^{\rm pert}(q^2,\alpha_s(\mu_\alpha), \overline m_c(\mu_m), \mu_\alpha, \mu_m) \, = \, \dfrac{1}{12\pi^2 Q_c^2}\sum_{n=0}^\infty q^{2n} M_n^{\rm pert} \,,\end{aligned}$$ we have for the perturbative moments $M_n^{\rm pert}$ $$\begin{aligned} \label{Mnpertfixedorder2} M_n^{\rm pert} & = & \frac{1}{(4\overline m^2_c(\mu_m))^{n}} \sum_{i,a,b} \left(\frac{\alpha_s(\mu_\alpha)}{\pi}\right)^i C^{a,b}_{n,i}\,\ln^a\left(\frac{\overline{m}^2_c(\mu_m)}{\mu^2_m}\right) \ln^b\left(\frac{\overline{m}^2_c(\mu_m)}{\mu^2_\alpha}\right).\end{aligned}$$ This is the standard fixed-order expression for the perturbative moments. At ${\cal O}(\alpha_s^3)$ the coefficients $C^{0,0}_{n,3}$ were recently determined for $n=1$ [@Chetyrkin:2006xg; @Boughezal:2006px], $n=2$ [@Maier:2008he], $n=3$ [@Maier:2009fz] and higher [@Hoang:2008qy; @Kiyo:2009gb; @Greynat:2010kx]. We refer to Ref. [@Boughezal:2006uu; @Maier:2007yn] for the coefficients at ${\cal O}(\alpha_s^2)$. We have summarized the numerical expressions for the $C^{a,b}_{n,i}$ coefficients of the first four moments in Tab. \[tabcfixedorder\]. The standard fixed-order expansion in Eq. (\[Mnpertfixedorder2\]) is the common way to represent the perturbative moments. However, written in this form the non-linear dependence on $\overline m_c$ does for some values of the experimental moments and the renormalization scales not yield numerical solutions for $\overline m_c$.[^3] [(b) Linearized expansion]{}\ Concerning the charm mass dependence, a more linear way to organize the perturbative expansion is to take the root of Eq. (\[Mnpertfixedorder2\]): $$\begin{aligned} \label{Mnpertlinearized1} \Big(M_n^{\rm th, pert}\Big)^{1/2n} & = & \frac{1}{2\overline{m}_c(\mu_m)} \,\sum_{i,a,b}\left(\frac{\alpha_s(\mu_\alpha)}{\pi}\right)^i \tilde C_{n,i}^{a,b} \ln^a\left(\frac{\overline{m}^2_c(\mu_m)}{\mu^2_m}\right) \ln^b\left(\frac{\overline{m}^2_c(\mu_m)}{\mu^2_\alpha}\right) ,\end{aligned}$$ or equivalently $$\begin{aligned} \label{Mnpertlinearized2} \overline m_c(\mu_m) & = & \frac{1}{2\Big(M_n^{\rm th, pert}\Big)^{1/2n}} \sum_{i,a,b}\,\left(\frac{\alpha_s(\mu_\alpha)}{\pi}\right)^i\tilde C_{n,i}^{a,b} \ln^a\left(\frac{\overline{m}^2_c(\mu_m)}{\mu^2_m}\right) \ln^b\left(\frac{\overline{m}^2_c(\mu_m)}{\mu^2_\alpha}\right) .\end{aligned}$$ The coefficients $\tilde C_{n,i}^{a,b}$ using again $\mu_\alpha$ for the renormalization scale in $\alpha_s$ and $\mu_m$ for the renormalization scale in the $\overline{\rm MS}$ charm mass are given in Tab. \[tabctildefixedorder\]. Although relation (\[Mnpertlinearized2\]) involves a non-linear dependence on ${\overline m}_c$, we find that it always has a numerical solution. [(c) Iterative linearized expansion]{}\ For the standard and the linearized expansions in Eqs. (\[Mnpertfixedorder2\]) and (\[Mnpertlinearized1\]) one searches for numerical solutions of the charm mass $\overline m_c(\mu_m)$ keeping the exact mass dependence on the respective RHS at each order in $\alpha_s$. An alternative way that is consistent within perturbation theory is to solve for $\overline m_c(\mu_m)$ iteratively order-by-order supplementing appropriate lower order values for $\overline m_c(\mu_m)$ in higher order perturbative coefficients. To be more explicit, we describe the method in the following. As the basis for the iterative expansion carried out in our analysis we use the linearized expansion of Eq. (\[Mnpertlinearized2\]). In the first step we determine $\overline m_c(\mu_m)$ employing the tree-level relation $$\begin{aligned} \label{Mnpertiterative1} \overline m_c^{(0)}(\mu_m) = \frac{1}{2\Big(M_n^{\rm th, pert}\Big)^{1/2n}} \,\tilde C_{n,0}^{0,0} \,,\end{aligned}$$ giving the tree-level charm mass $\overline m_c^{(0)}(\mu_m)$. In the next step one employs the relation $$\begin{aligned} \label{Mnpertiterative2} \overline m_c^{(1)}(\mu_m) &=& \frac{1}{2\Big(M_n^{\rm th, pert}\Big)^{1/2n}}\,\Biggr\{\, \tilde C_{n,0}^{0,0} \, + \, \frac{\alpha_s(\mu_\alpha)}{\pi}\bigg[\tilde C_{n,1}^{0,0}+\tilde C_{n,1}^{1,0} \ln\left(\frac{\overline{m}_c^{(0)\,2}(\mu_m)}{\mu^2_m}\right)\Bigg]\Biggr\} \,,\quad\end{aligned}$$ to determine the ${\cal O}(\alpha_s)$ charm mass $\overline m_c^{(1)}(\mu_m)$. In the ${\cal O}(\alpha_s)$ terms on the RHS of Eq. (\[Mnpertiterative2\]) the tree-level charm mass $\overline m_c^{(0)}$ is used, which is consistent to ${\cal O}(\alpha_s)$. At ${\cal O}(\alpha_s^2)$ for the determination of $\overline m_c^{(2)}(\mu_m)$ one uses $\overline m_c^{(0)}$ for the ${\cal O}(\alpha_s^2)$ coefficient and $\overline m_c^{(1)}(\mu_m)$ for the ${\cal O}(\alpha_s)$ correction, which in the strict $\alpha_s$ expansion yields $$\begin{aligned} \label{Mnpertiterative3} \overline m_c^{(2)} & = & \frac{1}{2\Big(M_n^{\rm th, pert}\Big)^{1/2n}}\,\Biggr\{\, \tilde C_{n,0}^{0,0} \, +\, \frac{\alpha_s(\mu_\alpha)}{\pi}\Bigg[\tilde C_{n,1}^{0,0}+\tilde C_{n,1}^{1,0} \ln\left(\frac{\overline{m}_c^{(0)\,2}(\mu_m)}{\mu^2_m}\right)\Bigg]+\nonumber\\ &&\left(\frac{\alpha_s(\mu_\alpha)}{\pi}\right)^2\Bigg[2\,\dfrac{\tilde C_{n,1}^{1,0}\,\tilde C_{n,1}^{0,0}}{\tilde C_{n,0}^{0,0}}+ 2\,\dfrac{(\tilde C_{n,1}^{1,0})^2}{\tilde C_{n,0}^{0,0}}\ln\left(\frac{\overline{m}_c^{(0)\,2}(\mu_m)}{\mu^2_m}\right)+\nonumber\\ &&\sum_{a,b}\,\tilde C_{n,2}^{a,b} \ln^a\left(\frac{\overline{m}_c^{(0)\,2}(\mu_m)}{\mu^2_m}\right) \ln^b\left(\frac{\overline{m}_c^{(0)\,2}(\mu_m)}{\mu^2_\alpha}\right)\Bigg]\Biggr\} \,.\end{aligned}$$ Here the second line contains the derivative of the ${\mathcal O}(\alpha_s)$ terms with respect to the charm mass. The determination of the ${\cal O}(\alpha_s^3)$ charm mass $\overline m_c^{(3)}(\mu_m)$ is then carried out in an analogous way involving the second (first) derivative with respect to the mass in the ${\cal O}(\alpha_s)$ (${\cal O}(\alpha_s^2)$) correction and using again $\overline m_c^{(0)}(\mu_m)$ for the ${\cal O}(\alpha_s^3)$ coefficient. In general we can write the iterative expansion as follows: $${\overline m}_c(\mu_m)\,=\,{\overline m}_c^{(0)} \sum_{i,a,b}\left(\frac{\alpha_s(\mu_\alpha)}{\pi}\right)^i \hat C_{n,i}^{a,b}\, \ln^a\left(\frac{{\overline m}_c^{(0)\,2}}{\mu^2_m}\right) \ln^b\left(\frac{{\overline m}_c^{(0)\,2}}{\mu^2_\alpha}\right), \label{eq:iterative-general}$$ where the numerical value of the coefficients $\hat C_{n,i}^{a,b}$ are collected in Tab. \[tab:chat\]. The iterative way to treat the perturbative series for the charm mass has the advantage that solving for the charm mass involves equations that are strictly linear in the charm mass at any order of the $\alpha_s$ expansion and thus always have solutions. In this way any possible influence on the analysis arising from a non-linear dependence is eliminated. [(d) Contour improved expansion]{}\ For the expansion methods (a)-(c) the moments and the charm quark mass are computed for a fixed choice of the renormalization scale $\mu_\alpha$ in the strong coupling $\alpha_s$. In analogy to the contour improved methods used for (see e.g. Refs. [@LeDiberder:1992te; @Pivovarov:1991rh; @Braaten:1991qm; @Narison:1988ni; @Braaten:1988ea; @Braaten:1988hc]) one can employ a path-dependent $\mu_\alpha$ in the contour integration that defines the perturbative moments [@Hoang:2004xm], see Fig. \[figcontour\], $$\begin{aligned} \label{Mnpertcontour1} M_n^{\rm c, pert} & = & \frac{6\pi Q_c^2}{i}\,\int_c\,\frac{{\rm d}s}{s^{n+1}} \Pi(q^2, \alpha_s(\mu_\alpha^c(s,\overline m_c^2)), \overline{m}_c(\mu_m), \mu_\alpha^c(s,\overline m_c^2), \mu_m) \,.\end{aligned}$$ ![ Path of integration in the complex for the computation of the moments. \[figcontour\] ](contour.eps){width="30.00000%"} Due to the independence of the moments on $\mu_\alpha$ and since no large logarithms are being generated anywhere for a path with distance of order $\overline m_c$ from the cut on the real axis, this method is a viable alternative to carry out the perturbative expansion. The different orders in the expansion of the contour improved moments $M_n^{\rm c, pert}$ are generated from the fixed-order $\alpha_s$ expansion of the vacuum polarization function $\Pi$ in Eq. (\[Mnpertcontour1\]). A useful path-dependent choice for $\mu_\alpha^c$ is given by [@Hoang:2004xm] $$\begin{aligned} \label{mualphacontour} (\mu_\alpha^c)^2(s,\overline m_c^2) & = & \mu_\alpha^2\,\bigg(\,1-\frac{s}{4\overline m_c^2(\mu_m)}\,\bigg) \,,\end{aligned}$$ which implements a modified weighting of threshold versus high energy contributions. It is straightforward to prove that the resulting moments $M_n^{\rm c, pert}$ can be obtained from the small-$q^2$ expansion of the perturbative vacuum polarization function using $\mu_\alpha^c$ as the renormalization scale of $\alpha_s$, $$\begin{aligned} \label{Mnpertcontour2} \Pi^{\overline {\rm MS}}\Big(q^2, \alpha_s(\mu_\alpha^c(q^2,\overline m_c^2)), \overline m_c(\mu_m), \mu_\alpha^c(q^2,\overline m_c^2), \mu_m\Big) & = & \sum\limits_{n=0}^\infty \, q^{2n}\,M_n^{\rm c, pert} \,.\end{aligned}$$ From Eq. (\[Mnpertcontour2\]) we can see that the $M_n^{\rm c, pert}$ can be derived from the expressions for the fixed-order moments $M_m^{\rm pert}$ with $m\le n$ given in Eq. (\[Mnpertfixedorder2\]).[^4] They also depend on the QCD and its derivatives, which arise in the small-$q^2$ expansion of $\alpha_s(\mu^c_\alpha(q^2,\overline m_c^2))$. Note that the that has to be employed must be exactly the same that is used for the contour integration in Eq. (\[Mnpertcontour1\]). Expanding the dependence of the on $\alpha_s$ strictly in fixed-order one recovers the fixed-order moments $M_n^{\rm pert}$. So the dependence of the contour improved moments $M_n^{\rm c, pert}$ on the fixed-order moments $M_m^{\rm pert}$ with $m<n$ is only residual due to the truncation of the $\alpha_s$ series, representing yet another alternative parametrization of higher order perturbative corrections. The contour improved moments do in particular have a residual dependence on the UV-subtraction scheme for the vacuum polarization function, i.e. on $\Pi(0)=M_0^{\rm pert}$. Using the “on-shell” scheme with $\Pi(0)=0$ one finds that $M_1^{\rm c,pert} = M_1^{\rm pert}$. For our analysis we employ the $\overline{\rm MS}$ scheme for $\Pi(0)$ defined for $\mu={\overline m}_c({\overline m}_c)$. Expressed in terms of $\alpha_s(\mu_\alpha)$ and ${\overline m}_c(\mu_m)$ it has the form [@Chetyrkin:2006xg] $$\begin{aligned} \label{Pi0msbar} \Pi^{\overline {\rm MS}}(0) & = & \sum_{i,a,b} \left(\frac{\alpha_s(\mu_\alpha)}{\pi}\right)^i C^{a,b}_{0,i}\,\ln^a\left(\frac{\overline{m}^2_c(\mu_m)}{\mu^2_m}\right) \ln^b\left(\frac{\overline{m}^2_c(\mu_m)}{\mu^2_\alpha}\right) \,.\end{aligned}$$ The numerical values for the coefficients $C^{a,b}_{0,i}$ can be found in Tab. \[tabcPi0\]. Using contour improved moments to determine $\overline m_c$ also involves non-linear relations, which implies that in some cases there is no solution. Again this can happen for $n=>1$ and for $\mu_m\sim 3$ GeV at any order. Gluon Condensate Contribution {#subsectioncondensate} ----------------------------- The dominant subleading contribution in the OPE for the theory moments $M_n^{\rm th}$ is arising from the gluon condensate. Writing $$\begin{aligned} \label{MnOPE1} M_n^{\rm th} & = & M_n^{\rm pert} + \Delta M_n^{\langle G^2\rangle}\,+\,\ldots \,,\end{aligned}$$ and neglecting any higher order power-suppressed condensate contributions of the OPE we have [@Novikov:1977dq; @Baikov:1993kc; @Broadhurst:1994qj] $$\label{DeltaMncondensate1} \Delta M_n^{\langle G^2\rangle} = \dfrac{1}{(4\overline{m}_c^2(\mu_m))^{n+2}}\Big\langle\frac{\alpha_s}{\pi} G^2\Big\rangle_{\rm RGI} \left\{ a^{0,0}_{n}+\dfrac{\alpha_{s}(\mu_\alpha)}{\pi}\left[a^{1,0}_{n}+a^{1,1}_n \log\dfrac{\overline{m}_c^2(\mu_m)}{\mu_m^{2}}\right]\right\}\,.$$ We have written Eq. (\[DeltaMncondensate1\]) using the renormalization group invariant (RGI) scheme for the gluon condensate [@Narison:1983kn] and the $\overline{\rm MS}$ charm quark mass. In Ref. [@Chetyrkin:2010ic] it is suggested that the Wilson coefficient of the gluon condensate should be expressed in terms of the pole rather than the $\overline{\mbox{MS}}$ mass based on the observation that the pole mass leads to a condensate correction that is numerically quite stable for higher moments. We disagree with this prescription due to the infrared renormalon issue in the pole mass and since it is expected that the condensate corrections for the moment increases with $n$. On the other hand, for a sum rule analysis of the first moment with $n=1$ this issue is absolutely irrelevant due to the small size of the gluon condensate contribution compared to the level of the remaining perturbative uncertainties. Numerical values for the coefficients $a_n^{i,j}$ are given in Tab. \[tabgluoncondensate\] for $n=1,2,3,4$. For the RGI gluon condensate we adopt [@Ioffe:2005ym] $$\begin{aligned} \label{condensatevalue1} \Big\langle\frac{\alpha_s}{\pi} G^2\Big\rangle_{\rm RGI} & = & 0.006\pm0.012\;\mathrm{GeV}^4\,.\end{aligned}$$ The overall contribution of the gluon condensate correction in Eq. (\[DeltaMncondensate1\]) in the charm quark mass analysis is quite small. Its contribution to the moments amounts to around $0.2\%$, $0.6\%$, $2\%$, and $3\%$ for the first four moments, respectively. For $n=1$ it leads to a correction in the $\overline{\rm MS}$ charm quark mass at the level of $1$ MeV and is an order of magnitude smaller than our perturbative uncertainty. We therefore ignore the condensates correction for the discussion of the perturbative uncertainties in Sec. \[subsectionmcerror\]. Its contribution is, however, included in the final charm mass results presented in Sec. \[sectionanalysis\]. Running Coupling and Mass {#subsectionrunning} ------------------------- The analysis of the charmonium sum rules naturally involves renormalization scales around the charm mass, $\mu\sim {\overline m}_c\sim 1.3$ GeV, which are close to the limits of a perturbative treatment. In fact, parametrically, the typical scale relevant for the perturbative computation of the moment $M_n^{\rm th}$ is of order $\mu\sim {\overline m}_c/n$ (see e.g. Ref. [@Hoang:1998uv]) because the energy range of the smearing associated to the weight function $1/s^{n+1}$ in Eq. (\[momentdef2\]) decreases with $n$. We will therefore use $n=1$ for our final numerical analysis. Moreover, it is common practice to quote the $\overline{\rm MS}$ charm mass $\overline m_c(\overline m_c)$, i.e. for the scale choice $\mu_m=\overline m_c$. It is therefore useful to have a look at the quality of the perturbative behavior of the renormalization group evolution of the strong $\overline{\rm MS}$ coupling $\alpha_s$ and the $\overline{\rm MS}$ charm quark mass. In Fig. \[fig:alphasevolutionexact\][^5] we have displayed $\alpha_s^{\rm N^3LL}(\mu)/\alpha_s^{\rm N^kLL}(\mu)$ using for $\alpha_s^{\rm N^kLL}$ the QCD and the respective exact numerical solution for $\alpha_s(3~\mbox{GeV})= 0.2535$ as the common reference point. We see that the convergence of the lower order results towards the 4-loop evolution is very good even down to scales of around $1$ GeV. The curves indicate that the remaining relative perturbative uncertainty in the 4-loop evolution might be substantially smaller than $1\%$ for scales down to $\overline m_c\sim 1.3$ GeV. It is also instructive to examine the evolution using a fixed-order expansion. In Fig. \[fig:alphasevolutionexpanded\] we display $\alpha_s^{\rm N^3LL}(\mu)/\alpha_s^{(m)}(\mu)$ where $\alpha_s^{(m)}(\mu)$ is the ${\cal O}(\alpha_s^{m+1})$ fixed-order expression for $\alpha_s(\mu)$ using the reference value $\alpha_s(3~\mbox{GeV})=0.2535$ as the expansion parameter. The convergence of the fixed-order expansion for $\alpha_s(\mu)$ towards the exact N${}^3$LL numerical solution $\alpha_s^{\rm N^3LL}(\mu)$ is somewhat worse compared to the renormalization group resummed results since the deviation of the ratio from one is in general larger. However, convergence is clearly visible. In particular there are not any signs of instabilities. It therefore seems to be safe to use renormalization scales down to the charm mass and associated renormalization scale variations as an instrument to estimate the perturbative uncertainties. In Figs. \[fig:mcevolutionexact\] and \[fig:mcevolutionexpanded\] an analogous analysis has been carried out for the $\overline{\rm MS}$ charm quark mass. In Fig. \[fig:mcevolutionexact\] $\overline m_c^{\rm N^3LL}(\mu)/\overline m_c^{\rm N^kLL}(\mu)$ is plotted for $k=0,1,2,3$ using the exact numerical solutions of the renormalization group equations and $\overline m_c(\mu=3~\mbox{GeV})$ as the respective reference value.[^6] Compared to the Fig. \[fig:alphasevolution\] we observe a very similar convergence. In Fig. \[fig:mcevolutionexpanded\], finally, we show $\overline m_c^{\rm N^3LL}(\mu)/\overline m_c^{(m)}(\mu)$, where $\overline m_c^{(m)}(\mu)$ is the ${\cal O}(\alpha_s^m)$ fixed-order expression for $\overline m_c(\mu)$ using $\alpha_s(3~\mbox{GeV})=0.2535$ as the expansion parameter. Again, the convergence towards the exact N${}^3$LL evolved result is very similar to the corresponding results for the strong coupling, and we find again no evidence for perturbative instabilities. Of course the corrections are somewhat larger when the fixed-order expansion is employed. We therefore conclude that perturbative evolution and renormalization scale variations for the $\overline{\mbox{MS}}$ charm quark mass can be safely used down to scales above $\overline m_c\sim 1.3$ GeV. One should of course mention that the lines in Fig. \[fig:mcevolutionexpanded\] also give an indication about the expected size of scale variations depending on the range of the variations. Scales above $2$ GeV can lead to sub-MeV variations, while scales down to the charm mass will result at best in percent precision (i.e. ${\cal O}(10~\mbox{MeV})$). Perturbative Uncertainties in the $\overline{\rm MS}$ Charm Mass {#subsectionmcerror} ---------------------------------------------------------------- In this section we discuss in detail the perturbative series for the determination of the $\overline{\rm MS}$ charm mass $\overline m_c$ and how to set up an adequate scale variation to estimate the perturbative uncertainty. In the previous subsections we have discussed four different ways to carry out the perturbative expansion and we presented the corresponding order-by-order analytic expressions. As described there, we can determine at each order of the perturbative expansion for the moments $M_n^{\rm th}$ a value for $\overline m_c(\mu_m)$ which also has a residual dependence on $\mu_\alpha$, the renormalization scale used for $\alpha_s$. To compare the different mass determinations we then evolve $\overline m_c(\mu_m)$ to obtain $\overline m_c(\overline m_c)$ using the 4-loop renormalization group equations for the mass and the strong coupling [@Tarasov:1980au; @Larin:1993tp; @vanRitbergen:1997va; @Chetyrkin:1997dh; @Vermaseren:1997fq].[^7] The obtained value of $\overline m_c(\overline m_c)$ thus has a residual dependence on the scales $\mu_m$ and $\mu_\alpha$, on the order of perturbation theory and on the expansion method.[^8] For the results we can therefore use the notation $$\begin{aligned} \label{mcnotation1} && \overline m_c(\overline m_c)[\mu_m,\mu_\alpha]^{i,n} \,,\end{aligned}$$ where $n=0,1,2,3$ indicates perturbation theory at ${\cal O}(\alpha_s^n)$ and $$\begin{aligned} \label{methodnotation1} i & = & \left\{ \begin{array}[c]{ll} a \qquad & \mbox{(fixed-order expansion),} \\ b & \mbox{(linearized expansion),} \\ c & \mbox{(iterative expansion),} \\ d & \mbox{(contour improved expansion).} \end{array}\right.\end{aligned}$$ To initiate the discussion, we show in Fig. \[fig:trumpet1\] results for $\overline m_c(\overline m_c)$ at ${\cal O}(\alpha_s^n)$ for expansions a–d using $\mu_m=\mu_\alpha$ (upper four graphs) and using $\mu_m=\overline m_c(\overline m_c)$ (lower four graphs). For each method and order we have displayed the range of $\overline m_c(\overline m_c)$ values for a variation of $2~\mbox{GeV}\le \mu_\alpha \le 4~\mbox{GeV}$, which corresponds to the scale variation employed in Refs. [@Chetyrkin:2006xg; @Boughezal:2006px; @Chetyrkin:2009fv; @Kuhn:2007vp]. Their analysis used the fixed-order expansion with the setting $\mu_m=\mu_\alpha$ and is represented by graph 1. We make several observations: - Choosing $\mu_m$ and $\mu_\alpha$ both larger than $2$ GeV makes the $\overline m_c(\overline m_c)$ value decrease with the order of perturbation theory. - Choosing $\mu_m$ smaller than $1.5$ GeV and $\mu_\alpha$ larger than $2$ GeV makes the $\overline m_c(\overline m_c)$ value increase with the order of perturbation theory. - For most choices of the scale setting and the expansion method the spread of the $\overline m_c(\overline m_c)$ values from the variation of $\mu_\alpha$ does not decrease in any substantial way with the order. However, viewing all methods and scale setting choices collectively a very good convergence is observed. We have checked that these statements apply also in general beyond the specific cases displayed in Fig. \[fig:trumpet1\]. Quite conspicuous results are obtained for the scale choice $\mu_m=\mu_\alpha$ for the fixed-order (graph 1) and linearized expansions (graph 2). Here, extremely small variations in $\overline m_c(\overline m_c)$ are obtained. They amount to $1.8$ MeV ($4$ MeV) and $0.6$ MeV ($1.4$ MeV) at order $\alpha_s^2$ and $\alpha_s^3$, respectively, for the fixed-order expansions (linearized expansions). We note that our scale variation for the fixed-order expansion at ${\cal O}(\alpha_s^3)$ is consistent with the corresponding numbers quoted in Ref. [@Chetyrkin:2006xg; @Boughezal:2006px], where the ${\cal O}(\alpha_s^3)$ corrections to the first moment were computed,[^9] but differs from the scale variations given in Refs. [@Chetyrkin:2009fv; @Kuhn:2007vp] which also quoted numerical results from the fixed-order expansion. Interestingly the ${\cal O}(\alpha_s^2)$ and ${\cal O}(\alpha_s^3)$ variations we find do not overlap and the ${\cal O}(\alpha_s^3)$ ranges appear to be highly inconsistent with the ${\cal O}(\alpha_s^3)$ results from the iterative method (graph 3). A visual display of scale variations obtained from the four expansion methods at ${\cal O}(\alpha_s^3)$ with different types of variation methods is given in Fig. \[fig:trumpet2\]. An illustrative way to demonstrate how a small scale variation can arise is given in Fig. \[fig:mccontour1fixed\] - \[fig:mccontour1improved\]. For all four expansion methods contour curves of constant $\overline m_c(\overline m_c)$ are displayed as a function of (the residual dependence on) $\mu_m$ and $\mu_\alpha$. For the fixed-order (a) and the linearized expansions (b) we see that there are contour lines closely along the diagonal $\mu_m=\mu_\alpha$. For the fixed-order expansion (a) this feature is almost exact and thus explains the extremely small scale-dependence seen in graph 1 of Fig. \[fig:trumpet1\]. For the linearized expansion (b) this feature is somewhat less exact and reflected in the slightly larger scale variation seen in graph 2 of Fig. \[fig:trumpet1\]. On the other hand, the contour lines of the iterative (c) and contour improved (d) expansions have a large angle with respect to the diagonal $\mu_m=\mu_\alpha$ leading to the much larger scale variations of $9$ and $5$ MeV, respectively, at ${\cal O}(\alpha_s^3)$ visible in graphs 3 and 4 of Fig. \[fig:trumpet1\]. The contour plots shown in Fig. \[fig:mccontour1\] also show the variation along the line $\mu_m$ or $\mu_\alpha=\overline m_c(\overline m_c)$ (border of gray shaded areas). Here, the contour lines are relatively dense leading to scale variation of around $15$ MeV at ${\cal O}(\alpha_s^3)$. Overall, we draw the following conclusions: - The small scale variations observed for the fixed-order and linearized expansions for $\mu_m=\mu_\alpha$ result from strong cancellations of the individual $\mu_m$ and $\mu_\alpha$ dependences that arise for this correlation. - Other correlations between $\mu_m$ and $\mu_\alpha$ that do not generate large logarithms do not lead to such cancellations. One therefore has to consider the small scale variations observed for $\mu_m=\mu_\alpha$ in the fixed-order and linearized expansions as accidental. - For an adequate estimate of perturbative uncertainties specific correlations between $\mu_m$ and $\mu_\alpha$ that are along contour lines of constant $\overline m_c(\overline m_c)$ have to be avoided. Moreover, adequate independent variations of $\mu_m$ and $\mu_\alpha$ should not induce large logarithms. As the outcome of this discussion we adopt for our charm mass analysis an independent and uncorrelated variation of $\mu_m$ and $\mu_\alpha$ in the range $$\begin{aligned} \label{muammvariation} \overline m_c(\overline m_c) & \le & \mu_m,\mu_\alpha \, \le \, 4~\mbox{GeV}\,,\end{aligned}$$ in order to estimate perturbative uncertainties. The excluded region $\mu_m,\mu_\alpha<\overline{m}_c(\overline{m}_c)$ in the plane in the contour plots of Fig. \[fig:mccontour1\] is indicated by the gray shaded areas. This two-dimensional variation avoids accidental cancellations from correlated variations, large logarithms involving ratios of the scales $\overline m_c(\mu_m)$, $\mu_m$, $\mu_\alpha$ and remains well in the validity ranges for the perturbative renormalization group evolution of the $\overline{\rm MS}$ charm quark mass and $\alpha_s$ (see Sec. \[subsectionrunning\]). The range of Eq. (\[muammvariation\]) is also consistent with a scale variation $\mu\sim 2\,\overline m_c\times (1/2,1,2)$ one might consider as the standard choice with respect to the to particle threshold located at $\sqrt{s}=2\,{\overline m}_c$. As a comparison the range $\mu=(3\pm 1)$ GeV corresponds to $\mu=2\,{\overline m}_c(0.8,1.2,1.6)$. We emphasize that the lower boundary $\overline m_c(\overline m_c)\sim 1.25$ GeV is also reasonable as it represents the common flavor matching scale where gauge coupling evolution remains smooth up to NLL order. We do not see any evidence for perturbation theory for the first moment $M_1$ being unstable at the charm mass scale, in contrast to claims made in Ref. [@Kuhn:2007vp]. In Fig. \[fig:mmcerror1\] we show the ranges of $\overline m_c(\overline m_c)$ at ${\cal O}(\alpha_s^{1,2,3})$ for the four expansion methods employing the scale variations of Eq. (\[muammvariation\]). We see that all four expansion methods now lead to equivalent results. The variations are also compatible with the overall variations shown in Fig. \[fig:trumpet1\]. At ${\cal O}(\alpha_s^3)$ we obtain a scale variation for $\overline m_c(\overline m_c)$ of around $20$ MeV. This is an order of magnitude larger than the perturbative uncertainties quoted in Refs. [@Chetyrkin:2006xg; @Boughezal:2006px; @Chetyrkin:2009fv; @Kuhn:2007vp]. In Fig. \[fig:m3cerror1\] we have displayed the corresponding results for $\overline m_c(3~\mbox{GeV})$. At ${\cal O}(\alpha_s^3)$ they also exhibit a scale variation of around $20$ MeV. Experimental Data {#sectiondata} ================= Data Collections {#subsectioncollections} ---------------- [Narrow resonances]{}\ Below the open charm threshold there are the $J/\psi$ and $\psi^\prime$ narrow charmonium resonances. Their masses, and electronic widths are taken from the PDG [@Nakamura:2010zzi] and are collected in Tab. \[tabpsidata\] together with the value of the $\overline{\rm MS}$ QED coupling at their masses. The total widths are not relevant since we use the narrow width approximation for their contributions to the moments. The uncertainty for the contribution to the moments coming from the masses can be neglected. $J/\psi$ $\psi^\prime$ -------------------------- ---------------- ---------------- $M$ (GeV) $3.096916(11)$ $3.686093(34)$ $\Gamma_{ee}$ (keV) $5.55(14)$ $2.48(6)$ $(\alpha/\alpha(M))^{2}$ $0.957785$ $0.95554$ : Masses and electronic widths [@Nakamura:2010zzi] of the narrow charmonium resonances and effective electromagnetic coupling [@Kuhn:2007vp]. $\alpha=1/137.035999084(51)$ stands for the fine structure constant, and $\alpha(M)$ stands for the $\overline{\mbox{MS}}$ electromagnetic coupling at the scale $M$.\[tabpsidata\] [Threshold and data continuum region]{}\ The open charm threshold is located at $\sqrt{s}=3.73$ GeV. We call the energies from just below the threshold and up to $5$ GeV the threshold region, and the region between $5$ GeV and $10.538$ GeV, where the production rate is dominated by multiparticle final states the data continuum region. In these regions quite a variety of measurements of the total hadronic cross section exist from BES [@Bai:1999pk; @Bai:2001ct; @Ablikim:2004ck; @Ablikim:2006aj; @Ablikim:2006mb; @:2009jsa], CrystalBall [@Osterheld:1986hw; @Edwards:1990pc], CLEO [@Ammar:1997sk; @Besson:1984bd; @:2007qwa; @CroninHennessy:2008yi], MD1 [@Blinov:1993fw], PLUTO [@Criegee:1981qx], and MARKI and II [@Siegrist:1976br; @Rapidis:1977cv; @Abrams:1979cx; @Siegrist:1981zp]. Taken together, the entire energy region up to $10.538$ GeV is densely covered with total cross section measurements from these 19 data sets.[^10] The measurements from BES and CLEO have the smallest uncertainties. They do, however, not cover the region between $5$ and $7$ GeV. Here CrystalBall and MARKI and II have contributed measurements albeit with somewhat larger uncertainties. The statistical and total systematical uncertainties of the measurements can be extracted from the respective publications. For some data sets the amount of uncorrelated and correlated systematical uncertainties is given separately (BES [@Bai:2001ct; @Ablikim:2006aj; @Ablikim:2006mb], CrystalBall [@Osterheld:1986hw; @Edwards:1990pc], CLEO [@:2007qwa], MARKI and II [@Siegrist:1981zp; @Abrams:1979cx], MD1 [@Blinov:1993fw]) while for all the other data sets only combined systematical uncertainties are quoted. All these data sets are shown in Figs. \[figdatacompilation\], where the displayed error bars represent the (quadratically) combined statistical and systematical uncertainties. Interestingly, none of the previous charm mass analyses, to the best of our knowledge, ever used the complete set of available data. As examples, Bodenstein et al. [@Bodenstein:2010qx] used data sets [@Bai:1999pk; @Bai:2001ct; @Ablikim:2006mb; @CroninHennessy:2008yi] from BES and CLEO. Jamin and Hoang [@Hoang:2004xm] used the data sets of Refs. [@Bai:2001ct], [@Blinov:1993fw] and [@Ammar:1997sk] from BES, MD1 and CLEO, covering the regions $2~\mbox{GeV} \leq E \leq 4.8~\mbox{GeV}$ and $6.964~\mbox{GeV} \leq E \leq 10.538~\mbox{GeV}$. Boughezal et al. [@Boughezal:2006px], Kuhn et al. [@Kuhn:2001dm], and Narison [@Narison:2010cg] use only one data set from BES [@Bai:2001ct]. Kuhn et al. [@Chetyrkin:2009fv; @Kuhn:2007vp] used the data sets of Refs. [@Bai:2001ct; @Ablikim:2006mb] from BES covering the energy region $2.6~\mbox{GeV} \leq E \leq 4.8~\mbox{GeV}$. We consider three different selections of data sets to study the dependence of the experimental moments on this choice: - The [minimal selection]{} contains all data sets necessary to cover the whole energy region between $2$ and $10.538$ GeV without any gaps and keeping only the most accurate ones. These 8 data sets are from BES [@Bai:1999pk; @Bai:2001ct; @Ablikim:2006aj; @:2009jsa], CrystalBall [@Edwards:1990pc], CLEO [@CroninHennessy:2008yi; @:2007qwa] and MD1 [@Blinov:1993fw] corresponding to the data sets 1, 2, 5, 6, 9, 12, 13, and 14 (see Tab. \[tab:datasets\] for references). - The [default selection]{} contains all data sets except for the three ones with the largest uncertainties. It contains 16 data sets and fully includes the minimal selection. It contains all data sets except for Mark I and II data sets 16, 17 and 19 from Refs. [@Siegrist:1981zp; @Rapidis:1977cv; @Siegrist:1976br]. - The [maximal selection]{} contains all 19 data sets. We use the default selection as our standard choice for the charm mass analysis, but we will also quote results for the other data selections. [Perturbative QCD region]{}\ Above $10.538$ GeV there are no experimental measurements of the total hadronic that might be useful for the experimental moments. In this energy region we will therefore use perturbative QCD to provide estimates for the charm production . As a penalty for not using experimental data we assign a $10$% total relative uncertainty to the contribution of the experimental moments coming from this region, which we then treat like an uncorrelated experimental uncertainty for the combination with the moment contribution from lower energies. As we see in Sec. \[subsectionmoments\] the energy region above $10.538$ GeV contributes only $(6, 0.4, 0.03, 0.002)\%$ to $M^{\rm exp}_{1,2,3,4}$. In the first moment $M_1^{\rm exp}$ the total contribution is about three times larger than the combined statistical and systematical (true) experimental uncertainties from the other energy regions. So the 10% penalty we assign to this approach represents a subleading component of the final quoted uncertainty.[^11] In the second and higher moments $M_{n\ge 2}^{\rm exp}$ the contributions from above $10.538$ GeV and the corresponding uncertainty are negligible compared to the uncertainties from the lower energy regions. As the theoretical formula to determine the moment contribution from the perturbative QCD region we use the ${\cal O}(\alpha_s^3)$ non-singlet massless quark cross section including charm mass corrections up to ${\cal O}(\overline m_c^4/s^2)$: $$\begin{aligned} R_{cc}^{\rm th}(s) & = & N_c \,Q_c^2\,R^{\rm ns}(s,\overline m_c^2(\sqrt{s}), n_f=4,\alpha_s^{n_f=4}(\sqrt{s}))\,, \label{Rcchighdef1}\end{aligned}$$ where $$\begin{aligned} \label{Rnsdef1} \lefteqn{R^{\rm ns}(s,\overline m_c^2(\mu), n_f=4,\alpha_s^{n_f=4}(\mu),\mu)}\\ & = &1+\dfrac{\alpha_{s}}{\pi} +\left(\dfrac{\alpha_{s}}{\pi}\right)^{2}\left[ 1.52453-2.08333 L_s\right] +\left(\dfrac{\alpha_{s}}{\pi}\right)^{3} \left[-11.52034+34028 L_s^2 -9.56052 L_s\right]\nonumber\\ & + & \dfrac{\overline m_c^{2}(\mu)}{s} \Big\{ 12\,\dfrac{\alpha_{s}}{\pi}+ \left(\dfrac{\alpha_{s}}{\pi}\right)^{2}\left[109.167-49 L_s\right] +\left(\dfrac{\alpha_{s}}{\pi}\right)^{3}\left[634.957-799.361 L_s+ 151.083 L_s^2\right]\!\!\Big\} \nonumber\\ & + & \dfrac{\overline m_c^4(\mu)}{s^2}\Big\{ \!\! -6+\dfrac{\alpha_{s}}{\pi}\left[-22+24 L_s\right] + \left(\dfrac{\alpha_{s}}{\pi}\right)^{2} \left[140.855+221.5 L_s-5.16667 L_m-73 L_s^2\right]\nonumber\\ && \qquad + \left(\dfrac{\alpha_{s}}{\pi}\right)^{3}\left[ 3776.94-509.226 L_s -174.831 L_m-1118.29 L_s^2\right.\nonumber\\ && \qquad\qquad + 10.3333 L_m^2+\left.42.1944 L_m L_s +198.722 L_s^3\right]\Big\} \,,\nonumber\end{aligned}$$ with $$L_s \, \equiv \, \ln\Big(\frac{s}{\mu^{2}}\Big)\,,\qquad L_m \, \equiv \, \ln\Big(\dfrac{\overline m_c^{2}(\mu)}{s}\Big)\,.$$ For the computation of the contribution to the experimental moments we determine $\overline m_c(\sqrt{s})$ and $\alpha_s(\sqrt{s})$ appearing in Eq. (\[Rcchighdef1\]) using $\overline m_c(\overline m_c)=1.3$ GeV and $\alpha_s(m_Z)=0.118$ as initial conditions. It is instructive to examine for the moment contributions from $\sqrt{s}>10.538$ GeV terms related to charm production that we do not account for in Eq. (\[Rnsdef1\]). In Tab. \[tabneglected\] the relative size with respect to the full first four moments (in percent) of the most important neglected contributions are given. In the second column the size of the mass corrections up to order $\overline m_c^4$, which we have included in $R^{\rm th}_{cc}$, are shown as a reference. The third column shows the contributions coming from secondary $c\bar c$ radiation through gluon splitting. The fourth column depicts the contributions from the ${\cal O}(\alpha_s^3)$ singlet corrections (including the mass corrections up to order $\overline m_c^4$), which one can take as an rough estimate for the actual contributions from the charm cut. Finally in the last column we show the size of the Z-boson exchange terms integrated from threshold to $10.538$ GeV. This contribution represents the Z-exchange contribution that is contained in the data, but - by definition - not accounted for in the theory moments. We see that at least for the first two moments, the contributions neglected are much smaller than the charm mass corrections we have accounted for in the nonsinglet production rate, which are already constituting a very small effect. Overall the numerical effect on the charm mass of all these contributions is tiny considering the scaling $\overline m_c\sim M_n^{1/2n}$. Since we assign a $10$% error on the moments’ contribution from the energy region $\sqrt{s}>10.538$ GeV where we use theory input, our approach to neglect subleading effects is justified. [Non-charm background]{}\ Experimentally only the total hadronic cross section is available. Although charm-tagged rate measurements are in principle possible [@CroninHennessy:2008yi] they have not been provided in publications. On the other hand, they would also exhibit sizable additional uncertainties related to the dependence on simulations of the decay of charmed mesons into light quark final states. So to obtain the charm production cross section from the data we have to subtract the non-charm background using a model based on perturbative QCD related to the production of $u$, $d$ and $s$ quarks. A subtle point is related to the secondary radiation of $c\bar c$ pairs off the $u$, $d$ and $s$ quarks from gluon splitting and to which extent one has to account theoretically for the interplay between real and virtual secondary $c\bar c$ radiation which involves infrared sensitive terms [@Hoang:1997ca]. Since in this work we define the moments from primary $c\bar c$ production (see Eq. (\[momentdef1\])), secondary $c\bar c$ production is formally counted as non-charm background. Thus for the model for the non-charm background for $\sqrt{s}>2\, \overline m_c$ we employ the expression $$\begin{aligned} \label{Rudsdef} R_{uds}(s) & = & N_c(Q_u^2 + 2 Q_d^2)\,R^{\rm ns}(s,0,n_f=3,\alpha_s^{n_f=4}(s))\\ &&+\, N_c\, Q_c^2\left(\dfrac{\alpha_s^{n_f=4}(s)}{\pi}\right)^{2}\left(\rho^{V}+\rho^{R}+\dfrac{1}{4}\,\log\dfrac{\overline m_{c}^{2}(\overline m_{c})}{s}\right) \,.\nonumber\end{aligned}$$ The second term on the RHS describes the contributions from real and virtual secondary $c\bar c$ radiation. The analytic expressions for $\rho^R$ and $\rho^V$ can be found in Eqs. (2) and (6) of Ref. [@Hoang:1994it]. We have checked that the numerical impact of real ($\rho^R$) and virtual ($\rho^V$) secondary radiation individually as well as the complete second term on the RHS of Eq. (\[Rudsdef\]) on the moments is negligible, see Tab. \[tabneglected\]. We use Eq. (\[Rudsdef\]) and fit the non-charm background including also data in the region $2~\mbox{GeV}\le E\le 3.73~\mbox{GeV}$ via the ansatz $R_{{\rm non}-c\bar c}(s) = n_{\rm ns}\,R_{uds}(s)$, where the constant $n_{\rm ns}$ represents an additional fit parameter. Data Combination {#subsectioncombination} ---------------- Combining different experimental measurements of the hadronic cross section one has to face several issues: (a) the measurements are given at individual separated energy points, (b) the set of measurements from different publications are not equally spaced, cover different, partly overlapping energy regions and have different statistical and systematical uncertainties, (c) the correlations of systematical errors are only known (or provided) for the data sets within each publication, (d) there are a number of very precise measurements at widely separated energies. In this section we discuss the combination of the experimental data from the threshold and the data continuum regions between $2$ and $10.538$ GeV using a method based on a fitting procedure used before for determining the hadronic vacuum polarization effects for $g-2$ [@Hagiwara:2003da]. In this work we extend this approach and also account for the subtraction of the non-charm background. [Combination method]{}\ The method uses the combination of data in energy bins (clusters) assuming that the within each cluster changes only very little and can thus be well approximated by a constant. Thus clusters for energies where $R$ varies rapidly need to be small (in this case the experimental measurements are also denser). The in each cluster is then obtained by a $\chi^2$ fitting procedure. Since each experimental data set from any publication covers an energy range overlapping with at least one other data set, the clusters are chosen such that clusters in overlapping regions contain measurements from different data sets. Through the fitting procedure correlations are then being communicated among different data sets and very accurate individual measurements can inherit their precision into neighbouring clusters. Both issues are desirable since the hadronic is a smooth function with respect to the sequence of clusters. To describe the method we have to set up some notation: - All measurements $R(E)$ are distinguished according to the energy $E$ at which they have been carried out. - Each such energy point having a measurement is written as $E_i^{k,m}$, where $k=1,\ldots,N_{\rm exp}$ refers to the $N_{\rm exp}$ data sets, $m=1,\ldots,N_{\rm cluster}$ runs over the $N_{\rm cluster}$ clusters and $i=1,\ldots,N^{k,m}$ assigns the of the $N^{k,m}$ measurements. - Each individual measurement of the is then written as $$\begin{aligned} \label{Rexp} R(E^{k,m}_i) & = & R_i^{k,m} \, \pm \, \sigma_i^{k,m} \, \pm \, \Delta_i^{k,m} \,,\end{aligned}$$ where $R_i^{k,m}$ is the central value, $\sigma_i^{k,m}$ the combined statistical and uncorrelated systematical uncertainty and $\Delta_i^{k,m}$ the correlated systematical experimental uncertainty. - For convenience we define $\Delta f_i^{k,m}=\Delta_i^{k,m}/R_i^{k,m}$ to be the relative systematical correlated uncertainty. As our standard choice concerning the clusters we use 5 different regions each having equidistant cluster sizes $\Delta E$. The regions are as follows: - [non-charm region]{}: has 1 cluster for $2~\mbox{GeV}\leq E \leq 3.73~\mbox{GeV}$ ($\Delta E=1.73$ GeV). - [low charm region]{}: has 2 clusters for $3.73~\mbox{GeV}< E \leq 3.75~\mbox{GeV}$ ($\Delta E=10$ MeV). - [$\psi(3S)$ region/threshold region 1]{}: has 20 cluster for $3.75~\mbox{GeV}< E \leq 3.79~\mbox{GeV}$ ($\Delta E=2$ MeV). - [resonance region 2]{}: has 20 cluster for $3.79~\mbox{GeV}< E \leq 4.55~\mbox{GeV}$ ($\Delta E=38$ MeV). - [continuum region]{}: has 10 cluster for $4.55~\mbox{GeV}< E \leq 10.538~\mbox{GeV}$ ($\Delta E=598.8$ MeV). We assign to this choice of $52+1$ clusters the notation (2,20,20,10) and later also examine alternative cluster choices demonstrating that the outcome for the moments does within errors not depend on them. The cluster in the non-charm region is used to fit for the normalization constant $n_{\rm ns}$ of the non-charm background contribution, see Eq. (\[Rudsdef\]). Our standard procedure to determine the central energy $E_m$ associated to each cluster is just the weighted average of the energies of all measurements falling into cluster $m$, $$\begin{aligned} \label{Eclusterdef} E_m & = & \dfrac{\sum_{k,i}\dfrac{E_{i}^{k,m}} {(\sigma_{i}^{k,m})^2+(\Delta_{i}^{k,m})^2}} {\sum_{k,i}\dfrac{1}{(\sigma_{i}^{k,m})^2+(\Delta_{i}^{k,m})^2}} \,.\end{aligned}$$ The corresponding for the charm cross section that we determine through the fit procedure described below is called $$\begin{aligned} \label{Rclusterdef} R_m & \equiv & R_{c\bar c}(E_m) \,. \end{aligned}$$ We note that using instead the unweighted average or simply the center of the cluster has a negligible effect on the outcome for the moments since the clusters we are employing are sufficiently small. [Fit procedure and ]{}\ We determine the charm cross section $R_{c\bar c}$ from a that accounts for the positive correlation among the systematical uncertainties $\Delta_i^{k,m}$ within each experiment $k$ and, at the same time, also for the non-charm background. To implement the correlation we introduce the auxiliary parameters $d_k$ ($k=1,\ldots, N_{\rm exp}$) that parametrize the correlated deviation from the experimental central values $R_i^{k,m}$ in units of the correlated systematical uncertainty $\Delta_i^{k,m}$, see Eq. (\[Rexp\]). In this way we carry out fits to $R_i^{k,m}+d_k \,\Delta_i^{k,m}$ and treat the $d_k$ as additional auxiliary fit parameters that are constraint to be of order one (one standard deviation) by adding the term $$\begin{aligned} \label{chi2totald} \chi^2_{\rm corr}(\{d_k\}) & = & \sum_{k=1}^{N_{\rm exp}} \, d_k^2\,,\end{aligned}$$ to the . To implement the non-charm background we assume that the relative energy dependence of the non-charm cross section related to primary production of $u$, $d$ and $s$ quarks is described properly by $R_{uds}$ given in Eq. (\[Rudsdef\]). We then parametrize the non-charm background cross section by the relation $$\begin{aligned} \label{Rncdef} R_{{\rm non}-c\bar c}(E) &= & n_{\rm ns}\,R_{uds}(E) \,,\end{aligned}$$ where the fit parameter $n_{\rm ns}$ is determined mainly from the experimental data in the first clusters below $3.73$ GeV by adding the term $$\begin{aligned} \label{chi2noncharm} \chi^2_{\rm nc}(n_{\rm ns},\{d_k\}) & = & \sum_{k=1}^{N_{\mathrm{exp}}} \sum_{i=1}^{N^{k,1}}\left(\dfrac{ R_i^{k,1}-(1+\Delta f_i^{k,1}\, d_k)\, n_{ns}\, R_{uds}(E_i^{k,1})}{\sigma_i^{k,1}}\right)^2\,,\end{aligned}$$ to the . The complete then has the form $$\begin{aligned} \label{chi2total} \chi^2(\{R_m\}, n_{\rm ns},\{d_k\}) & = & \chi^2_{\rm corr}(\{d_k\}) \, + \, \chi^2_{\rm nc}(n_{\rm ns},\{d_k\})\,+\,\chi^2_{\rm c}(\{R_m\}, n_{\rm ns},\{d_k\}) \,,\end{aligned}$$ where[^12] $$\begin{aligned} \label{chi2charm} &&\chi^2_{\rm c}(\{R_m\}, n_{\rm ns},\{d_k\}) = \\[2mm] &&\sum_{k=1}^{N_{\mathrm{exp}}} \sum_{m=2}^{N_{\mathrm{clusters}}}\sum_{i=1}^{N^{k,m}} \left(\dfrac{R_{i}^{k,m}-(1+\Delta f_i^{k,m}\, d_k)\, (R_m+n_{ns}\, R_{uds}(E_i^{k,m}))}{\sigma_i^{k,m}}\right)^{2} \,.\nonumber\end{aligned}$$ Note that in our approach the non-charm normalization constant $n_{\rm ns}$ is obtained from a combined fit together with the cluster values $R_m$. This form of the is an extended and adapted version of the ones used in Refs. [@D'Agostini:1993uj; @Takeuchi:1995xe]. A special characteristic of the in Eqs. (\[chi2noncharm\]) and (\[chi2charm\]) is that the relative correlated experimental uncertainties $\Delta f_i^{k,m}$ enter the by multiplying the fit value $R_m$ rather than the experimental values $R_i^{k,m}$. This leads to a non-bilinear dependence of the on the $d_m$ and the $R_m$ fit parameters and avoids spurious solutions where the best fit values for the $R_m$ are located systematically below the measurements. Such spurious solutions can arise for data points with substantial positive correlation when with strictly bilinear dependences are employed [@D'Agostini:1993uj; @Takeuchi:1995xe].[^13] We also note that the implementation of the non-charm background subtraction given in Eq. (\[chi2charm\]) leads to a partial cancellation of systematical uncertainties for the $R_m$ best fit values for the charm cross section. Moreover, it is interesting to mention that in the limit where each cluster contains exactly one measurement (except below threshold, in which we always keep one cluster) the decouples, after performing the change of variables $R'_m=R_{m}+n_{ns}\, R_{uds}(E^{k,m})$, into the sum of two independent , one containing data below threshold and depending only on $n_s$ and $d_k$, and another one containing data above threshold $R^{k,m}$ (we drop the label $i$ because having only one data per cluster it can take only the value $1$) and depending only on $R'_m$. After minimizing the first one can obtain the best fit values for $n_s$ and $d_k$, denoted by $n_s^{(0)}$ and $d_k^{(0)}$, respectively. The second $\chi^2$ has a minimal value of $0$ and the best fit parameters read $R_m^{(0)}=R^{k,m}/(1+d_k^{(0)}\Delta^{k,m})-n_s^{(0)}R_{uds}(E^{k,m})$. Close to the minimum the of Eq. (\[chi2total\]) can be written in the Gaussian approximation $$\begin{aligned} \label{chi2Gauss} \chi^2(\{p_i\}) & = & \chi^2_{\rm min}\,+\, \sum_{i,j} (p_i-p_i^{(0)})V_{i,j}^{-1}(p_j-p_j^{(0)}) \, + \, {\cal O}\Big((p-p^{(0)})^3\Big) \,,\end{aligned}$$ where $p_i=( \{R_m\}, n_{\rm ns}, \{d_k\})$ and the superscript $(0)$ indicates the respective best fit value. After determination of the correlation matrix $V_{ij}$ by numerically inverting $V^{-1}_{i,j}$ we can drop the dependence on the auxiliary variables $n_{\rm nc}$ and $d_k$ and obtain the correlation matrix of the $R_m$ from the of $V_{ij}$ which we call $V_{m m^\prime}^R$. In order to separate uncorrelated statistical and systematical uncertainties from correlated systematical ones we compute the complete $V_{m m^\prime}^R$ accounting for all uncertainties and a simpler version of the correlation matrix, $V_{m m^\prime}^{R,u}$ accounting only for uncorrelated uncertainties. The latter is obtained from dropping all correlated errors $\Delta_i^{k,m}$ from the (\[chi2total\]).[^14] The outcome of the fit for the sum of the charm and the non-charm cross section in the threshold and the data continuum region using the standard data set explained above is shown in Figs. \[fig:fitcompilation\](a)-(f) together with the input data sets. The red line segments connect the best fit values and the brown band represents the total uncertainty. The clusters are indicated by dashed vertical lines. For completeness we have also given all numerical results for the $R_m$ values in Appendix A. There we also give results for the minimal and maximal data set selections. Experimental Moments {#subsectionmoments} -------------------- [Narrow resonances]{}\ For the $J/\psi$ and $\psi^\prime$ charmonium contributions to the experimental moments we use the narrow width approximation, $$\begin{aligned} \label{Mnresonances} M_n^{\rm res} & = & \dfrac{9\,\pi\,\Gamma_{ee}}{\alpha(M)^{2}M^{2n+1}}\,,\end{aligned}$$ with the input numbers given in Tab. \[tabpsidata\]. We neglect the tiny uncertainties in the charmonium masses as their effects are negligible. [Threshold and data continuum region]{}\ For the determination of the moment contributions from the threshold and the continuum region between $3.73$ and $10.538$ GeV we use the results for the clustered $c\bar c$ cross section values $R_m$ determined in Sec. \[subsectioncombination\] and the trapezoidal rule. Using the relations ${\rm d}s={\rm d}E^2=2E{\rm d}E$ we thus obtain $$\begin{aligned} \label{Mndata} M_n^{\rm thr+cont} & = &\sum_{i=1}^{N_{\mathrm{clusters}}}\dfrac{R_{i}}{E_{i}^{2n+1}}(E_{i+1}-E_{i-1})+\dfrac{R_{0}}{E_{0}^{2n+1}}(E_{1}-E_{0})\\ &+&\dfrac{R_{N_{\mathrm{clusters}}+1}}{E_{N_{\mathrm{clusters}}+1}^{2n+1}}(E_{N_{\mathrm{clusters}}+1}-E_{N_{\mathrm{clusters}}})\,,\nonumber\end{aligned}$$ where $R_0$ and $E_0$ are the $R$ and energy values at the lower boundary of the smallest energy cluster, and $R_{N_{cl}+1}$ and $E_{M_{cl}+1}$ are the corresponding values of the upper boundary of the highest energy cluster. The values for $R_0$ and $R_{N_{cl}+1}$ are obtained from linear extrapolation using the $R_m$ values of the two closest lying clusters[^15] $m^\prime$ and $m^\prime+1$ with the formula $$\begin{aligned} \label{REextrapol} R(E) & = & \frac{R_{m^\prime+1}-R_{m^\prime}}{E_{m^\prime+1}-E_{m^\prime}}\, (E-E_{m^\prime})+ R_{m^\prime} \,.\end{aligned}$$ For the computation of the moment contributions from subintervals within the range between $3.73$ and $10.538$ GeV we also use Eq. (\[Mndata\]) using corresponding adaptations for the boundary values at $m=0$ and $m=N_{cl}+1$. [Perturbative QCD region]{}\ For the region above $10.538$ GeV where we use the perturbative QCD input described in Eqs. (\[Rcchighdef1\]) and (\[Rnsdef1\]) for the charm the contribution to the experimental moment is obtained from the defining equation (\[momentdef2\]) with a lower integration limit of $10.538$ GeV: $$\begin{aligned} \label{MnQCD} M_n^{\rm QCD} & = & \gamma \times\int_{(10.538\,\rm GeV)^2}^\infty {\rm d}s \,\dfrac{R_{cc}^{\rm th}(s)}{s^{n+1}} \,.\end{aligned}$$ The variable $\gamma$ is an auxiliary variable used to parametrize the $10\%$ uncertainty we assign to the perturbative QCD contribution, $$\begin{aligned} \label{gammadef} \gamma & = & 1.0 \,\pm \, 0.1 \,.\end{aligned}$$ [Correlations]{}\ The experimental moments are obtained from the sum of the resonance, threshold plus continuum and perturbative QCD contributions described just above, $$\begin{aligned} \label{Mnsum} M_n^{\rm exp} & = & M_n^{\rm res} \, + \, M_n^{\rm thr+cont} \, + \, M_n^{\rm QCD} \,.\end{aligned}$$ To determine the uncertainties we account for the errors in the $e^+e^-$ widths of $J/\psi$ and $\psi^\prime$ and in the cluster values $R_m$, and for the $10\%$ assigned uncertainty in $M_n^{\rm QCD}$. For the evaluation we use the usual error propagation based on a $\bar m\times\bar m$ correlation matrix with $\bar m=N_{cl}+3$. The correlation matrix of the experimental moments thus has the form $$\begin{aligned} \label{CMcorr} C_{nn^\prime}^{\rm exp} & = & \sum\limits_{i,j=1}^{N_{cl}+3} \Big(\frac{\partial M_n^{\rm exp}}{\partial \bar R_i}\Big)\, \Big(\frac{\partial M_{n^\prime}^{\rm exp}}{\partial \bar R_j}\Big) \, V_{ij}^{\bar R} \,,\end{aligned}$$ where we have $\bar R_i = (\{R_m\},\Gamma_{e^+e^-}(J/\psi),\Gamma_{e^+e^-}(\psi^\prime),\gamma)$. The entries of $V^{\bar R}$ in the $R_m$ subspace are just the entries of the correlation matrix $V^R$ obtained from the cluster fit. The diagonal entries in the $\Gamma_{e^+e^-}$ subspace are the combined statistical and systematical uncertainties of the $e^+e^-$ widths and the $\delta\gamma=0.1$ for $M_n^{\rm QCD}$, respectively. We treat the latter uncertainty as uncorrelated with all other uncertainties. So all non-diagonal entries of $V_{ij}^{\bar R}$ for $i$ or $j=N_{cl}+3$ are zero. For the uncertainty of the $e^+e^-$ widths we adopt a model where the (quadratic) half of the error is uncorrelated and the other (quadratic) half is positively correlated among the two narrow resonances and to the $R_n$ cluster values as well. Thus for the corresponding non-diagonal entries of $V_{i,j}^{\bar R}$ with $i\in\{1,N_{cl}\}$ and we have the entries $V_{ij}^{\bar R}=\delta\Gamma^{1,2}_{e^+e^-}\Delta R_m/\sqrt{2}$, and for $i=N_{cl}+1$ and $j=N_{cl}+2$ we have $V_{ij}^{\bar R}=\delta\Gamma^1_{e^+e^-}\delta\Gamma^2_{e^+e^-}/2$, where $\delta\Gamma^{1,2}_{e^+e^-}$ are the respective $e^+e^-$ width total uncertainties for $J/\psi$ and $\psi^\prime$, respectively, and $\Delta R_m$ the correlated uncertainty of the respective cluster $R_m$ value. The results for the moments showing separately the contributions from the resonances, various energy subintervals and their total sum using the defaults data set collection (see Sec. \[subsectioncollections\]) are given in Tab. \[tab:momres1\]. Using Eq. (\[CMcorr\]) it is straightforward to compute the correlation matrix of the moments, and we obtain $$\begin{aligned} \label{CMtotal} C^{\rm exp} & = &\left( \begin{array}{cccc} 0.250 & 0.167 & 0.147 & 0.142\\ 0.167 & 0.120 & 0.107 & 0.103\\ 0.147 & 0.107 & 0.095 & 0.092\\ 0.142 & 0.103 & 0.092 & 0.090 \end{array} \right)\,,\end{aligned}$$ for the total correlation matrix accounting for all correlated and uncorrelated uncertainties. We remind the reader that all numbers related to the moment $M_n^{\rm exp}$ are given in units of $10^{-(n+1)}\,\mbox{GeV}^{-2n}$. To quote correlated and uncorrelated uncertainties separately it is also useful to show the correlation matrix that is obtained when only uncorrelated uncertainties are accounted for. The result it $$\begin{aligned} \label{CMuncorr} C^{\rm exp}_{\rm uc} & = & \left( \begin{array}{cccc} 0.041 & 0.035 & 0.034 & 0.034\\ 0.035 & 0.034 & 0.034 & 0.035\\ 0.034 & 0.034 & 0.035 & 0.036\\ 0.034 & 0.035 & 0.036 & 0.037 \end{array}\right) \,.\end{aligned}$$ These results can be used to carry out combined simultaneous fits to several of the moments. This is, however, not attempted in this work. Examination {#subsectionexaminations} ----------- We conclude this section with an examination of some of the choices and assumptions we have implemented for the treatment of the experimental uncertainties. Our defaults choices include - treating one (quadratic) half of $J/\psi$ and $\psi^\prime$ $e^+e^-$ partial width uncertainties as uncorrelated and the other half as positively correlated among themselves and with the correlated uncertainties of the $R_m$ cluster values; - treating the entire systematical uncertainties of the measurements as correlated for the data sets where only total systematical uncertainties were quoted; - defining the cluster energies $E_m$ through the weighed average of measurement energies falling into the clusters, see Eq. (\[Eclusterdef\]); - using $N_{cl}=52+1$ clusters distributed in groups of (2,20,20,10) clusters in the energy ranges bounded by $(3.73, 3.75, 3.79, 4.55, 10.538)$ GeV (see Sec. \[subsectioncombination\]) and - using the default data set collection consisting of all data sets discussed in Sec. \[subsectioncollections\] except for sets 16, 17 and 19 as defined in Tab. \[tab:datasets\]. In Tab. \[tab:compcorr1\] alternative correlation models are being studied. The second column shows, as a reference, the first four moments with our default settings. The third and fourth columns show the moments treating the uncertainties of the $J/\psi$ and $\psi^\prime$ partial widths being minimally correlated with the $R_m$ values[^16] and completely uncorrelated, respectively. In the fifth column we display the moments treating, for data sets 1, 6, 13, 15, 16 and 17, one (quadratic) half of systematical uncertainties for the uncorrelated and the other half correlated. In the sixth column all $R_m$ values of all data sets are treated as completely uncorrelated. We see that the central values depend only weakly on the correlation model for those data were the corresponding information is unknown. In particular, for the determination of the uncertainties the ignorance about the composition of the systematical uncertainties in the from data sets 1, 6, 13, 15, 16 and 17 is not essential. However, for quoting the final uncertainties it is important to account for all (known) correlations since treating all errors uncorrelated underestimates the uncertainties considerably. In Tab. \[tab:compcorr2\] we examine the impact of modifying the definition of the cluster energy $E_m$ and of changes to the default cluster distribution (2,20,20,10). In the second column we display the resulting moments of the default setting. In the third and fourth columns we have shown the moments using for $E_m$ simply the mean of the energies and the center of the cluster, respectively. The resulting differences to the default definition is an order of magnitude smaller than the uncertainties and thus negligible. The fifth, sixth and seventh columns display the moments using some alternative cluster distributions. The deviations for the default choice illustrated in the table are much smaller than the uncertainties and typical for all modifications that satisfy the guidelines for viable cluster definitions we have formulated in Sec. \[subsectioncombination\]. This demonstrates that the choice of the cluster distribution does not result in a bias for the resulting experimental moments. Finally, we also examine the dependence of the moments on the data set collections as described in Sec. \[subsectioncollections\]. In Tabs. \[tab:moments-results-minimal\] and \[tab:moments-results-maximal\] the results for the moments are displayed using the minimal and the maximal collections (with default choices for all other settings). We see that the differences in the central values to the default collection are about half the size of the systematical correlated uncertainties for the first moment $M_1^{\rm exp}$. For the higher moments the differences are much smaller than the uncertainties. Using, instead of the default, the minimal and maximal collections affects the uncertainties of $M_1^{\rm exp}$ by only about $10\%$. For the higher moments the differences decrease strongly and basically disappear for the fourth moment. Again, the results show that having an increased or decreased redundancy in the data set collection only has a minor impact on the final numbers for the experimental moments. To summarize, we find that modifications to the choices and assumptions that go into the combined treatment of the experimental data from different publications and experiments lead to changes that are well within the experimental uncertainties we obtain from our combination method. We therefore consider these uncertainties as appropriate. An instructive comparison of the moments obtained in our analysis to those obtained in some previous publications is given in Tab. \[tab:mom-comparison\]. A graphical illustration of the results is shown in Fig. \[fig:Moment-comparison\]. Charm Quark Mass Analysis {#sectionanalysis} ========================= $\alpha_s(m_Z)$ ${\overline m_c}({\overline m_c})$ $\Delta_{\rm pert}$ $\Delta_{\rm stat}$ $\Delta_{\rm syst}$ ${\overline m_c}(3\,{\rm GeV})$ $\Delta_{\rm pert}$ $\Delta_{\rm stat}$ $\Delta_{\rm syst}$ ----------------- ------------------------------------ --------------------- --------------------- --------------------- --------------------------------- --------------------- --------------------- --------------------- $0.113$ $1.255$ $0.013$ $0.006$ $0.013$ $1.010$ $0.013$ $0.006$ $0.014$ $0.114$ $1.259$ $0.014$ $0.006$ $0.013$ $1.006$ $0.015$ $0.006$ $0.014$ $0.115$ $1.263$ $0.015$ $0.006$ $0.013$ $1.002$ $0.016$ $0.006$ $0.014$ $0.116$ $1.267$ $0.016$ $0.006$ $0.013$ $0.999$ $0.017$ $0.006$ $0.014$ $0.117$ $1.271$ $0.017$ $0.006$ $0.013$ $0.995$ $0.019$ $0.006$ $0.014$ $0.118$ $1.275$ $0.019$ $0.006$ $0.013$ $0.990$ $0.020$ $0.006$ $0.014$ $0.119$ $1.280$ $0.020$ $0.006$ $0.013$ $0.986$ $0.022$ $0.006$ $0.014$ : Results for the central values of ${\overline m_c}({\overline m_c})$ and ${\overline m_c}(3\,{\rm GeV})$ (second and sixth columns) and their perturbative (third and seventh columns), experimental statistical (fourth and eight columns) and experimental systematical (fifth and ninth columns) errors.[]{data-label="tabmcalphas"} Since it is theoretically most reliable, we use the first moment $M_1$ for our final numerical charm quark mass analysis. As ingredients for the analysis we use - the iterative expansion method for the perturbative contribution of the theoretical moment at ${\cal O}(\alpha_s^3)$, see Eq. (\[eq:iterative-general\]), - the gluon condensate correction with its Wilson coefficient determined at ${\cal O}(\alpha_s)$ as described in Sec. \[subsectioncondensate\], - the first experimental moment $$\begin{aligned} \label{M1exp} M_1^{\rm exp} & = & 0.2138\pm 0.0020_{\rm stat}\pm 0.0046_{\rm syst}\end{aligned}$$ using our default settings as discussed in Sec. \[sectiondata\]. One important source of uncertainty we have not yet discussed is the value of the strong $\overline{\mbox{MS}}$ coupling $\alpha_s$. Since in the recent literature [@Abbate:2010xh; @Frederix:2010ne; @Gehrmann:2009eh; @Blumlein:2006be; @Blumlein:2010rn] $\alpha_s$ determinations with a spread larger than the current world average [@Bethke:2009jm] have been obtained, we carry out our numerical analysis for values of $\alpha_s(m_Z)$ between $0.113$ and $0.119$.[^17] The outcome of our analysis is shown in Tab. \[tabmcalphas\]. In Fig. \[figmcalphas\] the central values, perturbative, statistical and systematical uncertainties are displayed graphically. For the central value and the perturbative uncertainty, which show a significant dependence on $\alpha_s$, we can present a linear fit. For the statistical and systematical uncertainties the variation with $\alpha_s$ is smaller than $1$ MeV and we only quote constant values. We thus obtain $$\begin{aligned} \label{mcfinalalphas} \overline m_c(\overline m_c)& = &(0.788 + 4.13\times\alpha_s(m_Z)) \, \pm \, (0.006)_{\rm stat} \, \pm \, (0.013)_{\rm syst}\\ && \, \pm \, (-0.127+1.23\times\alpha_s(m_Z))_{\rm pert} \, \pm \, (0.002)_{\langle GG\rangle} \,,\nonumber \\[2mm] \overline m_c(3~\mbox{GeV})& = &(1.462 - 4.00\times\alpha_s(m_Z)) \, \pm \, (0.006)_{\rm stat} \, \pm \, (0.014)_{\rm syst}\\ && \, \pm \, (-0.148+1.43\times\alpha_s(m_Z))_{\rm pert} \, \pm \, (0.002)_{\langle GG\rangle}\,.\nonumber\end{aligned}$$ Taking as an input $$\begin{aligned} \label{alphaswa} \alpha_s(m_Z) & = & 0.1184 \pm 0.0021\,,\end{aligned}$$ which is the current world Bethke average [@Bethke:2009jm] with a tripled uncertainty we obtain $$\begin{aligned} \label{mcfinalalphaswa} \overline m_c(\overline m_c) & = & 1.277 \, \pm \, (0.006)_{\rm stat} \, \pm \, (0.013)_{\rm syst} \, \pm \, (0.019)_{\rm pert} \, \pm \, (0.009)_{\alpha_s} \, \pm \, (0.002)_{\langle GG\rangle} \,,\nonumber \\[2mm] \overline m_c(3~\mbox{GeV}) & = & 0.998 \, \pm \, (0.006)_{\rm stat} \, \pm \, (0.014)_{\rm syst} \, \pm \, (0.022)_{\rm pert} \, \pm \, (0.010)_{\alpha_s} \, \pm \, (0.002)_{\langle GG\rangle} \,,\nonumber\\\end{aligned}$$ which represents, together with Eq. (\[mcfinalalphas\]), our final numerical result for the $\overline{\mbox{MS}}$ charm mass. Our result is in good agreement with other recent determinations of $\overline{m}_{c}(\overline{m}_{c})$. A summary of the numerical results is shown in Tab. \[tab:comparison-alpha\] and in Fig. \[fig:comparison-alpha\]. Compared to the analysis carried out in Refs. [@Chetyrkin:2009fv; @Kuhn:2007vp] our experimental uncertainty is larger by $5$ MeV and our perturbative uncertainty is a factor of $10$ larger. Compared to Refs. [@Chetyrkin:2006xg; @Boughezal:2006px] the discrepancy in the perturbative error estimate is even larger. We note that the almost head on agreement in the central value of Refs. [@Chetyrkin:2009fv; @Kuhn:2007vp] is accidental, since the central value we obtain is based on a different perturbative expansion (iterative expansion in our work versus fixed-order expansion in theirs), on the way how the central value is defined (average of the lowest and highest values in our work versus value for $\mu=3$ GeV) and on a different central value for the experimental moment ($M_1^{\rm exp}=0.2138$ in our work versus $M_1^{\rm exp}=0.2166$ in theirs). Whereas our definition of the central value makes $\overline m_c$ smaller, our smaller value for $M_1^{\rm exp}$ makes it bigger by almost the same amount. ![ Comparison of recent determinations of $\overline{m}_{c}(\overline{m}_{c})$. Red corresponds to our result, black and gray correspond to ${\mathcal O}\alpha_s^3)$ and ${\mathcal O}(\alpha_s^2)$ charmonimum sum rules analyses, respectively, green labels other kind of sum rule analyses (weighted finite energy sum rules [@Bodenstein:2010qx] and ratios of moments [@Narison:2010cg]), and blue stands for lattice simulations. \[fig:comparison-alpha\] ](alpha-comparison.eps){width="90.00000%"} Conclusions and Final Thoughts {#sectionconclusions} ============================== In this work we have used state-of-the-art ${\cal O}(\alpha_s^3)$ input from perturbative QCD to determine the $\overline{\mbox{MS}}$ charm quark mass from relativistic ($n=1$) charmonium sum rules using experimental data on the total hadronic cross section in $e^+e^-$ annihilation. The main aims were (i) to carefully reexamine perturbative uncertainties in the charm mass extractions from the moments of the charm vector current correlator and (i) to fully exploit the available experimental data on the hadronic cross section. We carried out this work having in mind recent ${\mathcal O}(\alpha_s^3)$ sum rule analyses [@Chetyrkin:2006xg; @Boughezal:2006px; @Chetyrkin:2009fv; @Kuhn:2007vp], where perturbative errors of $2$ MeV or smaller were quoted. Moreover in Refs. [@Chetyrkin:2009fv; @Kuhn:2007vp] an experimental uncertainty of $9$ MeV was quoted. Given these numbers we found it appropriate to reexamine their analysis. In their work the perturbative uncertainty estimate was achieved using a specific choice to arrange the perturbative expansion and by setting the renormalization scales in $\alpha_s$ ($\mu_\alpha$) and of the $\overline{\mbox{MS}}$ charm quark mass ($\mu_m$) equal. We demonstrated that this results in an accidental cancellation of $\mu_\alpha$ and $\mu_m$ scale variations that is not observed in other alternative ways to treat the perturbative expansion. Moreover, concerning the experimental input their work relied on perturbative QCD predictions instead of available data for energies $\sqrt{s}>4.8$ GeV which results in a rather strong dependence of the final experimental uncertainty on the model-uncertainty assigned to the theory input. Concerning a proper assessment of perturbative and experimental uncertainties we established in our work the following improvements: - We demonstrated that for achieving an estimate of perturbative uncertainties based on scale variations that is independent of the perturbative expansion method one needs to vary $\mu_\alpha$ and $\mu_m$ independently, albeit with ranges that avoid large logs. As a result the perturbative uncertainty estimates using different ways to carry out the expansion in $\alpha_s$ become equivalent, which is not the case for $\mu_\alpha=\mu_m$. - Using a data clustering method similar to Refs. [@D'Agostini:1993uj; @Takeuchi:1995xe; @Hagiwara:2003da] we combined available data on the total $e^+e^-$ hadronic cross section from many different experiments covering energies up to $\sqrt{s}=10.538$ GeV to fully exploit the existing experimental information for the experimental moments. This avoids a strong dependence of the experimental moments on ad-hoc assumptions on the “experimental” model uncertainty being associated to the QCD theory input used for the experimental moments. As a result we were also able to quantify the correlation between different experimental moments. Using $\alpha_s(m_Z)$ as an unspecified variable and the theoretically more reliable first moment $M_1$ for the fits we have obtained $$\begin{aligned} \overline m_c(\overline m_c)& = &(0.788 + 4.13\times\alpha_s(m_Z)) \, \pm \, (0.006)_{\rm stat} \, \pm \, (0.013)_{\rm syst}\\ && \, \pm \, (-0.127+1.23\times\alpha_s(m_Z))_{\rm pert} \, \pm \, (0.009)_{\alpha_s} \, \pm \, (0.002)_{\langle GG\rangle} \,,\nonumber \\[2mm] \overline m_c(3~\mbox{GeV})& = &(1.462 - 4.00\times\alpha_s(m_Z)) \, \pm \, (0.006)_{\rm stat} \, \pm \, (0.014)_{\rm syst}\\ && \, \pm \, (-0.148+1.43\times\alpha_s(m_Z))_{\rm pert} \, \pm \, (0.002)_{\langle GG\rangle} \,,\nonumber\end{aligned}$$ for the $\overline{\mbox{MS}}$ charm mass. At the level of uncertainties obtained in our work excellent convergence of perturbation theory was observed. Adopting $\alpha_s(m_Z) = 0.1184 \pm 0.0021$ we then obtain $$\begin{aligned} \overline m_c(\overline m_c) & = & 1.277 \, \pm \, (0.006)_{\rm stat} \, \pm \, (0.013)_{\rm syst} \, \pm \, (0.019)_{\rm pert} \, \pm \, (0.009)_{\alpha_s} \, \pm \, (0.002)_{\langle GG\rangle} \,,\nonumber \\[2mm] \overline m_c(3~\mbox{GeV}) & = & 0.998 \, \pm \, (0.006)_{\rm stat} \, \pm \, (0.014)_{\rm syst} \, \pm \, (0.022)_{\rm pert} \, \pm \, (0.010)_{\alpha_s} \, \pm \, (0.002)_{\langle GG\rangle} \,.\nonumber\\\end{aligned}$$ Our perturbative error of $20$ MeV is a factor of ten larger, and the experimental uncertainty of $14$ MeV is by $5$ MeV larger than in the most recent analysis of Ref. [@Kuhn:2007vp]. For estimating the perturbative error a range of scale variations between $\overline m_c(\overline m_c)$ and $4$ GeV was employed. Adding all uncertainties quadratically we obtain $$\begin{aligned} \label{conclusionmc} \overline m_c(\overline m_c) & = & 1.277 \, \pm \, 0.026~\mbox{GeV} \,,\nonumber \\[2mm] \overline m_c(3~\mbox{GeV}) & = & 0.998\,\pm \, 0.029~\mbox{GeV} \,,\end{aligned}$$ giving an uncertainty that is twice the size of the one obtained in Refs. [@Chetyrkin:2009fv; @Kuhn:2007vp]. This difference arises mainly from the more appropriate estimate of perturbative uncertainties obtained in this work. The agreement in the central value with Refs. [@Chetyrkin:2009fv; @Kuhn:2007vp] is a coincidence, arising from an interplay of differences in the experimental input and theoretical treatment. As a final thought one might ask which further improvements might be possible in the future. As can be seen from Tab. \[tab:momres1\], from the experimental side the biggest improvement could be made from more accurate measurements of the $J/\psi$ and $\psi^\prime$ electronic partial widths. The current relative uncertainties are $2.5$% and $2.4$%, respectively. Here some improvement might be conceivable with dedicated measurements. On theory side, viewing the uncertainties, it is not unreasonable to assume that the computation of ${\cal O}(\alpha_s^4)$ moments of the vector current correlator might further reduce the perturbative error below the level of $20$ MeV. Using the $\overline{\mbox{MS}}$ scheme for the charm mass the OPE states that the remaining perturbative infrared renormalon ambiguity is of order $\Lambda_{\rm QCD}^4/\overline m_c^3\sim {\cal O}(5 \mbox{\,-}15~\mbox{MeV})$. This expectation has also been confirmed by explicit bubble chain (large-$\beta_0$) computations [@Grozin:2004ez] and indicates that a further reduction of the perturbative uncertainty is not excluded. However, to throw in some words of caution, at the level of the present perturbative uncertainties one should also remind oneself about possible loopholes still left in the charmonium sum rule method. An issue we would like to mention concerns the separation of charm and non-charm hadronic production rates needed to carry out the charmonium sum rule. On the theory side the issue is conceptually subtle due to the singlet and secondary radiation contributions which arise at ${\cal O}(\alpha_s^3)$ and ${\cal O}(\alpha_s^2)$, respectively. In this work (as well as in Ref. [@Chetyrkin:2009fv]) both contributions have been considered as non-charm although they contain terms belonging to the $c\bar c$ final state. This treatment might be justified since the size of the corresponding terms is quite small (see Tab. \[tabneglected\]) and since it is the common approach to determine the experimental charm production rate in the continuum region by subtracting theoretical results (or models) for the non-charm rate from the measured total hadronic . In our method to determine the experimental moments this subtraction involves a normalization constant multiplying the theoretical non-charm that is fitted within our clustering method as well accounting for data below and above the charm threshold. The result (see Eq. (\[eq:ns-results\])) reveals a disparity of $4$% between the theoretical non-charm and the data. Setting, in contrast, the normalization constant to unity results in a shift in the charm mass by $-15$ MeV.[^18] Since this shift is compatible with the overall systematical uncertainty in the experimental data we have not treated it as an additional source of uncertainty. On the other hand, the size of the shift could also be considered as an inherent conceptual uncertainty related to separating the charm from the non-charm , which is based on theory considerations rather than on experimental methods and which apparently cannot be improved simply by higher order perturbative computations. As an alternative, one might avoid the separation of the charm and the non-charm contributions altogether and use the total hadronic cross section for the charm mass fits. Apart from the shift mentioned above such an approach would, however, also lead to a substantially larger dependence on the uncertainties in $\alpha_s$. Given these considerations we believe that a substantial reduction of the uncertainties also relies on a resolution of the disparity mentioned above. This might certainly involve more precise measurements in the charm threshold and below-threshold regions, but also some deeper conceptual insight. Until then a substantial reduction of the uncertainties shown in Eq. (\[conclusionmc\]) appears hard to achieve without imposing ad-hoc assumptions. This work was supported in part by the European Community’s Marie-Curie Research Networks under contract MRTN-CT-2006-035482 (FLAVIAnet), MRTN-CT-2006-035505 (HEPTOOLS) and PITN-GA-2010-264564 (LHCphenOnet), and by the U.S. Department of Energy under the grant FG02-94ER40818. V. Mateu has been partially supported by a DFG “Eigenen Stelle” under contract MA 4882/1-1 and by a Marie Curie Fellowship under contract PIOF-GA-2009-251174. S. Zebarjad thanks the MPI for hospitality while part of this work was accomplished. S. Zebarjad and V. Mateu are grateful to the MPI guest program for partial support. We thank S. Schutzmeier for confirmation of our numerical ${\cal O}(\alpha_s^3)$ fixed-order results. A. Hoang acknowledges discussion with H. Kühn and C. Sturm. Appendix A: Results of the Fit Procedure {#appendix-a-results-of-the-fit-procedure .unnumbered} ======================================== \[ap:FitOutcome\] In this appendix we present in some more detail the numerical results of our fit procedure. In Tabs. \[tab:FitFunction-Standard\], \[tab:FitFunction-Minimal\] and \[tab:FitFunction-Maximal\], the results for the cluster energies and the cluster charmed are shown for the standard, minimal and maximal selection of data sets, respectively, using our default setting for the correlations. We use the results for the standard data set selection for our final charm mass analysis. The numbers in the parentheses correspond to the statistical and systematical errors. The correlation matrices for the is available, but cannot be displayed due to lack of space. They can be obtained by the authors on request. For the three data selections, the fit gives the following minimal $\chi^2$ per degree of freedom, $$\dfrac{\chi_{\rm standard}^2}{\rm dof}=1.89\,, \qquad \dfrac{\chi_{\rm minimal}^2}{\rm dof}=1.86\,, \qquad \dfrac{\chi_{\rm maximal}^2}{\rm dof}=1.81\,, \label{eq:chi2-results}$$ and the following normalization constants for the non-charm background $$\begin{aligned} \!\!\!\!\!\!n^{\rm standard}_s&=&1.039 \pm 0.003_{\rm stat} \pm 0.012_{\rm syst},\; n^{\rm minimal}_s=1.029 \pm 0.003_{\rm stat} \pm 0.015_{\rm syst}, \label{eq:ns-results} \\ \!\!\!\!\!\!n^{\rm maximal}_s&=&1.023 \pm 0.003_{\rm stat} \pm 0.011_{\rm syst}.\nonumber\end{aligned}$$ Since the minimal $\chi^2/{\rm dof}$ values are not close to unity, one has to conclude that fit quality is not really very good. This is not at all visible from the agreement of the fit and the data for the total cross section (see Figs. \[fig:fitcompilation\]) and thus might be related to the disparity between the fits of charm versus non-charm production rates described in Sec. \[sectionconclusions\]. In Eq. (\[correlmatrices\]) we show for the correlation matrices of the first four experimental moments for the minimal and the maximal data set selection. The results for our standard selection are given in Eqs. (\[CMtotal\]) and (\[CMuncorr\]). All numbers are related to moments $M_n^{\rm exp}$ normalized to units of $10^{-(n+1)}\,\mbox{GeV}^{-2n}$. We show the results accounting for the full set of correlated and uncorrelated uncertainties and the correlation matrices accounting only for uncorrelated systematical and statistical uncertainties (subscript uc). $$\begin{aligned} \label{correlmatrices} C^{\rm exp}_{\rm min}&=&\left( \begin{array}{cccc} 0.304 & 0.194 & 0.165 & 0.157\\ 0.194 & 0.133 & 0.115 & 0.109\\ 0.165 & 0.115 & 0.100 & 0.096\\ 0.157 & 0.109 & 0.096 & 0.092\\ \end{array} \right)\,,\quad C^{\rm exp}_{\rm uc}\,=\,\left( \begin{array}{cccc} 0.047 & 0.038 & 0.035 & 0.035\\ 0.038 & 0.035 & 0.035 & 0.035\\ 0.035 & 0.035 & 0.035 & 0.036\\ 0.035 & 0.035 & 0.036 & 0.037 \end{array}\right)\,,\\\nonumber\\ C^{\rm exp}_{\rm max}&=&\left( \begin{array}{cccc} 0.193 & 0.138 & 0.125 & 0.122\\ 0.138 & 0.108 & 0.099 & 0.097\\ 0.125 & 0.099 & 0.091 & 0.090\\ 0.122 & 0.097 & 0.090 & 0.088\\ \end{array} \right)\,,\quad C^{\rm exp}_{\rm uc}\,=\,\left( \begin{array}{cccc} 0.037 & 0.034 & 0.033 & 0.034\\ 0.034 & 0.033 & 0.034 & 0.035\\ 0.033 & 0.034 & 0.035 & 0.036\\ 0.034 & 0.035 & 0.036 & 0.037 \end{array}\right)\,.\nonumber\end{aligned}$$ Appendix B: On the Equivalence of $\chi^2$-Functions {#appendix-b-on-the-equivalence-of-chi2-functions .unnumbered} ==================================================== In this appendix we demonstrate that a in which the auxiliary fit parameters $d_k$, which describe the correlated deviation off the experimental central value within experiment $k$, multiplies only the experimental systematical uncertainties,[^19] $$\chi^2 = \sum_k\left[d_k^2+\sum_{i,m}\left(\dfrac{R_i^{k,m}+d_k\, \Delta_i^{k,m}-R_m}{\sigma_i^{k,m}}\right)^2\right]\,, \label{additive}$$ is mathematically equivalent to the well known written solely in terms of the fit parameters $R_m$ and a correlation matrix, $$\bar \chi^2=\sum_{i,j,k,m,n}(R_i^{k,m}-R_m)(V_k^{-1})_{ij}^{mn}(R_j^{k,n}-R_n)\,, \label{traditional}$$ where there is no correlation among the different experiments $k$ and $k^\prime$, and the correlation matrix within one experiment $k$ has the form $$(V_k)_{ij}^{mn}=\sigma_i^{k,m\,2}\delta_{ij}\delta^{mn}+\Delta_i^{k,m}\Delta_j^{k,n} \,. \label{tradcorr}$$ The matrix Eq. (\[tradcorr\]) can be inverted analytically giving $$(V_k^{-1})_{ij}^{mn} = \dfrac{\delta_{ij}\delta^{mn}}{(\sigma_i^{k,m})^2}- \dfrac{1}{1+\dfrac{\left\langle \Delta^2\right\rangle_k}{\left\langle \sigma\right\rangle_k^2}} \dfrac{\Delta_i^{k,m}}{(\sigma_i^{k,m})^2}\dfrac{\Delta_j^{k,n}}{(\sigma_j^{k,n})^2}\,, \label{invtradcorr}$$ where we have defined the mean statistical error and the statistical average of the systematical error within one experiment as follows: $$\left\langle \sigma\right\rangle_k^2 \equiv \left(\sum_{i,m}\dfrac{1}{(\sigma_i^{k,m})^2}\right)^{-1},\qquad \left\langle \Delta^2 \right\rangle_k\equiv\left\langle \sigma\right\rangle_k^2\sum_{i,m} \dfrac{(\Delta_i^{k,m})^2}{(\sigma_i^{k,m})^2}\,. \label{averages}$$ In case of a sizable positive correlation between measurements in the same experiment (such that $\Delta_i^{k,m}/R_i^{k,m}$ is sizable and approximately constant) is it known that the form of the in Eq. (\[additive\]) leads to best fit values $R_m$ that are systematically below the measurements [@D'Agostini:1993uj]. Our proof demonstrates that the standard of Eq. (\[traditional\]) has the same property for correlation matrices with the form of Eq. (\[tradcorr\]). This motivates to use the so-called minimal-overlap correlation model where the second term on the RHS of Eq. (\[traditional\]) is replaced by $\mbox{min}^2(\Delta_i^{k,m},\Delta_j^{k,n})$. In general, within the minimal-overlap model, the correlations are sufficiently reduced such that the unphysical effect described above does not arise. We proceed by showing that one can “integrate out" auxiliary parameters $d_k$ in Eq. (\[additive\]) obtaining a new function $\tilde \chi^2(R^i)$ which yields the same results for the best fit for the $R_m$, as long as one works in the Gaussian approximation. The minimum of $\chi^2(R_i,d_k)$ is located at the best fit values (indicated by superscripts $(0)$) defined by the conditions $$\left.\dfrac{\partial\chi^{2}}{\partial R_{i}}\right\|_{R_i^{(0)},d_k^{(0)}}=\left.\dfrac{\partial\chi^{2}}{\partial d_{j}}\right\|_{R_i^{(0)},d_k^{(0)}}=0\,.\label{Mincond}$$ To invert the matrix of second derivatives we proceed in blocks $$\left(\begin{array}{cc} \dfrac{\partial^{2}\chi^{2}}{\partial R_{i}\partial R_{j}} & \dfrac{\partial^{2}\chi^{2}}{\partial R_{i}\partial d_{k}}\\ \dfrac{\partial^{2}\chi^{2}}{\partial d_{l}\partial R_{j}} & \dfrac{\partial^{2}\chi^{2}}{\partial d_{l}\partial b_{k}}\end{array}\right)_{\tilde{R},\tilde{b}}\left(\begin{array}{cc} c_{jm} & b_{jr}\\ b_{mk} & a_{kr}\end{array}\right)=\left(\begin{array}{cc} \delta_{im} & 0\\ 0 & \delta_{lr}\end{array}\right).$$ where $c_{ij}$ and $a_{kr}$ are $N_{\rm cluster}\times N_{\rm cluster}$ and $N_{\rm exp}\times N_{\rm exp}$ square matrices, respectively, and $b_{jr}$ is a $N_{\rm cluster}\times N_{\rm exp}$ rectangular matrix. We find the following four matrix relations $$\begin{aligned} &&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \sum_{j}\dfrac{\partial^{2}\chi^{2}}{\partial R_{i}\partial R_{j}}\, c_{jm}+\sum_{k}\dfrac{\partial^{2}\chi^{2}}{\partial R_{i}\partial d_{k}}\, b_{mk}=\delta_{im}\,, \quad \sum_{j}\dfrac{\partial^{2}\chi^{2}}{\partial R_{i}\partial R_{j}}\, b_{jr}+\sum_{k}\dfrac{\partial^{2}\chi^{2}}{\partial R_{i}\partial d_{k}}\, a_{kr}=0\,, \label{eq:general_block_1}\\ &&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \sum_{j}\dfrac{\partial^{2}\chi^{2}}{\partial d_{l}\partial R_{j}}\, c_{jm}+\sum_{k}\dfrac{\partial^{2}\chi^{2}}{\partial d_{l}\partial d_{k}}\, b_{mk}=0\,, \qquad\;\, \sum_{j}\dfrac{\partial^{2}\chi^{2}}{\partial d_{l}\partial R_{j}}\, b_{jr}+\sum_{k}\dfrac{\partial^{2}\chi^{2}}{\partial d_{l}\partial d_{k}} a_{kr}=\delta_{lr}\,.\label{eq:general_block_2}\end{aligned}$$ Combining Eqs. (\[eq:general\_block\_1\]a) and (\[eq:general\_block\_2\]a) we find the inverse of the upper left , $$(c^{-1})_{ij}=2(V^{-1})_{ij}=\dfrac{\partial^{2}\chi^{2}}{\partial R_{i}\partial R_{j}}-\sum_{kl}\dfrac{\partial^{2}\chi^{2}}{\partial d_{l}\partial R_{i}}\left[\dfrac{\partial^{2}\chi^{2}}{\partial d_{l}\partial d_{k}}\right]^{-1}\dfrac{\partial^{2}\chi^{2}}{\partial d_{k}\partial R_{j}}\,, \label{eq:noEOM_Rj_bk}$$ where $\left[\dfrac{\partial^{2}\chi^{2}}{\partial d_{l}\partial d_{k}}\right]^{-1}$ stands for the $(l,k)$ element of the inverse matrix of $\dfrac{\partial^{2}\chi^{2}}{\partial d_{l}\partial d_{k}}$ (and not the inverse of the element). Combining Eqs. (\[eq:general\_block\_1\]b) and (\[eq:general\_block\_2\]b) one can obtain relations for the $a$ and $b$ submatrices: $$\begin{aligned} b_{jr} & = & -\sum_{i,k}\left[\dfrac{\partial^{2}\chi^{2}}{\partial R_{j}\partial R_{i}}\right]^{-1}\dfrac{\partial^{2}\chi^{2}}{\partial d_{k}\partial R_{i}}\, a_{kr}\,,\\ \delta_{lr}&=&\sum_{k}\left(\dfrac{\partial^{2}\chi^{2}}{\partial d_{l}\partial d_{k}}-\sum_{i,j}\dfrac{\partial^{2}\chi^{2}}{\partial d_{l}\partial R_{j}}\left[\dfrac{\partial^{2}\chi^{2}}{\partial R_{j}\partial R_{i}}\right]^{-1}\dfrac{\partial^{2}\chi^{2}}{\partial d_{i}\partial R_{k}}\right)a_{kr}\,,\\ \delta_{im} & = & \sum_{j}\dfrac{\partial^{2}\chi^{2}}{\partial R_{i}\partial R_{j}}\, c_{jm}-\sum_{k,l,s}\dfrac{\partial^{2}\chi^{2}}{\partial d_{k}\partial R_{i}}\,\left[\dfrac{\partial^{2}\chi^{2}}{\partial R_{m}\partial R_{s}}\right]^{-1}\dfrac{\partial^{2}\chi^{2}}{\partial d_{l}\partial R_{s}}\, a_{lk}\,,\end{aligned}$$ where again $\left[\dfrac{\partial^{2}\chi^{2}}{\partial R_{j}\partial R_{i}}\right]^{-1}$ stands for the $(j,i)$ element of the inverse matrix of $\dfrac{\partial^{2}\chi^{2}}{\partial R_{j}\partial R_{i}}$. We “integrate out" the auxiliary parameters $d_k$ by substituting their minimum conditions $d^{(0)}_k=\tilde d_k(R^{(0)}_i)$ (which is analogous to using the equation of motion when integrating out heavy particles): $$\left.\dfrac{\partial\chi^{2}}{\partial d_{j}}\right\|_{d_k=\tilde d_k(R_i)}=0\,,\qquad \tilde \chi^2(R_i)=\chi^2(R_i,\tilde d_k(R_i))\,.\label{EOM}$$ Their first derivatives with respect to $R_{j}$ read $$\begin{aligned} \dfrac{\partial}{\partial R_{j}}\left.\dfrac{\partial\chi^{2}}{\partial d_{i}}\right\|_{\tilde d_m(R_{k})} & = & \left.\dfrac{\partial^{2}\chi^{2}}{\partial d_{i}\partial R_{j}}\right\|_{\tilde d_m(R_{k})}+\sum_{l}\left.\dfrac{\partial^{2}\chi^{2}}{\partial d_{i}\partial d_{l}}\right\|_{{\tilde d}_{m}(R_{k})}\dfrac{\partial {\tilde d}_{l}(R_{n})}{\partial R_{j}}=0\,,\nonumber \\ \dfrac{\partial \tilde d_i(R_{n})}{\partial R_{j}} & = & -\sum_l\left[\dfrac{\partial^{2}\chi^{2}}{\partial d_i\partial d_l}\right]_{\tilde d_m(R_{k})}^{-1}\,\left.\dfrac{\partial^{2}\chi^{2}}{\partial d_{l}\partial R_{j}}\right\|_{\tilde d_m(R_{k})} \,.\end{aligned}$$ The minimum of $\tilde \chi^2(R_i)$ is indeed located at $R^{(0)}_{i}$ because $$\dfrac{\partial\tilde\chi^2}{\partial R_{i}}=\sum_k\left.\dfrac{\partial\chi^{2}}{\partial d_k}\right\|_{d_k=\tilde d_k(R_i)}\dfrac{{\rm d} \tilde d_k}{{\rm d} R_i}+ \left.\dfrac{\partial\chi^{2}}{\partial R_i}\right\|_{d_k=\tilde d_k(R_i)}=\left.\dfrac{\partial\chi^{2}}{\partial R_i}\right\|_{d_k=\tilde d_k(R_i)} \,,\label{first_der}$$ and because the first term vanishes by the condition in Eq. (\[EOM\]). When evaluating Eq. (\[first\_der\]) for $R_i=R_i^{(0)}$ it vanishes because of Eq. (\[Mincond\]). Finally, let us calculate the inverse correlation matrix: $$\begin{aligned} &&\left.\dfrac{\partial^{2}\tilde{\chi}^{2}}{\partial R_{i}\partial R_{j}}\right\|_{\tilde{R}_{k}} = \left.\dfrac{\partial^{2}\chi^{2}}{\partial R_{i}\partial R_{j}}\right\|_{d^{(0)}_{m},R^{(0)}_{k}}+ \sum_{k}\left.\dfrac{\partial^2\chi^{2}}{\partial d_k\partial R_j}\right\|_{d^{(0)}_m,R^{(0)}_k} \left.\dfrac{\partial \tilde d_{k}(R_{i})}{\partial R_{i}}\right\|_{\tilde{R}_{k}}\nonumber \\ & = & \left.\dfrac{\partial^{2}\chi^{2}}{\partial R_{i}\partial R_{j}}\right\|_{d^{(0)},R^{(0)}_{k}}-\sum_{k,l}\left[\dfrac{\partial^{2}\chi^{2}}{\partial d_{k}\partial d_{l}}\right]_{d^{(0)}_{m},R^{(0)}_{k}}^{-1}\,\left.\dfrac{\partial^{2}\chi^{2}}{\partial d_{l}\partial R_{i}}\right\|_{d^{(0)}_{m},R^{(0)}_{k}}\left.\dfrac{\partial^{2}\chi^{2}}{\partial d_{k}\partial R_{j}}\right\|_{d^{(0)}_{m},R^{(0)}_{k}},\label{eq:EOM_Rj_bk}\end{aligned}$$ which agrees with Eq. (\[eq:noEOM\_Rj\_bk\]). We can now apply the previous results to Eq. (\[additive\]): $$\begin{aligned} \chi^{2} & = & \sum_{k}\left[d_k^2+\sum_{i,m}\left(\dfrac{R_i^{k,m}+d_k\, \Delta_i^{k,m}-R_m}{\sigma_i^{k,m}}\right)^2\right]\label{eq:with_bk_Rj}\\ & = & \sum_k\left\{ d_k^2\left[1+ \dfrac{\left\langle \Delta^2\right\rangle_k}{\left\langle \sigma\right\rangle _k^2}\right]+2\, d_k\sum_{i,m}\dfrac{(R_i^{k,m}-R_m)\Delta_i^k}{\sigma_{i}^{k,m\,2}}+\sum_{i,m} \left(\dfrac{R_i^{k,m}-R_{m}}{\sigma_i^{k,m}}\right)^2\right\} \,.\nonumber\end{aligned}$$ The equation of motion for $d_{k}$ reads $$d_k(R)=-\,\dfrac{1}{1+\dfrac{\left\langle \Delta^2\right\rangle_k}{\left\langle \sigma\right\rangle_k^2}} \sum_{i,m}\dfrac{(R_i^{k,m}-R_m)\Delta_i^{k,m}}{\sigma_i^{k,m\,2}} \,.$$ This renders for $\tilde\chi^2$ the form $$\begin{aligned} \tilde\chi^2 & = & \sum_k\left[d_k(R)\sum_{i,m}\dfrac{(R_i^{k,m}-R_m)\Delta_i^{k,m}}{\sigma_i^{k,m\,2}}+ \sum_{i,m}\left(\dfrac{R_i^{k,m}-R_m}{\sigma_i^{k,m}}\right)^2\right]\\ & = & \sum_k\left[\sum_{i,m}\left(\dfrac{R_i^{k,m}-R_m}{\sigma_i^{k,m}}\right)^2- \dfrac{1}{1+\dfrac{\left\langle \Delta^2\right\rangle_k}{\left\langle \sigma\right\rangle_k^2}}\sum_{i,j,m,n}\dfrac{(R_i^{k,m}-R_m)\Delta_i^{k,m}} {\sigma_i^{k,m\,2}}\dfrac{(R_j^{k,n}-R_n)\Delta_j^{k,n}}{\sigma_j^{k,n\,2}}\right]\,,\nonumber \end{aligned}$$ which reproduces Eq. (\[traditional\]), as we wanted to demonstrate. We conclude this appendix presenting an alternative way to write Eq. (\[chi2total\]) after using the equations of motion for $d_k$. We again concentrate on the simpler case without subtraction of the non-charm contribution: $$\begin{aligned} \chi^2 & = & \sum_k\left\{ d_k^{2}+\sum_{i,m}\left[\dfrac{R_i^{k,m}-(1+d_k\Delta f_k^{i,m})\, R_m}{\sigma_k^{i,m}}\right]^{2}\right\} \nonumber \\ & = & \sum_k\left\{ \left[1+\sum_{i,m} \left(\dfrac{\Delta f_k^{i,m}\, R_m}{\sigma_k^{i,m}}\right)^2\right]d_k^2-2d_k\sum_{i,m}\dfrac{\left(R_{k}^{i,m}-R_m\right) \Delta f_k^{i,m}\, R_m}{(\sigma_k^{i,m})^2}\right.\nonumber \\ &+&\left.\sum_{i,m}\left[\dfrac{R_k^{i,m}-R_m} {\sigma_k^{i,m}}\right]^2\right\} .\label{eq:chi2_auxiliary}\end{aligned}$$ The EOM for $d_k$ now reads $$\left[1+\sum_{i,m}\left(\dfrac{\Delta f_k^{i,m}\, R_m}{\sigma_k^{i,m}}\right)^2\right] \tilde d_k(R_n)-\sum_{i,m}\dfrac{\left(R_k^{i,m}-R_m\right)\Delta f_k^{i,m}\, R_m}{(\sigma_k^{i,m})^2}=0 \,,$$ which upon insertion into Eq. (\[eq:chi2\_auxiliary\]) renders $$\begin{aligned} \widetilde{\chi}^2 & = & \sum_k\left\{ \sum_{i,m}\left[\dfrac{R_k^{i,m}-R_m}{\sigma_k^{i,m}}\right]^2-\tilde d_k(R_n)\sum_{i,m}\dfrac{\left(R_k^{i,m}-R_m\right)\Delta f_k^{i,m}\, R_m} {(\sigma_k^{i,m})^2}\right\} \nonumber\\ & = & \sum_k\Biggr\{ \sum_{i,m}\left[\dfrac{R_k^{i,m}-R_m}{\sigma_k^{i,m}}\right]^2\nonumber\\ &-& \dfrac{1}{\left[1+\sum\left(\dfrac{\Delta f_k^{i,m}\, R_m}{\sigma_k^{i,m}}\right)^2\right]} \left[\sum_{i,m}\dfrac{\left(R_k^{i,m}-R_m\right)\Delta_k^{i,m}\, R_m} {(\sigma_k^{i,m})^2}\right]^2\Biggr\} \,.\label{eq:chi2_final} \end{aligned}$$ One can rewrite Eq. (\[eq:chi2\_final\]) in the matrix form $$\begin{aligned} \widetilde{\chi}^2 & = & \sum_k\left\{ \sum_{i,j,m,n}\left[R_{k}^{im}-R_m\right]\left[V_{k}^{-1}\right]^{mn}_{ij}\left[R_{k}^{jn}-R_n\right]\right\} \,,\label{eq:Matrix-Form} \\ \left[V_{k}^{-1}\right]^{mn}_{ij} & = & \dfrac{\delta_{ij}\delta^{mn}}{(\sigma_{k}^{i,m})^{2}}- \dfrac{\left(R_k^{i,m}-R_m\right)\Delta f_k^{i,m}\, R_m \left(R_k^{jn}-R_n\right)\Delta f_k^{jn}\, R_n} {\left[1+\sum\left(\dfrac{\Delta_k^{i,m}\, R_m}{\sigma_k^{i,m}}\right)^2\right]}\,.\nonumber\end{aligned}$$ In Eq. (\[eq:Matrix-Form\]) one can interpret the second term of the inverse correlation matrix, as a non-linear correlation among the measurements, where the correlation matrix itself depends on the value of the fit parameters. The total inverse correlation matrix is block diagonal, and the blocks correspond to $V_k^{-1}$. [^1]: As a word of caution we mention that, except for the contributions of the $J/\psi$ and $\psi'$ resonances, no experimental separation of the charm and non-charm contributions in the hadronic cross section has been provided in available data, although charm-tagged measurements are possible, see e.g. Ref. [@CroninHennessy:2008yi] (CLEO collaboration). So the charm pair production rate from above the $J/\psi$ and $\psi^\prime$ that enters the charmonium sum rules in Eq. (\[momentdef2\]) is usually obtained partly from the measured total with theory motivated subtractions of the non-charm rate, and partly by using theory predictions for the charm production rate. [^2]: Since the analyses of Refs. [@Chetyrkin:2006xg; @Boughezal:2006px] were based on outdated and less precise data for the $J/\psi$ and $\psi^\prime$ electronic partial widths [@Yao:2006px], we frequently only compare our numerical results with those of Refs. [@Chetyrkin:2009fv; @Kuhn:2007vp]. However the perturbative input of Refs. [@Chetyrkin:2006xg; @Boughezal:2006px; @Chetyrkin:2009fv; @Kuhn:2007vp] and ours is identical. [^3]: This can to happen at any order for $n>1$ and for $\mu_m\sim 3$ GeV. [^4]: This works in general as long as the path dependent $\mu_\alpha^c(q^2,\overline m_c^2)$ does not produce a spurious cut in $\alpha_s$ starting at $q^2=0$ and running towards $-\infty$. This condition is implemented into Eq. (\[mualphacontour\]) by $(\mu_\alpha^c)^2$ being negative along the physical cut of the vacuum polarization function above the charm pair threshold. [^5]: In this examination and throughout our other analyses we use the $\overline{\rm MS}$ renormalization group equations with $n_f=4$ active running flavors. [^6]: The numerical value of $\overline m_c(\mu=3~\mbox{GeV})$ is actually irrelevant, since the running involves $\overline m_c$ only in a linear way and exactly cancels in the ratio. [^7]: For the discussions in this section we use $\alpha_s^{(n_f=5)}(m_Z)=0.118$ ($\alpha_s^{(n_f=4)}(4.2~\mbox{GeV})=0.2245$) as an input using five-to-four flavor matching at $4.2$ GeV. For the first moment employed for the charm mass fits we use $M_1=0.2138\,{\rm GeV}^{-1}$, corresponding to the standard selection of data sets, as described in Sec. \[sectiondata\]. [^8]: Of course the extracted mass depends on the moment considered as well. Since in our analysis we focus on the first moment only, for simplicity we drop that label. [^9]: We are grateful to Thomas Schutzmeier for confirming agreement of the results of our ${\cal O}(\alpha_s^3)$ fixed-order code with theirs of Ref. [@Boughezal:2006px]. [^10]: There are 18 references quoted since Ref. [@Edwards:1990pc] provides results from two independent runs that we treat as two different data sets. [^11]: The situation is quite different if perturbative QCD is used already from $4.8$ GeV. Here the contribution to the moments $M_{1,2,3,4}^{\rm exp}$ from energies above $4.8$ GeV amounts to $(30, 10, 3, 1)\%$. For the first moment the $10$% penalty would then represent the largest source of uncertainty and correspond to an uncertainty in the charm mass of around $18$ MeV. [^12]: We have checked that the effect of using $R_{uds}(E_m)$ instead of $R_{uds}(E_i^{k,m})$ is totally negligible. [^13]: We prove in Appendix B the equivalence of a bilinear with fit auxiliary parameters to a bilinear without auxiliary fit parameters, but containing the standard correlation matrix. [^14]: In a very good approximation, it can be also obtained by dropping in $V^{-1}_{i,j}$ the rows and columns corresponding to $d_k$ and inverting the resulting matrix. After that one also drops the raw and column corresponding to $n_{\rm nc}$. We adopt this simplified procedure for our numerical fits. [^15]: For $R_0$ we have $m^\prime=1$ and for $R_{N_{cl}+1}$ we have $m^\prime=N_{cl}-1$. [^16]: We use a modified version of the minimal correlation model. The non-diagonal entries of the correlation matrix are filled in with $\Gamma_i\, R_m\, {\rm Min}^2\left\lbrace {\Delta{\Gamma_i}/\Gamma_i, \Delta R_m/R_m}\right\rbrace $. Here $\Delta\Gamma_i$ and $\Delta R_m$ represent the systematical uncertainties of the width of the narrow resonance and the $R$ value of the cluster, respectively, and $i=1,2$ refer to $J/\psi$ and $\psi^\prime$. [^17]: As our default we use $\alpha_s^{n_f=5}(m_Z)$ as the input, use the four-loop QCD beta-function for the renormalization group evolution and three-loop matching conditions to the $n_f=4$ theory at $\mu=4.2$ GeV. [^18]: The same disparity was found in Ref. [@Kuhn:2007vp]. In their analysis the corresponding effect is $-5$ MeV since experimental data were used in the experimental moments only for energies $\sqrt{s}<4.8$ MeV. In this approach, however, the moments are strongly dependent on the uncertainty one assigns to the theory input used for $\sqrt{s}>4.8$ MeV. [^19]: For the discussion in this appendix we do for simplicity of the presentation not account for fits of the non-charm contribution. It is, however, straightforward to generalize the presentation to this case.	ArXiv
--- abstract: 'We study the mechanism of the ‘pearling’ instability seen recently in experiments on lipid tubules under a local applied laser intensity. We argue that the correct boundary conditions are fixed chemical potentials, or surface tensions $\Sigma$, at the laser spot and the reservoir in contact with the tubule. We support this with a microscopic picture which includes the intensity profile of the laser beam, and show how this leads to a steady-state flow of lipid along the surface and gradients in the local lipid concentration and surface tension (or chemical potential). This leads to a natural explanation for front propagation and makes several predictions based on the tubule length. While most of the qualitative conclusions of previous studies remain the same, the ‘ramped’ control parameter (surface tension) implies several new qualitative results. We also explore some of the consequences of front propagation into a noisy (due to pre-existing thermal fluctuations) unstable medium.' author: - \| Peter D. Olmsted[^1] and F. C. MacKintosh\ [Department of Physics, University of Michigan, Ann Arbor MI 48109-2210]{} title: 'Instability and front propagation in laser-tweezed lipid bilayer tubules' --- =10000 5truemm PACS: 47.20.Dr, 47.20.-k, 47.20.Hw, 82.65.Dp 5truemm Short Title: [Dynamic instability in bilayer tubules.]{} [2]{} Introduction ============ A recent series of exciting experiments [@barziv94] demonstrated a dynamic instability induced on tubules of single lipid bilayers by application of laser ‘tweezers’, whereby the cylindrical tubule of radius $R_0$ modulates with a wavenumber given by $q^{\ast}R_0\simeq 0.8$. This instability has been attributed to an excess surface tension due to the gain in electrostatic energy when surfactant molecules, of higher dielectric constant than water, displace water in the electric field of the laser. The starting point for understanding this phenomenon is the Rayleigh instability [@rayleigh1; @rayleigh2; @tomotika] of a thin cylindrical thread of liquid with positive surface tension, whereby the thread can reduce its surface area at fixed volume by modulating and evolving towards a string of beads. Rayleigh calculated the preferred wavelength of a cylinder of fluid in air in the inviscid [@rayleigh1] and non-inertial (viscous) [@rayleigh2] limits, finding in the former case a characteristic non-zero wavenumber and in the latter case a preferred wavenumber of zero (or infinite wavelength). Later, Tomotika [@tomotika] calculated the instability for a viscous fluid surrounded by another viscous fluid, again in the non-inertial regime, finding that the change in boundary conditions restores a finite characteristic wavelength. See Olami and Granek [@granek95] for a discussion of this point. The present problem, however, requires a much different detailed dynamical analysis which relates the flow of lipid molecules in the interface to the bulk flow in the surrounding fluid. An important physical ingredient is a new conserved quantity, the lipid on the surface. At present there are (at least) two theoretical treatments of the experiments of Ref.[@barziv94]. Bar-Ziv and Moses [@barziv94] and Nelson and co-workers [@nelsontube95a] have proposed the picture that the surface tension rapidly equilibrates everywhere to an induced value $\Sigma_0$, and the instability proceeds from this state. In contrast, Granek and Olami [@granek95] have postulated that the correct treatment of the problem is to impose a constant rate at which lipid molecules are drawn into the trap from the tubule. This loss of lipid is accommodated by stretching out small wavelength surface fluctuations and the result is again a uniform surface tension $\Sigma_0$. Goldstein, [et al.]{} (GNPS) [@nelsontube95b] demonstrated quantitatively how the equilibration of the tension in the tube stays ‘ahead’ of a shape change, so that a treatment with a constant (in time) surface tension is reasonable; and argued that the primary loss of area is in the shape instability itself, rather than through the removal of small-scale wrinkles. We propose a slightly different picture of the steady state before the onset of the instability, which follows from consideration of the experimental configuration. The tubules, as formed, are several hundred microns long and are attached at either end to ‘massive lipid globules’ [@barziv94] of order $10\mu\hbox{m}$ in diameter. Hence, the tubules must be in contact with a reservoir which fixes the lipid chemical potential (or, equivalently, the surface tension). If we assume the system is equilibrated, it follows that the chemical potential for exchange between the tubule, reservoir, and solvent/lipid bath vanishes [@schulman61], and we may assume a reference chemical potential of zero or, equivalently, zero surface tension. This coincides with the experimental observation of visible thermal fluctuations on the tubules [@barziv94]. Now imagine applying a laser to the tubule. In the electric field of the chemical potential of a lipid molecule is lowered by an amount $\delta\varepsilon{\cal E\/} D a$, where $D$ is the molecular length, $\delta\varepsilon$ is the dielectric constant relative to water, $a$ the area of the lipid, and ${\cal E\/}$ the energy density deposited in the trap. Nelson [et al.]{} [@nelsontube95b] calculated that this yields an energy gain per area of bilayer of $\Sigma_0\sim 2 \cdot 10^{-3} \,\hbox{erg cm}^{-2}$, for a laser power of $50\,\hbox{mW}$. Hence there is a large reduction in the local chemical potential as the lipid suddenly finds it advantageous to move into the laser spot. The surface tension in the adjacent portion of the tubule increases as lipids start to move out of the surface. Since the other end of the tubule is in contact with a reservoir at zero chemical potential, the final state (prohibiting, for the moment, surface undulations) must be a non-equilibrium steady state in which: 1. Lipid is transported at constant velocity from the reservoir at zero chemical potential to the laser trap at a negative chemical potential. 2. The chemical potential drops linearly along the tubule, with a gradient that balances the frictional drag of the bulk fluid in steady state. 3. The local lipid concentration also varies linearly, since the two-dimensional lipid fluid membrane is compressible. This differs significantly from the treatments of Nelson [et al.]{} and Granek and Olami in that [lipid must flow out of the anchoring globules]{} and the chemical potential (or surface tension) never attains a non-zero constant over the duration of the experiment. In fact, prohibiting the shape instability, the boundary conditions specified by both Olami and Granek [and]{} Nelson [et al.]{} yield a tense final state as (a small amount of) area is drawn out of surface fluctuations, while the treatment of the anchoring globules as reservoirs yields the steady-state described below.[^2] Several consequences follow from this observation. First, a chemical potential gradient suggests a mechanism for front propagation [@nelsontube95b; @sarloos88]. The front starts at the laser spot where the surface tension is largest, and ‘propagates’ outward toward the anchoring globule simply because the amplitude of the instability grows at different rates along the tube. Our results predict a speed of front propagation which is inversely proportional to the length of the tube, and is largest near the laser spot, decreasing to zero somewhere near the anchoring reservoirs; and a characteristic wavenumber which also decreases (much slower, see Fig. \[fig:dispersion2\] below) away from the laser spot. The outline of this paper is as follows. In Section 2 we derive the linear concentration gradient in the absence of surface undulations. We predict a ‘ramped’, or spatially-varying control parameter, the effective surface tension, which is in fact the two-dimensional pressure whose gradient drives the flow of lipid against the viscous drag of the bulk fluid. In Section 3 we present a detailed microscopic picture of the uptake of surfactant by the trap, and argue that a competition between bending and compression energies modifies the effective surface tension of the trap. This leads to a prediction of a critical laser power for the onset of an instability. While this section may safely be omitted in reading this paper, it illuminates the nature of the instability by treating a realistic scenario for how the trap buckles to initiate flow. In Section 4 we discuss the implications of a slowly varying surface tension on the detailed calculation of Goldstein, [et al.]{} [@nelsontube95a; @nelsontube95b]. We also discuss front propagation within the picture of a surface tension gradient, which relates the problem to a large body of work on front propagation with ‘ramped’ parameters [@kramer82]. The issue of front propagation in this system is delicate [@nelsontube95b], and our results suggest at least two possibilities, which we briefly raise in this work and pose for further investigation. Depending on whether noise ([i.e.]{} existing thermal fluctuations in the tubule) is present, we expect front propagation which is either (a) characteristic of that predicted by the so-called Marginal Stability Criteria (MSC) [@nelsontube95b; @kramer82], or (b) dominated by amplification of existing ‘noise’, which can lead to behavior reminiscent of front propagation for a steep enough ramp. We conclude in Section 5 by recalling the relevant timescales and frequencies, and summarizing our predictions and the differences from previous treatments. Steady state ============ In this section we calculate the steady-state configuration of a tubule under the action of an applied laser intensity, assuming the laser supplies a chemical potential $- \Sigma_0$ at the laser spot. Note that this implies a reservoir in which to pack lipid molecules. In Section 3 we support this with a microscopic picture which leads to virtually the same results that we obtain in this section, with a prefactor of order one which depends on the laser shape. Note that there are several possible microscopic scenarios for initiating flow into the trap, and Section 3 addresses only one of these. Equations of motion ------------------- Changes in chemical potential $\delta\!\mu$ are related to changes in surface tension $\delta\Sigma$ by $\delta\mu = - \phi^{-1}\delta\Sigma $, where $\phi$ is the lipid concentration (and hence $\phi^{-1}=a$ is the area per lipid). Also, $$\Sigma=-p,$$ where $p$ is the 2-dimensional pressure of the fluid of lipid molecules. The geometry of the system is taken as shown in Figure \[fig:geom\], with the cylinder aligned parallel to the $z$-axis and $r$ the radial coordinate. The boundary conditions are $$\begin{aligned} p(z=0) = 0 && \qquad(\hbox{reservoir}) \\ p(z=L) = - \Sigma_0 && \qquad(\hbox{laser spot}),\end{aligned}$$ where $\Sigma_0$ is the surface tension induced by the laser. 20.5pc =3truein The Navier-Stokes and continuity equations for the 2D fluid of lipid molecules are $$\begin{aligned} \partial_t \phi&=& -\nabla\!\cdot\!(\phi{\rm\bf v}) \label{eq:continuity}\\ \rho_s (\partial_t + {\rm\bf v}\!\cdot\!\nabla){\rm\bf v} &=& \eta_s \nabla^2 {\rm\bf v} + (\case{1}{3}\eta_s + \gamma_s) \nabla (\nabla\cdot{\rm\bf v}) \label{eq:NS}\\ && - \nabla p + \Delta{\bf T}^b\!\!\cdot\!{\bf\hat{r}}. \nonumber\end{aligned}$$ Here $\eta_s$ and $\gamma_s$ are 2D shear and bulk viscosities, $\rho_s$ is 2D lipid mass density, and $\Delta{\bf T}^b\cdot{\bf\hat{r}}$ is the viscous drag acting on the surface from the dissipative stress tensor $T^b_{\alpha\beta} = \case12\eta(\nabla_{\alpha}u_{\beta} + \nabla_{\beta}u_{\alpha})$ in the surrounding fluid. This flow is established in a vorticity diffusion time $\tau_{v}$ which is much smaller than other times in the problem. We ignore drag from outside the cylinder for the moment, since this flow essentially moves with the surfactant molecules and contributes relatively little to the boundary stress[^3]. With the above approximation, the boundary stress is given by the shear stress in the tube. For a [uniform]{} flow of lipid ${\rm\bf v} = v {\bf\hat z}$, the interior flow is Poiseuille [@landaufluids], $${\rm\bf u}(r) = v {2r^2 - R_0^2\over R_0^2}{\bf\hat z},$$ where $R_0$ is the tube radius and we use the no-slip boundary condition $u(r=R_0)=v$. Hence the stress acting on the surface is $$\Delta{\bf T}^b\!\cdot\!{\bf\hat{r}} = -{2\eta v \over R_0}. \label{eq:stress}$$ Gradients of ${\rm\bf v}$ in the $z$-direction change the flow profile from simple Poiseuille, but this has only a very small effect on the dynamics of establishing the steady state, primarily in the region of the laser spot, which we ignore for now. The final ingredient we need is the compressibility of the film, through the constitutive relation $$p = p_0 - \chi^{-1} \delta a \label{eq:pressure}$$ where $p_0$ is the equilibrium pressure. Now we specialize to the problem at hand. We linearize the dynamic equations in ${\rm\bf v}$ and $\delta\phi = \phi - \phi_0$, assume a velocity of the form ${\rm\bf v} = v(z){\bf\hat z}$, and ignore the inertial term in the Navier-Stokes equation. Employing Eq. (\[eq:pressure\]), we obtain $$\begin{aligned} \partial_t \delta\phi&=& -\phi_0 \nabla_z v \label{eq:phidot}\\ 0 &=& \hat{\eta} \nabla_z^2 v - B \phi_0^{-1} \nabla_z \delta \phi + {2\eta\over R_0} v,\label{eq:vdot}\end{aligned}$$ where $\hat{\eta}=\case43 \eta_s + \gamma_s$. The boundary conditions are $$\begin{aligned} \delta \phi(z=0)&=& 0 \\ \delta \phi(z=L)&=& \phi_0\Sigma_0/B.\end{aligned}$$ where $B=\chi^{-1}\phi_0^{-1}$ is the two dimensional bulk modulus. Dynamics of equilibration ------------------------- Assigning $\delta\phi(z,t) = \delta\hat{\phi}(q,\omega) e^{i (q z - \omega t)}$ and similarly for $v(z,t)$, we obtain the following dispersion relation: $$\omega = {i B q^2 \over q^2 \hat{\eta} + 2\eta/R_0}. \label{eq:dispersion}$$ This yields $\omega\simeq i R_0 B q^2 / \eta$ for $q\ll q^{\ast}$ and $\omega \simeq B / \hat{\eta}$ for $q\gg q^{\ast}$, where $q^{\ast} = \sqrt{2 \eta / (R_0 \hat{\eta})}$. Hence, at long wavelengths we have diffusive behavior governed by the friction against the bulk fluid, while at short wavelengths the dynamics is dominated by the 2D viscosity. The crossover length is given by $1/q^{\ast} \sim 0.1\mu\hbox{m}$, where we have taken $\eta \sim 10^{-2} \,\hbox{g cm}^{-1}\hbox{s}^{-1}, \hat{\eta} \sim 10^{-6} \,\hbox{g s}^{-1}$, and $R_0\sim 0.5 \mu\hbox{m}$. Hence in most cases of interest we are in the regime dominated by bulk fluid dissipation and may ignore $\hat{\eta}$. We can now estimate (within linear response) the time to attain steady state after imposing the localized tension by the laser as, roughly, the relaxation time of the slowest mode given by the dispersion relation Eq. (\[eq:dispersion\]). Taking $q = 2\pi/L$, we have $$\tau_{\mit ss} \sim {2 L^2\eta\over (2\pi)^2 R_0 B}\sim 10^{-5}\,\hbox{s},$$ for $B\sim 150 \,\hbox{erg cm}^{-2}$ [@evans87] and $L\sim 100\mu\hbox{m}$. This estimate of $B$ is a zero-temperature estimate and ignores small-scale thermal fluctuations which soften this modulus considerably [@helfrich84; @nelsontube95b]. As we discuss in the conclusion and as shown in Reference [@nelsontube95b], this effect can reduce $B$ by up to three orders of magnitude, increasing $\tau_{\mit ss}$ accordingly, to of order $10^{-2}\,\hbox{s}$. We can compare this to the vorticity diffusion time, $$\tau_{\it v} = {\rho R_0^2 \over \eta} \sim 10^{-7}\hbox{s},$$ where we take $\rho=1\,\hbox{g cm}^{-3}$. Since $\tau_{\mit ss}>\tau_v$ our assumption above of a uniform shear stress is reasonable. Steady State ------------ To find the steady state we equate the left hand side of Eq. (\[eq:phidot\]) to zero, which yields a constant velocity $\bar{v}$. From Eq (\[eq:vdot\]) $\nabla_z \delta \phi$ is also a constant and, applying the boundary conditions and the pressure constitutive equation, Eq. (\[eq:pressure\]), we find the following steady-state profile $$\begin{aligned} \bar{v} &=& { R_0 \Sigma_0\over 2 \eta L} \label{eq:vbar}\\ \delta \phi &=& -\phi_0 {\Sigma_0 \over B} {z\over L} \\ \delta\Sigma &=& \Sigma_0 {z\over L} \label{eq:sigma}.\end{aligned}$$ Thus we find that the steady state, excluding modulations of the cylinder, is a non-equilibrium steady state, where the lipid molecules run down a chemical potential gradient and the molecular spacing increases to reflect this changing local potential. An estimate above yields $\bar{v} \sim 1 \mu\hbox{m s}^{-1}$, where we use $\Sigma_0 \sim 10^{-3} \hbox{erg cm}^{-2}$. The effective surface tension (or two dimensional pressure) is induced by the applied laser, and is non-zero [only in the presence of flow]{}. Microscopic Picture =================== We have shown how lipid flow, which is a necessary condition for an effective non-zero surface tension far from the trap, follows from a boundary condition of fixed chemical potential at the trap. In this section we argue that this boundary condition requires an instability in the trap, and we present detailed calculations for a possible scenario for the trap to initiate flow. We stress that there are several possible mechanisms, including buckling, ejection of micelles or bilayer structures, and growth of ‘cancerous’ membranes. This is surely not an exhaustive list. Basic Considerations -------------------- We first note that a laser spot centered at $z=L$ typically has a Gaussian intensity profile [@webb81], which leads to an energy gain per area of lipid ${\cal U\/}(z) = - \Sigma_0\,\zeta(z)$, with $$\zeta(z) = e^{-(z-L)^2/2\Delta^2}.$$ The spot radius was estimated to be $\Delta\simeq 0.15 \mu\hbox{m}$ in the experiments of Bar-Ziv and Moses [@barziv94]. We can envision two scenarios after applying the laser: - Lipid can be sucked into the trap until the electrostatic energy gain balances the cost of compressing the molecules in the bilayer. At this point the trap is full, flow stops, and the chemical potential (and surface tension) of the entire tube reverts back to zero. - For a critical tension $\Sigma^{\ast}$ (Eq. \[eq:deltabar0\]) we expect the compressed section of the tubule to become unstable with respect to buckling. For higher intensities the trap continues to fold to accommodate more lipid, initiating a flow along the tubule. This flow must be accompanied by a chemical potential (or surface tension) gradient, which drives the instability seen in the experiments. Our discussion suggests that the trap boundary condition should contain the physics that, at a certain distance from the center of the trap, lipid is incorporated into folds to relieve the in-layer compression. A reasonable choice is $$a(z=L-\bar{\Delta})=a_0, \label{eq:BCtrap}$$ where $\bar{\Delta}$ is a distance to be determined. This asserts that the area per head assumes its preferred equilibrium value at the point where the folding begins. In the next two subsections we derive the steady-state flow into the trap (prohibiting for the moment the ‘pearling’ shape change). We first obtain a general relation for the local area per lipid $a(z)$, which depends on the trap boundary condition. We then deduce a crude criterion for the position $z=L-\bar{\Delta}$ at which the trap buckles. Applying the [assumption]{} of Eq. (\[eq:BCtrap\]) at this position then yields the desired profile and steady-state flow. Detailed Steady State --------------------- Let us examine the steady state. The continuity equation, Eq. (\[eq:continuity\]), yields the condition $$v(z) = C a(z),$$ where $C$ is a constant to be determined. Hence the steady-state Navier-Stokes equation, ignoring the 2D viscosity, becomes $$0 = {2\eta C a(z) \over R} + \chi^{-1} \nabla_z a(z) - \nabla {\cal U\/}(z).$$ 20.5pc =3truein The solution of this equation with the boundary condition $a(0)=a_0$, which follows from contact with a reservoir at ambient pressure, is $$\begin{aligned} {\delta a \over a_0} &=& e^{\lambda z} - 1 - \gamma\int_0^z\!ds\, e^{-\lambda(s - z)} {\partial \over\partial s} \zeta(s) \label{eq:solution}\\ \lambda &=& {2\eta\chi C\over R_0}, \\ \gamma &=& {\Sigma_0\over B},\end{aligned}$$ where $\lambda$ is determined by the boundary condition at the trap. Note that $\gamma \sim 10^{-5}$, since $\Sigma_0\sim 10^{-3} \,\hbox{erg cm}^{-2}$ and $B\sim 150 \,\hbox{erg cm}^{-2}$. Using Eq. (\[eq:BCtrap\]), we find $$\lambda= {- \gamma \,\zeta(L-\bar{\Delta})\over L - \bar{\Delta} - \int_0^{L-\bar{\Delta}}\!ds\, \zeta(s)},$$ where we have expanded for small $\lambda$ and made use of $\zeta(0)=0$ far from the trap. For all practical purposes $\lambda= - \gamma \,\zeta(L-\bar{\Delta})/L$. Since $\lambda L \ll 1$ for most cases of experimental interest, the calculation of Section 2 applies in the region outside the trap (see Fig. \[fig:profiles\]) with the surface tension replaced by $$\Sigma_0 \rightarrow \bar{\Sigma} = \Sigma_0\,\zeta(L-\bar{\Delta}). \label{eq:barsig}$$ For a Gaussian shape $\zeta(L-\bar{\Delta}) \alt 1$ \[$\zeta(L-\bar{\Delta})\simeq 0.31$ in Fig. \[fig:buckle\]\], so the modification to the naive boundary condition is rather minimal. Trap Boundary Condition ----------------------- To complete our discussion we estimate the stability against folding inside the trap. This determines the position $\bar{\Delta}$ at which the boundary condition (\[eq:BCtrap\]) applies, as well as a critical effective surface tension parameter (or laser intensity) $\Sigma^{\ast}$ at which the system initiates flow. We imagine that the system has attained a steady state in the absence of buckling and flow, given by Eq. (\[eq:solution\]) with $\lambda=0$: $$a(z) = a_0\left[1 - \gamma\,\zeta\left(z\right)\right].$$ Here $\gamma\,\zeta\left(z\right)$ is a measure of the compression. Rather than calculating the stability of a patch with a non-uniform area per head $a(z)$, we calculate the stability against buckling of a patch with uniform $a=\psi a_0$, with $\psi= 1 - \epsilon $, and use the resulting critical strain $\epsilon^{\ast}$ to determine $\bar{\Delta}$ through $$\gamma\,\zeta(L-\bar{\Delta}) \equiv \epsilon^{\ast}. \label{eq:deltabar}$$ We consider perturbations $R(z) = R_0(1 + u(z))$ which preserve the volume of the fluid. This constraint yields the condition $\int [u(z)^2 + 2 R_0 u(z)] =0$ [@safran]. The free energy, which includes in-plane compression and bending, is [^4] $$2 F = \int d^2\!r \left[ B \left({\psi a - a_0 \over a_0}\right)^2 + \kappa H^2 \right],$$ where $H$ is the mean curvature, $a_0 = dz R_0$, and $a = \psi d^2\!r$. For the perturbation above [@granek95], $$\begin{aligned} d^2\!r &=& dz (R_0 + u(z)) \sqrt{1 + u'(z)^2} \\ H &=& {(1 + u'(z)^2)^{-1/2}\over R_0 + u(z)} - {u''(z)\over(1 + u'(z)^2)^{3/2}},\end{aligned}$$ where $u'(z) = du/dz$. To quadratic order in $u(z) = \sum_q \hat{u}(q) e^{iqz}$ the energy per unit area $A$ becomes [@granek95] $$\begin{aligned} {2 F\over A} &=& \sum_q \hat{u}(q)^2 \left[ {3\kappa\over R_0} - 5 B R_0 \epsilon\right. \nonumber\\ && - \hat{q}^2 \left.\left(3B R_0 \epsilon + {\kappa\over R_0}\right) + 2 \hat{q}^4 {\kappa\over R_0}\right],\end{aligned}$$ where $\hat{q}=q R_0$. The vanishing of the term in square brackets defines $\epsilon^{\ast}$, the minimum strain above which this energy is unstable to undulations. Combining this condition with our estimate for how $\epsilon(z)$ varies away from the trap, Eq. (\[eq:deltabar\]), we find the following relation which determines $\bar{\Delta}$: $$\begin{aligned} \sigma^{\ast}\equiv{\Sigma^{\ast} R_{0}^2 \over \kappa} &=& {1\over \zeta(L-\bar{\Delta})} {3 + 2 \bar{q}^4 - \bar{q}^2 \over 5 + 3 \bar{q}^2} \label{eq:deltabar0}\\ &\equiv& g(\bar{q})\end{aligned}$$ where $\bar{q} \simeq \pi R_0/\bar{\Delta}$. For $\sigma>\sigma^{\ast}$ the tubule should buckle inside the trap. Fig. \[fig:buckle\] shows $g(\bar{q})$ for the Gaussian laser spot. 20.5pc =3truein Our criterion depends on the trap shape, but not the compression modulus $B$. This happens because, while the strain induced in packing lipid in the trap varies as $1/B$, the critical strain at which buckling occurs is also inversely proportional to $B$, and the $B$ dependence cancels out. In fact, the same order of magnitude estimate emerges from a comparison of bending and effective surface tension. What we have gained, however, is a picture of the forces at play that induce the buckling. While our estimate apparently fails to predict buckling for typical values, relaxing a few approximations we have made should change this picture. First, we have assumed uniform coverage at the lowest value of the actual nonuniform coverage in the quiescent trap. Second, we have assumed an axisymmetric deformation. This is obviously not the case if the laser spot size is smaller that the tubule diameter. In addition, the volume constraint must be handled differently. Removing these approximations should, in both cases, result in a smaller $\Sigma^{\ast}$. For example, in the limit of small trap sizes we can ignore curvature and ask about the stability of a flat interface against buckling, for which the criterion above becomes $$\Sigma^{\ast} = {\kappa \bar{q}^2\over\zeta(L-\bar{\Delta})}.$$ demanding that the characteristic buckling wavevector $\bar{q}$ be roughly the inverse of the trap size $\bar{\Delta}$ determines the critical trap size and intensity. This is essentially the same estimate given by Bar-Ziv, Frisch, and Moses in a somewhat different context [@barziv95]. Dispersion Relation and Front Propagation ========================================= Dispersion Relation ------------------- We have shown thus far that, under steady-state conditions before any macroscopic shape instability occurs, the proper boundary conditions imply a surface tension gradient along the tubule which supports lipid flow. We now turn to the effects of this gradient. Rather than repeating the the analysis of GNPS [@nelsontube95b] with a non-uniform surface tension, we note that the characteristic wavenumber at which the instability occurs is typically $q^{\ast}R_0\simeq 0.8$. Since $R_0\ll L$, we suspect that the assumption of a locally constant surface tension along the tubule is a good first step. This allows us to transcribe the results of Ref. [@nelsontube95b]. The primary result of interest is the growth rate $\omega(q)$ of an undulation $u(q,t)$, where $q$ denotes a Fourier mode along the tubule. This frequency is defined through $$({\partial\over \partial t} + i q \bar{v}) u(q,t) = \omega(q) u(q,t),$$ where the convective term arises because the lipids have an average velocity. In the original instability calculation presented by Rayleigh [@rayleigh1], and as has been emphasized in Refs.[@barziv94; @nelsontube95a; @granek95; @nelsontube95b], the structure of $\omega(q)$ is as follows: $$\omega(q) = \Phi(q) T(qR_0).$$ The function $T(qR_0)$ is determined by the energetics of the problem and, in our case, is non-zero for $\Sigma R_0^2/\kappa$ greater than a critical value of order unity [@nelsontube95b; @granek95]. In the Rayleigh case the instability occurs for $\Sigma>0$. The function $\Phi(q)$ is determined by the dynamics of the problem, and it is here that much of the interesting and surprising physics lies. Energetics tells us that the most unstable modes are at low $q$, where undulations are the least ‘violent’, while dynamic considerations severely penalize the growth of modes in the limit $q\rightarrow 0$. Goldstein [et al.]{} [@nelsontube95b] calculated $\omega(q)$ for a uniform surface tension $\bar{\Sigma}$, including the effects of bending as well as friction between the two bilayer leaves. Changing the boundary conditions of their work to allow for flux from the reservoir adds a convective term to the dynamics, and, aside from our local approximation above, changes nothing else. A plot of $\omega(q)$ is shown in Figure \[fig:dispersion\] for several values of the surface tension $\sigma =\bar{\Sigma} R_0^2 / \kappa$, with values for the bilayer friction and bulk modulus taken as in GNPS. 20.5pc =3truein Since $\sigma$ is $z$-dependent (Eq. \[eq:sigma\]), the growth rate $\omega^{\ast}$ and wavenumber $q^{\ast}$ of the fastest growing mode are $z$-dependent, and are greatest near the laser spot, as in Fig. \[fig:dispersion2\]. A single Fourier modulation has, locally, the following form: [@nelsontube95b] $$u(z,t) = u_0 e^{i q (z - \bar{v} t) + \omega(q,z)t},$$ where $\omega(q,z)$ is the function plotted as $\omega(q)$ in Fig. 5 of Reference [@nelsontube95b] and is reproduced here in Fig. \[fig:dispersion\]. The $z$-dependence comes through the $z$-dependence of $\sigma$. Note that we rely [strongly]{} on the condition $q L \gg 1$ \[Note that in the experiments [@barziv94] with, say, $L\simeq 200\,\mu\hbox{m}$ and a diameter of $1\,\mu\hbox{m}$, this condition is easily satisfied, $qL\simeq 160$.\] Given a dispersion relation which depends on position, there are several immediate naive predictions: The local wavenumber and apparent growth rate of the pattern should decrease as the anchoring globules are approached, with the instability vanishing at a point close to the reservoir where the induced surface tension is not strong enough to overcome the barrier due to bending, $\Sigma^{\ast} \simeq \kappa/R_0^2$. Hence, in the experiments with $\sigma\simeq 20$ [@nelsontube95a], this occurs at $1/20th$ of the distance from the anchoring globule to the laser trap. At points closer to the globule any undulation is a decaying remnant of the pattern developed closer to the trap. 20.5pc =3truein Front Propagation ----------------- We now confront the issues of front propagation and wavelength selection. GNPS argued that the Marginal Stability Criterion hypothesis provides a reasonable estimate for both the propagation speed $v_f$ and the selected wavenumber $k^{\ast}$ [@nelsontube95b]. A naive extension of this calculation, again assuming a local dispersion relation, predicts a spatially varying front speed and selected wavenumber. However, this prediction and the qualitative picture relies on two assumptions: (1) the existence of a propagating front, and (2) the absence of noise—i.e., that the propagation occurs into a uniform, unstable medium with no thermal fluctuations. The latter assumption is obviously not correct in detail, as thermal fluctuations, including modes of wavelengths comparable to the most unstable modes, are apparent in the experiments prior to the onset of the instability. Here we discuss these issues in the context of a spatially varying control parameter. Because a full treatment of the problem does not yet exist in the literature and is beyond the scope of this paper, we limit ourselves in this work to some numerical experiments and suggestions which, we hope, will stimulate further research on both this specific problem and the general aspects of front propagation into spatially-varying media in the presence of noise. For the rest of this paper we refer to ‘noise’ as a set of random initial conditions which obey a Boltzmann distribution, and do not consider temporally fluctuating noise. Kramer [et al.]{} [@kramer82], followed by others [@buell86; @kramer85; @riecke87], showed that, in the presence of a ‘ramped’ control parameter that becomes subcritical at some point, as happens near the reservoir, the uniquely-selected wavenumber need not correspond to that determined by the MSC. This effect is expected to take precedence over the MSC-determined wavenumber at times after which non-linearities become important and the ‘phase’ of the pattern has time to diffuse of order the system size. However, we are concerned with the more fundamental issue of the [existence]{} of a propagating front. In the presence of ‘noise’ which, in the present experiment, corresponds to existing thermal fluctuations around the reference smooth cylindrical state, a propagating front can be expected to exist for times less than the characteristic growth times of existing fluctuations in the vicinity of the most unstable mode. Hence, given a quench into an unstable ‘ramped’ state with an initial perturbation near the laser spot, propagation away from the perturbation occurs for an initial period of time, followed by rapid growth all along the cylinder as the initial conditions (‘noise’ of unstable wavelengths) are amplified to visible length scales. An initial perturbation near the laser spot is natural because, in practice, the laser spot diameter is smaller than the tubule diameter and a ‘pinching’ effect results whereby surfactant flows around the circumference of the tubule (as well as along the cylinder diameter) to fill the trap. The effect of a ‘ramp’ in the control parameter should be most dramatic after the noise overwhelms the front propagation: for a flat control parameter (no ramp) the noise grows randomly everywhere, and the ‘front’ should break down when the noise has grown to visible amplitudes. However, for a steep enough ramp the non-uniform amplification of the noise could resemble front propagation. To check these conjectures we have employed a simple caricature of the tubule dynamics, specified by $$({\partial\over \partial t} + \bar{v} \partial_x) u(x,t) = \left[a(x) 2 k_0^2 \partial_x^2 + \partial_x^4 \right] u(x,t) - g u(x,t)^3, \label{eq:caricature}$$ where $a(x)$ is a spatially varying control parameter chosen to mimic the dispersion relation and position dependencies in Figs. \[fig:dispersion\] and \[fig:dispersion2\]. A choice which gives reasonable qualitative agreement is $$a(x) = {\|x-x_0\|^{\alpha+1}\over (x-x_0)}, \label{eq:trial}$$ where $x_0$ is the point at which the system is absolutely unstable. We emphasize that this is a toy model whose details do not correspond to the Bar-Ziv [et al.]{} experiments, but which we believe contains the essential physics of front propagation into an unstable inhomogeneous medium, as occurs in these experiments. For Fig. \[fig:dispersion2\], $x_0\sim 0.05 L$ and $\alpha=1/8$ are reasonable. Fig. \[fig:mimic\] shows the local dispersion relation $\omega^{\ast}(x)$ for various $\alpha$. 20.5pc =3truein We have chosen the simplest possible non-linearity to stabilize the system. One may choose a more physical non-linearity such as the driving force arising from terms of higher than quadratic order in the mean curvature [@nelsontube95b], but our purpose here is primarily to illustrate some qualitative behavior of a front propagating into a noisy, non-uniform media. Figs. \[fig:snapshots1\] and \[fig:snapshots2\] show snapshots in the evolution of the system, given by Eq. (\[eq:caricature\]), for an initial perturbation at the trap of $1\%$ of the final amplitude (as determined by the non-linear term $g$) and an initial condition (or ‘noise’) which is taken to be a superposition of $300$ harmonics weighted with a Boltzmann weight corresponding to a non-zero surface tension ([i.e.]{} with an energy proportional to $q^2$). We have chosen a system size of $150$ wavelengths, and arbitrarily chosen the vertical scale to fill the figures. The general features are as described above: a front ‘propagates’ for an initial time from the initial perturbation, after which the ‘noise’ takes over and a very irregular growth quickly overtakes the system. 20.5pc =3truein The steeper ramp ($\alpha=1/2$, Fig. \[fig:snapshots1\]) has a better defined growth in the ‘noise’ regime, and could almost be called a ‘front’. In contrast, the growth into the shallow ramp ($\alpha=1/8$, Fig. \[fig:snapshots2\]) is more ragged and it would be charitable to call this a front. The shallower ramp has a very slightly faster propagation speed in the initial regime, and an obviously faster ‘propagation’ speed in the noise-dominated regime. Both of these behaviors may be traced to the faster overall growth rate for a shallower ramp (where a larger fraction of the tubule is more unstable). 20.5pc =3truein Fig. \[fig:noise\] shows the the results of fixing the ramp and varying the noise amplitude. For a noisier system the effective propagation speed in the noise-dominated regime is faster, and the breakdown of the simple propagating regime occurs earlier. The propagation velocity before the noise takes over is independent of the noise amplitude. The delay before noise-dominance increases logarithmically with increasing noise amplitude, consistent with the simple argument that the propagating solution exists until the noise has grown to a given amplitude, since this initial growth is exponential. To summarize, we have performed exploratory numerical calculations to investigate some of the consequences of a ramped control parameter, with an initial localized perturbation and initial global ‘noise’ for an initial condition, finding: 1. At early times a front propagates away from the localized perturbation. We find a dimensionless front velocity of $v\,\omega(k_0)/k_0 = 3.3$, while the Marginal Stability Criteria [@sarloos88] predict $4.6$. A similar agreement was found in the simulations of GNPS [@nelsontube95b]. 20.5pc =3truein 2. In this first regime the propagation velocity is independent of noise amplitude. 3. After a time, which may be taken to be the time required for the ‘noise’ to grow to an observable amplitude, the unstable pattern rapidly develops everywhere on the tubule. 4. The speed and qualitative character of this growth depend on the noise and ramp characteristics. The growth is faster for a shallower ramp and/or stronger noise, and looks reminiscent of a front for a steeper ramp; and in all cases is much faster than the simple front propagation from the initial localized perturbation at the laser trap. We have also performed calculations with no initial localized perturbation but, as this is probably not physically relevant, we do not report the results here. This initial study raises several questions which we feel are worth pursuing. In cases where the noise is weak enough and an apparent front exists, can this be understood quantitatively in terms of the gradient in the control parameter, and how does this relate to previous investigations of ‘ramped’ control parameters [@kramer82; @buell86; @kramer85; @riecke87]? Can a ramp stabilize an advancing front? Conclusion ========== Physical Picture ---------------- We have given the following picture of the action of lipid tubules upon the application of laser tweezers. In the absence of buckling, the laser induces a local compression of lipid molecules in the laser spot. This takes place in a time of order $\tau_{\mit ss}\sim 10^{-5} \hbox{s}$. A sufficiently large laser intensity induces a local buckling of the membrane in the trap, which initiates flow down the tubule from the reservoir. We do not have an estimate of the delay time for the instability in the trap. In the absence of undulations [outside]{} the trap, this flow would build up to a steady-state value $\bar{v}\sim 1\,\mu\hbox{m s}^{-1}$ in a time order $\tau_{\mit ss}$. The physics of this flow is a balance between drag against the bulk fluid and a force due to the gradient imposed by the chemical potential drop between the reservoir and the trap. Given the steady-state tension profile within the membrane and the reasonable assumption that the gradient occurs over a length (L) much larger than the critical wavelength ($\sim R_0$), the analysis of GNPS [@nelsontube95b] leads to a Rayleigh-like instability to undulations. This instability initiates near the laser spot where the chemical potential (or surface tension) is lowest, and propagates away from the spot to a point along the tubule at which the local surface tension falls below the critical tension $\Sigma_{cr}=\kappa/R_0^2$ which characterizes the instability. Typical growth frequencies are $\omega\sim 25\,\hbox{s}^{-1}$ [@nelsontube95b], which corresponds to times $\tau_{\omega}\sim 10^{-1}\hbox{s}$. We note that the experiments find a significant time delay of order seconds before the instability, [@mosesprivate], a still-unexplained observation. The estimate above for $\tau_{\mit ss}$ assumes a bare $2D$ compression modulus $B$, while GNPS (see [@helfrich84]) pointed out that $B$ undergoes significant softening at the lengthscale of the tubule, due to the thermal fluctuations at low surface tension, and estimated a decrease of up to three orders of magnitude. This, correspondingly, would increase the value of $\tau_{\mit ss}$ to $10^{-2}\,\hbox{s}$, so that an accurate quantitative calculation must include the dynamics of the increase in surface tension. This leads to, effectively, a smaller applied tension $\sigma$ and hence a slower propagation speed. This spirit was followed in the approach of Granek and Olami [@granek95]. Our interpretation assumes that the trap accommodates material by folding, or some other means. Our analysis suggests that the proper boundary condition should be a fixed surface tension $\bar{\Sigma}<\Sigma_0$ at the laser spot, where the laser shape determines $\bar{\Sigma}$ through Eq. (\[eq:barsig\]) and $\bar{\Delta}$ through Eqs.(\[eq:deltabar\],\[eq:deltabar0\]). This reservoir picture suggests that, upon turning off the laser, the system can revert to the original tubule by unfolding or, if severe topological changes have occurred (by, for example, budding in the laser spot or the creation of metastable ‘pearls’ as seen in the experiments), attain some other long-lived metastable state. Discussion ---------- In this work we have made several assumptions. The assumption that we can treat the anchoring globules as reservoirs presupposes that any damping processes retarding the transfer of lipid to and from the globules is negligible relative to other dynamical processes. We expect this to arise from the same source as the two-dimensional surface viscosity, which we have argued in Section 2.2 to be negligible. We have given a simplified picture of the scenario of trap buckling, where we take a single characteristic buckling wavevector, and treat the trap as uniform. This ‘single-mode’ approximation may be naive, and preliminary calculations suggest that the system is in fact less stable than this simple analysis would suggest [@pdofcmunpub]. There are also several other possible modes of instability which we have only mentioned but which could certainly play a role. We have also specified a boundary condition at the trap whereby the the lipid relaxes to its preferred area per head group, which seems reasonable but is not otherwise justified. Finally, we have made a local approximation for the variation of the surface tension so that we may use the results of GNPS. This applies for sufficiently long tubules, $L/R_0\gg 1$. Front propagation and the detailed effects of propagating into a spatially-varying medium have only been touched upon in our numerical treatment. This study still leaves much to be resolved; one important question is how to accurately treat the non-linear regime. This has been treated in different ways by Olami and Granek [@granek95] who considered the non-linear effect of removing surfactant from the membrane in the [absence]{} of a gradient, and by Goldstein [et al.]{} [@nelsontube95b], who added the correct non-linear terms in the bending energy to examine the propagation of the pearls. The primary new ingredients in our theory are (1) our treatment of the anchoring lipid globules as reservoirs and (2) our exploratory treatment of the role of pre-existing thermal fluctuations (noise) in determining the ‘front-like’ characteristics of the instability. Both Nelson and co-workers [@nelsontube95a; @nelsontube95b] and Olami and Granek [@granek95] ‘turn off’ the reservoir. In the latter case material is drawn out of the existing thermal fluctuations, while Nelson and co-workers attribute the area change primarily to the shape instability itself. Olami and Granek impose a constant flux boundary condition at the trap, while Nelson and co-workers impose a fixed chemical potential $-\bar{\Sigma}$ at the trap which, fairly rapidly, reduces the chemical potential everywhere to $-\bar{\Sigma}$. Our picture essentially gives the same boundary condition at the trap, but the treatment of the globule as a reservoir changes the qualitative picture dramatically. Our theory differs from previous theories in several respects, and there are many consequences which may be checked experimentally. Obviously, we expect flow when an instability develops. This could be visualized by, for example, fluorescence spectroscopy with a very dilute fraction of labelled lipids. The inhomogeneous surface tension implies that the local dispersion relation is also spatially-dependent, as in Fig \[fig:dispersion2\], which implies that the velocity of front propagation $v_f$ (which is proportional to $\omega^{\ast}$ [@nelsontube95b]) and characteristic wavenumbers should decrease farther away from the laser spot. Note that the characteristic wavelength changes very gradually compared to the speed of propagation, and as such would be more difficult to detect. It would also be interesting to see, experimentally, whether fluctuations are actually strong enough to destroy the front-like character, or whether two characteristic regimes exist in the experiments, as indicated in Fig. \[fig:noise\]. Finally, we mention that the opportunity of using laser pulses to control flow within lipid and other systems presents amusing possibilities and applications. It is a pleasure to thank E. Moses, R. Granek, P. Nelson, T. Powers, C.-M. Chen, S. Milner, and W. van Saarloos for helpful conversations and correspondence. This work was supported in part by NSF Grant No. DMR 92-57544 and by The Donors of the Petroleum Research Fund, administered by the American Chemical Society. [10]{} R. Bar-Ziv and E. Moses, [Instability and “pearling”states produced in tubular membranes by competition of curvature and tension]{}, Phys. Rev. Lett. [73]{} (1994) 1392. , [On the instability of jets]{}, Proc. Lond. Math. Soc. [ 10]{} (1879) 4. , [On the instability of a cylinder of viscous liquid under capillary force]{}, Phil. Mag. [34]{} (1892) 145. S. Tomotika, [On the instability of a cylindrical thread of a viscous liquid surrounded by another viscous fluid]{}, Proc. Roy. Soc. Lond. [A150]{} (1932) 322. R. Granek and Z. Olami, [Dynamics of Rayleigh-like instability induced by laser tweezers in tubular vesicles of self-assembled membranes]{}, J. Phys. II (France) [5]{} (1995) 1349. P. Nelson, T. Powers, and U. Seifert, [Dynamical theory of the pearling instability in cylindrical vesicles]{}, Phys. Rev. Lett. [74]{} (1995) 3384. R. E. Goldstein, P. Nelson, T. Powers, and U. Seifert, [Front propagation in the pearling instability of tubular vesicles]{}, J. Phys. II (France) [ 6]{} (1996) 767. J. H. Schulman and J. B. Montagne, [Formation of microemulsions by amino alkyl alcohols]{}, Ann. N.Y. Acad. Sci. [92]{} (1961) 366. W. van Sarloos, [Front propagation into unstable states]{}, Phys. Rev. [ A37]{} (1988) 211. L. Kramer, E. Ben-Jacob, H. Brand, and M. C. Cross, [Wavelength selection in systems far from equilibrium]{}, Phys. Rev. Lett. [49]{} (1982) 1891. L. D. Landau and E. M. Lifschitz, [Fluid Mechanics]{} (Pergamon, Oxford, 1959). E. Evans and D. Needham, [Physical properties of surfactant bilayer membranes: thermal transitions; elasticity; rigidity; cohesion; and colloidal interactions]{}, J. Phys. Chem. [91]{} (1987) 4219. W. Helfrich and R.-M. Servuss, [Undulations steric interaction and cohesion of fluid membranes]{}, Nuovo Cim. [3D]{} (1984) 137. M. B. Schneider and W. W. Webb, [Measurement of submicron laser beam radii]{}, Appl. Opt. [20]{} (1981) 1382. S. A. Safran, [Statistical Thermodynamics of Surfaces, Interfaces, and Membranes]{} (Addison-Wesley, Reading, MA, 1994). R. Bar-Ziv, T. Frisch, and E. Moses, [Entropic expulsion in vesicles]{}, Phys. Rev. Lett. [75]{} (1995) 3481. J. B. Buell and I. Catton, [Wavenumber selection in ramped Rayleigh-[B]{}énard convection]{}, J. Fluid Mech. [171]{} (1986) 477. L. Kramer and H. Riecke, [Wavelength selection in Rayleigh-[B]{}énard convection]{}, Z. Phys. B - Cond. Matt. [59]{} (1985) 245. H. Riecke and H.-G. Paap, [Perfect wave-number selection and drifting patterns in ramped Taylor vortex flow]{}, Phys. Rev. Lett. [59]{} (1987) 2570. E. Moses, private communucation. P. D. Olmsted and F. C. MacKintosh, unpublished (1996). [^1]: e-mail: [[email protected]]{}; permanent address, Dept. of Physics, University of Leeds, Leeds LS2 9JT, UK [^2]: If we wait long enough the trap will ‘fill up’ with surfactant and the chemical potential return to zero everywhere. However, in the present case of strong laser power the surface instability will have occurred by this time. See Section 3.3. [^3]: We may include this as, for example, the Stokes drag on a cylinder, which increases the right hand side of Equation \[eq:stress\] by of order 10% [@landaufluids]. [^4]: We do not include a term involving interaction with the laser, since we are interested in an instability at fixed particle number on the membrane.	ArXiv
--- abstract: 'We propose an approach of open-ended evolution via the simulation of swarm dynamics. In nature, swarms possess remarkable properties, which allow many organisms, from swarming bacteria to ants and flocking birds, to form higher-order structures that enhance their behavior as a group. Swarm simulations highlight three important factors to create novelty and diversity: (a) communication generates combinatorial cooperative dynamics, (b) concurrency allows for separation of timescales, and (c) complexity and size increases push the system towards transitions in innovation. We illustrate these three components in a model computing the continuous evolution of a swarm of agents. The results, divided in three distinct applications, show how emergent structures are capable of filtering information through the bottleneck of their memory, to produce meaningful novelty and diversity within their simulated environment.' author: - Olaf Witkowski - Takashi Ikegami bibliography: - 'witkowski.bib' title: \| How to Make Swarms Open-Ended?\ Evolving Collective Intelligence Through a Constricted Exploration of Adjacent Possibles --- OEE === Life has been evolving on our planet over billions of years, undergoing several major transitions which transformed the way it stored, processed and transmitted information. All these transitions, from multicellularity to the formation of eusocial systems and the development of complex brains, seem to lead to the idea that the evolution of living systems is open-ended. In other words, life appears to be capable of increasing its complexity indefinitely Another formulation of open-endedness, echoed by Standish [@standish2003open] and Soros [@soros2014identifying], is that open-endedness depends fundamentally on the continual production of novelty. Life keeps uncovering new inventions, in a process which never seems to stop. Since the 1950’s, open-ended evolution (OEE) has been a central topic of research for artificial life approaches to the fundamental principles of life. Soon after, John Von Neumann [@von1966theory] has contributed to the issue as well, with his early model of self-reproducing automata. Since 2015, a series of workshops have been taking place at Artificial Life conferences [@taylor2016open], the last of which[^1] was a launchpad for the present special issue. In general, an evolutionary system is considered to be open-ended when it is able to endlessly generate diverse novel entities of growing complexity. Engineering open-ended systems in the lab is not easy, and the main obstacle is that the designed evolutionary systems are subject to a thermodynamic drift making them collapse into equilibrium states. Once local optima are reached, they do not produce novelty anymore, which bounds their complexity and diversity. Innovation seems to emerge from collective intelligence, a phenomenon which refers to groups or networks of agents that develop together the ability to enhance the group’s cognitive capacity or creativity. This is reminiscent of the ongoing innovative process of science, which does not have any other fixed objective than the production of new knowledge, but makes discoveries mostly through accidents. Stuart Kauffman advocated for the idea of the adjacent possible, claiming that a biosphere can be viewed as a secular or long-term trend and it can maximize the rate of exploration of the adjacent possible of an existing organization [@kauffman2000investigations; @kauffman2003adjacent]. Ikegami et al. (2017) [@ikegami2017life] builds on that idea to explain how, in terms of evolutionary transitions [@MaynardSmithSzathmary1997], a new stage (e.g. multicellular oranism) of evolution may be produced without any information being passed on from the previous stage (e.g. from single cells). Rather, structural properties are assembled, producing a stepping stone to the next level of innovation. These structural properties of a collective group can be compared to a bottleneck that acts as a filter on several levels of the system, implementing computation that is not present in any of its parts. Part of this idea is analog to Tishby and Polani (1999) [@tishby1999information], where the information is squeezed through a bottleneck, to extract relevant or meaningful information from an environment. The resulting “filtered” information through the bottleneck, retains only the features most relevant to general concepts. In this paper, we present three “C” factors that we deem important for novelty and diversity. We then introduce a swarm model to study these factors, applied in three different studies. We conclude with a discussion on open-endedness at large, framing the three factors in terms of the emergence of collective intelligence in swarm simulations. Conditions for OEE ================== The OEE literature has proposed various conditions that are supposed to lead to the successful production of OEE in a system. Number of studies have attempted to formalize necessary conditions for OEE [@holland1999echoing; @conrad1970evolution]. Taking a recent example, Soros and Stanley (2014) [@soros2014identifying] suggest four conditions at the scale of single reproducing individuals in the system, which should each fulfill some nontrivial minimal criterion, be able to create novelty, act autonomously, and dispose of access to unbounded memory. Such papers have typically been proposing their own model, to demonstrate the importance of the hypothesized conditions for the emergence of OEE within it. However, most evolutionary algorithms seem to either converge very quickly to a solution, or get stuck in a confined area of the search space. Either way, they don’t seem to be able to intrinsically generate the amounts of complexity and novelty we find in nature, even at a scale. Although this failure of simulated evolution to match open-ended properties found in natural evolution may be explained by shorter timescales, in principle one would have expected decades of efforts, and increasingly larger resources poured into research in evolutionary computation, to have unlocked more of its potential to create novelty. However, even the latest technologies don’t seem to keep their inventivity. In general, promising models [@lenski2003evolutionary; @lehman2011evolving; @goodfellow2014generative; @greenbaum2016digital; @silver2017mastering] that manage to demonstrate at least a few phases transitions or creative leaps – not necessarily with evolutionary computation – seem to have one common denominator of containing several structural bottlenecks which filter relevant information through them, as a catalyst of creativity. What seems to be missing to achieve general OEE? We choose to emphasize three “C” candidates which we see as worth pursuing – Communication, Concurrency, and Complexity: (a) Communication: constricted information flows between parts of the systems allow for synergy and cooperation effects. (b) Concurrency: the creation of separate space and time scales requires concurrent, nondeterministic, asynchronous models. (c) Complexity: mere system growth can boost novelty and diversity. These are the three C-factors on which we choose to concentrate, in this paper. Next, will expand on each of them a little further, before proposing how to apply them in concrete models. We will now expand on these three points, before presenting concrete examples, with Study 1, 2 and 3. Communication {#communication .unnumbered} ------------- This first point addresses synergies and coordination between components of the systems, using information transfers. One well known example of OEE is combinatorial creativity in human language, where syntactical rules are capable of producing infinite well-formed structures using recursion, thus making the number of potential sentences unbounded [@hauser2002faculty]. Although these may seem slightly dated remarks, at the advent of language studies based on artificial life systems [@kirby2002natural], it is promising to focus on the cultural layer of dynamics, that lives on top of the main layer of entities. For example, in the case of web services (social tagging networks), we can analyze how combinatioral complexity is effective in evoking OEE. In cellular automata, one may want to study the interactions between gliders or other patterns. In artificial chemistry, one may want to look at information flows between types of molecules or replicators. In agent-based modeling, perhaps establishments of protocols between agents or groups of agents can become a factor to focus on. Communication naturally adds relevant computing filters on unexploited information flows, effectively increasing the bandwidth of useful information flows within the system per clock cycle, communication offers a layer for metadynamics at a different timescale from the first-order dynamics. This induces a separation of timescales, thus doubling the system’s capacity to implement learning mechanisms. Designing information to circulate between a sub-entities of the system forces the creation of more structural bottlenecks. We propose information exchanges as a central mechanism promoting open-ended evolution. From information flows in groups of individuals, a system can boost its own production of creativity to achieve indefinite complexity. Examples are detailed in Study 1 and 2, below. Concurrency {#concurrency .unnumbered} ----------- In many situations, a system cannot scale up to larger space-time scales as it is. We need to add some ingredient to make it work in the larger scales. One such remedy is to give it asynchronous updates. Removing the global clock is needed to make larger systems function consistently without constantly checking local consistency. On the other hand, we know that cellular automata (CA) tend to lose their complexity by adopting asynchronous dynamics. Yet asynchrony is an original natural phenomena difficult to bring to artificial systems. According to Dave Ackley [@ackley2015artificial], models should be indefinitely scalable, ruling out deterministic, synchronous models (such as simple cellular automata), and suggesting nondeterministic, asynchronous ones. Bersini et al. (1994) [@bersini1994asynchrony] proposed that asynchronous rather than synchronous updating may be key factor in inducing stability in simulations. Although they were examining a variant of cellular automata, their results, based on an analysis of the Lyapunov exponent, indicated the responsibility of asynchrony for sensitivity of the update function. Ackley and Ackley (2015) [@ackley2015artificial] propose to use asynchrony. The concurrency is also closely related to the ability a system has to evolve separate timescales. Although highly contingent on other properties of a simulation, the capacity to develop heterogeneous timescales often constitutes a barrier to producing intrinsic novelty. Researchers used to separate lifetime learning and evolutionary learning, as two distinct mechanisms [@nolfi1990learning]. However, the effects of accumulating and filtering information into and out of one system’s memory occurs at a much more continuous range of different timescales. In nature, from phenotypic plasticity, to maternal effects, to sexual selection, or to gene flow, many events have their time scales intricately interlaced. We will address this in particular in Study 1 and 2. Complexity {#complexity .unnumbered} ---------- We have no grounds to argue that nature is its own only possible realization, since there would be no satisfactory explanation for that. One important feature of nature though, is its complexity, which can translate into both system size and landscape complexity. In the simplest of all cases, complexity can be reached merely with large population sizes. Ikegami et al. (2017) [@ikegami2017life] proposed that large groups of individuals, given the right set of structural characteristics, may be the main driver for emergence. They discussed this hypothesis in relation to large-scale boid swarm simulations [@reynolds1987flocks], in which the nucleation, organization and collapse dynamics were found to be more diverse in larger flocks than in smaller flocks. Collective behaviors can be qualitatively different by increasing the number of agents, i.e. a colony or group size. In the actual observation, e.g., the individual bees change their behaviors depending on the colony size. Also the fish change their performance of sensing the environmental gradient depending on the school size. In previous works [@mototake2015simulation], we simulated half a million birds flock using a boids model [@reynolds1987flocks; @toner1998flocks] and found that qualitatively different behavior emerges when the total number of individuals exceeds a few thousands or so. Flocks of different sizes and different shapes interact to diminish some flocks but to generate new ones. Different types of fluctuation become dominant in different size of flocks. A correlation of the local density fluctuation becomes dominant in the larger size flocks and that of the velocity fluctuation dominates in the smaller size flocks. An example of that is offered in Study 3. Stretching the argument on size, environmental complexity is definitely necessary to a certain extent to create complex behaviors. Only with richer environments, encompassing complex distributions of energy resources and ways for systems to survive, can emerging individuals explore a rich set of strategies and increasingly increasingly complex solutions. As mentioned earlier, evolutionary landscapes have become an important concept in biology to analyze the dynamics at play in an ecosystem. The picture to have is the one of a unit of selection (e.g. a gene, among many other options) being represented by a point in a multidimensional search space. That space is typically given as many dimensions as there are degrees of freedom for the entity to vary and evolve in the space (e.g. combination of nucleotide sequences). The search space is mapped onto an additional dimension, which is usually the reproductive success, or fitness. The shape of the fitness across all degrees of freedom of the system have a strong impact on the dynamics that it can achieve. Malan et al. (2013) [@malan2013survey] identifies eleven characteristics of fitness, which make them more or less difficult to solve. These characteristics include the degree of variable inter-dependency, noise, fitness distribution, fitness distribution in search space, modality, information to guide search and deception, global landscape structure, ruggedness, neutrality, symmetry, and searchability. In evolutionary systems, richer environments, benefiting of a complex distribution of energy sources and ways for systems to survive, give rise to richer sets of pathways. The larger the search spaces, the more complex fitness functions are potentially evolved. Another way is to make the environment a more complex function of time, which the agents will need to learn in order to extract more energy from it. In Study 1 and 2, we present results suggesting that simulations should be ensured to provide sufficient system complexity in terms of the environment of agents. Simulation time and memory, though not mentioned yet, are important components to consider. Computationally, the whole course of evolution on Earth is like a single run of a single algorithm that invented all of nature, and seems like it will never end. One obvious difference is the size of the systems, which might be the missing element to get ever-greater emergent complexity and novelty through very long time. However, we do not insist on this component in this paper, as we consider trivial that a system with too low computational power will not be able to achieve OEE to any extent. Similarly, there is point to be made about endo-OEE, producing novelty from within, against exo-OEE, which makes use of input from outside the system. Picbreeder [@secretan2008picbreeder], for example, explicitly requires external human input to function, which makes it a debatable generator of OEE. Nevertheless, OEE is not about new information, but rather inventions achieved by the system. In that respect, swarms are a promising model: without increasing ensemble size, they let us focus on how coordination patterns self-organize, generating intrinsic novelty. To give another example, even increasing the number of neurons in a neural network still requires neurons to differentiate themselves, and create coordinated networks before they get to foster innovative ideas. We exemplify the importance of size and complexity in Study 1 and 3, while discussing how to make simulations parallelizable, to save considerable amounts of time and memory by distributing them over many machines. Concurrent evolutionary neural boids model ========================================== The model we choose to present puts together the abovementioned series of features, as a means to promote the system’s open-endedness. We give some details here, and will go over the details of several applications of it in the next section. The evolutionary system is an agent-based simulation, based off Reynolds’ boids model [@reynolds1987flocks]. The boids model was based on simple rules computed locally, allowing to simulate flocks of agents moving through artificial environments. As with most artificial life simulations, boids showed emergent behavior, that is, the complexity of boids arises from the interaction of individual agents adhering to a set of simple rules of separation (steer to avoid crowding local flockmates), alignment (steer towards the average heading of local flockmates, and cohesion (steer to move toward the center of mass of local flockmates). In our model, like in Reynolds’ model, the population of agents moves around in a continuous three-dimensional space, with periodic boundary conditions (Figure \[fig:worldsim\]). However, instead of using fixed rules to control the boids’ motion, we allow agents to evolve their own controllers through concurrent evolutionary computation. Each agent, instead of responding to simple rules, is controlled by its own neural network. The parameters of the neural network are encoded in a genotype vector, which determines the individual’s sensorimotor choices at each moment in time. This corresponds to standard evolutionary robotics methodologies [@nolfi2000evolutionary], although we introduce the following variant. The genotype is evolved through the course of the simulation, via a continuous variant of an evolutionary algorithm [@witkowski2014a], that is, agents with high level of fitness are allowed at any point to replicate with mutation in the middle of the running simulation. This model also builds up on prior work on the effect of self-organized inter-agent communication and cooperative behavior on the performance of agents to solve tasks [@prokopenko2006evolving; @olson2013predator]. Previous research has shown the difficulty of using communication channels [@rasmusen1994games; @mitri2013using] but showed cooperative value of information transfers [@witkowski2016emergence]. This will be complemented by the results from previous information-theoretic analyses of learning systems, which managed to shed light on the role of information flows in learning [@tishby2015deep; @lin2017does; @tishby2011information]. Agents are given a certain energy, that also acts as their fitness. This will be specific to the study cases. Each agent comes with a set of 12 different sensors. The neural network (represented on Figure \[fig:ann\]) takes the information from those sensors as inputs, in order to decide the agent’s actions at every time step. The possible actions amount to the agent’s motion, and in the specific variant shown here, a Prisoner’s Dilemma action (cooperate or defect), as well as two output signals. The architecture is composed of a 12 input, 10 hidden, 5 output, and 10 context neurons connected to the hidden layer (see Figure \[fig:ann\]). The agents’ motion is controlled by $M_1$ and $M_2$, outputting two Euler rotation angles: $\psi$ for pitch (i.e. elevation) and $\theta$ for yaw (i.e. heading), with floating point values between $0$ and $\pi$. Even though the agents’ speed is fixed, the rotation angles still allow the agent to control its average speed (for example, if $\psi$ is constant and $theta$ equals zero, the agents will continuously loop on a circular trajectory, which results in an almost-zero average speed over 100 steps). The outputs $S_\text{out}^{(1)}$ and $S_\text{out}^{(2)}$ control the signals emitted on two distinct channels, which are propagated through the environment to the agents within a neighboring radius set to $50$. The choice for two channels was made to allow for signals of higher complexity, and possibly more interesting dynamics than greenbeard studies [@gardner2010]. The received signals are summed separately for each direction (front, back, right, left, up, down), and weighted by the squared inverse of the emitters distance. This way, agents further away have much less impact on the sensors than closer ones do. Every agent is able to receive signals on the two emission channels, from 6 different directions, totalling $12$ different values sensed per time step. For example, the input $S_\text{in}^{(6,1)}$ corresponds to the signals reaching the agent from the neighbors below. ![ [Architecture of the agent’s controller.]{} The network is composed of 12 input neurons, 10 hidden neurons, 10 context neurons and 5 output neurons.[]{data-label="fig:ann"}](witkowski-fig2-ann.png){width=".7\textwidth"} The evolution is performed continuously over the population. Agents with negative or zero energy are removed, while agents with energy above a threshold are forced to reproduce, within the limits of one infant per time step. The reproduction cost is low enough, considering the threshold, to not put the life of the agent at risk. Study cases =========== We go over the application of this model in three selected examples of studies. Each of them highlights a specific property for OEE. The first model shows how agents can form patterns to accelerate their search for energy, distributed over an n-dimensional space, collaborating via local signaling with their neighbors. The second study shows the invention of dynamical group strategies in a spatial prisoner’s dilemma, allowing for specific cooperation effects. The third example shows the impact of growth on the emergence of noise-canceling effects. OEE via collective search based on communication ------------------------------------------------ Since Reynold’s boids, coordinated motion has often been reproduced in number of artificial models, but the conditions leading to its emergence are still subject to research, with candidates ranging from obstacle avoidance to virtual leaders. The relation of spatial coordination and group cooperation has long been studied in game theory and evolutionary biology. We here apply our model of agents exchanging signals and moving in a three-dimensional environment, to a task of dynamical search for free energy in space [@witkowski2014a; @witkowski2016emergence]. Each agent’s movements are controlled by artificial networks, evolved through generations of an asynchronous selection algorithm. At the term of the evolution, the agents are able to communicate to produce cooperative, coordinated behavior. Individuals develop swarming using only their ability to listen to each other’s signals. The agents are selected based on their performance at finding invisible resources in space giving them fitness. The agents are shown to use the information exchanged between them via signaling to form temporary leader-follower relations allowing them to flock together. The swarmers outperform the non-swarmers at finding the resource, thus reaching a neutral evolutionary space which leads to a genetic drift. This work constructs an adaptive system to evolve swarming based only on individual sensory information and local communication with close neighbors. This addresses directly the problem of group coordination without central control, without being aware of the position direct neighbors, nor any use of the substrate where to deposit information (stigmergy) [@hauert2009evolved]. The approach has also the advantage of yielding original and efficient swarming strategies. A detailed behavioral analysis is then performed on the fittest swarm to gain insight as to the behavior of the individual agents. The results show that agents progressively evolve the ability to flock through communication to perform a foraging task. We observe a dynamical swarming behavior, including coupling/decoupling phases between agents, allowed by the only interaction at their disposal, that is signaling. Eventually, agents come to react to their neighbors’ signals, which is the only information they can use to improve their foraging. This can lead them to either head towards or move away from each other. While moving away from each other has no special effect, moving towards each other, on the contrary, leads to swarming. Flocking with each other may lead agents to slow down their pace, which for some of them may keep them closer to a food resource. This creates a beneficial feedback loop, since the fitness brought to the agents will allow them to reproduce faster, and eventually multiply this type of behavior within the total population. The algorithm converges to build a heterogeneous population, as shown on Figure \[fig:treehorizontal\]. The phylogeny is represented horizontally in order to compare it to the average number of neighbors throughout the simulation. The neighborhood becomes denser around iteration $400k$, showing a higher portion of swarming agents. This leads to a firstly strong selection of the agents able to swarm together over the other individuals, a selection that is soon relaxed due to the signaling pattern being largely spread, resulting in a heterogeneous population, as we can see on the upper plot, with numerous branches towards the end of the simulation. ![Top: average number of neighbors during a single run. Bottom: agents phylogeny for the same run. The roots are on the left, and each bifurcation represents a newborn agent.[]{data-label="fig:treehorizontal"}](witkowski-fig3-phylogeny.jpeg){width=".9\textwidth"} In this scenario, agents do not need extremely complex learning to swarm and eventually get more easily to the resource, but rather rely on dynamics emerging from their communication system to create inertia and remain close to goal areas. ![ Two principal components of a PCA on the genotypes of all agents of a typical run, over one million iterations. Each circle represents one agent’s genotype, the diameter representing the average number of neighbors around the agent over its lifetime, and the color showing its time of death ranging from bright green (at time step $0$, early in the simulation) to red (at time step $10^6$, when the simulation approaches one million iterations). []{data-label="fig:pca"}](witkowski-fig4-pca.png){width=".85\textwidth"} The simulated population displays strong heterogeneity due to the asynchronous reproduction schema, which can be seen in the phylogenetic tree (Figure \[fig:treehorizontal\]). The principal component analysis plotted on Figure \[fig:pca\] shows a large cluster (left side) in addition to a series of smaller ones (right side). The genotypes in the early stages of the simulation belong to the right clusters, but get to the left cluster later on, reaching a higher number of neighbors. The plot shows a diverse set of late clusters, which translates to numerous distinct behaviors in the late stage of the simulation. Such heterogeneity may suppress swarming but the evolved signaling helps the population to form and keep swarming. The simulations do not exhibit strong selection pressures to adopt specific behavior apart from the use of the signaling. Without high homogeneity in the population, the signaling alone allows for interaction dynamics sufficient to form swarms, which proves in turn to be beneficial to get extra fitness. These results represent an improvement on previous models using hard-coded rules to simulate swarming behavior, as they are evolved from very simple conditions. Our model also does not rely on any explicit information from leaders, like previously used in part of the literature [@cucker2008flocking; @su2009flocking]. It does not impose any explicit leader-follower relationship beforehand, letting simply the leader-follower dynamics emerge and self-organize. In spite of being theoretical, the swarming model presented in this paper offers a simple, general approach to the emergence of swarming behavior once approached via the boids rules. This simulation improves on previous work because agents naturally switch leadership and followership by exchanging information over a very limited channel of communication. Finally, our results also show the advantage of swarming for resource finding. It’s only through swarming, enabled by signaling behavior, that agents are able to reach and remain around the goal areas. In terms of cooperation, this model exemplifies a case of multilevel selection theory [@wilson1994reintroducing; @traulsen2006evolution], which models the layers of competition and evolution, within an ecological system. Our system shows the emergence of different levels which function cohesively to maximize reproductive success. The fitness value of the group level dynamics outweighs the competitive costs, resulting in individuals constantly innovating in ways they are cooperating in a non-trivial way, to create behaviors which are not centrally coded for. This study shows swarming dynamics emerging from a communication system between agents, immersed in a simulated environment with spatial distribution of energy resource. The concurrent evolution scheme, running at the same time as the simulation itself, led to decentralized leader-follower interactions, which allowed for collective motion patterns, which in turn significicantly improved the groups’ fitnesses. This model encodes the stochastic evolution of a controller that maps sensory input onto motor output, to optimize the performance on a task, as framed broadly by Nolfi and Floreano (2000) [@nolfi2000evolutionary]. We capture the fight against a difficult wall [@schmickl2016sooner], which simulations typically fail at because they suddenly hit a so-called “wall of complexity”: trivial tasks are solved easily, but it’s hard to jump to solving difficult ones. If we take the no-free-lunch argument from Wolpert and Macready (1997) [@wolpert1997no] that no optimisation algorithm is at all times superior to others, it is natural that the more specific the algorithm, the more it is likely to fail with new problems. Our results suggest that novelty can be produced by the asynchronous evolution of a heterogeneous community of agents, which through their mixture of strategies, may achieve open-ended, uninformed learning. The heterogeneity present in the model also offers an extension to the advantages of particle swarm optimization (PSO) [@eberhart1995new]. While PSO only offers one unique objective function to optimize, each agent in the swarm effectively runs its own function, which are combined into a swarm behavior. Although these results suggest open-endedness, it is worth noting we do not bring a proof that the phenomenon is truly open-ended, which may require the emergence of ever-complexifying communication, or an uninterrupted sequence of evolutionary innovations. The information flows were a focus of the original work [@witkowski2016emergence]. From these flows, one can notice three main bottlenecks. The evolutionary computation contains a bottleneck effect, as a result from the selection based on the agents’ performance on the task. Another bottleneck can be found between the sensory inputs of each agent, and its motor outputs, as the neural controller acts as a filter for the information. The agents’ signaling also naturally contains a bottleneck effect, as the information transmitted from agent to agent is constrained by the physical communication bandwidth. The combination of these three bottlenecks allows for relevant information to be filtered into the swarm, which is able learn certain behaviors (see also next section). OEE via cooperative flocking ---------------------------- The evolution of cooperation is studied in game theory, and stretches have been made to include spatial dimensions. This problem is often tackled by using simple models, such as considering interactions to be a game of Prisoner’s Dilemma (PD). We examined a variation of the model with a distinct fitness function in a separate study, based this time on the agents playing a spatial version of the Prisoner’s Dilemma [@witkowski2014b]. We study the impact of the movement control on optimal strategies, and show that cooperators rapidly join into static clusters, creating favorable niches for fast replications. It is also noted that, while remaining inside those clusters, cooperators still keep moving faster than defectors. The system dynamics are analyzed further to explain the stability of this behavior. This work presents, in an even more explicit fashion than the previous study, a model aimed at showing emergent levels of selection for cooperative behavior [@wilson1994reintroducing]. At every time step, agents are playing a N-player version of the prisoner’s dilemma with their surrounding, meaning that they make a single decision that affects all agents around them. They get reward and/or punishment based on the number of cooperator around them. Their decision is one of the outputs of their neural network. Effectively, the payoff matrix we used is an extension of Chiong and Kirley (2012) [@chiong2012], where we added the distance to take into account the spatial continuity. Based on the outcome of the match, agents can choose a new direction, which is similar to leaving the group in the walk away strategy [@aktipis2004], the main difference being that, in our case, it is also possible for groups to split. It is also similar in another aspect: there is a cost to leaving a group, as a lone agent may need time to meet others. At the beginning of each run, the environment is seeded with random agents. Since all weights in their neural network are set at random, roughly half of the agents initially choose to cooperate while the other half choose to defect. This leads to a fast extinction of cooperators, until approximately 50000 time steps), until a group emerges strong enough to survive. The second phase follows, in which cooperators are quickly increasing in number due to the autocatalytic nature of this strategy. A third step happens eventually, where defectors invade the cluster, followed either by the survival of the cluster due to cooperators running away or a reboot of the cycle. In case of survival, oscillations in the proportion of cooperators can be observed. However, this phenomenon is averaged away over multiple runs, since period and phase of the oscillations are not correlated from one experiment to the other. Were a defector to appear near a cluster of cooperators, the cluster would react by “reproducing away”. However, the chances to be overtaken by the defectors is much higher than in the dynamic case. From this three-dimensional model of agents playing the Prisoner’s Dilemma, the first result is that cooperators, when they are present, quickly evolve to form clusters as they represent a favorable pattern. The clustering behavior can be interpreted as a degenerated version of the simulations presented above, since the cooperating agents present the same capacities of information exchange as that model. We note that this solution is evolved through a longer time scale, as it is not always viable locally, depending on the distribution and behavioral thresholds of defectors. While the clustering itself can be expected, it is interesting to observe that their overall movement rate is still higher than defectors. This is even more surprising considering that those clusters do not seem to move fast. Instead, analysis shows that cooperators are moving quickly inside the cluster, which may be a way to adapt to an aggressive environment. In addition, comparison with the static case showed that movement made the emergence of cooperators harder, but more stable in the long run. Since it is harder for defectors to overtake a cluster of cooperators, our systems often show a soft bistability, meaning that they will eventually switch from one state to the other. It is even possible to observe a sort of symbiosis, where cooperators are generating more energy than necessary, which is in turn used by peripheral defectors. In this case, replacement rates allow cooperators to stay ahead, keeping this small ecosystem stable. This cohesion among cooperators seems to be enhanced by signaling, even though signals might attract defectors. Additional investigation on the transfer entropy, for instance, could be a promising next step. Another result is found in the choice of actions, generated by the neural networks without consideration of the past actions. We notice the emergence of a dynamical memory effect, that otherwise requires to be encoded in each agent, here emerging from the agents’ motion in space. Since the Prisoner’s Dilemma game has become a common model used in evolutionary biology to study the outcomes depending on the costs that characterize an ecosystem, this model, with a fitness based on the results of such game, showed the emergence of spatial coordination based on a the exchange of signals between agents. The signals remained very simple, and the environment was fixed in time. This model’s evolutionary computation reached solutions composed of different parts, including soft bistable strategies, different radiuses of clusters, as well as the use of dynamical patterns to improve their fitness. The solutions were also distributed over different timescales. The communication between agents also allowed them to converge on these behaviors more quickly. These elements refer back to our 3-C arguments earlier, for the discovery of novel solutions to a simple PD game. Lastly, we note that many different neural architectures may coexist as only a part of the neural architecture is used to implement flocking. This neural heterogeneity is something we’d like to insist in the context of OEE. Additionally, communication is important to filter out the neural architecture heterogeneity, which potentially holds the heterogeneity in a community (i.e. agents can stay in a community if they can communicate to each other). The communication may therefore indirectly help preserving the heterogeneity. OEE via large scale swarms -------------------------- Studying flocking models can also lead to the emergence of OEE, by focusing on emergent phenomena as macroscopic layers of patterns and structures that appear as a result of cooperative phenomena between autonomously behaving elements. A group of elements creates a self-organizing structure, which governs the individual micro rules and creates a new macro structure. Therefore, consecutive micro–macro recurrent self-organization is defined as an emergent phenomenon. Here, we describe the contribution of the same swarming simulation, scaled up, showcasing the effect of size on emergence of open-endedness [@drozd2016critical]. Before that, we start by presenting a degenerated version of that model, which shows large-scale dynamics in the less computationally costly case of agents that don’t preserve any internal state other than position and velocity [@ikegami2017life]. Starting with this simpler stateless model, we observe a noticeable change when the total number of boids increases from 2048 to 524,288, while the density is kept constant (Figure \[fig:largeboids\]). In order to compute large swarming behaviour, we parallelized the computational steps using the general-purpose computing on graphics processing units (GPGPU) method. The next step was to extend it to stateful agents. ![Visualization of swarming behavior, simulated by a large scale stateless boids model [@ikegami2017life]. The total number of boids in each panel is (a) 2048, (b) 16 384, (c) 131 072 and (d) 524 288, respectively. Some flocks are composed of a very large number of boids with narrow filament patterns. The initial velocity of each boid is set at random, and the density of the total number of boids is kept constant at 16,384 (number per cubic unit). The minimum and the maximum speed are set at 0.001 and 0.005 (unit per step), respectively. []{data-label="fig:largeboids"}](witkowski-fig6-largeboids.jpg){width=".7\columnwidth"} We explore the effect of reaching a critical mass, and how it impact the efficiency of the swarm’s foraging behavior. In particular, we study the problem of maintaining the swarm’s resilience to noisy signals from the environment. To do so, we look at stateful boids, i.e. moving agents controlled by neural network controllers, which we evolve through time in order to explore further the emergence of swarming, like in the previous two sections. However, we now ground our model in a more realistic setting where information about the resource location made partly accessible to the agents, but only through a highly noisy channel. The swarming is shown to critically improve the efficiency of group foraging, by allowing agents to reach resource areas much more easily by correct individual mistakes in group dynamics. As high levels of noise may make the emergence of collective behavior depend on a critical mass of agents, it is crucial to reach in simulation sufficient computing power to allow for the evolution of the whole set of dynamics. Because this type of simulations based on neural controllers and information exchanges between agents is computationally intensive, it is critical to optimize the implementation in order to be able to analyze critical masses of individuals. In this work, we address implementation challenges, by showing how to apply techniques from astrophysics known as treecodes to compute the signal propagation, and efficiently parallelize for multi-core architectures. The results confirm that signal-driven swarming improves foraging performance. The agents overcome their noisy individual channels by forming dynamic swarms. The measured fitness is found to depend on the population size, which suggests that large scale swarms may behave qualitatively differently. The minimalist study presented in this paper together with crucial computational optimizations opens the way to future research on the emergence of signal-based swarming as an efficient collective strategy for uninformed search. Future work will focus on further information analysis of the swarming phenomenon and how swarm sizes can affect foraging efficiency. In this model, we specifically focus on the addition of noise to the food detection sense that the agents possess, and hypothesize that it can be overcome by the emergence of a collective behavior involving sufficiently large groups of agents. Many systems, from atomic piles to swarms, seem to work towards preserving a precarious balance right at their critical point [@bak2013nature]. An atomic pile is said to be “critical” when a chain reaction of nuclear fission becomes self-sustaining. A minimal amount of fissionable material has to be compacted together to keep the dynamics from fading away. The notion of critical mass as a crucial factor in collective behavior has been studied in various areas of application [@marwell1993critical; @oliver2001whatever]. Similarly, the size of the formed groups of agents may be crucial, in order to reach a critical mass in swarms, enough to overcome very noisy environments. Part of the focus will therefore be on the optimization of the computer simulation itself, as large-scale swarms may qualitatively differ in behavior from regular-sized ones. The model extends the original setup described before, which proposed an asynchronous simulation evolving a swarming behavior based on signaling between individuals. However, unlike the original model, where the individuals don’t perceive directly either the food patches or the other agents around them, here we give a sense of vision to every agent, allowing them to detect nearby resources. However, we add a high level of noise to make this information highly imperfect. We used an agent-based simulation to show how signal-driven swarming, emerging in an evolutionary simulation such as in Witkowski and Ikegami (2014) [@witkowski2014a], allow agents to overcome noisy information channels an improve their performance in a resource finding task. Our first contribution is the very introduction of noise, demonstrating that the algorithm performs well against noises filling up channels of information almost up to their full capacity, in the inputs of agents. The individuals, by means of a swarming behavior helped by basic signaling, manage to globally filter out the noise present in the information from their sensory inputs, to reach the food sites. We proposed a hierarchical method based on the Barnes-Hut simulation in computational physics and its parallel implementation. We achieved a performance improvement of a few orders of magnitude over the previous implementation [@witkowski2014a]. This implementation is crucial to achieve the simulation of a sufficient number of agents to test for large-scale swarms (i.e. involving a very large number of individuals), which have been suggested to generate qualitatively different dynamics. The optimization of the fitness acquired by phenotypes using efficient patterns of behavior (motion and signaling), which themselves are encoded in the weights of agents’ neural networks. The real optimization therefore occurs at the higher level of the darwinian-like process in the genotypic search space. Efficient genotypes are selected by the asynchronous genetic algorithm throughout a simulation run. We observed that signaling improves the foraging of agents (see Figure \[fig:boidsefficiency\] for plots from Drozd et al. [@drozd2016critical] of efficiency or fitness against simulation time), the average resource retrieved per agent per iteration as a measure of the population’s fitness. Without noise, the agents using signaling are less efficient than their silent counterpart, which we found is not due to the cost of signaling, but rather because of the excess of noise brought by the signal inputs. The difference remains very small between signaling and non-signaling agents. ![Agents’ efficiency plot with and without signal, from a stateful (neural-network-controlled) boids model in the original work [@drozd2016critical], with mean (central line) and standard deviation range (area plot) over 10 runs. The plots correspond to noiseless (top), constant noise 20 (middle), and constant noise 40 (bottom), respectively.[]{data-label="fig:boidsefficiency"}](witkowski-fig7-efficiency.jpeg){width=".7\columnwidth"} We find however that from a certain noise level, the cost to signal is fully compensated by the benefits of signaling, as it helps the foraging of agents. The average fitness becomes even higher as we increase the noise level, which suggest that the signaling behavior increases in efficiency for high levels of noise, allowing the agents to overcome imperfect information by forming swarms. We also observe scale effects in the influence of the signal propagation on the average fitness of the population. For a smaller population, only middle values of signal propagation seem to bring about fitter behaviors, whereas this is not the case for larger sizes of population. On the contrary, larger populations are most efficient for lower levels of signal propagation. This may suggest a phase transition in the agents’ behavior for large populations, eventually in the way the swarming itself helps foraging. Understanding criticality seems strongly related to a broad, fundamental theory for the physics of life as it could be, which still lacks a clear description of how it can arise and maintain itself in complex systems. The effects of criticality have recently been investigated futher by one of the authors, using a similar setup [@khajehabdollahi2018critical]. The results showed exploratory dynamics at criticality in the evolution of foraging swarms, and the tension between local and global scale adaptation. Through this work, by increasing the number of simulated boids that maintain their own states, we may introduce more than the mere number. By allowing for many information exchanges between computing agents, the simulation can effectively take leaps of creativity. In Stanley and Lehman’s 2015 book [@stanley2015greatness], objective functions are presented as a distraction, as novelty and diversity might not be achieved by hard-coding the arrival point. Here, in contrast, we have many evolvable objective functions cooperate in reaching a solution, as a stepping stone to reach the search for novelty. By letting the swarm grow, we see the emergence of collective intelligence, which corresponds to the invention of signal-based error correction. By exchanging signals, the agents are able to correct the error induced by the noise we injected in the simulation. Like for the large scale boids simulation, the invention happens after a critical mass of agents is reached, suggesting similar dynamics with stateful agents. Discussion ========== OEE is the everlasting innovative processes found in human technologies and biological evolutions, and we barely observe open-endedness outside these examples. Yet, some artificial systems demonstrate close-to-OEE phenomena, which we have discussed earlier in this paper. Achieving real OEE remains an open challenge, and at this point all works in the literature works fall short of that objective. Although that may be the case with the swarm models presented above, it was one of our goals to emphasize the importance of maintaining the evolvability of a system. In an adversarial game theoretical setup for example, reaching an ESS or a local attractor may keep the system from inventing new solutions. In this sense, explicitly stopping the system from learning too much may allow the system to avoid being stuck in such attractor, and possibly keep innovating forever. In this paper, we propose collective intelligence as a driving force towards open-ended evolution, suggesting that collective groups can develop the ability to be more innovative. Instead of aiming at optimizing one fixed objective function, a collective swarm of agents works with as many competing objectives as there are agents in the swarm. Through information exchanges between a certain number of agents, these objectives, embodied in the agent’s behaviors, can collaborate to implement a search for novelty. All agents contribute to the search in behavioral space, as one whole organization, by exploring the adjacent possible. Each novel discovery in the system, or emergent level of organization, can be reached from an adjacent state where the system was previously. The way one moves from one point to the next, which should retain information accumulated in the past, is constrained by the structure of the swarm, in a bottleneck effect. We discuss several instances of bottlenecks in this paper. One is a task or environmental condition which each agent must overcome. In the case of foraging environment, organizing swarming turned out to be a critical step. So it became a major transition from non-swarming to swarming agents. Swarming behavior was obtained by organizing a hill side function by the neural controller. After the swarming behavior has been achieved, other properties (e.g. individual pattern) start to evolve. So for the task, swarming was a necessary behavior to organize and was a bottleneck for the entire evolutionary process. In other words, OEE emerges by setting up a right environmental condition. In case of a game-theoretic situation, such as Study 2, the communication system among all agents constituted a bottleneck to achieve mutual cooperation. With the emergence of niche construction, the door opens for regulating mechanisms such as cooperation, reciprocal altruism or social punishment, to get implemented. In this example, the OEE in terms of the invention of cooperation mechanisms, can only evolve as a secondary structure once the swarming structure is already established. In the case of large swarm models, the bottleneck is twofold. One is scale itself and also its CPU resource. We discussed the evolution of swarm by increasing its size, showing there is a critical size where the different kinds of fluctuation dominates in larger flocks (i.e. heading direction fluctuation to the density fluctuation). If such a transition occurs at the larger size simulation (we expect it can happen in each 3-4 order difference in sizes), we say that OEE is caused by increasing the size. In addition to this point, 3D swarm models require a huge computational power and we need to elaborate programming for a large scale systems. In Study 3, each boid has a list of neighbors and it is updated periodically. This speeds up the calculation of the distance from the one to its neighboring boids. In study 1, each bird can listen to the sound sent unidirectionally from the other boids, so that we don’t have to calculate the exact distance. Real birds will never measure the distance to the other individuals. So measuring the distance is an unnecessary bottleneck due to the computational model. Here the OEE is the new computational techniques to overcome this computational bottlenecks. The computation of a swarm displays a bottleneck effect, in the sense that the emergent properties of the swarm and its embodiment in a simulated environment may constrain the way the information (communication, lineage information) flows within the system, and the way relevant information (strategies, motion patterns) is progressively retained[^2] through time in its structure. Nevertheless, the simplicity of such information flows may be limiting, more complex information transfer protocols may need to emerge from bottlenecks to bootstrap OEE. For open-endedness, bottlenecks are crucial to, perhaps counterintuitively, act against learning. We observe examples of such bottlenecks in systems like Picbreeder [@secretan2008picbreeder], where one must find a way to avoid the system from assuming that the current apparent goal is the ultimate goal, as this would preclude further innovations. Picbreeder-like systems present similarities with our signal-based swarms, as they have communication between many agents filter information to let innovations come about. As suggested in the beginning of the paper, bottlenecks can be caused by different components: an explicit communication system, a concurrent evolutionary system, and a greater complexity. These three components are highlighted in the studies described above, and we propose them as the principal ones to create novelty and heterogeneity in solutions. First, the communication between agents is shown to catalyze swarming and cooperation strategies. In previous work of turn-taking interaction between two agents installed with neural networks [@iizuka2003adaptive], we noticed that performing democratic turn-taking offers novel styles of motion evolutionarily. Accordingly, here, the local interactions between agents in a flock allow for the swarm to take particular shapes (Study 1), invent an explicit cooperative protocol (Study 2), and implement a noise-canceling policy (Study 3). To reach OEE, perhaps more than mere signaling, higher complexity levels of language may need to emerge. Second, the concurrent evolution algorithm essentially selects for meaningful information in behavioral space, by squeezing noisy behavior through a selective bottleneck. However, instead of using one unique objective function, the selection is distributed asynchronously in space and time. Differential timescales also helps accelerating the learning, which should happen as fast as possible, while retaining the way to generate the best found patterns discovered in the past. Lastly, once past the selection bottleneck, heterogeneity seems to increase considerably in genotypic space. Third, in terms of complexity, given the population size is large enough, with a consequently large number of degrees of freedom, we notice the swarm dynamics significantly change, in various ways. The flock’s surface curvature may vary for large or small flocks, as well as the attraction and repulsion induced by the exchange of different signals. The motor responses may be amplified, since the input signals may significantly increase, given a higher density of neighboring agents, as seen in Study 1. Similarly, smaller flocks may display a more ordered behavior, with the trade-off however of being more sensitive to noise, since the critical mass is not reached to implement noise-canceling effects, as demonstrated in Study 3. Larger flocks can also be a source of individual behavioral differentiation, when a higher order of organization emerges. The key is not the size nor the amount of new information, but rather the system promoting the invention of new coordination patterns within itself. We have shown how collective intelligence has the ability to augment the creation of new and diverse solutions in a swarm, when given limited channels of communication, a concurrent evolution bottleneck and a large number of constrainted degrees of freedom. It come as an inspiration for scientists: a good way to build an open-ended system, able to indefinitely discover new inventions, seems not to reside in centralized computation, but rather in distributed systems, composed of large collectives of communicating agents. Acknowledgements ================ The authors would like to thank their collaborators who contributed partially to this work: Nathanael Aubert-Kato, Aleksandr Drozd, Yasuhiro Hashimoto, Norihiro Maruyama, Yoh-ichi Mototake and Mizuki Oka. [^1]: at ALIFE 2018, in Tokyo [^2]: A swarm can be shown to act as a collective memory, either explicitly/statefully [@witkowski2016emergence] or dynamically/statelessly [@couzin2002collective].	ArXiv
--- author: - 'by Bill Baritompa, Rainer Löwen, Burkard Polster, and Marty Ross' title: Mathematical Table Turning Revisited --- [Abstract]{} We investigate under which conditions a rectangular table can be placed with all four feet on a ground described by a function $\mathbb R^2\to \mathbb R$. We start by considering highly idealized tables that are just simple rectangles. We prove that given any rectangle, any continuous ground and any point on the ground, the rectangle can be positioned such that all its vertices are on the ground and its center is on the vertical through the distinguished point. This is a mathematical existence result and does not provide a practical way of actually finding a balancing position. An old, simple, beautiful, intuitive and applicable, but not very well known argument guarantees that a square table can be balanced on any ground that is not “too wild”, by turning it on the spot. In the main part of this paper we turn this intuitive argument into a mathematical theorem. More precisely, our theorem deals with rectangular tables each consisting of a solid rectangle as top and four line segments of equal length as legs. We prove that if the ground does not rise by more than $\arctan\left (\frac{1}{\sqrt 2}\right) \approx 35.26^\circ$ between any two of its points, and if the legs of the table are at least half as long as its diagonals, then the table can be balanced anywhere on the ground, without any part of it digging into the ground, by turning the table on the spot. This significantly improves on related results recently reported on in [@Martin] and [@Polster1] by also dealing with tables that are not square, optimizing the allowable “wobblyness” of the ground, giving minimal leg lengths that ensure that the table won’t run into the ground, and providing (hopefully) a more accessible proof. Finally, we give a summary of related earlier results, prove a number of related results for tables of shapes other than rectangles, and give some advice on using our results in real life. %&\$\#!!! ========= You sit down at a table and notice that it is wobbling, because it is standing on a surface that is not quite even. What to do? Curse, yes, of course. Apart from that, it seems that the only quick fix to this problem is to wedge something under one of the feet of the table to stabilise it. However, there is another simple approach to solving this annoying problem. Just turn the table on the spot! More often than not, you will find a position in which all four legs of the table are touching the ground. This may seem somewhat counterintuitive. So, why and under what conditions does this trick work? Balancing Mathematical Tables—a Matter of Existence =================================================== In the mathematical analysis of the problem, we will first assume that the ground is the graph of a function $g:\mathbb R^2\to \mathbb R$, and that a [mathematical table]{} consists of the four vertices of a rectangle of diameter 2 whose center is on the $z$-axis. What we are then interested in is determining for which choices of the function $g$ can a mathematical table be [balanced locally]{}: that is, when can a table be moved such that its center remains on the $z$-axis, and all its vertices end up on the ground. We first observe that it is not always possible to balance a mathematical table locally. Consider, for example, the reflectively symmetric function of the angle $\theta$ about the $z$-axis with $$g(\theta)= \left \{ \begin{array}{l} 2 \quad \mbox{ if } 0\leq \theta < \frac{\pi}{2} \mbox{ or } \pi \leq \theta < \frac{3\pi}{2}, \\ 1 \quad \mbox{ otherwise}. \end{array} \right.$$ So, the ground consists of four quadrants, two at height 1 and two at height 2; see Figure \[cliff\]. It is not hard to see that a square mathematical table cannot be balanced locally on such a clifflike piece of ground. On the other hand, we can prove the following theorem: A mathematical table can always be balanced locally, as long as the ground function $g$ is continuous. This result is a seemingly undocumented corollary of a theorem by Livesay [@Livesay], which can be phrased as follows: [For any continuous function $f$ defined on the unit sphere, we can position a given mathematical table with all its vertices on the sphere such that $f$ takes on the same value at all four vertices.]{} Note that since our mathematical table has diagonals of length 2, its four vertices will be on the unit sphere iff the centers of the table and the sphere coincide. Choose as the continuous function the [vertical distance]{} from the ground, $$f:\mathbb S^2 \to \mathbb R:(x,y,z) \mapsto z-g(x,y).$$ Note that here and in everything that follows the vertical distance of a point in space from the ground is really a signed vertical distance; depending on whether the point is above, on or below the ground its vertical distance is positive, zero or negative, respectively. Now, we are guaranteed a position of our rectangle with center at the origin such that all its vertices are the same vertical distance from the ground. This means that we can balance our mathematical table locally by translating it this distance in the vertical direction. Easy! Balancing Real Tables...by Turning the Tables ============================================= So, one of our highly idealized tables can be balanced locally on any continuous ground. However, being an existence result, Theorem 1 is less applicable to our real-life balancing act than it appears at first glance. Here are two problems that seem worth pondering: 1. [Mathematical vs. Real Tables.]{} A real table consists of four legs and a table top; our theorem only tells us that we can balance the four endpoints of the legs of this real table. However, balancing the whole real table in this position may be physically impossible, since the table top or other parts of the legs may run into the ground. To deal with this complication, we define a [real table]{} to consist of a solid rectangle with diameters of length 2 as [top]{}, and four line segments of equal length as [legs]{}. These legs are attached to the top at right angles, as shown in Figure \[wobble\]. The end points of the legs of a real table form its [associated mathematical table]{}. We say that a real table is balanced locally if its associated mathematical table is balanced locally, and if no point of the real table is below the ground. 2. [Balancing by Turning.]{} A second problem with our analysis so far is that Theorem 1, while guaranteeing a balancing position, provides no practical method for finding it. After all, although we restrict the center of the table to the $z$-axis, there are still four degrees of freedom to play with when we are actually trying to find a balancing position. The following rough argument indicates how, by turning a table on the spot in a certain way, we should be able to locate a balancing position, as long as we are dealing with a square table and a ground that is not “too crazy”. Unlike most other real-world applications of the Intermediate Value Theorem, it seems that this neat argument is not as well-known as it deserves. We have not been able to pinpoint its origin, but from personal experience we know that the argument has been around for at least thirty five years and that people keep rediscovering it. In terms of proper references in which variations of the argument explicitly appear, we are only aware of [@gardner], [@gardner1], [@gardner2] (Chapter 6, Problem 6), [@Hunzinker], [@Kraft], [@Martin], [@Polster], [@Polster1] and [@Polster2]; the earliest reference in this list, [@gardner], is Martin Gardner’s [Mathematical Games]{} column in the May 1973 issue of [Scientific American]{}. Note that an essential ingredient of the argument is the simple fact that a quarter-turn around its centre takes a square into itself—to move the table from the initial position to the end position takes roughly a quarter-turn around the $z$-axis. Closely related well-documented quarter-turn arguments date back almost a century; see, for example, Emch’s proof that any oval contains the vertices of a square in [@Emch] or [@Mayerson], Section 4. At any rate, we definitely do not claim to have invented this argument. At first glance, the above argument appears reasonable and, if true, would provide a foolproof method for balancing a square table locally by turning. However, for arbitrary continuous ground functions, it appears just about impossible to turn this intuitive argument into a rigorous proof. In particular, it seems very difficult to suitably model the rotating action, so that the vertical distance of the hovering vertices depends continuously upon the rotation angle, and such that we can always be sure to finish in the end position. As a second problem, it is easy to construct continuous grounds on which real tables cannot be balanced locally. For example, consider a real square table with short legs, together with a wedge-shaped ground made up of two steep half-planes meeting in a ridge along the $x$-axis. Then it is clear that the solid table top hitting the ground will prevent the table from being balanced locally on this ridge. By restricting ourselves to grounds that are not too wild, we can prove that [balancing locally by turning]{} really works. Suppose the ground is described by a Lipschitz continuous function[^1] with Lipschitz constant less than or equal to $\frac{1}{\sqrt 2}$. Then a real table with ratio $$r=\frac{\mbox {length short side}}{\mbox{length long side}}$$ can be balanced locally on this ground by turning if its legs have length greater than or equal to $\frac{1}{\sqrt {1+r^2}}$. Since $0<r\leq 1$, the maximum of $\frac{1}{\sqrt {1+r^2}}$ in this range is 1, while all our tables have diagonals of length 2. Thus we conclude that any real table whose legs are at least half as long as its diagonal can be balanced locally by turning on any “good” ground. If we are dealing with a square table, then this table can definitely be balanced locally by turning if its legs are at least half as long as its sides. Because of the half-turn symmetry of rectangles, we can be sure to reach a balancing position of a rectangular table whilst turning it 180 degrees on the spot. As we indicated earlier, to balance a square table we never have to turn it much more than 90 degrees. For an outline of the following proof for the special case of square tables, aimed at a very general audience, see [@Polster1]. Furthermore, it has just come to our attention that André Martin has also recently published a proof of this result in the special case of square tables and Lipschitz continuous ground functions with Lipschitz constant less than $2-{\sqrt 3}$. In terms of angles, Martin’s Lipschitz constant corresponds to 15 degrees and ours, which is optimal for local turning, to approximately 35.26 degrees. [Proof.]{} We again start by considering a mathematical table $ABCD$ with diameters of length 2 and centre $O$. Our approach is to bound the wobblyness of our ground by a suitable Lipschitz condition such that putting the two opposite vertices $A$ and $C$ on the ground, and wobbling the table about $AC$ until $B$ and $D$ are at equal vertical distance from the ground, are unique operations. This ensures that everything in sight moves continuously, as we turn the table on the spot. Following this, it is easy to conclude that we can balance the table locally by turning it. Our intuition tells us that to successfully place the four corners, we need four degrees of freedom, four separate motions of the table. Putting our intuition into effect, we approach our balancing act as a succession of four Intermediate Value Theorem (IVT) arguments, taking one ÒdimensionÓ at a time. FIRST VERTEX: $A$ Start out with the table hovering horizontally above the ground so that $OA$ lies above the positive $x$-axis, and lower the table until $A$ touches the ground. SECOND VERTEX: $C$ We now show that since the ground function $g$ is continuous, $A$ can be slid along the ground, towards the $z$-axis (with $O$ sliding up or down the $z$-axis), so that $C$ also touches the ground. To do this consider the function $$D(t)= \|(t,0,g(t,0))-(-t,0,g(-t,0))\|^2=4t^2+(g(t,0)-g(-t,0))^2\, .$$ Since our table has diagonals of length 2, we want a value of $t\leq \frac{2}{2}=1$ such that $D(t)=2^2=4$. Since $D$ is continuous, $D(0)=0$ and $D(1)\geq 4$, this follows trivially from IVT. UNIQUENESS OF $C$ Assuming that $g$ is Lipschitz with $Lip(g)\leq 1$, we show that the above positioning of $C$ on the ground is unique. This follows from the fact that the function $D$ is strictly monotonic; this can be seen by differentiating $D$, noting that $\|g(t,0)-g(-t,0)\|\leq 2t$ with equality only if the function $g(t,0)$ is linear with slope $\pm 1$ in the interval under consideration. (Lipschitzness is enough for this differentiation argument to work, but a direct algebraic argument is also easy). EQUAL HOVERING POSITION We now rotate the table through an angle $\theta\in [-\frac{\pi}{2}, \frac{\pi}{2}]$ about the diagonal $AC$. We choose the direction so that rotating the table through the angle $-\frac{\pi}{2}$ brings the table into a vertical position with $B$ lying above $AC$. We want to prove the existence of a $\theta$ for which the points $B$ and $D$ are at an equal vertical distance from the ground: we call such a position an [equal hovering position]{}. To show that there is such a special position, we first choose $\theta=-\frac{\pi}{2}$. The table is now vertical, with $B$ above $AC$ and $D$ below $AC$. Since the segments $AB$ and $BC$ are orthogonal, one of the slopes[^2] of these segments will be greater than or equal to 1. Hence, since $Lip(g)\leq 1$ and since both $A$ and $C$ are on the ground, we conclude that $B$ is above or on the ground; similarly, we conclude that $D$ is below or on the ground. If we now rotate the table about $AC$ until $\theta=\frac{\pi}{2}$, then $B$ is below or on the ground and $D$ is above or on the ground. Now, a straightforward application of IVT guarantees a value of $\theta$ for which $B$ and $D$ are an equal vertical distance from the ground. UNIQUENESS OF THE EQUAL HOVERING POSITION We now fix $k\leq 1$ and take the ground to have Lipschitz constant at most $k$. We show there exists a choice of $k$ which guarantees the uniqueness of the hovering position. Take $A$ and $C$ to be touching the ground as above, with $AC$ then inclined at an angle $\phi$. In the following, we sometimes need to express the various objects as functions of $\theta$, the rotation angle about $AC$ (when assuming the inclination angle $\phi$ to be fixed, which is the case when we are referring to a particular ground); then, for example, $AB$ would be expressed as $AB(\theta)$. At other times, we need to express the objects as functions of $\phi$ and $\theta$ (when we are not referring to a particular ground); $AB$ would then be expressed as $AB(\phi, \theta)$. Here $\phi\in [-\frac{\pi}{4}, \frac{\pi}{4}]$ and $\theta\in [-\frac{\pi}{2}, \frac{\pi}{2}]$. We first note that for any equal hovering position the slopes of both $AB$ and $BC$ must be at most $k$ in magnitude. To see this, suppose $AB$ has slope greater than $k$. Then, clearly, $B$ is either above or below the ground. Since $CD$ is parallel to $AB$, it has the same slope as $AB$; further, if $B$ is higher than $A$, then $D$ is lower than $C$, and vice versa. Therefore, if $B$ is above the ground, then $D$ is below the ground, and vice versa. It follows that equal hovering is impossible. Second, let $tangentB(\theta)$ and $tangentD(\theta)$ be the tangent vectors to the semi-circles swept out by the points $B(\theta)$ and $D(\theta)$, and let $vertB(\theta)$ and $vertD(\theta)$ be respectively the vertical distances of $B$ and $D$ to the ground. Note that we have an equal hovering position iff $vertB(\theta)-vertD(\theta)=0$. It is easy to see that in the $\theta$-interval where the slope of $tangentB(\theta)$ is greater than or equal to $k$, then $vertB(\theta)$ is strictly decreasing. And also, since $tangentB(\theta)=-tangentD(\theta)$, $vertD(\theta)$ is strictly increasing in this interval. Thus, $vertB(\theta)-vertD(\theta)$ is strictly decreasing. Now, let’s choose $$k = \min_{\phi,\theta}\,\max\{slopeAB(\phi, \theta), slopeBC(\phi,\theta), slopetangentB(\phi,\theta)\}.$$ Of course, $slopeAB(\phi, \theta)$, $slopeBC(\phi,\theta)$, and $slopetangentB(\phi,\theta)$ denote the slopes of $AB$, $BC$ and the tangent vector at $B$, respectively, and the minimum is taken over all choices of $\phi\in [-\frac{\pi}{4}, \frac{\pi}{4}]$ and $\theta\in [-\frac{\pi}{2}, \frac{\pi}{2}]$. Also, because of compactness, the minimum above is actually achieved. Given this choice of $k$, we shall show that in the interval where equal hovering is possible the slope of the tangent is at least $k$. So, in this interval, $vertB(\theta)-vertD(\theta)$ is strictly decreasing and thus the equal hovering position must be unique. We first show that $k=\frac{1}{\sqrt{2}}$. Note that the vectors $AB(\phi, \theta), BC(\phi,\theta),$ and $tangentB(\phi,\theta)$ are mutually orthogonal. If we then write $(0,0,1)$ in terms of this orthogonal frame and take norms, it immediately follows that $$1 =\sin^2\beta_1 + \sin^2\beta_2 +\sin^2\beta_3\, ,$$ where $\beta_1$, $\beta_2$ and $\beta_3$ are the angles the three vectors make with the $xy$-plane. Therefore, at least one of the $\sin^2\beta_j$ is at least $\frac{1}{{3}}$, and thus the vertical slope ($=\|\tan\beta_j\|$) of the corresponding vector must be at least $\frac{1}{\sqrt{2}}$. It follows that $k \geq \frac{1}{\sqrt{2}}$. To demonstrate the minimum $k=\frac{1}{\sqrt{2}}$ is achieved, we show any table can be oriented in the critical position, with all three slopes equal to $\frac{1}{\sqrt{2}}$, and with associated tilt angle $\phi$ between $-\frac{\pi}{4}$ and $\frac{\pi}{4}$. To do this, consider the tripod formed from three edges of a cube tilted to have vertical diagonal shown in the left diagram in Figure \[check\]. These edges are mutually orthogonal, and one easily calculates that the slopes of all three edges are $\frac{1}{\sqrt{2}}$. Notice that every table is similar to one of the grey rectangles, shown in the right diagram, created by moving the point $A'$ from $P$ to $B'$. Furthermore, it is clear that the slope of the diagonal $A'C'$ is less than the slope of $B'C'$, which is equal to $\frac{1}{\sqrt 2}$, guaranteeing the tilt angle $\phi$ is in the desired range. By scaling and translating the rectangle suitably, and relabelling the vertices $A',B',$ and $C'$ as $A,B$, and $C$, respectively, we arrive at the desired orientation of our table. It remains to show that $k=\frac{1}{\sqrt 2}$ implies that the slope of the tangent is at least $k$ in the interval where equal hovering is possible. Note that in this interval, $slopeAB$ and $slopeBC$ are at most $\frac{1}{\sqrt 2}$. Then the equation $1 =\sin^2\beta_1 + \sin^2\beta_2 +\sin^2\beta_3\, $ implies that $slopetangentB$ is at least $\frac{1}{\sqrt 2}$. CONTINUITY OF $A$, $B$, $C$, AND $D$ All of the above calculations were performed with $OA$ projecting to the positive $x$-axis. We now consider rotating the table about the $z$-axis (while of course being willing to tilt the table as we rotate). So let $\gamma$ be the angle the projection of $OA$ makes with the positive $x$-axis. By our Lipschitz hypotheis, for any $\gamma$ there is a unique equal hovering position (with the projection of $OA$ making the angle $\gamma$, and tilting the table around $AC$ an angle between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$ in a fixed direction). We need to show that the positions of the four vertices $A$, $B$, $C$ and $D$, of the equally hovering table are continuous functions of $\gamma$. To do this, consider a sequence $\gamma_n \to \gamma$, with corresponding corner positions $A_n$, $B_n$, $C_n$, and $D_n$ and $A,B,C,$ and $D$. We want to show $A_n\to A$, $B_n\to B$, $C_n\to C$, and $D_n\to D$. By compactness, we can take a subsequence so that the corners converge to something: $A_n\to A^\ast$, $B_n\to B^\ast$, etc. But, by continuity of everything in sight, $A^\ast$ and $C^\ast$ are touching the ground $B^\ast$ and $D^\ast$ are an equal vertical distance from the ground, all four points are corners of the kind of table we are considering, with the projection of $OA^\ast$ making an angle $\gamma$ with the positive $x$-axis. By uniqueness, we must have $A^\ast=A$, $B^\ast=B$, $C^\ast =C$, and $D^\ast =D$, as desired. BALANCING POSITION With the uniqueness of the equal hovering position for a given $\gamma$, and with the continuous dependence of this position of the table upon $\gamma$, we conclude that the distance that $B$ and $D$ are hovering above the ground is also a continuous function of $\gamma$. If we are dealing with a square table, we can now finish the proof using IVT one more time, as described in the intuitive table turning argument presented at the beginning of this section. For a general rectangle, let the [initial position]{} be an equal hovering position for which the $z$-coordinate of the center of the table is a minimum, and let the [end position]{} be an equal hovering positions for which the $z$-coordinate of the center of the table is a maximum. Note that the hovering vertices in the initial position must be on or below the ground: if not, we could create a lower equal hovering position, contradicting minimality, by pushing vertically down on the table until the hovering vertices touch the ground. Similarly, in the end position the hovering vertices are on or above the ground. Now, IVT can be applied to guarantee that among all the equal hovering positions there is at least one balancing position. BALANCING REAL TABLES To balance a real table of side lengths ratio $r$, we determine a balancing position of the associated mathematical table, as described above. We now show that legs of length at least $\frac{1}{\sqrt{1+r^2}}$ guarantee that, balanced in this position, none of the points of the real table are below the ground. We give the complete argument for a square table, and then describe how things have to be modified to give the result for arbitrary rectangular tables. In the following, we will refer to the four vertices of the table top as $A',B',C'$ and $D'$, corresponding to the vertices $A,B,C$ and $D$, respectively. We first convince ourselves that no matter how long are the legs of our table, no part of a leg of the balanced table will be below the ground. Let’s consider the orthogonal tripod consisting of $AB$, $AD$ and the leg at $A$. Since the Lipschitz constant of our ground is at most $\frac{1}{\sqrt 2}$, the slopes of $AB$ and $AD$ are less than or equal to this value; thus, arguing as above, we see that the leg must have slope at least $\frac{1}{\sqrt 2}$. This implies that no leg of our balanced table will dip below the ground.[^3] It remains to choose the length of the legs such that no point of the table top of our balanced table will ever be below the ground. First, fix the length of the legs and consider the inverted [solid]{} circular cone, whose vertex is one of the vertices of our mathematical table, whose symmetry axis is vertical, and whose slope is $\frac{1}{\sqrt 2}$. Intersecting this cone with the plane in which the table top lies gives a conic section which is either an ellipse, a parabola or a hyperbola.[^4] Note that since we intersect the plane with a solid cone this conic section will be “filled in”. We can be sure that a point in this plane is not below the ground if it is contained in the conic section. Therefore, what we want to show is that the union of the four conic sections associated with the four vertices of our mathematical table contains the whole table top. It is clear that the four conic sections are congruent and that any two of them can be brought into coincidence via a translation. Furthermore, given any point of one of these conic sections, this point and the respective points in the other three conic sections form a square that is congruent to our table top. Finally, since the legs have slope of at least $\frac{1}{\sqrt 2}$, the end point of a leg of our table on the plane is contained in the conic section associated with the other end point of this leg. To show that we need legs of length at least $\frac{1}{\sqrt 2}$, consider a special ground with Lipschitz constant $\frac{1}{\sqrt 2}$. This ground coincides with the $xy$-plane outside the unit circle, and above the unit circle it is the surface of the cone with vertex $(0,0,\frac{1}{\sqrt 2})$ and base the unit circle. Since the diagonals of our table are of length 2, the mathematical table will balance locally on this ground iff its vertices are on the unit circle. This means that the length of the legs have to be at least as long as the cone is high if we want to ensure that no point of the table top is below the ground; it follows that we have to choose the length of our legs to be at least $\frac{1}{\sqrt 2}$. If the length of the legs is equal to $\frac{1}{\sqrt 2}$ and the table is balanced on this ground, then the four conic sections are circles that intersect in the center of the table top as shown in Figure \[intersect\]. As you can see, the table top is indeed contained in the union of these four circles. If we make the legs longer, the circles will overlap more. If we make the legs shorter, the circles will no longer overlap in the middle. Now, consider any ground, and take the legs to be of length $\frac{1}{\sqrt 2}$; clearly, if we can show that this table does not dip below the ground, then the same is true for any table with longer legs. When we tilt the table away from the horizontal position, the intersection pattern of the conic sections gets more complicated. The critical observation is that tilting the table results in the conic sections getting larger: it can be shown that each conic section contains a copy of one of the circles in Figure \[intersect\].[^5] Since, given any point of one of these conic sections, it and the corresponding points in the other three conic sections form a square that is congruent to our table top, the union of these conic sections will contain a possible translated image of the union of the circles, that we encountered before; see Figure \[intersect1\]. So, in a way our previous picture has just grown a little bit and been translated. (Note, however, it is not immediate that the conic sections together cover the table top rather than some translation of the table top). Using this fact and the simple possible convex shapes of the conic sections that we are dealing with, we can conclude that no matter how we tilt the table, the union of these conic sections will always be a simply connected domain. This means that we can be sure that the table top is contained in this union if we can show that the boundary of the table top is contained in it. We proceed to show that for all possible positions of our table in space the sides of the table top never dip below the ground. Clearly, it suffices to show this for one of the sides of the table top, say $A'B'$. For this we consider the possible positions of the rectangle with vertices $A, A',B$ and $B'$ in space. We start with the rectangle vertical and $AB$ horizontal. Draw lines of slope $\frac{1}{\sqrt 2}$ ending in $A$ and $B$; see Figure \[thales\] (left). Since the point of intersection of these two lines is not above $A'B'$, no point of this segment can be below the ground when the rectangle is positioned in such a way. Now rotate the rectangle around its center, keeping it in a vertical plane, and keeping the slope of $AB$ less than or equal to $\frac{1}{\sqrt 2}$. Again, draw lines of slope $\frac{1}{\sqrt 2}$ ending in $A$ and $B$; see Figure \[thales\] (middle). Again, the position of the point at which these two lines intersect tells you whether $A'B'$ can possibly touch the ground with the rectangle in this position. Since the two lines always intersect in the same angle, we know that the points of intersection are on a circle segment through $A$ and $B$; see Figure \[thales\] (right). Now, tilt the original rectangle around $AB$. We repeat everything that we have done so far to end up with another circle segment. However, the apex of this circle segment will the be closer to $A'B'$ than the one we encountered before. In fact, the more we tilt, the closer we will get; see Figure \[tiltrec\]. Since the apex of one of these circles is the point closest to $A'B'$, and since the apex corresponds to $AB$ being horizontal, we now calculate just how close this apex gets when we tilt around $AB$ though a maximal possible angle. This maximal possible angle is attained if $AD$ (which is orthogonal to $AB$) has slope $\frac{1}{\sqrt 2}$. It is a routine exercise to check that in this position the slope of the line connecting $A$ with the midpoint of $A'B'$ is $\frac{1}{\sqrt 2}$. This means that in this position the apex will be contained in $A'B'$. We conclude that if we choose the legs of our square table to be at least $\frac{1}{\sqrt 2}$ long, then the boundary of the table top, and hence also the table top itself, will not dip below the ground. For tables that are not square, the same arguments apply up the point where we start tilting the rectangle $AA'B'B$ around $AB$. We now have to worry about two different rectangles corresponding to the longer and shorter sides of the table top. If $r$ is the ratio of the lengths of the sides of the table, then it is easy to see that the critical length of the legs that we need to avoid running into the ground is the length that makes the longer of the two rectangles similar to the rectangle that we considered in the square case. This critical length is $\frac{1}{\sqrt{ 1+r^2}}$.$\Box$ Other Balancing Acts ==================== Horizontal balancing {#horizontal-balancing .unnumbered} -------------------- When we balance a table locally, the table will usually not end up horizontal, and a beer mug placed on the table may still be in danger of sliding off. It would be great if we could arrange it so that the table is not only balanced but also horizontal, maybe by moving the center of the table off the $z$-axis and balancing it somewhere else on the ground. Just imagine the ground to be a tilted plane, and you can see that this will not be possible in general. However, Fenn [@Fenn] proved the following result: [If a continuous ground coincides with the $xy$-plane outside a compact convex disc and if the ground never dips below the $xy$-plane inside the disk, then a given square table can be balanced horizontally such that the center of the table lies above the disk.]{} Let’s call the special kind of ground described here a [Fenn ground]{} and the part of this ground inside the distinguished compact disk its [hill]{}. The problem of horizontally balancing tables consisting of plane shapes other than squares on Fenn grounds has also been considered. Here ‘horizontal balancing on a Fenn ground’ means that in the balancing position some interior points of the shape are situated above the hill. It has been shown by Zaks [@Zaks] that a triangular table can be balanced on any Fenn ground. In fact, he showed that if we start out with a horizontal triangle somewhere in space and mark a point inside the triangle, then we can balance this triangle on any Fenn ground, with the marked point above the hill, by just translating the triangle. Fenn also showed that tables with four legs that are not concircular and those forming regular polygons with more than four legs cannot always be balanced horizontally on Fenn grounds. Zaks mentions an unpublished proof by L.M. Sonneborn that any polygon table with more than four legs cannot always be balanced horizontally on Fenn grounds. It is not known whether any concircular quadrilateral tables other than squares can always be balanced horizontally on Fenn grounds. See [@Kronheimer], [@Mayerson], [@Mayerson1], and [@Mayerson2] for further results relating to this line of research. Local Balancing of Exotic Mathematical Tables {#local-balancing-of-exotic-mathematical-tables .unnumbered} --------------------------------------------- Taking things to different mathematical extreme, we can consider a table consisting of $n\geq 3$ [leg points]{} in 3-space together with an additional [center point]{}. We then ask whether, given any continuous ground, it is possible to always balance this table locally, that is, move this configuration of $n+1$ points into a position in which the $n$ leg points are on the ground, and the center is on the $z$-axis. The example of a plane ground shows that the leg points of an always locally balancing table have to be coplanar. Let’s consider the example of a ground that contains part of a sphere that is large enough to ensure that all legs of our table end up on this part of the sphere whenever the table is locally balanced on this ground. Then intersecting this sphere with the plane that the leg points are contained in gives a circle that all leg points are contained in. Hence the leg points of the table are concircular. Now, let’s consider a ground that includes part of an ellipsoid which does not contain a copy of the circumcircle of the leg points of the table; moreover, we choose the ellipsoid large enough so that all leg points of our table end up on the ellipsoid whenever the table is locally balanced on this ground. Then intersecting the ellipsoid with the plane containing the leg points gives an ellipse that is different from the circumcircle of the leg points. However, this is impossible if the table contains more than four leg points because five points on an ellipse determine this ellipse uniquely. We conclude that an always locally balancing table must have three or four leg points and that these points are concircular. Note that requiring concircularity in the case of three points is not superfluous since we need to exclude the case of three collinear points. Livesay’s theorem, which made the proof of Theorem 1 so easy, has a counterpart for triangles, due to Floyd [@Floyd]. It is a straightforward exercise to apply this result to prove the following theorem: If the ground function is continuous, a triangular table whose three leg points are contained in a sphere around its center can be balanced locally. Of course, one should be able to prove a lot more when it comes to balancing triangular tables! In the case that the center and the (three or four) leg points of an always locally balancing table are coplanar, we can say a little bit more about the location of the center point with respect to the leg points. Begin by balancing the table in the $xy$-plane and drawing the circles around the center that contain leg points; if one of the leg points coincides with the center, then also consider this point as one of the circles. Now it is easy to see that there cannot be more than two such circles. Otherwise a ground that coincides with the $xy$-plane inside the third smallest circle and that lies above the plane outside this circle would clearly thwart all local balancing efforts. Therefore, if we want to check whether our favorite set of three or four concircular points is the set of leg points of a locally balancing table, there are usually very few positions of the center relative to the leg points which need to be considered. Perhaps the most natural choice for the center is the center of the circle that the leg points are contained in. As a corollary to the above theorem for triangles, we conclude that a triangular table with this natural choice of center is always locally balancing. In the case of four concircular points with this natural choice of center we do not know whether any tables apart form the rectangular ones are always locally balancing. However, a result worth mentioning in this context is Theorem 3 in Mayerson’s paper [@Mayerson] (see also the concluding remarks in Martin’s paper [@Martin]). It can be phrased as follows: [Given a continuous ground and one of these special four-legged tables in the $xy$-plane, the table can be rotated in the $xy$-plane around its center such that in this new position the four points on the ground above the leg points are coplanar.]{} The quadrilateral formed by the coplanar points on the ground will be congruent to the table if and only if the plane it is contained in is horizontal, in which case we have actually found a balancing position for our table. In all other cases, the quadrilateral on the ground is a deformed version of the table. Still, if the ground is not too wild, both quadrilaterals will be very similar, and lifting the table up onto the ground should result in the table not wobbling too much. Livesay’s theorem is a generalization of a theorem by Dyson [@Dyson], which only deals with the square case. A higher-dimensional counterpart of Dyson’s theorem arises as a special case of results of Joshi [@Joshi], Theorem 2 and Yang [@Yang], Theorem 3: [Given a continuous real-valued function defined on the $n$-sphere, there are $n$ mutually orthogonal diameters of this sphere such that the function takes on the same value at all $2n$ end points of these diameters.]{} Note that the endpoints of $n$ mutually orthogonal diameters of the $n$-sphere are the vertices an $n$-dimensional orthoplex, one of the regular solids in $n$-dimensions. (For example, a 1-dimensional orthoplex is just a line segment and a 3-dimensional orthoplex is an octahedron.) Using the same simple argument as in the case of Livesay’s theorem, we can prove the following theorem: An $(n-1)$-dimensional orthoplex-shaped table in $\mathbb R^n$ can be balanced locally on any ground given by a continuous function $g:\mathbb R^{n-1}\to \mathbb R$. For other closely related results see [@de; @Mira], [@Fenn1], [@Hadwiger], [@Yamabe], [@Yang1], [@Yang2], and [@Yang3]. Balance Everywhere {#balance-everywhere .unnumbered} ------------------ Imagine a square table with diameter of length 2 suspended horizontally high above some ground, with its center on the $z$-axis. Rotate it a certain angle about the $z$-axis, release it, and let it drop to the ground. It is easy to identify continuous grounds such that all four leg of the table will hit the ground simultaneously, no matter what release angle you choose. Of course, any horizontal plane will do, and so will any ground that contains a vertical translate of the unit circle. We leave it as an exercise for the reader to construct a ground that is not of this type but admits horizontal balancing for any angle. Also, the reader may wish to convince themselves that the following is true: we are dealing with a ground as in Theorem 2. If the center of the table has the same $z$-coordinate in all its equal hovering positions (positions in which $A$ and $C$ touch the ground and $B$ and $D$ are at equal vertical distance from the ground), then in fact the table is balanced in all these positions. Some Practical Advice ===================== Short Legs and Tiled Floors {#short-legs-and-tiled-floors .unnumbered} --------------------------- Note that if you shorten one of the legs of a real-life square table, this table will wobble if you set it down on the plane, and no turning or tilting will fix this problem. In real life rectangular tables the ends of whose legs do not form a perfect rectangle are not uncommon and, as our simple example shows, those uneven legs may conspire to make our anti-wobble tactics fail. Considering our example of a discontinuous ground at the beginning of this article, it should be clear that a wobbling table on a tiled floor may also defy our table turning efforts. How to Turn Tables in Practice {#how-to-turn-tables-in-practice .unnumbered} ------------------------------ In practice, it does not seem to matter how exactly you turn your table on the spot, as long as you turn roughly around the center of the table. Notice that you needn’t actually establish the equal hovering: as you rotate towards the correct balancing position, there will be less and less wobble-room until, at the correct rotation, the balancing position is forced. With a square table, you can even go for a little bit of a journey, sliding the table around in your (continuous) backyard. As long as you aim to get back to your starting position, incorporating a quarter turn in your overall movement, you can expect to find a balancing position. [99]{} de Mira Fernandes, A. Funzioni continue sopra una superficie sferica. [Portugaliae Math]{} [4]{} (1943), 69–72. Dyson, F. J. Continuous functions defined on spheres. [Ann. of Math.]{} [54]{} (1951), 534–536. Emch A. Some properties of closed convex curves in a plane, [Amer. J. Math.]{} XXXV (1913), 407–412. Fenn, Roger. The table theorem. [Bull. London Math. Soc.]{} [2]{} (1970), 73–76. Fenn, Roger. Some applications of the width and breadth of a closed curve to the two-dimensional sphere. [J. London Math. Soc.]{} [10]{} (1975), 219–222. Floyd, E. E. Real-valued mappings of spheres. [Proc. Amer. Math. Soc.]{} [6]{} (1955), 957–959. Gardner, Martin. Mathematical Games column in [Scientific American]{} (May 1973), page 104. Gardner, Martin. Mathematical Games column in [Scientific American]{} (June 1973), 109–110. Gardner, Martin. [Knotted Doughnuts and Other Mathematical Entertainments]{}. W.H. Freeman and Company, New York, 1986. Hadwiger, H. Ein Satz über stetige Funktionen auf der Kugelfläche. [Arch. Math.]{} [11]{} (1960), 65–68. Hunziker, Markus. The Wobbly Table Problem. [In Summation]{} Vol. 7 (April 2005), 5–7 (Newsletter of the Department of Mathematics, Baylor University). Joshi, Kapil D. A non-symmetric generalization of the Borsuk-Ulam theorem. [Fund. Math.]{} [80]{} (1973), 13–33. Kraft, Hanspeter. The wobbly garden table. [J. Biol. Phys. Chem.]{} 1 (2001), 95–96. Kronheimer, E. H. and Kronheimer, P. B. The tripos problem. [J. London Math. Soc.]{} [24]{} (1981), 182–192. Livesay, George R. On a theorem of F. J. Dyson. [Ann. of Math.]{} [59]{} (1954), 227–229. Martin, André. On the stability of four feet tables. http://arxiv.org/abs/math-ph/0510065 Meyerson, Mark D. Balancing acts. The Proceedings of the 1981 Topology Conference (Blacksburg, Va., 1981). [Topology Proc.]{} [6]{} (1981), 59–75. Meyerson, Mark D. Convexity and the table theorem. [Pacific J. Math.]{} [97]{} (1981), 167–169. Meyerson, Mark D. Remarks on Fenn’s “the table theorem” and Zaks’ “the chair theorem”. [Pacific J. Math.]{} [110]{} (1984), 167–169. Polster, Burkard; Ross, Marty and QED (the cat). Table Turning Mathsnack in [Vinculum]{} 42(2), June 2005 (Vinculum is the quarterly magazine for secondary school teachers published by the Mathematical Association of Victoria, Australia. Also available at www.mav.vic.edu.au/curres/mathsnacks/mathsnacks.html). Polster, Burkard; Ross, Marty and QED (the cat). Turning the Tables: feasting from a mathsnack. [Vinculum]{}, 42(4), 4 November 2005, 6–9 (also available at www.mav.vic.edu.au/curres/mathsnacks/mathsnacks.html). Vinculum, Editorial Board. Mathematical inquiry—from a snack to a meal. [Vinculum]{}, 42(3), September 2005, 11–12 (also available at www.mav.vic.edu.au/curres/mathsnacks/ $\mbox{mathsnacks.html}$). Yamabe, Hidehiko and Yujobô, Zuiman. On the continuous function defined on a sphere. [Osaka Math. J.]{} [2]{} (1950), 19–22. Yang, Chung-Tao. On theorems of Borsuk-Ulam, Kakutani-Yamabe-Yujobô and Dyson. I. [Ann. of Math.]{} [60]{} (1954), 262–282. Yang, Chung-Tao. On theorems of Borsuk-Ulam, Kakutani-Yamabe-Yujobô and Dyson. II. [Ann. of Math.]{} [62]{} (1955), 271–283. Yang, Chung-Tao. Continuous functions from spheres to euclidean spaces. [Ann of Math.]{} [62]{} (1955), 284–292. Yang, Chung-Tao. On maps from spheres to euclidean spaces. [Amer. J. Math.]{} [79]{} (1957), 725–732. Zaks, Joseph. The chair theorem. Proceedings of the Second Louisiana Conference on Combinatorics, Graph Theory and Computing (Louisiana State Univ., Baton Rouge, La., 1971), pp. 557–562. Louisiana State Univ., Baton Rouge, La., 1971. [^1]: Recall that for the ground function $g$ to be Lipschitz continuous means that there exists a $k$ such that the slope of the line segment connecting any two points on the ground is at most $k$: $\|g(P)-g(Q)\|\leq k\|P-Q\|$ for any $P,Q\in {\mathbb R}^2$. The Lipschitz constant of $g$ is then defined to be the optimal (smallest) choice of $k$. Also, recall that any Lipschitz continuous function is automatically continuous. [^2]: We emphasize that the slope of a line in space is always nonnegative. [^3]: For certain grounds with Lipschitz constant $\frac{1}{\sqrt 2}$ it is possible that a leg of a balanced table may lie along the ground. [^4]: Parabolas and hyperbolas can occur since the plane that the table top is contained in can have maximum slope greater than $\frac{1}{\sqrt 2}$. [^5]: To see this note that what we are looking at are the possible intersections of a given cone with planes that are a fixed distance from the vertex of the cone.	ArXiv
--- abstract: 'We introduce the quantum stochastic walk (QSW), which determines the evolution of generalized quantum mechanical walk on a graph that obeys a quantum stochastic equation of motion. Using an axiomatic approach, we specify the rules for all possible quantum, classical and quantum-stochastic transitions from a vertex as defined by its connectivity. We show how the family of possible QSWs encompasses both the classical random walk (CRW) and the quantum walk (QW) as special cases, but also includes more general probability distributions. As an example, we study the QSW on a line, the QW to CRW transition and transitions to genearlized QSWs that go beyond the CRW and QW. QSWs provide a new framework to the study of quantum algorithms as well as of quantum walks with environmental effects.' author: - 'César A. Rodríguez-Rosario' - 'James D. Whitfield' - 'Alán Aspuru-Guzik' bibliography: - 'refs2008.bib' title: 'Quantum stochastic walks: A generalization of classical random walks and quantum walks' --- Many classical algorithms, such as most Markov-chain Monte Carlo algorithms, are based on classical random walks (CRW), a probabilistic motion through the vertices of a graph. The quantum walk (QW) model is a unitary analogue of the CRW that is generally used to study and develop quantum algorithms [@Farhi98a; @Venegas08a; @Ambainis08a]. The quantum mechanical nature of the QW yields different distributions for the position of the walker, as a QW allows for superposition and interference effects [@Aharonov03a]. Algorithms based on QWs exhibit an exponential speedup over their classical counterparts have been developed [@childs01; @Watrous01a; @Childs03a]. QWs have inspired the development of an intuitive approach to quantum algorithm design [@Shenvi03a], some based on scattering theory [@FarhiGoldstone2008]. They have recently been shown to be capable of performing universal quantum computation [@Childs09a]. The transition from the QW into the classical regime has been studied by introducing decoherence to specific models of the discrete-time QW [@Brun03a; @Kendon07a; @Romanelli05a; @Love05a]. Decoherence has also been been studied as non-unitary effects on continuous-time QW in the context of quantum transport, such as environmentally-assisted energy transfer in photosynthetic complexes [@Rebentrost08a; @Mohseni08a; @Plenio08a; @Caruso09a; @Rebentrost09a] and state transfer in superconducting qubits [@Strauch08a; @Strauch09a]. For the purposes of experimental implementation, the vertices of the graph in a walk can be implemented using a qubit per vertex (an inefficient or unary mapping) or by employing a quantum state per vertex (the binary or efficient mapping). The choice of mapping impacts the simulation efficiency and their robustness under decoherence [@Hines07a; @Drezgic08a; @Strauch09b]. The previous proposed approaches for exploring decoherence in quantum walks have [added]{} environmental-effects to a QW based on computational or physical models such as pure dephasing [@Rebentrost09a] but have not considered walks where the environmental effects are constructed axiomatically from the underlying graph. In this work, we define the quantum stochastic walk (QSW) using a set of axioms that incorporate unitary and non-unitary effects. A CRW is a type of classical stochastic processes. From the point of view of the theory of open quantum systems, the generalization of a classical stochastic process to the quantum regime is known to be a quantum stochastic process [@Sudarshan61a; @Kossakowski72a; @Lindblad76a; @Gorini76a; @Rodriguez08; @Mohseni08a] which is the most general type of evolution of a density matrix, not simply the Hamiltonian process proposed by the QW approach. The main goal of this paper is to introduce a set of axioms that allow for the construction of a quantum stochastic process constrained by a graph. We call all the walks that follow these axioms QSWs. We will show that the family of QSWs includes both the CRW and the QW as limiting cases. The QSW can yield new distributions that are not found either in the CRW or the QW. The connection between the three types of walks discussed in this manuscript is summarized in Fig. \[venn\]. For clarity, we focus on continuous-time walks, but also sketch the corresponding procedure for the discrete-time walks. The QSW provides the fundamental tools to study the quantum-to-classical transition of walks, as well as new methods of control by the application of non-unitary operations on a quantum system. The CRW describes probabilistic motion over a graph. The dynamics of the probability distribution function is given by the master equation for a classical stochastic process, $$\label{cw} \frac{d}{dt} p_a=\sum_b M_{b}^a\, p_b,$$ where the vector element $p_b$ is the probability of being found at the vertex $b$ of the graph. The matrix $M$ is the generator for the evolution; its structure is constrained by axioms derived from the connectivity of the graph. For example, if vertices $a$ and $b$ are connected, $M_{b}^a =-\gamma$, if they are not, $M_{b}^a =0$, and $M_a^a=d_a\gamma$ where $d_a$ is the degree of vertex $a$. In analogy to the CRW, the QW [@Farhi98a] has been defined so that the probability vector element $p_a$ is replaced with $\langle a \vert \psi \rangle$, which evolves according to the Schrödinger equation, $$\label{qw} \frac{d}{dt} \langle a \vert \psi \rangle=-i\sum_b \langle a \vert H \vert b \rangle \langle b \vert \psi \rangle,$$ where $H$ is a Hamiltonian to be defined based on axioms coming from the graph. A choice of this definition is $\langle a \vert H \vert b \rangle = M_{b}^a$. This unitary evolution effectively rotates populations into coherences and back [^1]. The QW fails to completely capture the stochastic nature of CRW. The random aspect of the QW comes solely from the probabilistic nature of measurements performed on the wavefunction. Since a classical stochastic process can be generalized to the quantum regime by means of a quantum stochastic process, a CRW should be generalized to a QSW derived from the graph [^2]. For the generalization, we identify the probability vector with elements $p_a$ with a density matrix with elements $\rho_{a\alpha}$, and generalize the evolution to a quantum stochastic process, $\frac{d}{dt} \rho=\mathcal{M}\big[\rho\big]$, where $\mathcal{M}$ is a superoperator [@Sudarshan61a; @Kossakowski72a; @Gorini76a]. To make this evolution look similar to Eq. (\[cw\]), we write the density matrix in terms of its indices, $\rho=\sum_{a,\alpha}\rho_{a\alpha}\vert a \rangle \langle \alpha \vert$, and the quantum stochastic master equation becomes, $$\label{qsw} \frac{d}{dt} \rho_{a\alpha}=\sum_{b,\beta}\mathcal{M}^{a\alpha}_{b\beta}\;\rho_{b\beta},$$ with the tensor $\mathcal{M}^{a\alpha}_{b\beta}= \langle a \vert \,\mathcal{M}\big[\; \vert b \rangle \langle \beta \vert\; \big]\, \vert \alpha \rangle$. This correspondence was pointed out by Mohseni et al. in the context of energy transfer [@Mohseni08a]. For a quantum stochastic process to be related to a walk, the superoperator $\mathcal{M}$ must reflect the graph. The connectivity of the vertices will impose conditions on the transition rates of $\mathcal{M}$. Since the quantum stochastic process is more general than both the classical stochastic process and the Schrödinger equation, the correspondence of the connectivity of the graph to the rules imposed on $\mathcal{M}$ should include and go beyond the connectivity axioms for each of those. For a vertex $m$ connected to vertices that include vertex $n$, we define a processes $\vert m \rangle \leftrightarrows \vert n \rangle$ which occurs at some rate that can evolve $\vert m \rangle$ to and from $ \vert n \rangle$. Transition rates for vertices that are not connected are defined to be zero. We employ these connectivity rules as the main principle for defining valid QSWs from a given graph. To further explore the connection from the QSW to the CRW and QW as well as more general behaviors, we discuss the different limiting cases. For the classical case, the allowed transitions come from incoherent population hopping of the form $\vert m \rangle \langle m \vert \leftrightarrows \vert n \rangle \langle n \vert,$ and, for completeness, $\vert m \rangle \langle m \vert \leftrightarrows \vert m \rangle \langle m \vert$. These conditions constrain $\mathcal{M}$ to operate only on the diagonal elements of $\rho$, like Eq. (\[cw\]). In other words, the QSW transition tensor must have the property $\sum_{\alpha,\beta}\delta_{a\alpha}\,\mathcal{M}_{b\beta}^{a\alpha}\,\delta_{b\beta}= M_b^a$. The QSW should also recover the QW, where evolution among vertices happens through coherences developed by a Hamiltonian. These transitions include terms of the form, $\vert m \rangle \langle m \vert \leftrightarrows \vert m\rangle \langle n \vert$, that exchange populations to coherences with the connected vertices. For completeness, we also consider transitions of the form $\vert m \rangle \langle n \vert \leftrightarrows \vert m\rangle \langle n \vert$. If $m$ is also connected to another vertex $l$, additional transitions can happen. One transition exchanges populations into coherences among the two vertices connected to $m$, $\vert m \rangle \langle m \vert \leftrightarrows \vert l\rangle \langle n \vert$; the other exchanges coherences between $m$ and a connected vertex into coherence among two vertices connected to $m$, $\vert m \rangle \langle n \vert \leftrightarrows \vert l\rangle \langle n \vert$. These conditions allow for the recovery of Eq. (\[qw\]). By including the conjugates of these, we have now exhausted all the possibilities of the transitions that can happen following the connectivity of vertex $m$. These rules can be applied to vertices with any number of connections, and serve as the basis for the correspondence of the graph to the QSW. To complete the generalization, we need an equation of motion for a quantum stochastic process. We choose the Kossakowski-Lindblad master equation [@Kossakowski72a; @Lindblad76a; @Gorini76a], $\frac{d}{dt}\rho= \mathcal{L}\big[ \rho \big]=\sum_k-i\left[H,\rho \right]- \frac{1}{2}L_k^\dagger L_k\rho-\frac{1}{2}\rho L_k^\dagger L_k+L_k \rho L_k^\dagger$, which evolves the density matrix as a quantum stochastic process inspired by environmental effects under the Markov approximation. The Hamiltonian term describes the coherent evolution (Schrödinger equation) while the rest describe a stochastic evolution. We now set $\mathcal{M}\rightarrow \mathcal{L}$ in Eq. (\[qsw\]), where, $$\begin{aligned} \label{kossak} \mathcal{L}^{a\alpha}_{b\beta} &= \sum_k\delta_{\alpha\beta} \langle a \vert \left( -iH-\frac{1}{2}L_k^\dagger L_k\right) \vert b \rangle \\ &+\delta_{ab} \langle \beta \vert \left( iH-\frac{1}{2}L_k^\dagger L_k\right) \vert \alpha \rangle + \langle a \vert L_k \vert b \rangle \langle \beta \vert L_k^\dagger \vert \alpha \rangle. \nonumber\end{aligned}$$ ------------- ------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Axiom Matrix elements’ connectivity Transition rate \[0.5ex\] 1 $\vert m \rangle \langle m \vert \leftrightarrows \vert m \rangle \langle m \vert$ $\mathcal{L}^{mm}_{mm}=\;\langle m \vert L_k\vert m \rangle \langle m \vert L_k^\dagger \vert m \rangle -\langle m \vert L_k^\dagger L_k\vert m \rangle$ 2 $\vert m \rangle \langle m \vert \leftrightarrows \vert n \rangle \langle n \vert$ $\mathcal{L}_{mm}^{nn}=\; \langle n \vert L_k\vert m \rangle \langle m \vert L_k^\dagger \vert n \rangle $ \[1ex\] 3 $\vert m \rangle \langle m \vert \leftrightarrows \vert m \rangle \langle n \vert$ $\mathcal{L}^{mn}_{mm}=\; \langle m \vert L_k\vert m \rangle \langle m \vert L_k^\dagger \vert n \rangle + i\langle m \vert H\vert n \rangle-\frac{1}{2}\langle m \vert L_k^\dagger L_k\vert n \rangle$ 4 $\vert m \rangle \langle n \vert \leftrightarrows \vert m \rangle \langle n \vert $ $\mathcal{L}^{mn}_{mn}=\;\langle m \vert L_k\vert m \rangle \langle n \vert L_k^\dagger \vert n \rangle-i\langle m \vert H\vert m \rangle+i\langle n \vert H\vert n \rangle-\frac{1}{2}\langle m \vert L_k^\dagger L_k\vert m \rangle -\frac{1}{2}\langle n \vert L_k^\dagger L_k\vert n \rangle$ 5 $\vert m \rangle \langle n \vert \leftrightarrows \vert l \rangle \langle n \vert $ $\mathcal{L}^{ln}_{mn}=\; \langle l \vert L_k\vert m \rangle \langle n\vert L_k^\dagger \vert n \rangle-i\langle l \vert H\vert m \rangle-\frac{1}{2}\langle l \vert L_k^\dagger L_k\vert m \rangle$ \[1ex\] 6 $\vert m \rangle \langle m \vert \leftrightarrows \vert l \rangle \langle n \vert$ $\mathcal{L}^{ln}_{mm}=\; \langle l \vert L_k\vert m \rangle \langle m \vert L_k^\dagger \vert n \rangle$ \[1ex\] ------------- ------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- The connectivity conditions between a vertex $m$ and some connected vertices $n$ and $l$ and the corresponding non-zero transition rates according to Eq. (\[kossak\]) can be summarized in Table \[connectivity\] [^3]. The Axioms from Table \[connectivity\] capture the behavior sketched in Fig. (\[venn\]). To recover the CRW, it suffices to consider Axioms (1) and (2). These Axioms are a classical subset of the transition rates of the QSW. On the other hand, the QW is obtained by making all the rates of each element of $\{L_k\}$ zero. In this case, only the subset of Axioms (3), (4) and (5) from Table \[connectivity\] are relevant, corresponding to the Hamiltonian, and the QW is recovered. If the rates of $\{L_k\}$ are nonzero, these Axioms contain behavior beyond the QW. Finally, Axiom (6) has properties that have no equivalent in either the CRW or the QW; it is a type of transition that appears exclusively in the QSW and leads to different distributions. The choice of $H$ or $\{L_k\}$ is not uniquely determined by the Axioms. For example, a CRW is equivalent to a QSW with the choice of each element of $\{L_k\}$ to be associated to $M$ in the following manner. First, since connections are defined between two vertices, it is easier to write the index $k$ in terms of two indices $k\equiv(\kappa,\kappa^\prime)$. Using this notation, we enforce the graph by choosing $L_{(\kappa,\kappa^\prime)}=M_\kappa^{\kappa^\prime}\vert \kappa \rangle\langle \kappa^\prime\vert$. The Hamiltonian rates of $H$ are set to zero. This choice ensures all the transition rates to be zero except Axiom (2) from Table \[connectivity\], thereby recovering the CRW. An interpolation between the CRW and the QW conditions can be used to obtain the classical-to-quantum transition of any graph by changing the relative rates of the unitary and non-unitary processes. Other sets of $\{L_k\}$ can yield behavior different from both the classical and quantum walks. To illustrate the difference between these choices, we study a simple example: the walk on an infinite line, where a vertex $j$ is only connected to its nearest neighbors. The conditions for the dynamics of the walk on the line can be obtained from Table \[connectivity\] by making $j=m$, $j-1=l$ and $j+1=n$. To obtain behavior that is different from both the CRW and the QW, we specify a set $\{L_k\}$ with only one member that is $L=\sum_{\kappa,\kappa^\prime}M_\kappa^{\kappa^\prime}\vert \kappa \rangle\langle \kappa^\prime\vert$. This choice makes Axiom (6) nonzero, a type of transition that cannot be interpreted either as the CRW or the QW. We illustrate the resulting distribution, and compare it to CRW and the QW, in Fig. \[fig1\]. The QSW provides a general framework to study the transition between the CRW and the QW. In Fig. \[fig1\]a, we use the parameter $\omega=[0,1]$ to interpolate between the QW and the CRW: $\mathcal{L}_\omega \left[ \rho\right]=-(1-\omega)\,i\left[ H,\rho \right]+\omega\sum_k\left(- \frac{1}{2}L_k^\dagger L_k\rho-\frac{1}{2}\rho L_k^\dagger L_k+L_k \rho L_k^\dagger\right)$. By combining the unitary evolution and the evolution due to the $\{L_{(\kappa,\kappa^\prime)}\}$ terms, the transition from the QW to the CRW can be studied as one quantum stochastic walk. This procedure can be done in general to study the transition from any QW to a CRW for any graph [^4]. To highlight the difference between the QSW and the QW, we show the transition between them with $L=\sum_{\kappa,\kappa^\prime}M_\kappa^{\kappa^\prime}\vert \kappa \rangle\langle \kappa^\prime\vert$ on Fig. \[fig1\]b. Although in this paper we focused on continuous-time walks, a parallel argument holds for discrete-time walks. A discrete-time CRW evolves each time step by a Markov chain with a stochastic matrix $\mathcal{S}$ following $p_a=\sum_b \mathcal{S}_b^a\,p_b$. The quantum analogue is to use a quantum stochastic map $\mathcal{B}$ [@Sudarshan61a] by means of $\rho^\prime_{a\alpha}=\sum_{b\beta} \mathcal{B}_{b\beta}^{a\alpha}\,\rho_{b\beta}$, or equivalently as a superoperator, $\rho^\prime=\mathcal{B}\left[\rho\right]\equiv\sum_k\mathcal{C}_k\rho\mathcal{C}^\dagger_k$. A similar set of Axioms to Table \[connectivity\] can be computed by using $ \mathcal{B}^{a\alpha}_{b\beta}=\sum_k\langle a \vert\mathcal{C}_k\vert b \rangle \langle \beta \vert\mathcal{C}^\dagger_k\vert \alpha \rangle$ [^5]. The connection between the discrete-time QW and the continuous-time QW has been studied by Strauch [@Strauch06a]. In conclusion, we introduced an axiomatic approach to define the quantum stochastic walk, which behaves according to a quantum stochastic process as constrained by a particular graph. This walk recovers the classical random walk and the quantum walk as special cases, as well as the transition between them. The quantum stochastic walk allows for the construction of new types of walks that result in different probability distributions. As an example, we studied the walk on a line. The quantum stochastic walk provides a framework to study quantum walks under decoherence. Reexamination of previous work that considered environmental effects from physical motivations might suggest that, if interpreted as a quantum stochastic walk, the environment is changing the effective graph. Since the quantum stochastic walk is more general than the quantum walk, it is therefore universal for quantum computation. Its quantum to classical transiton can also be used to examine decoherence phenomena in quantum computation and classical-to-quantum phase transitions. New quantum stochastic distributions might suggest new kinds of quantum algorithms or even new classes of quantum algorithms based on this model. [Acknowledgments]{}: We thank F. Strauch for many insightful discussions and comments. We also thank I. Kassal, L. Vogt, S. Venegas-Andraca, P. Rebentrost, K. Modi, S. Jordan and E.C.G. Sudarshan for their comments on the manuscript. A.A.G. and J.D.W. thank the Army Research Office under contract W911NF-07-1-0304 and C.A.R. thanks the Mary-Fieser Postdoctoral Fellowship program. [^1]: The Hermitian form of the Hamiltonian eliminates directed graphs from consideration, motivating work on pseudo-Hermitian Hamiltonians [@Salimi09a]. [^2]: Just like a classical stochastic process includes, but is not limited, to bistochastic processes, a quantum stochastic processes includes but is not limited to unitary processes [@Sudarshan61a]. [^3]: For compactness, we assume that all connections are equally weighted; the general case can be obtained by having coefficients that depend on the vertices. [^4]: Our decoherence model comes from the graph, while the ones discussed in Ref. [@Kendon07a] were inspired by a physical model that localizes the position and thus give different distributions. We propose studies like Ref. [@Kendon07a] could be reinterpreted as a QSW, where the environmental part, as interpreted from Table \[connectivity\], affects the connectivity of the graph and thus generates different distributions. [^5]: There is no need for an explicit coin, as the stochastic map is the most general evolution of a density matrix, including random processes. However, if desired, a coin could be implemented as an environment from the open quantum system interpretation of the map.	ArXiv
--- abstract: 'In this paper we consider left-invariant pseudo-Kähler structures on six-dimensional nilpotent Lie algebras. The explicit expressions of the canonical complex structures are calculated, and the curvature properties of the associated pseudo-Kähler metrics are investigated. It is proved that the associated pseudo-Kähler metric is Ricci-flat, that the curvature tensor has zero pseudo-Riemannian norm, and that the curvature tensor has some non-zero components that depend only on two or, at most, three parameters. The pseudo-Kähler structures obtained give basic models of pseudo-Kähler six-dimensional nilmanifolds.' --- **** N. K. Smolentsev Canonical pseudo-Kähler structures on six-dimensional nilpotent Lie groups[^1] Preface {#Preface} ======= A left-invariant pseudo-Kähler (or indefinite Kähler) structure $(J, \omega)$ on a Lie group $G$ with Lie algebra ${{\mathfrak g}}$ consists of a nondegenerate closed left-invariant 2-form $\omega$ and a left-invariant complex structure $J$ on $G$ which are compatible, i.e. $\omega(JX, JY) = \omega(X, Y)$, for all $X, Y \in {{\mathfrak g}}$. Given a pseudo-Kähler structure $(J, \omega)$ on ${{\mathfrak g}}$, there exists an associated nondegenerate symmetric 2-tensor $g$ on ${{\mathfrak g}}$ defined by $g(X, Y) = \omega(X,JY)$ for $X, Y \in {{\mathfrak g}}$. It is well-known [@BG] that if the Lie algebra ${{\mathfrak g}}$ is nilpotent, then the associated metric $g$ for any compatible pair $(J, \omega)$ cannot be positive definite unless ${{\mathfrak g}}$ is abelian. Therefore $g$ is a pseudo-Riemannian metric. We shall say that $g$ is a left-invariant pseudo-Kähler metric on the Lie group $G$. A classification of left-invariant complex structures on six-dimensional nilpotent Lie groups is given in the work of Salamon [@Sal-1]. The classification of left-invariant symplectic structures on six-dimensional nilpotent Lie groups is established in the paper by Goze, Khakimdjanov and Medina [@Goze-Khakim-Med]. In the paper by Cordero, Fernández and Ugarte [@CFU2], the pseudo-Kähler structures in a six-dimensional nilpotent Lie group are studied. It is shown that for a pseudo-Kähler structure $(J, \omega)$ on a six-dimensional nilpotent Lie group, the complex structure $J$ will be nilpotent, and on some Lie groups abelian. Let $\{e^1,\dots,e^n\}$ be a basis of left-invariant 1-forms on $G$. In the following theorem we write $\mathfrak{g}$ as an $n$-tuple $(0, 0, de^3, . . . , de^n)$, abbreviating $e^{ij}=e^ {i} \wedge e^ {j}$ further to $ij$. For example, the $n$-tuple $(0,0,0,0,12,34)$ designates a Lie algebra with the equations: $de^1=0$, $de^4=0$, $de^5 =e^1\wedge e^2$ and $de^6 =e^3\wedge e^4$. Besides, for each algebra its number in the classification list for symplectic Lie algebras given in [@Goze-Khakim-Med] is also specified. \[[@CFU2]\] \[PsKahl\_6\] Let $\mathfrak {g}$ be a (nonabelian) six-dimensional nilpotent Lie algebra. Then $\mathfrak {g}$ possesses a compatible pair $(J, \omega)$ if and only if $\mathfrak {g}$ is isomorphic to one of the following Lie algebras: $$\begin{array}{ll} {{\mathfrak g}}_{21} = (0, 0, 0, 0, 12, 14 + 25), & {{\mathfrak g}}_{24} = (0, 0, 0, 0, 12, 34), \\ {{\mathfrak g}}_{14} = (0, 0, 0, 12, 13, 14), & {{\mathfrak g}}_{17} = (0, 0, 0, 0, 12, 14 + 23), \\ {{\mathfrak g}}_{13} = (0, 0, 0, 12, 13, 14 + 23), & {{\mathfrak g}}_{16}=(0, 0, 0, 0, 13 + 42, 14 + 23),\\ {{\mathfrak g}}_{15} = (0, 0, 0, 12, 13, 24), & {{\mathfrak g}}_{23} = (0, 0, 0, 0, 12, 13), \\ {{\mathfrak g}}_{11} = (0, 0, 0, 12, 13 + 14, 24), & {{\mathfrak g}}_{18} = (0, 0, 0, 12, 13, 23), \\ {{\mathfrak g}}_{10} = (0, 0, 0, 12, 14, 13 + 42), & {{\mathfrak g}}_{25} = (0, 0, 0, 0, 0, 12), \\ {{\mathfrak g}}_{12} = (0, 0, 0, 12, 13 + 42, 14 + 23). & \\ \end{array}$$ \[[@CFU2]\] In dimension 6, the Lie algebra ${{\mathfrak g}}$ has compatible pairs $(J,\omega)$ if and only if it admits both symplectic and nilpotent complex structures. For all the Lie algebras listed, the compatible complex structure $J$ is nilpotent, and for algebras ${{\mathfrak g}}_{21}$, ${{\mathfrak g}}_{12}$, ${{\mathfrak g}}_{16}$ and ${{\mathfrak g}}_{25}$ $J$ is abelian. For each Lie algebra in this list an example of a nilpotent complex structure is given in [@CFU2], and the compatible symplectic forms for it are presented. It is more natural to start with the classification list of Goze, Khakimdjanov and Medina [@Goze-Khakim-Med] in which all symplectic 6-dimensional Lie algebras are presented and where it is shown that each nilpotent Lie algebra is symplectoisomorphic to one of the algebras on this list. Therefore we will consider the Lie algebras from theorem \[PsKahl\_6\] with the symplectic structure from the list in [@Goze-Khakim-Med] and from these we will search for all compatible complex structures. Generally speaking, for a given symplectic structure $\omega$ there is a multiparametrical set of complex structures. For example, for the abelian group ${\mathbb{R}}^6$ with symplectic structure $\omega = e^1\wedge e^2 +e^3\wedge e^4 +e^5\wedge e^6$, there is a 12-parametrical set compatible with $\omega$ complex structures $J=(\psi_{ij})$. However the curvature of the associated metric $g(X,Y)=\omega(X,JY)$ will be zero for any complex structure $J$. Therefore, from the geometrical point of view, there is no sense in considering the general compatible complex structure $J$. It is much more natural to choose the elementary one: $J(e_1)=e_2$, $J(e_3)=e_4$, $J(e_5)=e_6$. We will adhere to this point of view for all Lie algebras g. After finding a multiparametrical set of complex structures $J=(\psi_{ij})$, compatible with the symplectic structure $\omega$ on ${{\mathfrak g}}$, we will consider all free parameters $\psi_{ij}$ on which the curvature of the associated metric does not depend to be zero. We will call such structures canonical. We will obtain explicit expressions for the canonical complex structures, and we will investigate the curvature properties of the associated pseudo-Kähler metrics. We will prove that the associated pseudo-Kähler metric is Ricci-flat, that the scalar square $g(R,R)$ of the curvature tensor $R$ is equal to zero and that the curvature tensor has some non-zero components that depend only on two or, at the most, three parameters. For some Lie groups in this paper, to find the compatible complex structure $J=(\psi_{ij})$ we will use the Magnin complex structures discovered in [@Mag-3], and we will solve only the compatible condition $\omega(JX,JY)=\omega(X,Y)$. For the remaining Lie groups the compatible complex structure $J=(\psi_{ij})$ is a solution of three equations: the compatible condition, $\omega \circ J+J^t\circ \omega=0$; the condition of an almost complex structure, $J^2=-Id$; the integrability $N_J(X,Y)=0$. First we solve a linear set of equations under the compatible condition. If there are difficulties in finding a solution to the equations for the integrability, we calculate a tensor of curvature of the associated metric of the almost complex structure, and we find the parameters on which it depends. Setting the remaining free parameters equal to zero, we solve the equations for the integrability, and we discover the required complex structure and the associated metric. All computations are fulfilled in system of computer mathematics Maple. Formulas for computations are specified in the end of this paper. Pseudo-Kähler structures on nilpotent Lie groups ================================================ Let $G$ be a real Lie group of dimension $n$ with Lie algebra $\mathfrak{g}$. In this paper we will study a left-invariant pseudo-Kähler structure $(J, \omega)$ on a Lie group $G$. A left-invariant pseudo-Kähler (or indefinite Kähler) structure $(J, \omega)$ on a Lie group $G$ with Lie algebra ${{\mathfrak g}}$ consists of a nondegenerate closed left-invariant 2-form $\omega$ and a left-invariant complex structure $J$ on $G$ which are compatible, i.e. $\omega(JX, JY) = \omega(X, Y)$, for all $X, Y \in {{\mathfrak g}}$. Given a pseudo-Kähler structure $(J, \omega)$ on ${{\mathfrak g}}$, there exists an associated left-invariant pseudo-Kähler metric $g$ on ${{\mathfrak g}}$ defined by $$g(X, Y) = \omega(X,JY), \mbox{ for } X, Y \in {{\mathfrak g}}.$$ As the symplectic structure $(J, \omega)$ and the complex structure $J$ are left-invariant on $G$, they are defined by their values on the Lie algebra ${{\mathfrak g}}$. Therefore from now on we will deal only with the Lie algebra ${{\mathfrak g}}$ and we will define $\omega$ and $J$ as the symplectic and, respectively, the complex structures on the Lie algebra ${{\mathfrak g}}$. An almost complex structure on a Lie algebra ${{\mathfrak g}}$ is an endomorphism $J:{{\mathfrak g}}\rightarrow {{\mathfrak g}}$ satisfying $J^2 = -I$, where $I$ is the identity map. The integrability condition of a left-invariant almost complex structure $J$ on $G$ is expressed in terms of the Nijenhuis tensor $N_J$ on ${{\mathfrak g}}$: $$\label{Nij1} N_J(X,Y) = [JX, JY] - [X,Y] - J[JX,Y] - J[X, JY], \mbox{ for all } X, Y \in {{\mathfrak g}}.$$ An almost complex structure $J$ on ${{\mathfrak g}}$ is called integrable if $N_J \equiv 0$. In this case $J$ is called a complex structure on $G$. The descending central series $\{C^k({g})\}$ of $\mathfrak{g}$ is defined inductively by $$C^0({{{\mathfrak g}}})={g},\quad C^k({{{\mathfrak g}}})=[\,{{{\mathfrak g}}},C^{k-1}({{{\mathfrak g}}})],\ k>0.$$ The Lie algebra $\mathfrak{g}$ is said to be nilpotent if $C^k({{\mathfrak g}}) =0$ for some $k$. The Lie group is said to be nilpotent if its Lie algebra is nilpotent. If ${{\mathfrak g}}$ is $s$-step nilpotent, then the ascending central series $\mathfrak{g}_0=\{0\}\subset \mathfrak{g}_1 \subset \mathfrak{g}_2\subset\dots \subset \mathfrak{g}_{s-1}\subset \mathfrak{g}_s =\mathfrak{g}$ of ${{\mathfrak g}}$ is defined inductively by $$\mathfrak{g}_k=\{X\in \mathfrak{g} \|\ [X,\mathfrak{g}]\subseteq \mathfrak{g}_{k-1}\}, \ k\geq 1.$$ It is well-known that the sequence $\{{{\mathfrak g}}_k\}$ increases strictly until ${{\mathfrak g}}_s$ and, in particular, that the ideal ${{\mathfrak g}}_1$ is the center $\mathcal{Z}$ of Lie algebra ${{\mathfrak g}}$. Since the spaces $\{{{\mathfrak g}}_k\}$ are not, in general, $J$-invariant, the sequence is not suitable for working with $J$. We introduce a new sequence $\{\mathfrak{a}_l(J)\}$ having the property of $J$-invariance [@CFGU01]. The ascending series $\{\mathfrak{a}_l(J)\}$ of the Lie algebra ${{\mathfrak g}}$, compatible with the left-invariant complex structure $J$ on $G$, is defined inductively as follows: $\frak{a}_0(J) = \{0\}$, $$\mathfrak{a}_l(J) =\{X\in \mathfrak{g}\ \|\ [X,\mathfrak{g}]\subseteq \mathfrak{a}_{l-1}(J) \mbox { and } [JX,\mathfrak{g}]\subseteq \mathfrak{a}_{l-1}(J) \},\ l\ge 1.$$ It is clear that each $\mathfrak{a}_l(J)$ is a $J$-invariant ideal in ${{\mathfrak g}}$ and $\mathfrak{a}_k(J) \subseteq \mathfrak{g}_{k}$ for $k\ge 1$. It must be noted that the terms $\mathfrak{a}_l(J)$ depend on the complex structure $J$ considered on $G$. Moreover, this ascending series, in spite of ${{\mathfrak g}}$ being nilpotent, can stop without reaching the Lie algebra ${{\mathfrak g}}$, that is, it may happen that $\mathfrak{a}_l(J) \neq {{\mathfrak g}}$ for all $l$. The following definition is motivated by this fact. The left-invariant complex structure $J$ on $G$ is called nilpotent if there is a number $p$ such that $\mathfrak{a}_p(J)=\mathfrak{g}$. It is obvious that the ideal $\mathfrak{a}_1(J)$ lies in the center $\mathcal{Z}$ of the Lie algebra ${{\mathfrak g}}$. If the nilpotent Lie algebra has the two-dimensional center $\mathcal{Z}$ for any left-invariant complex nilpotent structure $J$, the ideal $\mathcal{Z}$ is $J$-invariant. If the nilpotent Lie algebra has an increasing central sequence of ideals $\mathfrak{g}_k$, $k=0,1,\dots, s$, for which the dimension increases each time by two units for any left-invariant complex nilpotent structure $J$, then the equalities $\mathfrak{g}_k =\frak{a}_k(J)$, $k=0,1,\dots, s$ are fulfilled. If the basis of ${{\mathfrak g}}$ is chosen so that $\mathfrak{g}_1 =\{e_{2n-1},\, e_{2n}\}$, $\mathfrak{g}_2 =\{e_{2n-3},\, e_{2n-2},\, e_{2n-1},\, e_{2n}\}$, …, then the complex structure $J$ has the following block type (for example, for a six-dimensional case): $$\label{J6Bl} J_0= \left( \begin {array}{cccccc} \psi_{11}&\psi_{12}&0&0&0&0 \\ \psi_{21}&\psi_{22}&0&0&0&0\\ \psi_{31}&\psi_{32}&\psi_{33}&\psi_{34} &0&0\\ \psi_{41}&\psi_{42}&\psi_{43}&\psi_{44} &0&0\\ \psi_{51}&\psi_{52}&\psi_{53}&\psi_{54} &\psi_{55}&\psi_{56}\\ \psi_{61}&\psi_{62}&\psi_{63}&\psi_{64} &\psi_{65}&\psi_{66} \end {array} \right).$$ The remaining parameters in (\[J6Bl\]) are not free, because they are restricted by the integrability conditions $N_J=0$ and $J^2=-1$. Let us establish some properties of a symplectic nilpotent Lie algebra $({{\mathfrak g}},\omega)$ with a compatible almost complex structure $J$. \[C1ortZ\] Let $\mathcal{Z}$ be the Lie algebra center, then for any symplectic form $\omega$ on $\mathfrak{g}$, the equality $\omega (C^1\mathfrak {g}, \mathcal {Z}) =0$ is fulfilled This follows at once from the formula $d\omega(X,Y,Z)=\omega([X,Y],Z) -\omega([X,Z],Y) +\omega([Y,Z],X)=0$, $\forall X,Y\in\mathfrak{g}$, $\forall Z\in\mathcal{Z}$. If the ideal $C^k{{\mathfrak g}}$, $k\ge 1$, is $J$-invariant, then any vector $X\in\mathcal {Z} \cap C^k{{\mathfrak g}}$ is isotropic for the associated metric $g(X, Y) = \omega(X,JY)$. In particular, the associated metric is pseudo-Riemannian. $\omega (C^1\mathfrak{g} \oplus J (C^1\mathfrak {g}), \mathfrak{a}_1(J)) =0$. \[1.6\] For any (pseudo) Kähler structure $(\mathfrak{g}, \omega, g, J) $, the ideal $\mathfrak{a}_1(J) \subset \mathcal{Z} $ is orthogonal to a subspace $C^1\mathfrak{g}\oplus J(C^1\mathfrak {g})$: $$g(C^1\mathfrak{g}\oplus J(C^1\mathfrak{g}),\mathfrak{a}_1(J)) =0.$$ From the formula $$2g (\nabla_X Y, Z) {=} g ([X, Y], Z) {+} g ([Z, X], Y) {+} g (X, [Z, Y])$$ for a covariant derivative $\nabla$ of left-invariant vector fields, the following observations are at once implied: - if the vectors $X $ and $Y $ lie in the center $\mathcal{Z}$ of the Lie algebra ${{\mathfrak g}}$, then $ \nabla_X Y=0$ for any left-invariant (pseudo) Riemannian metric $g$ on the Lie algebra; - if the vector $X$ lie in the center $\mathcal{Z}$ of the Lie algebra ${{\mathfrak g}}$, then $\nabla_X Y = \nabla_Y X$. \[1.7\] If the vector $X$ lies in an ideal $\mathfrak{a}_1(J)\subset \mathcal{Z}$, then $\nabla_X Y=\nabla_Y X=0$, $\forall Y\in \mathfrak{g}$. Let $X\in \mathfrak{a}_1(J)\subset \mathcal{Z}$ and $Z,Y\in \mathfrak{g}$. Then corollary \[1.6\] implies that $2g(\nabla_X Y,Z) = g(X,[Z,Y])=0$. Let $R(X,Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z -\nabla_{[X,Y]} Z$ be the curvature tensor. \[1.8\] If $X\in \mathfrak{a}_1(J) \subset \mathcal{Z}$, then $R(X,Y)Z = R(Z,Y) X=0$, for all $Y, Z\in \mathfrak {g} $. \[ParRiem\] Let $J$ be a compatible almost complex structure on the symplectic nilpotent Lie algebra $({{\mathfrak g}}, \omega)$, and let $g=\omega \cdot J$ be the associated (pseudo) Riemannian metric. Let us choose a basis $\{e_1, e_2,\dots e_{2p}, e_{2p+1},\dots, e_{2m-1}, e_{2m}\}$ of the Lie algebra ${{\mathfrak g}}$, such that $\{e_{2p},e_{2p+1}\dots e_{2m-1},e_{2m}\}$ is a basis of the ideal $\mathfrak{a}_1(J) \subset \mathcal{Z}$. Let $J = (\psi_{ij})$ be the matrix of $J$ in this basis. Then for all $X,Y\in {{\mathfrak g}}$, the covariant derivative $\nabla_X Y$ does not depend on the free parameters $\psi_{ij}$, where $i=2p,\dots 2m$, and for all $j$. In particular, the curvature tensor does not depend on these parameters $\{\psi_{ij}\}$, $i=2p,\dots 2m,\, j=1,\dots, 2m$. For all $X\in {{\mathfrak g}}$, let $JX = J_1X +J_aX$, where $J_1X \in \mathbb{R}\{e_1, e_2,\dots, e_{2p-1}\}$ and $J_aX\in \mathbb{R}\{e_{2p}, e_{2p+1},\dots, e_{2m-1}, e_{2m}\} = \mathfrak{a}_1(J)\subset \mathcal{Z}$. From the covariant derivative formula it follows that for all $Z\in {{\mathfrak g}}$: $$2g(\nabla_X Y, Z)=2\omega(\nabla_X Y, JZ)= \omega([X,Y],JZ) + \omega([Z,X],JY) +\omega([Z,Y],JX),$$ $$2\omega(\nabla_X Y, Z)= \omega([X,Y],Z) - \omega([JZ,X],JY) -\omega([JZ,Y],JX)=$$ $$= \omega([X,Y],Z) + \omega([J_1Z+J_aZ,X],J_1Y+J_aY) +\omega([J_1Z+J_aZ,Y],J_1X+J_aX)=$$ $$=\omega([X,Y],Z) + \omega([J_1Z,X],J_1Y) +\omega([J_1,Y],J_1X).$$ As the component $J_1$ does not depend on the parameters $\{\psi_{ij}\}$, $i=2p,\dots 2m,\, j=1,\dots, 2m$, it is also true that the covariant derivative $\nabla_X Y$ does not depend on them. \[ParRiem6\] For the six-dimensional case, if the nilpotent complex structure $J$ compatible with $\omega$ is of the form in (\[J6Bl\]), then the curvature tensor $R(X,Y)Z=\nabla_X \nabla_Y Z -\nabla_Y \nabla_X Z -\nabla_{[X,Y]}Z$ of the associated metric does not depend on the free parameters $\psi_{5j}$, $\psi_{6j}$, $j=1,\dots, 6$. Remark 1. The parameters $\psi_{ij}$ of the complex structure $J$ are restricted by three conditions: a compatibility condition, an integrability condition and the fact that $J^2 =-1$. Therefore, from what is specified above, some of the parameters can be express through others. In corollaries \[ParRiem\] and \[ParRiem6\] it this is a question of free parameters, i.e., parameters which remain independent. Curvature does not depend on them. For a six-dimensional nilpotent Lie algebra ${{\mathfrak g}}$ which possesses a pseudo-Kähler structure, the dimensions of its increasing central sequence $\mathfrak{g}_k$ can be: (2,4,6), (2,6), (3,6), (4,6) and 6. We will call the sequence of these dimensions the Lie algebra type. In the list of Lie algebras in theorem \[PsKahl\_6\], the Lie algebras with type (2,4,6) are at the beginning, and are the first seven Lie algebras. Let us consider the nilpotent Lie algebras for which the sequence of ideals $\mathfrak{g}_1\subset \mathfrak{g}_2\subset\mathfrak{g}_3 =\mathfrak{g}$ has the dimensions (2,4,6). It is easy to see that such a Lie algebra of type (2,4,6) is decomposed into the direct sum of two-dimensional subspaces: $$\mathfrak {g} = A \oplus B \oplus \mathcal {Z},$$ with properties: - $\mathcal{Z} = \mathfrak{g}_1$, the Lie algebra center, - $B \oplus \mathcal{Z} = \mathfrak{g}_2$, - $[A, A] \subset B \oplus \mathcal{Z}$, $ [A, B] \subset \mathcal{Z}$. Let us consider further that in $\mathfrak{g}$ the basis $e_1,\dots, e_6$ is chosen so that $\{e_1,e_2\}$, $\{e_3,e_4\}$ and $\{e_5,e_6\}$ are the bases of the subspaces $A$, $B$ and $\mathcal{Z}$ respectively. For any nilpotent complex structure $J $ on algebra of type $ (2,4,6), $ the sequence of ideals $\mathfrak {a}_k(J) $ coincides with $\mathfrak {g}_k $, $k=1,2,3$ and the matrix $J$ has a block shape (\[J6Bl\]). For this complex structure $J$ we also have $C^1\mathfrak{g}\oplus J(C^1\mathfrak{g})=B\oplus \mathcal{Z}=\mathfrak{g}_2$. The subspace $W\subset \mathfrak {g}$ is called $\omega$-isotropic if and only if $\omega(W, W) =0$. We will call subspaces $U, V \subset \mathfrak{g}$ $\omega$-dual if, for any vector $X\in U$ there is a vector $Y\in V$ such that $\omega (X, Y) \ne 0$ and, on the contrary, $\forall Y\in V$, $\exists X\in U$, such that $\omega (X, Y) \ne 0$. \[246\] Let the six-dimensional symplectic Lie algebra $(\mathfrak{g}, \omega)$ have type (2,4,6) and $$\label{abz} \mathfrak {g} = A \oplus B \oplus \mathcal {Z},$$ where $B \oplus \mathcal{Z} = \mathfrak{g}_2$ is an abelian subalgebra. We will assume that the subspaces $A$ and $\mathcal{Z}$ are $\omega$-isotropic and $\omega$-dual, and that on the subspace $B$ the form $\omega$ is nondegenerate. Then for any nilpotent complex structure $J$ compatible with $\omega$, and the Levi-Civita connection $\nabla$ of the associated pseudo-Riemannian metric $g=\omega\cdot J$, the following properties are fulfilled: - $\nabla_X Y\in B \oplus \mathcal{Z},\quad \forall\, X,Y\in A$, - $\nabla_X Y, \nabla_Y X\in \mathcal{Z}$,for all $X\in A,\, Y\in B$, - $\nabla_X Y = \nabla_Y X =0$,for all $X\in A,\, Y\in \mathcal{Z}$, - $\nabla_X Y=0,\quad \forall \, X,Y\in B\oplus \mathcal{Z}$. Let $X, Y\in A$. If $\nabla_X Y$ has a nonzero component from $A$ then there is a vector $JZ\in \mathcal {Z} $, such that $ \omega (\nabla_X Y, JZ) \ne 0$. On the other hand, $2\omega (\nabla_X Y, JZ) = 2g(\nabla_X Y, Z) = g([X, Y], Z) + g([Z,X], Y) + g([Z, Y], X) = \omega ([X, Y], JZ) =0$, as, from lemma \[C1ortZ\], $ \omega (C^1\mathfrak {g}, \mathfrak {a} _1 (J)) =0$. Let now $X\in A$ and $Y\in B$. It can be shown in an exactly similar way that $\nabla_X Y$ has a zero component from $A$. We will assume that $\nabla_X Y $ has a non-zero component from $B$. Then there is a vector $Z\in B\oplus \mathcal{Z}$ such that $JZ\in B$ and such that $\omega (\nabla_X Y, JZ) \ne 0$. At the same time, $2\omega(\nabla_X Y,JZ) = 2g(\nabla_X Y,Z) = g([X,Y],Z) +g([Z,X],Y) = \omega([X,Y],JZ)+ \omega([Z,X],JY) =0$. The last equality follows from the commutativity of $B\oplus \mathcal{Z}$, and from this that $Y,JY,Z,JZ\in B \subset C^1\mathfrak{g} \oplus J(C^1\mathfrak{g})$, so then $[X,Y],[Z,X] \in \mathcal{Z}$ and $\omega(C^1\mathfrak{g} \oplus J(C^1\mathfrak{g}),\mathcal{Z})=0$. $\nabla_Y X =\nabla_X Y - [X, Y]\in \mathcal{Z}$. Let us consider the third statement. Let $X\in A$ and $Y\in \mathcal{Z}$. Then for any $Z\in \mathfrak{g}$, $2g(\nabla_X Y,Z) = g([X,Y],Z) +g([Z,X],Y) +g([Z,Y],X) = \omega([Z,X],JY) =0$ by the same arguments as for the previous point. Let us consider the last statement. Let $X,Y\in B \oplus \mathcal{Z}$. Then for any $Z\in \mathfrak{g}$, $2g(\nabla_X Y,Z) = g([X,Y],Z) +g([Z,X],Y) +g([Z,Y],X) = \omega([Z,X],JY)+ \omega([Z,Y],JX) =0$ by the same arguments as for the previous point. \[3.3\] Under the suppositions of theorem \[246\], if the vector $X$ lies in an ideal $\mathfrak{a}_2(J)=B\oplus\mathcal{Z}$, then $R(X, Y)Z = R(Z,Y)X =0$, for all $Y, Z\in \mathfrak{g}$. \[Ricci-zero\] Under the suppositions of theorem \[246\], for any $X, Y, Z\in \mathfrak{g}$, the condition $R(X, Y)Z\in \mathcal{Z}$ is fulfilled. Therefore the pseudo-Riemannian norm of the curvature tensor is equal to zero. According to the expansion $\mathfrak{g}=A\oplus B\oplus \mathcal{Z}$ we choose the bases $\{e_1, e_2\}$, $\{e_3, e_4 \}$ and $\{e_5, e_6 \}$. Then the curvature tensor can have within its symmetries only four non-zero components, $R_{1,2,1}^5$, $R_{1,2,1}^6$, $R_{1,2,2}^5$, $R_{1,2,2}^6$. In particular, the Ricci tensor is equal to zero. \[Par3.3\] Under the suppositions of theorem \[246\], if the compatible almost complex structure $J$ has a block type (\[J6Bl\]), then the curvature tensor of the associated metric $g=\omega \cdot J$ does not depend on the free parameters $\psi_{i1}$ and $\psi_{i2}$,for $i=3,4,5,6$. Let us notice that parameters $\psi_{ij}$ of the complex structure $J$ are connected by three conditions: a compatibility condition, an integrability condition and the fact that $J^2=-1$. Therefore from what has been specified above some of the parameters can be expressed through others. If, as a result, among $\psi_{i1}$ and $\psi_{i2}$, $i=3,4,5,6$, there were independent parameters, it would be possible to set them at zero as the curvature does not depend on them. We remember that, according to corollary \[ParRiem6\], the curvature $R(X,Y)$ of the associated metric also does not depend on the free parameters $\psi_{51}$, $\psi_{52}$, $\psi_{53}$, $\psi_{54}$, $\psi_{61}$, $\psi_{62}$, $\psi_{63}$, $\psi_{64}$. Similar statements are true for type (2,6) Lie algebras. A type (4,6) Lie algebra is the direct product of a four-dimensional Lie algebra and $\mathbb{R}^2$. The case (3,6) is the most complicated. Remark 2. There is an obvious generalization of theorem \[246\] to the case where $\dim {{\mathfrak g}}> \, 6$. It is necessary to assume that there exist sequences of ideals ${{\mathfrak g}}_1\subset {{\mathfrak g}}_2\subset \dots \subset {{\mathfrak g}}_n = {{\mathfrak g}}$ with dimensions of $2,4, \dots, n $, invariant under $J$. Choosing additional two-dimensional subspaces $A_i$ to these ideals, we obtain the expansion ${{\mathfrak g}}=A_1\oplus A_2\oplus \dots \oplus A_n ={{\mathfrak g}}_1 =\mathcal{Z}$. The form of $\omega$ should be such that on each subspace $A_i$ (except, maybe, on $A_{n/2}$) it is degenerated, and also the subspaces $A_i$ and $A_{n-i}$ are $\omega$-dual. Remark 3. All calculations are made in the Maple system using the formulas specified at the end of the paper. Lie algebras of type $(2,4,6)$ ============================== Let us consider all six-dimensional nilpotent Lie algebras of type (2,4,6). According to theorem \[PsKahl\_6\], there are seven such Lie algebras which possess a pseudo-Kähler structure. We remember that the number of each algebra corresponds to its number in the classification list in [@Goze-Khakim-Med]. The Lie group $G_{14}$ ---------------------- Let us consider a six-dimensional Lie group $G_{14}$ which has a Lie algebra ${{\mathfrak g}}_{14}$ with non-trivial Lie brackets (see [@Goze-Khakim-Med]: $[X_{1},X_{2}]=X_{4}$, $[X_{1},X_{4}]=X_{6}$, $[X_{1},X_{3}]=X_{5}$. The algebra ${{\mathfrak g}}_{14}$ has (see [@Goze-Khakim-Med]) three symplectic structures which are noted using the dual base $\{\alpha^{i}\}$ as follows: $\omega _{1}=\alpha^{1}\wedge \alpha^{6}+\alpha^{2}\wedge \alpha^{4}+\alpha^{3}\wedge \alpha^{5}$, $\omega _{2}=\alpha^{1}\wedge \alpha^{6}-\alpha^{2}\wedge \alpha^{4} +\alpha^{3}\wedge \alpha^{5}$, $\omega _{3}=\alpha^{1}\wedge \alpha^{6}+\alpha^{2}\wedge \alpha^{5}+\alpha^{3}\wedge \alpha^{4}$.\ Left-invariant complex structures on this group are found in an explicit form in the work of Magnin [@Mag-3] (the algebra $M1$). We will use Magnin’s results, so we will make the replacement: $X_2=-e_1$, $X_1=e_2$, $X_3=e_3$, $X_4=e_4$, $X_6=e_5$, $X_5=e_6$. Then the non-trivial brackets are given in [@Mag-3]: $[e_1,e_2]=e_4$, $[e_2,e_3] = e_6$, $[e_2,e_4] = e_5$,\ and the symplectic forms become: $\omega _{1}=-e^{1}\wedge e^{4}+e^{2}\wedge e^{5} +e^{3}\wedge e^{6}$, $\omega _{2}=e^{1}\wedge e^{4} +e^{2}\wedge e^{5}+e^{3}\wedge e^{6}$, $\omega _{3}=-e^{1}\wedge e^{6}+e^{2}\wedge e^{5}+e^{3}\wedge e^{4}$. In [@Mag-3] it is shown that the Lie group $G_{14}$ has a 10-parametrical set of left-invariant complex structures. A direct check of the compatible property $\omega(JX,Y)+\omega(X,JY)$, $\forall \, X,Y\in {{\mathfrak g}}$, shows that for first two symplectic forms there are no compatible complex structures on the group $G_{14}$. For the form $\omega_3$ the compatible complex structure depends on 6 parameters and is of the form: $$J = \left(\begin{array}{cccccc} \psi_{11} & \psi_{12} & 0 & 0 & 0 & 0 \\ -\frac{\psi_{11}^2+1}{\psi_{12}} & -\psi_{11} & 0 & 0 & 0 & 0 \\ \frac{\psi_{42}(\psi_{11}^2+1) -2\psi_{41}\psi_{12}\psi_{11}}{\psi_{12}^2} & -\psi_{41} & -\psi_{11} & -\frac{\psi_{11}^2+1}{\psi_{12}} & 0 & 0 \\ \psi_{41} & \psi_{42} & \psi_{12} & \psi_{11} & 0 & 0 \\ \psi_{51} & J_{52} & \psi_{42} & \psi_{41} & \psi_{11} & \psi_{12}\\ \psi_{61} & -\psi_{51} & -\psi_{41} & \frac{\psi_{42}(\psi_{11}^2+1)- 2\psi_{41}\psi_{12}\psi_{11}}{\psi_{12}^2} & -\frac{\psi_{11}^2+1}{\psi_{12}}& -\psi_{11} \\ \end{array}\right),$$ where $J_{52}= \frac{-2\psi_{11}\psi_{12}(\psi_{42}\psi_{41} -\psi_{12}\psi_{51}) +\psi_{42}^2(\psi_{11}^2 +1) +\psi_{12}^2(\psi_{41}^2 +\psi_{12}\psi_{61})}{(\psi_{11}^2+1)\psi_{12}}$. The curvature tensor of the metric $g(X,Y)=\omega_3(X,JY)$ is equal to zero for all values of the parameters. Therefore we choose the elementary pseudo-Kähler structure with zero values of the free parameters and $\psi_{12}=-1$. Then the canonical pseudo-Kähler structure is set as follows: $J(e_1) = e_2,\quad J(e_3) = e_4,\quad J(e_5) = e_6$, $g=-e^1\,e^5 -e^2\,e^6 -(e^3)^2 -(e^4)^2.$ The Lie group $G_{21}$ ---------------------- The Lie algebra ${{\mathfrak g}}_{21}$ is defined by: $[e_1,e_2] = e_4$, $[e_1,e_4] = e_6$, $[e_2,e_3] = e_6$. This Lie algebra has two symplectic structures [@Goze-Khakim-Med]: $\omega_1 = e^1\wedge e^6 +e^2\wedge e^4 -e^3\wedge e^4 -e^3\wedge e^5,$ $\omega_2 = e^1\wedge e^6 +e^2\wedge e^5 -e^3\wedge e^4.$ A direct check shows that for the first structure $\omega_1 = e^1\wedge e^6 +e^2\wedge e^4 -e^3\wedge e^4 -e^3\wedge e^5$ there are no compatible complex structures. We will consider the second symplectic structure $\omega_2$. There is a multiparametrical set of compatible complex structures. Taking into account the results of theorem \[246\] and corollaries \[1.8\], \[ParRiem6\] and \[Ricci-zero\], we find by direct evaluation that the tensor of curvature of the associated metric $g(X,Y)=\omega(X,JY)$ depends on two parameters $\psi_{11}$ and $\psi_{12}\ne 0$ and has the following non-zero components:$R_{1, 2, 1}^6~=~1~+~\psi_{11}^2$,$R_{1, 2, 2}^6 = \psi_{12} \psi_{11}$, $R_{1, 2, 1}^5 = \psi_{12}\psi_{11}$,$R_{1, 2, 2}^5 = \psi_{12}^2$. Therefore the semicanonical complex structure is set as follows: $J(e_1) = \xi_{11}\, e_1 -\frac{\xi_{11}^2+1}{\xi_{12}} \, e_2,\qquad J(e_2) = \psi_{12}\, e_1 -\psi_{11}\, e_2$, $J(e_3) = \psi_{11}\, e_3 +\frac{\xi_{11}^2+1}{\xi_{12}}\, e_4,\qquad J(e_4) = -\xi_{12}\, e_3 -\psi_{11}\, e_4$, $J(e_5) = \psi_{11}\, e_5 +\frac{\xi_{11}^2+1}{\xi_{12}}\, e_6, \qquad J(e_6) = -\psi_{12}\, e_5 -\psi_{11}\, e_6$.\ The corresponding associated metric is: $$g {=} \left[\!\!\begin{array}{cccccc} 0 & 0 & 0 & 0 & \frac{\psi_{11}^2+1}{\psi_{12}} & -\psi_{11} \\ 0 & 0 & 0 & 0 & \psi_{11} & -\psi_{12} \\ 0 & 0 & -\frac{\psi_{11}^2+1}{\psi_{12}} & \psi_{11} & 0 & 0 \\ 0 & 0 & \psi_{11} & -\psi_{12} & 0 & 0 \\ \frac{\psi_{11}^2+1}{\psi_{12}} & \psi_{11} & 0 & 0 & 0 & 0 \\ -\psi_{11} & -\psi_{12} & 0 & 0 & 0 & 0 \\ \end{array}\!\!\right].$$ On omitting the index of the curvature tensor, it turns out that there is only one (within symmetries) non-zero component of the tensor of curvature $R_{1, 2, 1, 2} =-\psi_{12}$. Then, supposing $\psi_{12}=-a\ne 0$ and $\psi_{11}=0$, we find the following canonical complex structure and pseudo-Kähler metric with curvature $R_{1, 2, 1, 2}=a$ on a Lie algebra $\mathfrak{h}_{21}$: $$J(e_2) = -a\, e_1,\ J(e_4) = a\, e_3,\ J(e_6) = a\, e_5,$$ $$g= -\frac 2a\, e^1\cdot e^5 +2a\, e^2\cdot e^6 -\frac 1a\, (e^3)^2+ a\, (e^4)^2.$$ The Lie group $G_{13}$ ---------------------- The Lie algebra $\mathfrak{g}_{13}$ is defined by: $[e_1,e_2] =e_4$, $[e_1,e_3]=e_5$, $[e_1,e_4]=e_6$, $[e_2,e_3]=e_6$. This Lie algebra has three symplectic structures [@Goze-Khakim-Med]. Left-invariant complex structures on this group are discovered in an explicit form in the work of Magnin [@Mag-3] (algebra $M6$). In order to use the outcomes of Magnin’s work, we will rename the base vectors $e_3:=-e_3$, $e_5:=-e_5$, and also find the Lie brackets from Magnin’s paper [@Mag-3]: $[e_1,e_2] =e_4$, $[e_1,e_3]=e_5$, $[e_1,e_4]=e_6$, $[e_2,e_3]=-e_6$. The symplectic structures are: $\omega_1 = e^1\wedge e^6 -\lambda e^2\wedge e^5 - (\lambda-1)e^3\wedge e^4$, $\omega_2 =e^1\wedge e^6 +\lambda e^2\wedge e^4 -e^2\wedge e^5 +e^3\wedge e^5$, $\omega_3 = e^1\wedge e^6 +e^2\wedge e^4 -\frac 12 e^2\wedge e^5 +\frac 12 e^3\wedge e^4$. First case. We will consider the form $\omega_1= e^1\wedge e^6 -\lambda e^2\wedge e^5 -(\lambda-1)e^3\wedge e^4$. There is a multiparametrical set of the compatible complex structures. Taking into account the outcomes of theorem \[246\] and corollaries \[ParRiem6\] and \[Ricci-zero\], we find by direct evaluation that the tensor of curvature of the associated metric $g_1(X,Y)=\omega_1(X,J_1Y)$ depends on two parameters $\psi_{11}$ and $\psi_{12}\ne0$ and also has following non-zero components:$R_{1, 2, 2}^6 =\frac {(3\lambda -1) \psi_{12} \psi_{11}}{\lambda -1}$,$R_{1, 2, 2}^5 =-\frac{(1 + \lambda)(3\lambda -1)\psi_{12}^2}{\lambda (\lambda - 1)}$,$R_{1, 2, 1}^6 =\frac{(3 \lambda -1)(1 + \psi_{11}^2)}{\lambda^2 -1}$,$R_{1, 2, 1}^5 =-\frac{(3 \lambda - 1) \psi_{12} \psi_{11}}{\lambda (\lambda - 1)}$. After omitting the index, one non-zero component turns out to be $R_{1, 2, 1, 2} = -\frac{(3 \lambda -1) \psi_{12}}{\lambda -1}$. Therefore the semicanonical complex structure is set out as follows: $J_1(e_1) = \psi_{11}\,e_1 -\frac{1+{\psi_{11}}^2}{(1+\lambda)\psi_{12}}\, e_2,\quad J_1(e_2) = (1+\lambda)\psi_{12}\, e_1 -\psi_{11}\, e_2$, $J_1(e_3) = \psi_{11}\, e_3 -\frac{1+{\psi_{11}}^2}{\psi_{12}}\, e_4$,$J_1(e_4) = \psi_{12}\, e_3 -\psi_{11}\, e_4$, $J_1(e_5) = \psi_{11}\, e_5 -\frac{\lambda(1+{\psi_{11}}^2)}{(1+\lambda)\psi_{12}}\, e_6,\quad J_1(e_6) = \frac{(1+\lambda)\psi_{12}}{\lambda}\, e_5 -\psi_{11}\, e_6$. The corresponding pseudo-Kähler metric comes from the formula $g_1=\omega_1\circ J_1$. Suppose that $\psi_{12}=a\ne 0$ and $\psi_{11}=0$. We find the following canonical complex structure and the pseudo-Kähler metric of curvature $R_{1, 2, 1, 2}=-\frac{(3 \lambda -1) a}{\lambda -1}$ on a Lie algebra $\mathfrak{h}_{13}$: $$J_1(e_2) = (1+\lambda)a\, e_1,\quad J_1(e_4) = a\, e_3,\quad J_1(e_6) = \frac{(1+\lambda)a}{\lambda}\, e_5,$$ $$g_1= \left[ \begin {array}{cccccc} 0&0&0&0&-{\frac {\lambda}{ \left( 1+ \lambda \right) a}}&0\\ \noalign{\medskip}0&0&0&0&0&- \left( 1+\lambda \right) a\\ \noalign{\medskip}0&0&{\frac {\lambda-1}{a}}&0&0&0\\ \noalign{\medskip}0&0&0& \left(\lambda-1 \right) a&0&0\\ \noalign{\medskip}-{\frac {\lambda}{(1+\lambda )a}}&0&0&0&0&0\\ \noalign{\medskip}0&- \left( 1+\lambda \right)a&0&0&0&0 \end {array} \right].$$ Second case. For the symplectic form $\omega_2 =e^1\wedge e^6 +\lambda e^2\wedge e^4 -e^2\wedge e^5 +e^3\wedge e^5$ there are no compatible complex structures. Third case. The symplectic structure is: $\omega_3 = e^1\wedge e^6 +e^2\wedge e^4 -\frac 12 e^2\wedge e^5 +\frac 12 e^3\wedge e^4$. For any compatible complex structure and its associated metric, the curvature tensor depends on two parameters $\psi_{11}$ and $\psi_{12}\ne 0$: $R_{1, 2, 2}^5 =\frac {4 \psi_{12}^2}{3}$, $R_{1, 2, 1}^5 =\frac{4 \psi_{12} \psi_{11}}{3}$, $R_{1, 2, 2}^6 =-\frac{2 \psi_{12} \psi_{11}}{3}$,$R_{1, 2, 1}^6 =-\frac{2(1+ \psi_{11}^2)}{3}$. Therefore the semicanonical pseudo-Kähler structure is as follows: $$J_3 = \left[ \begin {array}{cccccc} \psi_{11}&\psi_{12}&0&0&0&0 \\ \noalign{\medskip}-{\frac {{\psi_{11}}^{2}+1}{\psi_{12}}}&- \psi_{11}&0&0&0&0\\ \noalign{\medskip}0&0&\psi_{11}&2\,\psi_{12}/3&0&0\\ \noalign{\medskip}0&0&-3\,{\frac {{\psi_{11}}^{2}+1}{2\, \psi_{12}}}&-\psi_{11}&0&0\\ \noalign{\medskip}0&0&-3\,{\frac {{\psi_{11}}^{2}+1}{ \psi_{12}}}&-4\,\psi_{11}&\psi_{11}&2\,\psi_{12}\\ \noalign{\medskip}0&0&0&{\frac {{\psi_{11}}^{2}+1}{\psi_{12}}} &-{\frac {{\psi_{11}}^{2}+1}{2\,\psi_{12}}}&-\psi_{11} \end {array} \right],$$ $$g_3= \left[ \begin {array}{cccccc} 0&0&0&{\frac {{\psi_{11}}^{2}+1}{ \psi_{12}}}&-{\frac {{\psi_{11}}^{2}+1}{2\,\psi_{12}}} & -\psi_{11}\\ \noalign{\medskip}0&0&0&\psi_{11}&-\psi_{11}/2& -\psi_{12}\\ \noalign{\medskip}0&0&-3\,{\frac {{\psi_{11}}^{2}+1}{4\, \psi_{12}}}&-\psi_{11}/2&0&0\\ \noalign{\medskip}{\frac {{\psi_{11}}^{2}+1}{ \psi_{12}}}&\psi_{11}&-1/2\,\psi_{11}&-\psi_{12}/3&0&0\\ \noalign{\medskip}-{\frac {{\psi_{11}}^{2}+1}{2\, \psi_{12}}}&-\psi_{11}/2&0&0&0&0\\ \noalign{\medskip}-\psi_{11}& -\psi_{12}&0&0&0&0\end {array} \right]$$ On omitting the index of the curvature tensor, it turns out that there is only one (within symmetries) non-zero component of the tensor of curvature $R_{1, 2, 1, 2} =\frac {2\psi_{12}}{3}$. Then, supposing $\psi_{12}= -a\ne 0$ and $\psi_{11}=0$, we find the following canonical complex structure and pseudo-Kähler metric of curvature $R_{1, 2, 1, 2}=-\frac {2\,a}{3}$ on a Lie algebra $\mathfrak{h}_{13}$ with $J_3$-invariant 2-planes $\{e_1,e_2\}$ and $\{e_5,e_6\}$: $J_3(e_2) = -a\, e_1$,$J_3(e_3) = \frac {3}{2\,a}\, e_4 +\frac{3}{a}\, e_5$, $J_3(e_4) = -\frac{2a}{3}\, e_3 -\frac{1}{a}\, e_6$, $J_3(e_6) = -2\,a\, e_5$, $$g_3=\left[ \begin {array}{cccccc} 0&0&0&-\frac 1a&\frac{1}{2a}&0\\ 0&0&0&0&0&a\\ 0&0&\frac{3}{4a}&0&0&0\\ -\frac 1a&0&0&\frac{a}{3}&0&0\\ \frac{1}{2a}&0&0&0&0&0\\ 0&a&0&0&0&0\end {array} \right].$$ The Lie group $G_{15}$ ---------------------- The Lie algebra $\mathfrak{g}_{15}$ is defined in [@Goze-Khakim-Med]: $[X_{1},X_{2}]=X_{4}$, $[X_{1},X_{4}]=X_{6}$, $[X_{2},X_{3}]=X_{5}$. The Lie algebra has two symplectic structures: $\omega _{1}=-\alpha^{1}\wedge \alpha^{5}+\alpha^{1}\wedge \alpha^{6}+\alpha^{2}\wedge \alpha^{5}+\alpha^{3}\wedge \alpha^{4},$ $\omega _{2}=\alpha^{1}\wedge \alpha^{6}+\alpha^{2}\wedge \alpha^{4}+\alpha^{3}\wedge \alpha^{5}.$ This is a Lie algebra $M7$ for which Magnin, in [@Mag-3], discovered the complex structures in an explicit form. To use these outcomes, we will make the replacement: $X_1:=e_2$, $X_2:=-e_1$, $X_5:=-e_6$, $X_6:=e_5$. Then the Lie bracket relations and symplectic structures become: $[e_1,e_2]=e_{4}$, $[e_2,e_{4}]=e_5$, $[e_1,e_{3}]=e_6$, $\omega _{1}=e^1\wedge e^6+ e^2\wedge e^6+e^2\wedge e^5 +e^{3}\wedge e^{4},$ $\omega _{2}=-e^1\wedge e^{4}+e^2\wedge e^5-e^{3}\wedge e^6.$ First case. The symplectic structure is: $\omega_1 = e^2\wedge e^6 +e^2\wedge e^5 +e^1\wedge e^6 +e^3\wedge e^4$. For any compatible complex structure, the curvature tensor of the associated metric depends on two parameters $\psi_{11}$ and $\psi_{12}\ne 0$. After omitting the index there is one non-zero component: $$R_{1, 2, 1, 2} =-\frac{\psi_{11}^4+\psi_{11}^3\,\psi_{12} +2\,\psi_{11}^2-2\,\psi_{12}^2\,\psi_{11}^2+ \psi_{11}\,\psi_{12}+1}{\psi_{12}\,(-2\,\psi_{11}\psi_{12}+1+\psi_{11}^2)}.$$ Setting the remaining free parameters $\psi_{ij}$ to zero, we find the canonical complex structure $J_1$: $J_1(e_1) = \psi_{11}\, e_1 -{\frac {1+{\psi_{11}}^{2}}{\psi_{12}}}\, e_2$, $J_1(e_2) = \psi_{12}\, e_1 -\psi_{11}\, e_2$, $J_1(e_3) = -{\frac {{\psi_{11}}^{3}- {\psi_{11}}^{2}\psi_{12}+\psi_{11}+\psi_{12}}{-2\,\psi_{11} \,\psi_{12}+1+{\psi_{11}}^{2}}}\, e_3 -{\frac { \left( {\psi_{12}}^{2}-2\,\psi_{11}\,\psi_{12}+1+{\psi_{11}}^ {2} \right) \psi_{12}}{-2\,\psi_{11}\,\psi_{12}+1+{\psi_{11}}^ {2}}}\, e_4$, $J_1(e_4) = \frac {1+2\,{\psi_{11}}^{2} +{\psi_{11}}^{4}}{\psi_{12}\, (-2\,\psi_{11}\,\psi_{12}+1+{ \psi_{11}}^{2})}\, e_3 +\frac {{\psi_{11}}^{3}-{\psi_{11}}^{2}\psi_{12}+\psi_{11}+ \psi_{12}}{-2\,\psi_{11}\,\psi_{12}+1+{\psi_{11}}^{2}}\, e_4$, $J_1(e_5) = -\frac {-\psi_{11}\,\psi_{12}+1+ {\psi_{11}}^{2}}{\psi_{12}}\, e_5 +\frac {1+{\psi_{11}}^{2}}{\psi_{12}} \, e_6$, $J_1(e_6) = -{\frac {{\psi_{12}}^{2}-2\, \psi_{11}\,\psi_{12}+1+{\psi_{11}}^{2}}{\psi_{12}}}\, e_5 +{\frac {-\psi_{11}\,\psi_{12}+1+{\psi_{11}}^{2}}{\psi_{12}}} \, e_6$.\ The metric tensor of the pseudo-Kähler structure is easily found using the formula $g_1=\omega_1\circ J_1$. Second case. The symplectic structure $\omega _{2}=-e^1\wedge e^{4}+e^2\wedge e^5-e^{3}\wedge e^6$ does not admit a compatible complex structure. The Lie group $G_{11}$ ---------------------- The Lie algebra $\mathfrak{g}_{11}$ is defined by: $[e_1,e_2] = e_4$, $[e_1,e_4] = e_5$, $[e_2,e_3] = e_6$, $[e_2,e_4] = e_6$.\ Its symplectic structure comes from the list in [@Goze-Khakim-Med]: $\omega = e^1\wedge e^6 + e^2\wedge e^5 - e^3\wedge e^4 + \lambda e^2\wedge e^6.$ This is the Lie algebra $M8$, considered in the work of Magnin [@Mag-3] and for which he found the complex structures in an explicit form. There is a multiparametrical set of the compatible complex structures. For any compatible complex structure $J$ and its associated metric $g=\omega \cdot J$, the curvature tensor depends on three parameters $\lambda$, $\psi_{11}$ and $\psi_{12}\ne 0$. The canonical compatible complex structure has $J$-invariant 2-plane $\{e_1,e_2\}$, $\{e_3,e_4\}$ and $\{e_5,e_6\}$, however its shape is complicated: $J(e_1) = \psi_{11}\, e_1 -{\frac {1+{\psi_{11}}^{2}}{\psi_{12}}}\, e_2,\quad J(e_2)=\psi_{12}\, e_1 -\psi_{11} \, e_2$, $J(e_3)=-{\frac {2\,{\psi_{12}}^{2}-3\,\psi_{11}\,\lambda\,\psi_{12} +{\lambda}^{2}(1+{\psi_{11}}^{2})}{\lambda \,\psi_{12}}}\, e_3+{\frac {{\psi_{12}}^{2}-2\,\psi_{11}\,\lambda\,\psi_{12} +\lambda^{2}(1+\psi_{11}^{2})}{\lambda\, \psi_{12}}}\, e_4$, $J(e_4)=-{\frac {{\lambda}^{2}({\psi_{11}}^{2}+1)-4\,\psi_{11}\,\lambda\,\psi_{12} +4\,{\psi_{12}}^{2}}{\psi_{12}\,\lambda}}\, e_3+{\frac {{ \lambda}^{2}({\psi_{11}}^{2}+1)-3\,\psi_{11}\,\lambda\,\psi_{12}+2\, {\psi_{12}}^{2}}{\psi_{12}\,\lambda}}\, e_4$, $J(e_5)={\frac {\psi_{11}\,\psi_{12}- \lambda(1+\psi_{11}^{2})}{\psi_{12}}}\,e_5 +{\frac {1+ {\psi_{11}}^{2}}{\psi_{12}}}\, e_6, $ $J(e_6)=-\frac {{\psi_{12}} ^{2}-2\,\psi_{11}\,\lambda\,\psi_{12}+{\lambda}^{2}+{\lambda}^{2}{ \psi_{11}}^{2}}{\psi_{12}}\,e_5 +\frac {-\psi_{11}\,\psi_{12}+ \lambda+{\psi_{11}}^{2}\lambda}{\psi_{12}}\, e_6. $ The curvature tensor of associated metric $g=\omega \cdot J$ has the following non-zero component: $$R_{1, 2, 1, 2} = -\frac{\lambda^2(\psi_{11}^2+1) -5\lambda \psi_{12}\,\psi_{11} +4\psi_{12}^2}{\lambda \psi_{12}}.$$ The Lie group $G_{10}$ ---------------------- The Lie algebra $\mathfrak{g}_{10}$ is defined by: $[e_1,e_2] = e_4$, $[e_1,e_4] = e_5$, $[e_1,e_3] = e_6$, $[e_2,e_4] = e_6$.\ Its symplectic structure comes from the list in [@Goze-Khakim-Med]: $\omega = e^1\wedge e^6 + e^2\wedge e^5 - e^2\wedge e^6 - e^3\wedge e^4 .$\ For any compatible complex structure $J$ and its associated metric $g=\omega \cdot J$, the curvature tensor depends on two parameters $\psi_{11}$ and $\psi_{12}\ne 0$. The canonical compatible complex structure has $J$-invariant 2-plane $\{e_1,e_2\}$, $\{e_3,e_4\}$ and $\{e_5,e_6\}$, however its shape is complicated: $J(e_1) = \psi_{11}\, e_1 -{\frac {1+{\psi_{11}}^{2}}{\psi_{12}}}\, e_2,\quad J(e_2)=\psi_{12}\, e_1 -\psi_{11} \, e_2$, $J(e_3)=-{\frac {\psi_{12}+3\,{ \psi_{11}}^{2}\psi_{12}+2\,{\psi_{12}}^{2}\psi_{11}+\psi_{11} +{\psi_{11}}^{3}}{2\,{\psi_{12}}^{2}+2\,\psi_{12}\,\psi_{11}+1 +{\psi_{11}}^{2}}}\, e_3 -{\frac {({\psi_{12}}^{2}+2\,\psi_{12}\,\psi_{11}+1+{{ \psi_{11}}}^{2}) \psi_{12}}{2\,{\psi_{12}}^{2}+2\,\psi_{12}\,\psi_{11}+1+{\psi_{11}}^{2}}}\, e_4$, $J(e_4)={\frac { \left( {\psi_{11}}^{2}+4\,\psi_{12} \,\psi_{11}+4\,{\psi_{12}}^{2}+1 \right) \left( 1+{\psi_{11}}^{ 2} \right) }{\psi_{12}\, \left( 2\,{\psi_{12}}^{2}+2\,\psi_{12} \,\psi_{11}+1+{\psi_{11}}^{2} \right) }}\, e_3 +{\frac {\psi_{12}+3\,{\psi_{11}}^{2}\psi_{12}+2\,{\psi_{12}}^{2}\psi_{11}+\psi_{11} +{\psi_{11}}^{3}}{2\,{\psi_{12}}^{2}+2\,\psi_{12}\,\psi_{11}+1 +{\psi_{11}}^{2}}}\, e_4$, $J(e_5)={\frac {\psi_{12}\,\psi_{11}+1+{\psi_{11}}^{2}}{\psi_{12}}}\,e_5 +{\frac {1+\psi_{11}^{2}}{\psi_{12}}}\, e_6$. $J(e_6)=-{\frac {{\psi_{12}}^{2}+2\,\psi_{12}\,\psi_{11}+1+{\psi_{11}}^{2}}{\psi_{12}}}\,e_5 -{\frac {\psi_{12}\,\psi_{11}+1+{\psi_{11}}^{2}}{\psi_{12}}}\, e_6$. The curvature tensor of associated metric $g=\omega \cdot J$ has the following non-zero component: $$R_{1, 2, 1, 2} = {\frac {{\psi_{11}}^{4}+4\,\psi_{12}\,{\psi_{11}}^{3}+3\, {\psi_{11}}^{2}{\psi_{12}}^{2}+2\,{\psi_{11}}^{2}+4\,\psi_{11}\, \psi_{12}-2\,\psi_{11}\,{\psi_{12}}^{3}+3\,{\psi_{12}}^{2}+1-2 \,{\psi_{12}}^{4}}{\psi_{12}\, \left( 2\,{\psi_{12}}^{2}+2\, \psi_{11}\,\psi_{12}+1+{\psi_{11}}^{2} \right) }}.$$ The Lie group $G_{12}$ ---------------------- The Lie algebra $\mathfrak{h}_{12}$ is defined by: $[X_1,X_2] = X_4$, $[X_1,X_4] = X_5$, $[X_1,X_3] = X_6$, $[X_2,X_3] = -X_5$, $[X_2,X_4] = X_6$. Its symplectic structure comes from the list in [@Goze-Khakim-Med]: $\omega = \lambda \alpha^1\wedge \alpha^5 +\alpha^2\wedge \alpha^6 +(\lambda+1)\alpha^3\wedge \alpha^4, \qquad \lambda\ne 0; -1.$ This is the Lie algebra $M10$, considered in the work of Magnin [@Mag-3] and for which he found the complex structures in an explicit form. To use these outcomes, we will make the replacements: $X_1=-e_1$, $X_2=e_2$, $X_3=-e_4$, $X_4=-e_3$, $X_5=e_5$, $X_6=e_6$. Then the Lie bracket relations and symplectic structures become: $[e_1,e_2] = e_3$, $[e_1,e_3] = e_5$, $[e_1,e_4] = e_6$, $[e_2,e_4] = e_5$, $[e_2,e_3] = -e_6$. $\omega = -\lambda e^1\wedge e^5 +e^2\wedge e^6 -(\lambda+1)e^3\wedge e^4, \qquad \lambda\ne 0; -1.$ In [@Mag-3] it is shown that on the given group there are several types of complex structures from which we will choose the ones that are compatible with $\omega$. First type: $\psi_{12}\ne \psi_{34}$.\ For any compatible complex structure of this type and its associated metric, the curvature tensor depends on three parameters $\psi_{11}$, $\psi_{33}$ and $\psi_{12}\ne 0$. The elements of the tensor of curvature have very complicated expressions which also depend on the parameter $\lambda$. We will reduce the expressions for the compatible complex structure $J_1$ and the corresponding metric $g_1$ using the zero parameters which do not appear in the curvature and adding the side condition: $\psi_{11}=0$, $\psi_{33}=0$. $$J_1(e_2)=\psi_{12}\, e_1,\qquad J_1(e_4)={\frac { \left( \lambda-1 \right) \psi_{12}}{{\lambda\,\psi_{12}}^{2}-1}}\, e_3,\qquad J_1(e_6)=-{\frac {1}{\lambda\,\psi_{12}}}\, e_5.$$ $$g_1= \left[ \begin {array}{cccccc} 0&0&0&0&0&{\psi_{12}}^{-1}\\ \noalign{\medskip}0&0&0&0&\lambda\,\psi_{12}&0 \\ \noalign{\medskip}0&0&-{\frac { \left( \lambda+1 \right) \left( {\psi_{12}}^{2}\lambda-1 \right) }{ \left( \lambda-1 \right) \psi_{12}}}&0&0&0\\ \noalign{\medskip}0&0&0&-{\frac {\psi_{12}\, \left( {\lambda}^{2}-1 \right) }{{\psi_{12}}^{2}\lambda-1}}&0&0 \\ \noalign{\medskip}0&\lambda\,\psi_{12}&0&0&0&0 \\ \noalign{\medskip}{\psi_{12}}^{-1}&0&0&0&0&0\end {array} \right]$$ The curvature tensor has the following non-zero components: $$R_{1, 2, 2}^6 = -\frac{\lambda^3\psi_{12}^2+3\lambda^2\psi_{12}^2 -3\lambda - 1}{\lambda^2-1},\quad R_{1, 2, 1}^5 = \frac{\lambda^3\psi_{12}^2+3\lambda^2\psi_{12}^2 -3\lambda - 1}{\lambda(\lambda^2-1)\psi_{12}^2}.$$ After omitting the index, there remains one component: $$R_{1, 2, 1, 2}=\frac{\lambda^3\psi_{12}^2+3\lambda^2\psi_{12}^2 -3\lambda - 1}{(\lambda^2-1)\psi_{12}^2}.$$ Second type: $\psi_{12}=\psi_{34}$, $\psi_{11}=\psi_{33}=0$.\ The compatible complex structure of this type is of the form: $$J_2= \left[ \begin {array}{cccccc} 0&-1&0&0&0&0\\ \noalign{\medskip}1&0&0&0 &0&0\\ \noalign{\medskip}\psi_{31}&\psi_{41}&0&-1&0&0 \\ \noalign{\medskip}\psi_{41}&-\psi_{31}&1&0&0&0 \\ \noalign{\medskip}\psi_{51}&\psi_{52}&{\frac {\psi_{41}\, \left( \lambda+1 \right) }{\lambda}}&-{\frac {\psi_{31}\, \left( \lambda+1 \right) }{\lambda}}&0&{\lambda}^{-1}\\ \noalign{\medskip}- \lambda\,\psi_{52}&J_{61}&\psi_{31} \,(\lambda+1)&\psi_{41}\,(\lambda+1)&-\lambda&0 \end {array} \right],$$ where $J_{61}=\lambda\,\psi_{51}-({\psi_{41}}^{2}+{\psi_{31}}^{2})(\lambda +1)$. For any such complex structure and its associated metric, the curvature tensor does not depend on the parameters $\psi_{ij}$, and depends only on $\lambda$. The non-zero elements of the tensor of curvature are: $R_{1, 2, 2}^6 = -\frac{\lambda^2+4\lambda +1}{\lambda+1}$,$R_{1, 2, 1}^5 = \frac{\lambda^2+4\lambda +1}{\lambda(\lambda+1)}$. After omitting the index, there remains one component: $R_{1, 2, 1, 2}=-\frac{\lambda^2+4\lambda +1}{\lambda+1}$. From our point of view, it is possible to consider that all the remaining parameters are zero. Then we find the canonical expressions for the complex structure and the pseudo-Riemannian metric: $$J_{20}(e_1)=e_2,\quad J_{20}(e_3)=e_4,\quad J_{20}(e_5)=-\lambda\, e_6,$$ $$g_{20}=-2\,e^1\cdot e^6 -2\,\lambda\,e^2\cdot e^5 -(\lambda+1)\, (e^3)^2 -(\lambda+1)\, (e^4)^2.$$ Third type: $\psi_{34}=\psi_{12}$, $\psi_{33}\ne \psi_{11}$.\ For any compatible complex structure of this type and the associated metric, the curvature tensor does not depend on the parameters $\psi_{ij}$. Converting the free parameters to zero, we find the following canonical expressions for the complex structure and the associated pseudo-Kähler metric: $$J_{30}= \left[ \begin {array}{cccccc} 1&\sqrt {2}&0&0&0&0\\ \noalign{\medskip}-\sqrt {2}&-1&0&0&0&0\\ \noalign{\medskip}0&0&-{ \frac {\lambda+1}{\lambda-1}}&\sqrt {2}&0&0\\ \noalign{\medskip}0&0&-{ \frac { \left( {\lambda}^{2}+1 \right) \sqrt {2}}{ \left( \lambda-1 \right) ^{2}}}&{\frac {\lambda+1}{\lambda-1}}&0&0 \\ \noalign{\medskip}0&0&0&0&-1&-{\frac {\sqrt {2}}{\lambda}} \\ \noalign{\medskip}0&0&0&0&\sqrt {2}\lambda&1\end {array} \right],$$ $$g_{30}= \left[ \begin {array}{cccccc} 0&0&0&0&\lambda&\sqrt {2}\\ \noalign{\medskip}0&0&0&0&\sqrt {2}\lambda&1\\ \noalign{\medskip}0&0 &{\frac { \left( \lambda+1 \right) \left( {\lambda}^{2}+1 \right) \sqrt {2}}{ \left( \lambda-1 \right) ^{2}}}&-{\frac { \left( \lambda+1 \right) ^{2}}{\lambda-1}}&0&0\\ \noalign{\medskip}0&0&-{\frac { \left( \lambda+1 \right) ^{2}}{\lambda-1}}& \left( \lambda+1 \right) \sqrt {2}&0&0\\ \noalign{\medskip}\lambda&\sqrt {2}\lambda&0&0&0&0 \\ \noalign{\medskip}\sqrt {2}&1&0&0&0&0\end {array} \right].$$ The curvature tensor has the following non-zero components: $$R_{1, 2, 2}^6 = -\frac{2(\lambda^4+4\lambda^3 -2\lambda^2 +4\lambda +1)}{(\lambda-1)^2(\lambda+1)},\quad R_{1, 2, 1}^5 = \frac{2(\lambda^4+4\lambda^3 -2\lambda^2 +4\lambda +1)}{(\lambda+1)\lambda(\lambda-1)^2}.$$ After omitting the index, there remains one component: $$R_{1, 2, 1, 2}=\frac{\sqrt{2}(\lambda^4+4\lambda^3 -2\lambda^2 +4\lambda +1)}{(\lambda-1)^2(\lambda+1)}.$$ Lie algebras of type $(2,6)$ ============================ There are three Lie algebras of type $(2,6)$ admitting a pseudo-Kähler structure. The Lie group $G_{24}$ ---------------------- The Lie algebra $\mathfrak{h}_{24}$ is defined by: $[e_1,e_4] = e_6$, $[e_2,e_3] = e_5$.\ This Lie algebra is a direct product two three-dimensional Lie algebras of Heisenberg: $\mathfrak{g}_{24} =\mathfrak{h}_3\times\mathfrak{h}_3$ The symplectic structure is: $\omega = e^1\wedge e^6 +e^2\wedge e^5 +e^3\wedge e^4$. For any compatible complex structure $J$ and its associated metric $g=\omega \circ J$, the curvature tensor depends on two parameters $\psi_{11}\ne 0$ and $\psi_{12}\ne 0$. Therefore the semicanonical pseudo-Kähler structure is as follows: $J(e_1) = \psi_{11}\, e_1 -\frac {1+\psi_{11}^{2}}{\psi_{12}}\, e_2,\quad J(e_2) = \psi_{12}\, e_1 -\psi_{11}\, e_2$, $J(e_3) = {\frac {{\psi_{11}}^{2}-1}{2\psi_{11}}}\, e_3 -{\frac {2\,{\psi_{11}}^{2}+ {\psi_{11}}^{4}+1}{2\psi_{11}\,{\psi_{12}}^{2}}}\, e_4,\quad J(e_4) = \frac {\psi_{12}^{2}}{2\psi_{11}}\, e_3 -\frac {\psi_{11}^{2}-1}{2\,\psi_{11}}\, e_4$, $J(e_5) = \psi_{11}\, e_5 +\frac {1+\psi_{11}^{2}}{\psi_{12}}\, e_6,\quad J(e_6) = -\psi_{12}\, e_5 -\psi_{11}\, e_6.$ $$g= \left[ \begin {array}{cccccc} 0&0&0&0&{\frac {1+{\psi_{11}}^{2}}{\psi_{12}}}&-\psi_{11}\\ \noalign{\medskip}0&0&0&0&\psi_{11}&-\psi_{12}\\ \noalign{\medskip}0&0&-{\frac {2\,{\psi_{11}}^{2}+ {\psi_{11}}^{4}+1}{2\,{\psi_{12}}^{2}\psi_{11}}}&-{\frac {{ \psi_{11}}^{2}-1}{2\,\psi_{11}}}&0&0\\ \noalign{\medskip}0&0&-{\frac {{\psi_{11}}^{2}-1}{2\,\psi_{11}}}&-{\frac {{\psi_{12}}^{2}}{2\,\psi_{11}}}&0&0\\ \noalign{\medskip}{\frac {1+{\psi_{11}}^{2}}{\psi_{12}}}&\psi_{11}&0&0&0&0\\ \noalign{\medskip}-\psi_{11}&-\psi_{12}&0&0&0&0\end {array} \right].$$ For any compatible complex structure and its associated metric, the curvature tensor depends on two parameters $\psi_{11}\ne 0$ è $\psi_{12}\ne 0$: $R_{1, 2, 2}^6 = -\psi_{11}^2$,$R_{1, 2, 1}^6 =-\frac{(1 +\psi_{11}^2)\psi_{11}}{\psi_{12}}$,$R_{1, 2, 2}^5 =-\psi_{12} \psi_{11}$,$R_{1, 2, 1}^5 = -\psi_{11}^2$.On omitting the index of the curvature tensor, it turns out that there is only one (within symmetries) non-zero component of the tensor of curvature $R_{1, 2, 1, 2} = \psi_{11}$. The Lie group $G_{17}$ ---------------------- The Lie algebra $\mathfrak{h}_{17}$ is defined by: $[e^1,e^3] = e^5$, $[e^1,e^4] = e^6$, $[e^2,e^3] = e^6$,\ Its symplectic structure comes from the list in [@Goze-Khakim-Med]: $\omega = e^1\wedge e^6 + e^2\wedge e^5 + e^3\wedge e^4$ For any compatible complex structure and its associated metric, the curvature tensor depends on two parameters $\psi_{11}$ and $\psi_{12}\ne 0$: Setting the remaining free parameters $\psi_{ij}$ to zero, we find the canonical complex structure $J$ and associated metric $g=\omega \circ J$: $J(e_1) = \psi_{11}\, e_1 -{\frac {1+{\psi_{11}}^{2}}{\psi_{12}}}\, e_2,\qquad J(e_2) = \psi_{12}\, e_1 -\psi_{11}\, e_2$, $J(e_3) = \psi_{11}\,e_3 +2\,{\frac {1+{\psi_{11}}^{2}}{\psi_{12}}}\, e_4,\qquad J(e_4) = -\frac {\psi_{12}}{2}\, e_3 -\psi_{11}\, e_4$, $J(e_5) = \psi_{11}\, e_5 +{\frac {1+{\psi_{11}}^{2}}{\psi_{12}}}\, e_6,\qquad J(e_6) = -\psi_{12}\, e_5 -\psi_{11}\, e_6$. $$g=\left[ \begin {array}{cccccc} 0&0&0&0&{\frac {1+{\psi_{11}}^{2}}{ \psi_{12}}}&-\psi_{11}\\ \noalign{\medskip}0&0&0&0&\psi_{11}&-\psi_{12}\\ \noalign{\medskip}0&0&2\,{\frac {1+{\psi_{11}}^{2}}{\psi_{12}}}&-\psi_{11}&0&0\\ \noalign{\medskip}0&0&-\psi_{11}&\psi_{12}/2&0&0\\ \noalign{\medskip}{\frac {1+{\psi_{11}}^{2}}{\psi_{12}}}&\psi_{11}&0&0&0&0\\ \noalign{\medskip}-\psi_{11}&-\psi_{12}&0&0&0&0\end {array} \right]$$ The curvature tensor has the following non-zero components: $R_{1, 2, 1}^5 = \psi_{11}\psi_{12}$,$R_{1, 2, 1}^6 = 1+\psi_{11}^2$,$R_{1, 2, 2}^5 = \psi_{12}^2$,$R_{1, 2, 2}^6 = \psi_{11}\psi_{12} $. After omitting the index, there remains one component: $R_{1, 2, 1, 2} = -\psi_{12}$. The Lie group $G_{16}$ ---------------------- The Lie algebra $\mathfrak{g}_{16}$ is defined by: $[X_1,X_2] = X_5$, $[X_1,X_3] = X_6$, $[X_2,X_4] = X_6$, $[X_3,X_4] =-X_5$. It is complex Heisenberg Lie algebra. The algebra $\mathfrak{g}_{16}$ has (see [@Goze-Khakim-Med]) two symplectic structures which are noted using the dual base $\alpha^i$ as follows: $\omega_1 = \alpha^1\wedge \alpha^6 +\alpha^2\wedge \alpha^3 -\alpha^4\wedge \alpha^5$, $\omega_2 = \alpha^1\wedge \alpha^6 -\alpha^2\wedge \alpha^3 +\alpha^4\wedge \alpha^5$.\ Left-invariant complex structures on this group are discovered in an explicit form in the work of Magnin [@Mag-3] (algebra $M5$). We will use Magnin’s results, so we will make the replacement: $X_1=e_1$, $X_2=e_3$, $X_3=e_4$, $X_4=e_2$, $X_5=e_5$, $X_6=e_6$. Then the non-trivial brackets are given in [@Mag-3]: $[e_1,e_3] = e_5$, $[e_1,e_4] = e_6$, $[e_2,e_3] =-e_6$, $[e_2,e_4] =e_5$.\ The symplectic forms become: $\omega_1 = e^1\wedge e^6 +e^3\wedge e^4 -e^2\wedge e^5$, $\omega_2 = e^1\wedge e^6 -e^3\wedge e^4 +e^2\wedge e^5$. Let $z_1=\frac{1}{2}(e_1+ie_2)$, $z_2=\frac{1}{2}(e_3-ie_4)$, $z_3=\frac{1}{2}(e_5-ie_6)$, then Lie algebra $\mathfrak{g}_{16}$ is defined by: $[z_1,z_2]=z_3$. If $J_0$ is complex structure of the complex Lie algebra $M5$, then: $J_0(e_1)=-e_2$, $J_0(e_3)=e_4$, $J_0(e_5)=e_6$. The complex structure $J_0$ is not $\omega_1$-compatible, but compatible with $\omega_2$. Thus, the metric $g_0$ of pseudo-Kähler structure $(J_0,\omega_2,g_0)$ is as follows: $$g_{0}=2\,e^1\cdot e^6 -2\,e^2\cdot e^5 -\, (e^3)^2 -\, (e^4)^2.$$ The curvature tensor has the following non-zero components: $R_{1, 2, 1}^6 =1$, $R_{1, 2, 2}^5 =1$, $R_{1, 2, 1, 2} =-1$. First case. The symplectic structure is: $\omega_1 = e^1\wedge e^6 -e^2\wedge e^5 +e^3\wedge e^4$. For any compatible complex structure $J$, the curvature tensor of the associated metric $g=\omega_1 \circ J$ depends on two parameters $\psi_{11}$ and $\psi_{12}\ne 0$. Setting the remaining free parameters $\psi_{ij}$ to zero, we find the canonical complex structure $J_1$ and pseudo-Kähler metric $g_1=\omega_1 \circ J_1$: $J_1(e_1) = \psi_{11}\, e_1 -{\frac {1+{\psi_{11}}^{2}}{\psi_{12}}}\, e_2,\qquad J_1(e_2) = \psi_{12}\, e_1 -\psi_{11}\, e_2$, $J_1(e_3) = -{\frac {\psi_{11}\, (1+{\psi_{11}}^{2}-{\psi_{12}}^{2})}{ {\psi_{12}}^{2}+1+{\psi_{11}}^{2}}}\, e_3 -{\frac {2( 1+{\psi_{11}}^{2}) \psi_{12}}{{\psi_{12}}^{2}+1+{\psi_{11}}^{2}}}\, e_4$, $J_1(e_4) = {\frac {2\,{\psi_{11}}^{2} +{\psi_{11}}^{4}-2\,{\psi_{11}}^{2}{\psi_{12}}^{2} +{\psi_{12}}^{4}+2 \,{\psi_{12}}^{2}+1}{2\psi_{12}\, ({\psi_{12}}^{2} +1 +{\psi_{11}}^{2}) }}\, e_3 +{\frac {\psi_{11}\, (1+{\psi_{11}}^{2}-{\psi_{12}}^{2}) }{{\psi_{12}}^{2}+1+{\psi_{11}}^{2}}}\, e_4$, $J_1(e_5) = \psi_{11}\, e_5 -{\frac {1+{\psi_{11}}^{2}}{\psi_{12}}}\, e_6,\qquad J_1(e_6) = \psi_{12}\, e_5 -\psi_{11}\, e_6$.\ The curvature tensor has the following non-zero components: $$R_{1, 2, 1}^5 =\frac{\psi_{11}(1+\psi_{11}^2+\psi_{12}^2)}{\psi_{12}},\quad R_{1, 2, 2}^5=1+\psi_{11}^2+\psi_{12}^2,$$ $$R_{1, 2, 1}^6 =-\frac{(1+\psi_{11}^2)(1+\psi_{11}^2+\psi_{12}^2)}{\psi_{12}^2},\quad R_{1, 2, 2}^6=-\frac{\psi_{11}(1+\psi_{11}^2+\psi_{12}^2)}{\psi_{12}}.$$ After omitting the index, there remains one component: $$R_{1, 2, 1, 2} =\frac{1+\psi_{11}^2+\psi_{12}^2}{\psi_{12}}.$$ Second case. The symplectic structure is: $\omega_2 = e^1\wedge e^6 +e^2\wedge e^5 -e^3\wedge e^4$. For any compatible complex structure $J$, the curvature tensor of the associated metric $g=\omega_2 \circ J$ depends on two parameters $\psi_{34}$ and $\psi_{12}=\pm 1$. Let $\psi_{12}=1$. Setting the remaining free parameters $\psi_{ij}$ to zero, we find the canonical complex structure $J_2$ and pseudo-Kähler metric $g_2=\omega_2 \circ J_2$: $$J_2(e_2) = e_1,\quad J_2(e_4) = \psi_{34}\, e_3,\quad J_2(e_6) = -e_5,$$ $$g_{2}=2\,e^1\cdot e^5 -2\,e^2\cdot e^6 +\frac{1}{\psi_{34}}\, (e^3)^2 +\psi_{34}\, (e^4)^2.$$ The curvature tensor has the following non-zero components: $R_{1, 2, 1}^6 = -{\rm sign}(\psi_{12})\psi_{34}$,$R_{1, 2, 2}^5 = -{\rm sign}(\psi_{12})\psi_{34}$. After omitting the index, there remains one component: $R_{1, 2,1, 2} = {\rm sign}(\psi_{12})\psi_{34}$. The complex structure $J_2$ is biinvariant (then $G_{16}$ is complex Lie group) if and only if $\psi_{34} =-1$, that is $J_2=J_0$. Lie algebras of type $(4,6)$ ============================ In this class there is only one algebra. The Lie group $G_{25}$ ---------------------- The Lie algebra $\mathfrak{h}_{25}$ is defined by $[e_1,e_2] =e_3$. This Lie algebra is a direct product of a three-dimensional nilpotent Lie algebra of Heisenberg $\mathfrak{h}_3$ and $\mathbb{R}^3$. The symplectic structure is: $\omega=e^1\wedge e^3+e^2\wedge e^4+e^5\wedge e^6$ In this case there is an 8-parametrical set of the compatible complex structures and pseudo-Kähler metrics. All the metrics are flat. Therefore we will specify only the simplest expressions, without parameters: $$J(e_1) = e_2,\quad J(e_3) = e_4,\quad J(e_5) = e_6.$$ $$g=\left[ \begin {array}{cccccc} 0&0&0&-1&0&0\\ 0&0&1&0&0&0\\ 0&1&0&0&0&0\\ -1&0&0&0&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1\end {array} \right].$$ Lie algebras of type $(3,6)$ ============================ There are two Lie algebras of type (3,6) admitting a pseudo-Kähler structure. The Lie group $G_{18}$ ---------------------- The Lie algebra $\mathfrak{h}_{18}$ is defined by: $[e_{1},e_{2}] =e_{4}$, $[e_{1},e_{3}]=e_{5}$, $[e_{2},e_{3}] =e_{6}$. This Lie algebra has three symplectic structures [@Goze-Khakim-Med]: $\omega _{1}(\lambda )=e^{1}\wedge e^{6}+\lambda e^{2}\wedge e^{5}+\left( \lambda -1\right) e^{3}\wedge e^{4}$, $\quad \lambda \ne 0,\,1$, $\omega _{2}(\lambda ) = e^{1}\wedge e^{5}{+}\lambda e^{1}\wedge e^{6}{-}\lambda e^{2}\wedge e^{5}{+}e^{2}\wedge e^{6}{-}2\lambda e^{3}\wedge e^{4}$, $\quad \lambda \ne 0$, $\omega _{3}= -e^{1}\wedge e^{6} +e^{2}\wedge e^{5}+2e^{3}\wedge e^{4}+e^{3}\wedge e^{5}$.\ Left-invariant complex structures on this group are discovered in an explicit form in the work of Magnin [@Mag-3] (algebra $M3$). First case. The symplectic structure is: $\omega_1 = e^1\wedge e^6 +\lambda e^2\wedge e^5 +(\lambda-1) e^3\wedge e^4$. The compatible complex structures exist only at $\lambda =-1$, therefore $\omega_1 = e^1\wedge e^6 - e^2\wedge e^5 -2\, e^3\wedge e^4$.\ For any compatible complex structure and its associated metric, the curvature tensor depends on three parameters $\psi_{11}$, $\psi_{12}$ and $\psi_{34}$. The curvature tensor has the following non-zero components: $R_{1, 2, 1}^6 =-\frac{2\psi_{34}(1 + \psi_{11}^2 )}{\psi_{12}}$,$R_{1, 2, 2}^6 =-2\psi_{11} \psi_{34}$,$R_{1, 2, 2}^5~=~2\psi_{12} \psi_{34}$,$R_{1, 2, 1}^5 =2\psi_{11} \psi_{34}$. After omitting the index, there remains one component: $R_{1, 2, 1, 2} =2\psi_{34}$. Setting the remaining free parameters $\psi_{ij}$ to zero, we find the canonical complex structure $J_1$ and the pseudo-Kähler metric $g_1 =\omega_1 \circ J_1$: $J_1(e_1) = \psi_{11}\, e_1 -{\frac {1+{\psi_{11}}^{2}}{\psi_{12}}}\, e_2,\qquad J_1(e_2) = \psi_{12}\, e_1 -\psi_{11}\, e_2$, $J_1(e_4) = -{\frac {1}{\psi_{34}}}\, e_4,\qquad J_1(e_4) = \psi_{34}\, e_3$, $J_1(e_5) = \psi_{11}\, e_5 -{\frac {1+{\psi_{11}}^{2}}{\psi_{12}}}\, e_6,\qquad J_1(e_6) = \psi_{12}\, e_5 -\psi_{11}\, e_6$. $$g_1= \left[ \begin {array}{cccccc} 0&0&0&0&-{\frac {1+{\psi_{11}}^{2}}{\psi_{12}}}&-\psi_{11}\\ \noalign{\medskip}0&0&0&0&-\psi_{11}&-\psi_{12}\\ \noalign{\medskip}0&0&{\frac {2}{\psi_{34}}}&0&0&0\\ \noalign{\medskip}0&0&0&2\,\psi_{34}&0&0\\ \noalign{\medskip}-{\frac {1+{\psi_{11}}^{2}}{\psi_{12}}}&-\psi_{11}&0&0&0&0\\ \noalign{\medskip}-\psi_{11}&-\psi_{12}&0&0&0&0\end {array} \right]$$ Second case. The symplectic structure is: $\omega_2 = e^1\wedge e^5 +e^2\wedge e^6 +\lambda e^1\wedge e^6 -\lambda e^2\wedge e^5 -\\ -2\lambda e^3\wedge e^4$. The compatible complex structures exist only for a case $\psi_{16}=0$, $\psi_{25}=0$ and $\psi_{12}^2=1$. We take a case $\psi_{12}=1$. For any compatible complex structure and its associated metric, the curvature tensor depends on one parameter $\psi_{34}$. The curvature tensor has the following non-zero components: $R_{1, 2, 2}^6 =-\frac{2 \lambda \psi_{34}}{\lambda^2 +1}$,$R_{1, 2, 1}^6 = -\frac{2 \lambda^2 \psi_{34}}{\lambda^2 +1}$,$R_{1, 2, 1}^5 = -\frac{2 \lambda \psi_{34}}{\lambda^2 +1}$,$R_{1, 2, 2}^5 =\frac{2 \lambda^2 \psi_{34}}{\lambda^2 +1}$. After omitting the index, there remains one component: $R_{1, 2, 1, 2} = 2\lambda \psi_{34}$.Setting the remaining free parameters $\psi_{ij}$ to zero, we find the canonical complex structure $J_2$ and the pseudo-Kähler metric $g_2 =\omega_2 \circ J_2$: $$J_2(e_2) =e_1,\quad J_2(e_4) = \psi_{34}\, e_3,\quad J_2(e_6) = e_5,$$ $$g_2=\left[ \begin {array}{cccccc} 0&0&0&0&-\lambda&1\\ 0&0&0&0&-1&-\lambda\\ 0&0&\frac {2\,\lambda}{\psi_{34}}&0&0&0\\ 0&0&0&2\,\lambda\,\psi_{34}&0&0\\ -\lambda&-1&0&0&0&0\\ 1&-\lambda&0&0&0&0\end {array} \right].$$ Third case. The symplectic structure is:\ $\omega_3 = -e^1\wedge e^6 +e^2\wedge e^5 +2e^3\wedge e^4 +e^3\wedge e^5 $. For any compatible complex structure and its associated metric, the curvature tensor depends on two parameters $\psi_{25}\ne 0$ and $\psi_{46}\ne 0$. The curvature tensor has the following non-zero components: $R_{1, 2, 1}^6 = -\frac{6 \psi_{25}}{\psi_{46}}$,$R_{1, 2, 2}^5 = 54 \psi_{46} \psi_{25}$,$R_{1, 2, 3}^5 = 18 \psi_{46} \psi_{25}$,$R_{1, 2, 3}^4 = -6 \psi_{46} \psi_{25}$,$R_{1, 3, 1}^6 =-\frac{2 \psi_{25}}{\psi_{46}}$,$R_{1, 3, 2}^5 = 18 \psi_{46} \psi_{25}$,$R_{1, 2, 2}^4 =-18 \psi_{46} \psi_{25}$,$R_{1, 3, 2}^4 = -6 \psi_{46} \psi_{25}$,$R_{1, 3, 3}^5 = 6 \psi_{46} \psi_{25}$,$R_{1, 3, 3}^4 = -2 \psi_{46} \psi_{25}$. After omitting the index, there remains three components: $R_{1, 2, 1, 3} = 6 \psi_{25}$,$R_{1, 3, 1, 3} = 2 \psi_{25}$,$R_{1, 2, 1, 2} = 18 \psi_{25}$. Setting the remaining free parameters $\psi_{ij}$ to zero, we find the canonical complex structure $J_3$ and the pseudo-Kähler metric $g_3 =\omega_2 \circ J_3$: $$J_3= \left[ \begin {array}{cccccc} 0&-3\,\psi_{46}&-\psi_{46}&0&0&0\\ \noalign{\medskip}0&0&0&3\,\psi_{25}&\psi_{25}&0\\ \noalign{\medskip}{\psi_{46}}^{-1}&0&0&-9\,\psi_{25}&-3\,\psi_{25}&0\\ \noalign{\medskip}0&-{\psi_{25}}^{-1}&0&0&0&\psi_{46} \\ \noalign{\medskip}0&2\,{\psi_{25}}^{-1}&0&0&0&-3\,\psi_{46}\\ \noalign{\medskip}0&0&0&2\,{\psi_{46}}^{-1}&{\psi_{46}}^{-1}&0 \end {array} \right],$$ $$g_{3}=\left[ \begin {array}{cccccc} 0&0&0&-2\,{\psi_{46}}^{-1}&-{\psi_{46}}^{-1}&0\\ \noalign{\medskip}0&2\,{\psi_{25}}^{-1}&0&0&0&-3\,\psi_{46}\\ \noalign{\medskip}0&0&0&0&0&-\psi_{46}\\ \noalign{\medskip}-2\,{\psi_{46}}^{-1}&0&0&18\,\psi_{25}&6\,\psi_{25}&0\\ \noalign{\medskip}-{\psi_{46}}^{-1}&0&0&6\,\psi_{25}&2\,\psi_{25}&0\\ \noalign{\medskip}0&-3\,\psi_{46}&-\psi_{46}&0 &0&0\end {array} \right].$$ The Lie group $G_{23}$ ---------------------- The Lie algebra $\mathfrak{h}_{23}$ is defined by: $[e_1,e_2] = e_5$, $[e_1,e_3] = e_6$. There are, according to [@Goze-Khakim-Med], three different symplectic structures: $\omega_1 = e^1\wedge e^6 +e^2\wedge e^5 +e^3\wedge e^4$, $\omega_2 = e^1\wedge e^4 +e^2\wedge e^6 +e^3\wedge e^5$ and $\omega_3 = e^1\wedge e^4 +e^2\wedge e^6 - e^3\wedge e^5$. For the first two symplectic structures there are no compatible complex structures. For the third symplectic structure $\omega_3$, there is a set of compatible complex structures that depend on several parameters. It will be convenient to enumerate basis vectors as follows: $e_2:=e_1$, $e_3:=-e_2$, $e_1:=e_3$, then $[e_1,e_3] = -e_5$, $[e_2,e_3] = e_6$\ and $\omega_3 = e^1\wedge e^6 + e^2\wedge e^5+ e^3\wedge e^4.$ It is easy to see that the given Lie algebra comes from the $\mathbb{R}^4= \mathbb{R}\{e_1, e_2, e_5, e_6\}$ semidirect product with $\mathbb{R} e_3$ and then a direct product with $\mathbb{R}e_4$, $\mathfrak{g}_{23}=\mathbb{R}^4 \rtimes \mathbb{R} e_3\times \mathbb{R} e_4$. The set of complex structures whose parameters influence the curvature operates on the invariant 2-planes $\{e_1, e_2\}$, $\{e_3, e_4\}$ è $\{e_5, e_6\}$ as follows: $$J(e_2) = \psi_{12}\, e_1 -\psi_{11}\, e_2,\quad J(e_4) = \psi_{34}\, e_3,\quad J(e_6) = -\psi_{12}\, e_5 -\psi_{11}\, e_6.$$ The curvature tensor depends on three parameters and has the following non-zero components: $R_{1, 2, 1}^6 =\frac{\psi_{34} (1 + \psi_{11}^2 )}{\psi_{12}}$,$R_{1, 2, 1}^5 = \psi_{11} \psi_{34}$,$R_{1, 2, 2}^5~=~\psi_{12} \psi_{34}$. After omitting the index, there remains one component $R_{1, 2, 1, 2} =-\psi_{34}$. Supposing $\psi_{34}=-a$, $\psi_{12}=1$, $\psi_{11}=0$ and $\psi_{33}=0$, we find the canonical pseudo-Kähler structure of curvature $R_{1, 2, 1, 2} =a$: $$J(e_2) = e_1,\quad J(e_4) = -a\, e_3 ,\quad J(e_6) = -e_5.$$ $$g=\left[ \begin {array}{cccccc} 0&0&0&0&1&0\\ 0&0&0&0&0&-1\\ 0&0&{a}^{-1}&0&0&0\\ 0&0&0&a&0&0\\ 1&0&0&0&0&0\\ 0&-1&0&0&0&0 \end {array} \right].$$ Formulas for evaluations ======================== We now present the formulas which were used for the evaluations (on Maple) of the Nijenhuis tensor and curvature tensor of the associated metrics. Let $e_1,\ldots,e_{2n}$ be a basis of the Lie algebra $\mathfrak g$ and $C_{ij}^k$ a structure constant of the Lie algebra in this base: $$[e_i,e_j]=\sum_{k=1}^{2n}C_{ij}^{k}e_k, \label{strukt}$$ 1. Nijenhuis tensor. Let $J^{k}_{i}$ be a matrix of a left-invariant almost complex structure $J$ in basis $\{e_i\}$, $Je_i=J^{k}_{i}e_k$. The Nijenhuis tensor is defined by formula (\[Nij1\]). For the basis vectors we obtain: $N(e_i,e_j) = N_{ij}^k e_k$, $$N(e_i,e_j)= [Je_i,Je_j] -[e_i,e_j] -J[Je_i,e_j] -J[e_i,Je_j]=$$ $$=\left(J_i^l J_j^m C_{lm}^k -J_i^l J_m^k C_{lj}^m -J_j^l J_m^k C_{il}^m -C_{ij}^k \right)\,e_k,$$ $$N_{ij}^k =J_i^l J_j^m C_{lm}^k -J_i^l J_m^k C_{lj}^m -J_j^l J_m^k C_{il}^m -C_{ij}^k.$$ 2. Compatible condition. This is the condition that $\omega(JX,Y) + \omega(X, JY) =0$, $\forall \, X, Y\in {{\mathfrak g}}$. For the basis vectors we have: $\omega (J (e_i), e_j) + \omega (e_i, J (e_j)) =0$, $\omega (J^k_i e_k, e_j) + \omega (e_i, J^s_j e_s) =0$. $$\omega _ {k, j} J^k_i + \omega _ {i, s} J^s_j=0.$$ 3. Connection components. These are the components $\Gamma_{ij}^{k}$ in the formula $\nabla _ {e _ {i}} e_j =\Gamma_{ij}^{k}e_k.$ For left-invariant vector fields we have: $2g ({\nabla}_{X} Y, Z) =g ([X, Y], Z) +g ([Z, X], Y)-g ([Y, Z], X)$. For the basis vectors we have: $$2g (\nabla _ {e _ {i}} e_j, e_k) =g ([e_i, e_j], e_k) +g ([e_k, e_i], e_j) +g (e_i, [e_k, e_j]),$$ $$2g _ {lk} \Gamma _ {ij} ^ {l} =g _ {pk} C _ {ij} ^ {p} +g _ {pj} C _ {ki} ^ {p} +g _ {ip} C _ {kj} ^ {p},$$ $$\Gamma_{ij}^{n}=\frac{1}{2}g^{kn}\left(g_{pk}C_{ij}^{p}+g_{pj}C_{ki}^{p} +g_{ip}C_{kj}^{p}\right).$$ 4. Curvature tensor. The formula is: $R(X, Y)Z =\nabla_{X} \nabla_{Y} Z -\nabla_{Y}\nabla_{X}Z -\nabla_{[X,Y]}Z$. For the basis vectors we have: $R (e_i, e_j) e_k=R _ {ijk} ^s e_s $, $$R(e_i,e_j)e_k=\nabla_{e_{i}}\nabla_{e_{j}}e_{k}-\nabla_{e_{j}}\nabla_{e_{i}}e_{k} -\nabla_{[e_{i},e_{j}]}e_{k}.$$ Therefore: $$R_{ijk}^{s}=\Gamma_{ip}^{s}\Gamma_{jk}^{p}-\Gamma_{jp}^{s}\Gamma_{ik}^{p} -C_{ij}^{p}\Gamma_{pk}^{s}.$$ [999]{} Benson C. and Gordon C.S. Kähler and symplectic structures on nilmanifold. Topology, Vol. 27, no. 4, 513-518, 1988. Cordero L.A., Fernandez M., Gray A., Ugarte L. Nilpotent complex structures. Rev. R. Acad. Cien. Serie A. Mat. Vol. 95(1), 2001, 45–55. Cordero L. A., Fernández M. and Ugarte L. Pseudo-Kähler metrics on six dimensional nilpotent Lie algebras. J. of Geom. and Phys., Vol. 50, 2004, 115 - 137. Goze M., Khakimdjanov Y., Medina A. Symplectic or contact structures on Lie groups. Differential Geom. Appl. Vol. 21, no. 1, 41-54, 2004, (arXiv:math/0205290v1 \[math.DG\]). Kobayashi S. and Nomizu K. Foundations of Differential Geometry, Vol. 1 and 2. Interscience Publ. New York, London. 1963. Magnin L. Complex structures on indecomposable 6-dimensional nilpotent real Lie algebras. Intern. J. of Algebra and Computation, vol. 17, Nr 1, 2007, p. 77–113. (http://monge.u-bourgogne.fr/lmagnin/artmagnin1.ps) Salamon S.M. Complex structure on nilpotent Lie algebras. J. Pure Appl. Algebra, Vol. 157, 311-333, 2001. (arXiv:math/9808025v2 \[math.DG\]) Smolentsev N.K. Canonical pseudo-Kähler metrics on six-dimensional nilpotent Lie groups. Bull. of Kemerovo State Univ. Vol. 3/1(47), 2011, P. 155–168. Ovando G. Invariant pseudo Kaehler metrics in dimension four. J. of Lie Theory, Vol. 16 (2), 2006, 371–391. (arXiv:math/0410232v1 \[math.DG\]) [^1]: The work was partially supported by RFBR 12-01-00873-a and by Russian President Grant supporting scientific schools SS-544.2012.1	ArXiv
--- abstract: 'We report the Suzaku/XIS results of the Galactic oxygen-rich supernova remnant (SNR), G292.0$+$1.8, a remnant of a core-collapse supernova. The X-ray spectrum of G292.0$+$1.8 consists of two type plasmas, one is in collisional ionization equilibrium (CIE) and the other is in non-equilibrium ionization (NEI). The CIE plasma has nearly solar abundances, and hence would be originated from the circumstellar and interstellar mediums. The NEI plasma has super-solar abundances, and the abundance pattern indicates that the plasma originates from the supernova ejecta with a main sequence of 30–35[[M]{}]{}$_{\odot}$. Iron K-shell line at energy of 6.6keV is detected for the first time in the NEI plasma.' author: - 'Fumiyoshi <span style="font-variant:small-caps;">Kamitsukasa</span>, Katsuji <span style="font-variant:small-caps;">Koyama</span>, Hiroshi <span style="font-variant:small-caps;">Tsunemi</span>, Kiyoshi <span style="font-variant:small-caps;">Hayashida</span>, Hiroshi <span style="font-variant:small-caps;">Nakajima</span>, Hiroaki <span style="font-variant:small-caps;">Takahashi</span>, Shutaro <span style="font-variant:small-caps;">Ueda</span>, Koji <span style="font-variant:small-caps;">Mori</span>, Satoru <span style="font-variant:small-caps;">Katsuda</span>, and Hiroyuki <span style="font-variant:small-caps;">Uchida</span>' title: 'Suzaku Discovery of Fe K-Shell Line from the O-Rich SNR G292.0$+$1.8' --- Introduction ============ X-ray spectra of optically-thin hot plasmas in supernova remnants (SNRs) provide key information on the nucleosynthesis during the stellar evolution and the supernova (SN) explosion. Since iron (Fe) is the final product of the major nuclear reaction network in a massive star and a SN, it is particularly important element. So far, Fe abundances of some SNRs have been estimated using the Fe L-shell lines, which are dominant in the low energy band around 1keV. This energy band is complex with other strong emission lines such as oxygen (O) and neon (Ne). Moreover the Fe L-lines consist of many emission lines in this narrow energy band, and hence conventional X-ray detectors such as X-ray CCD cannot resolve these many lines. The Fe K-shell lines at 6.4–6.7keV are simpler and their emission model is more reliable than those of the L-shell lines. However, due to the limited line fluxes and the sensitivity in the energy band above 6keV, the Fe K-shell lines have been reported from only a limited number of X-ray bright SNRs. Core-collapse (CC) SNe eject less Fe than those of Type Ia SNe (@Nomoto1984; @Iwamoto1999, and references therein). Most of the Fe in the center of the progenitor star is collapsed into a central compact object (neutron star or black hole). Thus observationally, Fe K-shell lines would become weaker than those in Type Ia. The flux of Fe may come from the combined effect of the elemental abundances and the thermal state of the gas. Thus the differences between Type Ia and CC SNRs may be related to the initial condition of the circumstellar medium (CSM) and an explosion mechanism. Fe K-shell lines have been reported from other candidates of CC SNRs (e.g., W49B: @Ozawa2009; IC443: @Yamaguchi2009; G349.7+0.2: @Slane2002 and G350.1-0.3: @Gaensler2008; 3C397: @Chen1999). However, some of them are controversial whether they are CC SNRs or not. The most reliable criteria of CC SN are the presence of a neutron star (pulsar), a pulsar wind nebula (PWN), and oxygen-rich (O-rich) knots, because O is largely enhanced in the ejecta of a CC SN. At present, three O-rich remnants, Cas A, PuppisA and G292.0$+$1.8, have been reported in our Galaxy (see @Vink2012, and references therein). Cas A is the most luminous, and hence the Fe K-shell line was firstly detected at 6.6keV (@Tsunemi1986). The spatial distribution is not a simple stratified structure (@Willingale2002). Thus the SN explosion would be highly asymmetric. PuppisA has also the interesting distribution of the ejecta, in that the ejecta is found only in the east, mostly north-east portion (@Winkler1985; @Hwang2008; @Katsuda2010), while a neutron star is propelled in the opposite direction (e.g., @Becker2012 and references therein). Such a recoil between SN ejecta and a neutron star is expected in a recent SN explosion model (@Scheck2006), and would be examined by asymmetric Fe distribution. However, no Fe K-shell line has been observed from PuppisA. For the study of the Fe K-shell line in CC SNRs, we observed G292.0$+$1.8. The discovery of a pulsar and a PWN in G292.0$+$1.8 is further confirmation of a CC SNR (@Hughes2001, [-@Hughes2003]; @Camilo2002; @Gaensler2003). The pulsar is located about 0.9southeast from the geometrical center of the SNR. The distance of G292.0$+$1.8 is estimated to be 6kpc (@Gaensler2003), and the age is likely 2990$\pm$60years (@Winkler2009). The morphology of G292.0$+$1.8 consists of many small knots and the central belt-like filaments running from the east to the west (@Park2002). The central filaments have a normal solar-type composition, suggesting that these are the shocked CSM. @Lee2010 reproduced the intensity profile of the outer CSM region by a slow wind from a red supergiant (RSG) star with the total mass of the wind of 15–40[[M]{}]{}$_\odot$. The implied progenitor mass ($M >$ 20[[M]{}]{}$_\odot$) was in plausible agreement with previous estimates (@Hughes1994; @Gonzalez2003; @Park2004). The knots have an enhanced metallicity; Si is enhanced in north-northeast, O is enhanced primarily in southeast, Ne is in northwest and southeast, and Mg is in northwest (@Park2002). These knots are probably ejecta origin. The asymmetric distribution of the ejecta elements is interpreted to be non-uniform thermodynamic conditions of the X-ray-emitting ejecta (@Park2007). In spite of these extensive studies, no significant Fe K-shell line has been detected. @Park2004 proposed that the ejecta are strongly stratified by composition and the reverse shock has not propagated to the Fe rich-zone yet. However, the X-ray spectra reported so far are faint in the hard band (except for that of PWN), and hence observed lines have been limited up to sulfur (S) K-shell line. In this paper, we report the Suzaku discovery of an Fe K-shell line in a high temperature plasma ($kT_e$ = 2–3keV) extending to $E$ = 10keV. K-shell lines of argon (Ar) and calcium (Ca) are also reported. Based on the wide band spectral analysis, we discuss the nature of G292.0$+$1.8. We adopt the solar abundances of @Anders1989. Unless otherwise specified, all errors represent 1$\sigma$ confidence levels. Observation and Data Reduction ============================== The Suzaku satellite (@Mitsuda2007) observed G292.0$+$1.8 on 2011 July 22-23 (ObsID: 506062010, PI: K. Koyama) with the X-ray Imaging Spectrometer (XIS, @Koyama2007). The XIS consists of four X-ray CCD cameras placed on the focal plane of the X-Ray Telescope (XRT). All four XRTs are co-aligned to image the same region of the sky. The field of view (FOV) of the XIS is $\times$. Details of Suzaku, the XIS and the XRT are given in @Mitsuda2007, @Koyama2007 and @Serlemitsos2007, respectively. Three of the XIS (XIS0, XIS2, and XIS3) have front-illuminated (FI) CCDs, sensitive in the 0.4–14keV energy band, and the other (XIS1) has a back-illuminated (BI) CCD, with high sensitivity down to 0.2keV. XIS2 has been out of function from 2006 November 9 and a small fraction of the XIS0 area has not been available from 2009 June 23, both due to the damage by micro-meteorites. Data reduction and analysis were performed by the HEAsoft version 6.9. The XIS data were processed with the Suzaku pipe-line software version 2.7. We combined the 3$\times$3 and 5$\times$5 event files. The response functions were generated by using the CALDB 2012-10-09. After removing hot and flickering pixels, we compiled the data using the ASCA-grade 0, 2, 3, 4, and 6 data. We excluded the data obtained at the South Atlantic Anomaly, during the earth occultation, at the elevation angle from the earth rim below 5$^{\circ}$ (night earth) and 20$^{\circ}$ (day earth). The exposure time after these screenings was 44ks. The spectral resolution has been degraded due to the radiation of cosmic particles 5 years after the launch, and restored by the spaced-row charge injection (SCI) technique; the charge traps are filled by the artificially injected electrons through CCD readouts. Details of the SCI technique are given in @Nakajima2008 and @Uchiyama2009. Analysis and Result =================== Combined Analysis of SNR and PWN -------------------------------- Figure \[xis\_image\] (a)-(c) shows X-ray images in the 0.3–0.8, 0.8–6 and 6–8keV energy bands. In the high energy band above 6keV (figure \[xis\_image\] (c)), we see a compact X-ray source at ($\alpha , \delta$) = (, ). This source corresponds to the pulsar/PWN (@Hughes2001, [-@Hughes2003]; @Camilo2002; @Gaensler2003). We make two source spectra, one is from the solid circle (radius of ) as shown figure \[xis\_image\] (c) (here PWN region). The other is from the solid circle (radius of ) excluding the dashed circle (radius of ) as shown in figure \[xis\_image\] (b) (here SNR region). In figure \[xis\_image\] (a), we show the background (BG) region: the whole FOV of the XIS (solid square), excluding the dashed circle (radius of ). This larger radius than that of the SNR region is employed to avoid the contamination from the SNR. We also exclude the region of calibration sources, which are shown by the dashed circles in the XIS corner. For all the regions of PWN, SNR and BG, we separately make the non X-ray background (NXB) spectra using [[xisnxbgen]{}]{} in the HEAsoft package (@Tawa2008). We make an X-ray background (XB) spectrum from the BG region by subtracting the relevant NXB. The spectra from the SNR and PWN regions are also made by subtraction of relevant NXBs for these regions. From these spectra, we subtract the XB spectrum assuming the uniform distribution within the FOV of the XIS after the correction of the vignetting effect. The resulting spectra of the SNR and the PWN regions in figure \[SNR-PWN\_spec\] show many emission lines. From the center energies of these lines, we identify them to be O Ly$\alpha$, Ne He$\alpha$, Ne Ly$\alpha$, Mg He$\alpha$, Mg Ly$\alpha$, Si He$\alpha$ and S He$\alpha$. Thus the spectra should be composed of an optically thin hot plasma with the temperature $kT_e \sim$1keV (@Gonzalez2003; @Park2004; @Lee2010). In the spectrum of the SNR region, we find line-like features at 3.1, 3.9 and 6.6 keV, which are likely K-shell transition lines of Ar, Ca and Fe. The Fe line at 6.6 keV is particularly clear. Therefore, in addition to the 1keV plasma (low-$kT_e$ plasma), a higher temperature plasma (high-$kT_e$ plasma) to emit K-shell lines of Ar, Ca and Fe should be prevailing in the SNR region. We call these two plasmas the SNR components. The X-rays of the SNR-components distribute not only in the SNR region but also in the PWN region. On the other hand, according to the observation with Chandra, the PWN is compact of – (@Gonzalez2003), and the spectrum is fitted with a power-law model of $\Gamma\sim$ 1.7 (@Hughes2001), indicating its non-thermal nature. We call this power-law emission the PWN-component. The spectrum extracted from the PWN region contains both the SNR-components and the PWN-component. The spectrum from the SNR region, on the other hand, is contaminated by X-rays of the PWN component due to the large point spread function (beam size) of the Suzaku XRT. We hence simultaneously fit the spectra in the SNR and the PWN regions with the combined model, SNR plus PWN-components. The ancillary response files (ARFs) employed in the fit are generated with [[xissimarfgen]{}]{} (@Ishisaki2007). The ARF for the PWN-component is generated from the Chandra image in the 4–7keV band, while that for the SNR-component is made using the thermal emission of the Chandra image (0.6–2.0keV), where the emissions of the PWN-component are excluded. The energy ranges of the PWN and the SNR regions are 1–9keV and 0.6–9keV, respectively. The former energy band is selected because the contamination of the SNR-component in the PWN region becomes large below 1keV. Considering the background level, we also ignore the energy band upper than 9keV for the FI, while 8keV for the BI. @Park2004 and @Gonzalez2003 reported that the spectra are significantly different from position to position. Therefore, the integrated spectrum from the entire SNR cannot be described by any single component model fit. We thus search for a many-components model, starting from one-component model then adding another component one by one, monitoring how much $\chi^2$ is reduced. We use VPSHOCK (@Borkowski2001) to represent multi-$n_e t$ non equilibrium ionization (NEI) plasma, where $n_e$ and $t$ are the plasma density and elapsed time after the shock heating. In order to fine-tune the calibration errors, between XIS0, 1 and 3, the gains and normalizations are set to be independent parameters for each XIS. A 1-VPSHOCK model fails with extreme large $\chi^2 / $d.o.f. of 15006/2071 = 7.25. A 2-VPSHOCK model is largely improved the fitting with $\chi^2 / $d.o.f. of 5400/2060 = 2.62, but still unacceptable. We thus add the third VPSHOCK component (VPSHOCK 1, 2 and 3), then $\chi^2 / $d.o.f. is improved to 5076/2049 = 2.48. Although the decrease of $\chi^2 / $d.o.f. is only $\delta$ = 0.14, the decrease of $\chi^2$ is 324, which is statistically highly significant. In fact, we check the significance using an F-test tool in the Xspec package, then this process is significant with better than 0.01% level. Although errors are large, the best-fit abundances in VPSHOCK 1 and 2 are the same with each other. We hence link the abundances in these two VPSHOCK components. Also abundances of Ni and Ca are linked to Fe and Ar, respectively. Since $n_e t$ of VPSHOCK 3 is 10$^{12-13}$ cm$^{-3}$s$^{-1}$, we replace this model by an APEC model (collisional ionization equilibrium plasma model; CIE). The $\chi^2 / $d.o.f. of this fit is 5184/2062 = 2.51, leaving large residuals in the low energy band. We thus added another APEC component linking the abundances to the APEC component in the 3-component model. This another APEC improve the $\chi^2 / $d.o.f. to 4838/2060 = 2.35, the F-test statistical significant is even better than the previous process. Though this $\chi^2 / $d.o.f is still large from a statistical point of view, its value would be due to non-negligible systematic errors. In fact, we find line-like residuals at about 0.82, 1.2, 1.3 and 1.8keV for both FI and BI, and 1.7keV for BI. The 1.7 and 1.8keV line structures are due to the well-known problem of the response function near the neutral Si K-edge energy at 1.84keV (@Yamaguchi2009). The other line structures would be due to the incompleteness of the VPSHOCK model code. The lines at 0.82, 1.2 and 1.3keV correspond to Fe-L complex (@Uchida2013; @Nakashima2013). We thus added extra 5 Gaussians to compensate these line-like residuals. The normalization factors of these Gaussians are linked between FI and BI, but that of the 1.7 and 1.8keV lines are treated as an independent parameters between FI and BI (@Suchy2011). The calibration errors of the contamination on the optical blocking filter has some problems in the low energy band (http://www.astro.isas.jaxa.jp/suzaku/doc/suzaku\_td/). For a fine-tuning of the cross errors between the FI and BI CCDs, we allow the $N_{\rm H}$ value in the BI CCD to be independent from the FI CCDs. The results of the combined fits by this model are shown in figure \[SNR-PWN\_spec\]. The best-fit parameters are given in table \[SNR-PWN\_para\]. We finally improve the $\chi^2 / $d.o.f. to 2872/2059 = 1.39. The F-test statistical significant is better than 0.01%. Thus, we regard that this model (2-VPSHOCK $+$ 2-APEC $+$ PL $+$ 5-Gaussians) is a reasonable approximation for the SNR and PWN spectra, and apply in the following analysis and discussion. Spatial Analysis of SNR ----------------------- In order to examine spatial asymmetry of the elements in the SNR, we make spatially-resolved spectra. Since the spatial resolution of Suzaku is limited compared to the size of G292.0$+$1.8 ($\sim$$\times$), we crudely divide the SNR into 3: the center, north and south regions as shown in figure \[N-C-S\_region\] by the solid lines. The spectra are given in figure \[each\_region\_spec\]. We find no significant differences, except a hint of Fe K$\alpha$ flux variations among the 3 regions. For quantitative estimate, we fit with the same model and the method given in subsection 3.1, but $N_{\rm H}$s are fixed to the best-fit values in table \[SNR-PWN\_para\]. We obtain nice fit with $\chi^2 / $d.o.f. of 1283/973 = 1.32, 1391/1055 = 1.32 and 1380/1022 = 1.35, for the north, center and south regions, respectively. The best-fit spectra are given in figure \[each\_region\_spec\] by the solid histograms. The best-fit parameters, including abundances, show no spatial-difference within their large errors (typical errors for the abundances are $\sim$50%). The only spatial-difference is found in the Fe abundances of the NEI plasma (2-VPSHOCK), which are 1.7$\pm$0.5, 0.75$\pm$0.22 and 1.0$\pm$0.4, for the north, center and south regions, respectively. The Fe abundance in the NEI plasma may be affected by the flux of the power-law component (PWN component), because the continuum emission of the ejecta is equal or even less than the power-law emission except the north region (see figure \[each\_region\_spec\]). We therefore re-fit the spatially-resolved spectra, changing the normalizations of the PWN component by $\pm$5% of the value in table \[SNR-PWN\_para\], and fixing the photon index of $\Gamma$ = 1.91. The fit gives no essential changes of the Fe abundances from those of the original value. Discussion ========== In the plasma evolution in SNRs, the X-ray emissions have two different components: the CSM (plus ISM) heated by the blast wave, and the ejecta from the progenitor star heated by the reverse shock. In the spectral fitting of G292.0$+$1.8, we find two type plasmas in CIE and NEI conditions; the 2-APEC ($kT_e\sim$0.2 and 0.7keV) and the 2-VPSHOCK ($kT_e\sim$1 and 2.5keV) plasmas. We call these two type plasmas, the low-$kT_e$ plasma and the high-$kT_e$ plasma, respectively. Since the low-$kT_e$ has nearly solar abundances for all elements and the high-$kT_e$ has super-solar abundances (see table \[SNR-PWN\_para\]), these would be the CSM plus ISM and the ejecta origin of a CC SN, respectively. Chandra spectra from many selected regions of bright small spots are described by 1-VPSHOCK model with super-solar abundances (@Park2004), while those from the faint outer-most shell are 1 or 2-VPSHOCK model with sub-solar to solar abundances (@Gonzalez2003; @Lee2010). These results of “no-CIE” plasma are in contrast to the existence of CIE ($\sim$ solar) components in the Suzaku spectra. Since the Chandra results are from selected spots or filaments and those of Suzaku are from the entire SNR region, we suspect that our CIE plasmas are prevailing over the entire SNR except the outer-most shell, while the bright spots are dominated by the VPSHOCK plasma. We discover Fe K-shell line at 6.6 keV in the eject plasma for the first time. The energy indicates that ionization state of Fe is around B-like. This medium ionization state is similar to another young CC SNR, Cas A, but is in contrast to nearly Ne-like states in young well known Type Ia SNRs, Tycho, Kepler, and SN1006. Figure \[ejecta\_abundance\_pattern\] is the abundance plot of the ejecta for O, Ne, Mg, Si, S, Ar and Fe relative to Si (from table \[SNR-PWN\_para\]) together with those of the CC SN model in various progenitor masses (@Woosley1995). We see that the observed abundance pattern is globally in agreement with the model of 30–35[[M]{}]{}$_\odot$. These mass range of the progenitor star confirm the previous report of 30–40[[M]{}]{}$_\odot$ (@Gonzalez2003), which was estimated based on the limited information of non-detection of the explosively synthesized heavy elements such as Ar, Ca and Fe. One may argue that CC SN of a massive progenitor 30–35[[M]{}]{}$_\odot$ may leave a black hole rather than a neutron star. However, other observations show that a neutron star can be still formed from even these massive progenitor stars (@Muno2006). Although @Park2004 and @Gonzalez2003 reported significant spatial variations in the sub-arcmin scale, we find no significant and systematic variations in the spatial scale over arcmin. In fact, the best-fit abundances of most of the heavy elements show no variations within their typical errors of 50%. Nevertheless, we find marginal evidence of spatial variation of Fe in the ejecta; the north region is enhanced compared to that of the center region. Since the position of the neutron star (PWN) is off-set to southeast from the geometrical SNR center (@Park2007), it would be conceivable that Fe from the core region would be ejected to the opposite northwest direction. Our observational result of the Fe variation is marginal to support this off-set effect due to large errors. To establish this kick-off scenario, we need higher quality observations. The best-fit spectral parameters of the PWN, the photon index and unabsorbed flux (4–8keV) are 1.91$\pm$0.03 and (3.80$\pm$0.18)$\times$10$^{-4}$ photons s$^{-1}$ cm$^{-2}$, respectively. The photon index is steeper than that of the pulsar (1.6–1.7, @Hughes2001, [-@Hughes2003]). Probably the index increases as the distance from the central pulsar increases (e.g. @Slane2000). The PWN flux is 52% of the total flux (4–8keV) from the whole SNR ((7.30$\pm$0.06)$\times$10$^{-4}$ photons s$^{-1}$ cm$^{-2}$). This ratio is slightly smaller than 66%, determined with the high spatial resolution observation of Chandra (Hughes et al. 2001). This difference, however, would be within uncertainty range due possibly to the NXB and CXB subtraction[^1], and/or other systematic cross errors including different data reduction processes between Suzaku and Chandra. Thus our simultaneous fitting analysis properly estimates the flux and spectra of both the SNR and the PWN, although the spatial resolution of Suzaku is limited to completely separate the emissions from these two sources. Summary ======= We have analyzed Suzaku/XIS data obtained from G292.0$+$1.8. The results are summarized as follows: 1. We confirm that the thermal X-ray emission from G292.0$+$1.8 consists of two type plasmas in CIE and NEI conditions. 2. The NEI plasma includes K-shell line from B-like Fe, with super solar abundances for O, Ne, Mg, Si, S, Ar, and Fe. Therefore this plasma is likely the ejecta origin of the CC SNR. 3. Using the abundance pattern of the ejecta, we confirm the progenitor mass to be 30–35[[M]{}]{}$_{\odot}$. 4. The CIE plasma has nearly solar abundances for all the relevant elements, and hence is likely the CSM and ISM origin. Acknowledgments {#acknowledgments .unnumbered} =============== We thank all members of the Suzaku operation and calibration teams. This work is supported by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Number 24540229 (Katsuji Koyama), 23000004 (Hiroshi Tsunemi), 23340071 (Kiyoshi Hayashida), 24684010 (Hiroshi Nakajima), 12J01194 (Hiroaki Takahashi), 12J01190 (Shutaro Ueda), 24740167 (Koji Mori), 25800119 (Satoru Katsuda), 11J00535 (Hiroyuki Uchida). S.K. is also supported by the Special Postdoctoral Researchers Program in RIKEN. (160mm,100mm)[figure1.eps]{} (80mm,50mm)[figure2a.eps]{} (80mm,50mm)[figure2b.eps]{} (80mm,50mm)[figure2c.eps]{} (80mm,50mm)[figure2d.eps]{} [@l@l@c@]{} Component & Parameter & Value\ Absorption & $N_{\rm H}$ (10$^{21}$ cm$^{-2}$) & 4.44$\pm$0.19 (FI)\ & & 4.11$\pm$0.19 (BI)\ Power-Law & photon index & 1.91$\pm$0.03\ & Absorbed flux& 3.71$\pm$0.17\ & Unabsorbed flux& 3.80$\pm$0.18\ APEC 1 & $kT_e$ (keV) & 0.17$\pm$0.04\ & O & 0.58$\pm$0.30\ & Ne & 0.74$\pm$0.56\ & Mg & 1.69$\pm$0.27\ & Si & 0.80$\pm$0.16\ & S & 0.83$\pm$0.50\ & Ar (=Ca) & $<$ 1.21\ & Fe (=Ni) & 0.36$\pm$0.09\ & VEM (10$^{11}$ cm$^{-5}$)& 230$\pm$160\ APEC 2 & $kT_e$ (keV) & 0.72$\pm$0.01\ & VEM (10$^{11}$ cm$^{-5}$)& 105$\pm$15\ VPSHOCK 1 & $kT_e$ (keV) & 1.07$\pm$0.19\ & O & 8.5$\pm$3.5\ & Ne & 17.8$\pm$6.1\ & Mg & 6.3$\pm$2.4\ & Si & 3.1$\pm$1.0\ & S & 2.9$\pm$1.4\ & Ar (=Ca) & 5.2$\pm$2.4\ & Fe (=Ni) & 1.7$\pm$0.5\ & $n_e t$ (10$^{11}$ cm$^{-3}$ s) & 3.0$\pm$2.6\ & VEM (10$^{11}$ cm$^{-5}$)& 6.3$\pm$2.3\ VPSHOCK 2 & $kT_e$ (keV) & 2.67$\pm$0.41\ & $n_e t$ (10$^{11}$ cm$^{-3}$ s) & 0.86$\pm$0.19\ & VEM (10$^{11}$ cm$^{-5}$)& 5.2$\pm$2.0\ $\chi^2 / $d.o.f. & & 1.39 (2872/2059)\ (53mm,53mm)[figure3.eps]{} (80mm,50mm)[figure4a.eps]{} (80mm,50mm)[figure4b.eps]{} (80mm,50mm)[figure4c.eps]{} (80mm,80mm)[figure5.eps]{} [^1]: The 4-8 keV band fluxes of the BG region are about 8% and 70% of the whole SNR regions for the data of Suzaku and Chandra, respectively	ArXiv
--- abstract: 'There is evidence that Newton and Einstein’s theories of gravity cannot explain the dynamics of a universe made up solely of baryons and radiation. To be able to understand the properties of galaxies, clusters of galaxies and the universe on the whole it has become commonplace to invoke the presence of dark matter. An alternative approach is to modify the gravitational field equations to accommodate observations. We propose a new class of gravitational theories in which we add a new degree of freedom, the Aether, in the form of a vector field that is coupled covariantly, but non-minimally, with the space-time metric. We explore the Newtonian and non-Newtonian limits, discuss the conditions for these theories to be consistent and explore their effect on cosmology.' author: - 'T.G Zlosnik$^{1}$, P.G [F]{}erreira$^{1}$, G.D. Starkman$^{1,2}$' title: 'Modifying gravity with the Aether: an alternative to Dark Matter' --- Introduction ============ Despite the tremendous successes of contemporary cosmology, there is a nagging problem that refuses to go away. If we try to measure the total gravitational field in the universe it far surpasses what we would expect from the baryonic mass we can see. This is true on a wide range of scales. On Kiloparsec scales it is well known that the velocity of objects in the outer reaches of galaxies are moving around the central core at much greater speeds than what one expect from Keplerian motion due to the stars and gas. On Megaparsec scales it has been established that the random motion of galaxies in clusters is too large for these systems to remain gravitationally bound due simply to the visible mass. And on tens to hundreds of Megaparsecs there is evidence for structure in the distribution of galaxies which should in principle have been erased by dissipational damping at recombination, when the universe was a few hundred thousand years old. There is a solution to this problem. One can invoke the existence of an exotic form of matter that does not couple to light. It is cold and clumps easily to form bound structures. The dark matter [@DM] will enhance the energy density of galaxies and clusters and can be modeled to fit almost all observations. It will also sustain gravitational potential wells through recombination and reinforce structure on large scales. A cosmological theory based on the existence of dark matter has emerged over the past twenty years with remarkable successes and predictive power [@Peebles; @spergel]. Laboratory searches are under way to find tangible evidence for dark matter candidates which go beyond their gravitational effects. One can take a different point of view. At the moment, all evidence for dark matter comes from its dynamical effect on visible objects. We see dark matter through its gravitational field. Could it be that our understanding of the gravitational field is lacking? This possibility has been mooted before. It has been proposed that the Newton-Poisson equation, $\nabla^2\Phi=4\pi G\rho$ (where $\Phi$ is the gravitational potential, $\rho$ is the energy density and $G$ is Newton’s constant) should be modified to ${\nabla}\cdot [f(\|{\nabla \Phi}\|/a_0){\nabla}\Phi]=4\pi G\rho$ where $f(x)=1$ in the strong field regime and $f(x)\simeq x$ in the weak field regime. In regions of low acceleration, gravity is boosted above the standard Newtonian prediction and an $f$ can be chosen to fit galactic rotation curves [@MOND]. Such a theory, dubbed Modified Newtonian Dynamics (MOND) has proven very effective and it has recently has been proposed that such a behaviour can emerge from the low energy, non-relativistic limit of a fully covariant theory (see [@Bekenstein2004; @models] for various approaches). MOND is not without problems. It has been shown that it is less effective at resolving the missing mass problem on the scale of clusters of galaxies. Indeed it has been shown by Sanders [@Sanders] that to correctly account for the mismatch between luminous and dynamical mass in clusters one must invoke a small fraction of massive neutrinos, with a mass of approximately $2$ eV. This result has recently been reconfirmed with weak lensing data presented by Clowe [et al]{} [@Clowe] and the subsequent analysis by Angus [et al]{} [@Angus].Given that neutrinos exist, are massive and the mass required falls within the allowed range constrained by laboratory measurements, this solution to the missing mass problem in clusters is not outlandish. In this paper we show that it is possible to modify gravity by introducing a dynamical [Aether]{} (or time-like vector field) with non-canonical kinetic terms. Our proposal builds on the extensive analysis of Einstein-Aether theories undertaken by Jacobson, Mattingly, Carroll, Lim and collaborators [@AE],[@CL] and follows along a long series of proposals by others [@Vector]. As the Aether vector field has a non-vanishing expectation value it will dynamically select a preferred frame at each point in spacetime (i.e. the frame in which the time co-ordinate basis vector $\partial_{t}$ aligns with the direction of the Aether field $\textbf{A}$). This violates local Lorentz invariance (and gauge invariance). Consequently, Aether theories traditionally have been used as phenomenological probes of possible Lorentz violation in quantum gravity. As there has been recent interest in modifying gravity by using additional scalar and vector fields it is worth initially comparing our approach to other attempts. Another group of theories retains local Lorentz invariance but introduces a vector field, the action of which breaks U(1) symmetry through a non-gauge invariant potential or kinetic term. These theories have variously been found to be able to model inflation [@FORD] and dark energy [@TRIAD], the latter invoking a set of three identical vectors along mutually orthogonal spacial directions (the “cosmic triad"). Such a vector field, coupled to scalar fields, has also been considered in the context of alternatives to dark matter [@MOFFT]. Quite distinctly, it has been suggested that the recent acceleration of the universe could be accounted for by allowing for non-canonical kinetic terms in the action of the electromagnetic field [@NONLIN]. Recently the possibility of a scalar field coupled to the Aether as a cause of inflation has been examined [@SODA]. We will first lay out the formalism for our proposal, with the full field equations. We will then proceed to analyze them in the non-relativistic regime and show that it is possible to naturally obtain modifications to Newtonian gravity. The physical consistency of the theory is discussed in the weak field regime as are constraints from the Solar system. We then briefly explore the possible impact on the expansion of the Universe, showing that this modification of gravity can lead to accelerated expansion at different stages of the evolution of the universe. A specific proposal for such a theory is presented and we conclude by discussing a series of open problems. The Theory ========== A general action for a vector field, A coupled to gravity can be written in the form $$\begin{aligned} S=\int d^4x \sqrt{-g}\left[\frac{R}{16\pi G_N}+{\cal L}(g,A)\right] +S_{M} \label{genaction}\end{aligned}$$ where g is the metric, $R$ the Ricci scalar of that metric, $S_M$ the matter action and $\cal{L}$ is constructed to by generally covariant and local. $S_M$ only couples to the metric, g and [not]{} to A. We shall use the metric signature (-,+,+,+) throughout. For most of this paper we will restrict ourselves consider a Lagrangian that only depends on covariant derivatives of $A$ and we will consider a A that is time-like. Such a theory can be written in the form $$\begin{aligned} \label{eq:Lagrangian} {\cal L}(A,g)&=&\frac{M^2}{16\pi G_N} {\cal F}({\cal K}) +\frac{1}{16\pi G_N}\lambda(A^\alpha A_\alpha+1) \nonumber \\ {\cal K}&=&M^{-2}{\cal K}^{\alpha\beta}_{\phantom{\alpha\beta}\gamma\sigma} \nabla_\alpha A^{\gamma}\nabla_\beta A^{\sigma} \nonumber \\ {\cal K}^{\alpha\beta}_{\phantom{\alpha\beta}\gamma\delta}&=&c_1g^{\alpha\beta}g_{\gamma\sigma} +c_2\delta^\alpha_\gamma\delta^\beta_\sigma+ c_3\delta^\alpha_\sigma\delta^\beta_\gamma\end{aligned}$$ where $c_i$ are dimensionless constants and $M$ has the dimension of mass. $\lambda$ is a non-dynamical Lagrange-multiplier field with dimensions of mass-squared. Note that it is possible to construct a more complicated ${\cal K}$ by including different powers in A and its derivatives. Indeed it is possible to show that Bekenstein’s theory of modified gravity [@Bekenstein2004] is formally equivalent to a theory with such an extended ${\cal K}$ (though with a more exotic method of achieving a non-vanishing vacuum-expectation value for A). We allow for these different possibilities by deriving a general form for the field equations below. We will comment on these models in the discussion. The gravitational field equations for this theory, obtained by varying $g^{\alpha\beta}$ (see [@CL] but also [@HEHL]) are $$G_{\alpha\beta}=\tilde{T}_{\alpha\beta}+8\pi GT^{matter}_{\alpha\beta} \label{fieldI}$$ where the stress-energy tensor for the vector field is given by $$\begin{aligned} \tilde{T}_{\alpha\beta} &=& \frac{1}{2}\nabla_{\sigma} ({\cal F}'(J_{(\alpha}^{\phantom{\alpha}\sigma}A_{\beta)}- J^{\sigma}_{\phantom{\sigma}(\alpha}A_{\beta)}-J_{(\alpha\beta)}A^{\sigma})) \nonumber \\ && -{\cal F}'Y_{(\alpha\beta)} +\frac{1}{2}g_{\alpha\beta}M^{2}{\cal F}+\lambda A_{\alpha}A_{\beta} \nonumber\\ {\cal F}' &=& \frac{d{\cal F}}{d{\cal K}} \nonumber \\ J^{\alpha}_{\phantom{\alpha}\sigma} &=& (\cal{K}^{\alpha\beta}_{\phantom{\alpha\beta}\sigma\gamma}+ \cal{K}^{\beta\alpha}_{\phantom{\beta\alpha}\gamma\sigma})\nabla_{\beta}A^{\gamma}\end{aligned}$$ Brackets around indices denote symmetrization and $Y_{\alpha\beta}$ is the functional derivative $$\begin{aligned} Y_{\alpha\beta} = \nabla_{\sigma}A^{\eta}\nabla_{\gamma}A^{\xi} \frac{\delta(\cal{K}^{\sigma\gamma}_{\phantom{\sigma\gamma}\eta\xi})}{\delta g^{\alpha\beta}} \nonumber\end{aligned}$$ The equations of motion for the vector field, obtained by varying $A^{\beta}$ are $$\begin{aligned} \label{eq:veceq} \nabla_{\alpha}({\cal F}'J^{\alpha}_{\phantom{\alpha}\beta}) +{\cal F}'y_{\beta}&=&2\lambda A_{\beta} \label{vectoreom}\end{aligned}$$ where once again we define the functional derivative $$\begin{aligned} y_{\beta}=\nabla_{\sigma}A^{\eta}\nabla_{\gamma}A^{\xi} \frac{\delta(\cal{K}^{\sigma\gamma}_{\phantom{\sigma\gamma}\eta\xi})} {\delta A^\beta} \nonumber\end{aligned}$$ Variations of $\lambda$ will fix $A^\mu A_\mu=-1$. By inspection, transvecting both sides of (\[eq:veceq\]) with $A^{\beta}$ leads to a solution for $\lambda$ in terms of the the vector field and its covariant derivatives. These equations allow us to study a general theory of the form presented in equation \[genaction\] with a time-like vector field. For our particular, restricted choice of ${\cal K}$ we have $Y_{\alpha\beta}=-c_{1}\left[ (\nabla_{\nu}A_{\alpha})(\nabla^{\nu}A_{\beta})-(\nabla_{\alpha}A_{\nu})(\nabla_{\beta}A^{\nu})\right]$ and $y_\beta=0$. The non-relativistic regime: Newtonian and MONDian limit ======================================================== Having established our general theory, we can now explore its properties in various different regimes. We start in the static, weak field, non-relativistic limit. We must expand both the metric and vector field around a fixed, Minkowski space background. The constraint equation very clearly fixes the zeroth order contribution of the vector field to be $A^{\alpha(0)}=\delta^{\alpha}_0$. Hence we have $$\begin{aligned} g_{\alpha\beta}&=&\eta_{\alpha\beta}+\epsilon h_{\alpha\beta} \nonumber\\ A^{\alpha}&=&\delta^{\alpha}_0+\epsilon B^{\alpha} \nonumber\end{aligned}$$ We will only consider terms which are linear in our parameter $\epsilon$ and restrict ourselves to the Poisson gauge (so $h_{00}=-2\Phi$,$h_{ij}=-2\Psi \delta_{ij}$). We have that $$\begin{aligned} \nabla \bf{A} = \epsilon {\Sigma} + O(\epsilon ^{2})\end{aligned}$$ where $\Sigma_{ij}=0$, $\Sigma_{0\mu}=\partial_\mu\Phi$ and $\Sigma_{i0}=\partial_{i}(\Phi+B^{0})$ and $_{(i)}$ indexes the three coordinates, $(x^1,x^2,x^3)$. Terms in $\nabla A\nabla A$ and ${\vec B}$ may be consistently regarded as being of $O(\epsilon^{2})$ or greater and so can be ignored in what follows. An immediate consequence of this is that, to first order, the $ij$th component of the Aether’s stress energy tensor disappears. The $ij$th component of the Einstein equations then yields $\Phi=\Psi$. The equality of scalar potentials is true of General Relativity in the absence of anisotropic stress and simplifies the equations dramatically. The modified Newton-Poisson and vector equations then are $$\begin{aligned} \label{eq:npe} 2\nabla^{2}\Phi - (c_{1}-c_{3})\nabla.({\cal F}'\nabla B^{0})- \lambda &=& 8\pi G\rho \nonumber \\ 2c_{3}\nabla.({\cal F}'\nabla\Phi)+2c_{1}\nabla.({\cal F}'\nabla(\Phi+B^{0}))&=&-2\lambda\end{aligned}$$ The constraint fixes ${\cal K}$ to be $$\begin{aligned} {\cal K}= -c_{1}\frac{(\nabla\Phi)^{2}}{M^{2}}\end{aligned}$$ Taking $c_{1}<0$ ensures that $\cal{K}$ is positive. We can manipulate the system of equations (\[eq:npe\]) so as to obtain the field equation for $\Phi$: $$\label{eq:ss} \nabla.((2+c_{1}{\cal F}')\nabla\Phi)=8\pi G\rho$$ If such a theory is to have $\nabla.(\left\|\nabla\Phi\right\|\nabla\Phi)\propto \rho$ in the limit of small $\nabla\Phi$ we require that: $$\lim_{\|\nabla\Phi\| << M}[2+c_{1}{\cal F}'] \propto {\cal K}^{\frac{1}{2}}$$ Integrating then we have that ${\cal F} = \alpha{\cal K}+\beta {\cal K}^{\frac{3}{2}}$ where $\alpha$ and $\beta$ are constants. Hence we can construct a theory with ‘MONDian’ limit on galactic scales, this holding as long as we identify $M$ with something of the order of $a_{0}$ [@MOND]. The limit as ${\cal K}\rightarrow 0$ is worth considering in more detail. Considering a test particle a distance $r$ away from an isolated source, the Modified Poisson Equation dictates that for $\|\nabla\Phi\|<<M$ the gravitational force will vary as $1/r$. This combined with the geodesic equation suggests that in this regime the metric component $g_{00}$ grows approximately as $+\ln (r)$ and therefore becomes ill defined in the limit of $r\rightarrow \infty$. This has been a generic feature of past metric theories of MOND. It was noted in [@WoodardNonlocal] that the value of this growing term will typically vary by around only $10^{-6}$ from the radius of onset of MOND in a system to the present Hubble radius ($> 10^{27}$cm). Therefore, the weak field approximation is unlikely to be seen to break down. The situation becomes more complicated if the theory is such that MOND only arises in a cosmological background (for instance if $A^{2}$ is not fixed and the ‘scale’ $M$ is contingent on its value at the particular cosmological era). It would only be consistent then to formulate the weak field limit in a FRW background rather than Minkowski spacetime and the asymptotic form of the metric may be expected to differ. The Solar System can supply us with stringent constraints on the weak field limit of these theories. Accelerations are typically substantially larger than $M$ (again assumed to be $\sim a_{0}$) and so requiring concordance with observations could constrain the possible form of ${\cal F}$ for large ${\cal K}$. Poisson’s equation $\nabla^{2}\Phi=4\pi G \rho$ is an excellent approximation in the Solar System and thus we shall expect the contribution of $c_{1}{\cal F}'$ in (\[eq:ss\]) to be small. We may expect then that ${\cal F}'$ in this limit can be expanded as a power series in inverse powers of ${\cal K}^{\frac{1}{2}}$. That is: $$\label{eq:limiting} \lim_{\|\nabla\Phi\| >> M}[{\cal F}'({\cal K})]= \sum_{i=1}^{\infty}\frac{\eta_{i}}{{\cal K}^{\frac{i}{2}}}$$ where $\eta_{i}$ are constants. Consider the leading term $\eta_{1}$/${\cal K}^{\frac{1}{2}}$. We would expect such a term in spherical symmetry to result in a constant anomalous acceleration equal to $\frac{\sqrt{-c_{1}\eta_{1}^{2}}M}{2}$. The existence of such a term is particularly constrained by observed bounds on the variation of Kepler’s constant $GM_{\odot}$. For instance, observations between Earth and Mars restrict the acceleration to be less than approximately $10^{-9}$ms$^{-1}$ [@sanders]. We shall see that $\eta$ and $c$ values are typically of order unity so the above is rather restrictive. Even under the assumption of spherical symmetry, the field equations for the theory are enormously complicated. The inherent nonlinearity in the weak field limit provided by ${\cal F}'$ presents a considerable challenge [@BMSS]. We note however that the fixed norm constraint on $\textbf{A}$ will, in the weak field limit, force terms of the form $\nabla\textbf{A}\nabla\textbf{A}$ and $\nabla\nabla\textbf{A}$ to be at most of the order of terms comprising the components of the Einstein tensor $\textbf{G}$. For the limiting form of ${\cal F}'$ given in (\[eq:limiting\]) we expect terms in the Aether stress energy tensor to be schematically of the form $\frac{\alpha_{i}}{{\cal K}^{\frac{i}{2}}}c_{j} [\nabla\nabla \textbf{A}$and$\nabla\textbf{A}\nabla\textbf{A}]$ and higher order derivative terms suppressed relatively by factors of $1$/${\cal K}^{\frac{i}{2}}$. At Mercury, the ratio $\|\nabla\Phi\|/ M$ is of order $10^{8}$. Provisionally neglecting terms in ${\cal F}'$ of ${\cal K}^{-\frac{1}{2}}$ (see above), we then expect corrections to terms in $\textbf{G}$ in the inner solar system to be of order $10^{-16}c_{j}\alpha_{i} /c_{k}$. It is tempting to conclude that two ‘Parameterized Post Newtonian’ (PPN) parameters measurable by inner solar system effects, $\beta$ and $\gamma$, would generally be expected to deviate from the predictions of General Relativity by a similar order. The complete set of PPN coefficients have been obtained for the case ${\cal F}({\cal K})\propto {\cal K}$ ([@AE], 3rd reference) . In particular the coefficients describing ‘preferred frame’ effects were found to be only consistent with experimental bounds for specific combinations of the $c_{i}$; they are expected to be a particularly strong test of more general forms of ${\cal F}$. Additionally we note that the asymptotic behaviour of ${\cal F}$ is consistent with our assumption that the term $M^{2}\frac{{\cal F}}{2}\textbf{g}$ is second order in perturbations. We see that for ${\cal K}<<1$, ${M^{2}\cal F} \rightarrow \|\nabla\Phi\|^{2}$, and for ${\cal K} >> 1$, ${M^{2}\cal F}$ shall be at most $\alpha_{1}M\|\nabla\Phi\|$. Finally we briefly consider the effect of allowing for first-order spatial components of A, $A^{i}$. It has been found [@DODLIG] that in Bekenstein’s theory of modified gravity the growth of large scale structure is necessarily accompanied by growth in $A^{i}$ so it is not unreasonable to expect that such behaviour shall be present in the model considered here. It may be readily checked that in the static weak-field limit that an $A^{i}$ of order $\epsilon$ will only contribute to $\tilde{T}_{00}$ and $\tilde{T}_{ii}$ at order two in $\epsilon$ and above. However, as the numerator in ${\cal K}$ is a second order quantity, it shall generally be affected. If $A^{i}$ has only a radial component (i.e. $A^{r}\neq 0$), it may be checked that ${\cal K}$ is modified as: $$\begin{aligned} {\cal K} = \frac{-c_{1}(\nabla\Phi)^{2}+c_{2}(\nabla_{r}A^{r})^{2}}{M^{2}} \end{aligned}$$ where the covariant derivative $\nabla_{r}A^{r}$ is a of order two or greater in $\epsilon$. We shall see later that $c_{2}$ is preferably the same sign as $c_{1}$. The effect of gradients of the radial component of the vector field then is to decrease ${\cal K}$ for a given $\|\nabla\Phi \|$. Recall that the onset of MONDian behaviour coincides with ${\cal K}<<1$. Therefore an $A^{r}(r)$ in this model will generally hasten the onset of this limit, the effect being to further increase the gravitational field for a given $\rho$ (see (\[eq:ss\])). It requires further work to see whether this can appreciably counter MOND’s problems on the scale of clusters of galaxies. Further Constraints =================== Recall that ${\cal F}$ tends to $\alpha {\cal K}+\beta {\cal K}^{\frac{3}{2}}$ for small ${\cal K}$ (i.e. far from a source). Therefore, in general, when considering classical perturbations one is effectively considering a theory with ${\cal F}\sim \cal{K}$, as in the Einstein-Aether theory with minimal couplings. This can be used to study the consistency of these theories in the perturbative regime. Lim [@LIM] has considered the dominant term in the limit where metric and vector field perturbations decouple. The vector field propagates in flat spacetime and allows for a decomposition of perturbations into spin-0 and spin-1 components. The requirement that the perturbations can be consistently quantized, that the spin-0 component propagates subluminally and nontachyonically when quantized, and that metric perturbations do not propagate superluminally place the following restrictions on the $c_{i}$: $$\begin{aligned} c_{1} &<& 0 \\ c_{2} &\leq& 0\\ c_{1}+c_{2}+c_{3} &\leq& 0.\end{aligned}$$ Additional constraints can be obtained via astroparticle physics. The observation of ultra-high energy cosmic rays implies a lack of energy loss via gravitational Cherenkov radiation. This radiation is expected when gravitational waves propagate subluminally. With the Aether, the usual transverse traceless modes exist along with three coupled Aether-metric modes. The speeds of propagation of each mode are functions of the $c_{i}$ and have been calculated in [@AE] for perturbations in Minkowski spacetime. The squared speed $s^{2}_{tt}$ of the transverse-traceless mode is: $$\begin{aligned} s^{2}_{tt}=1/(1+(c_{1}+c_{3})) \nonumber \nonumber\end{aligned}$$ and the squared speeds of the transverse Aether and trace Aether-metric modes ($s^{2}_{tA}$ and $s^{2}_{tr}$ respectively) are given by: $$\begin{aligned} s^{2}_{tA} &=& \frac{(c_{1}+\frac{c_{1}^{2}}{2}-\frac{c_{3}^{2}}{2})}{(c_{1}(1+c_{13}))} \nonumber\\ s^{2}_{tr} &=& \frac{(c_{123}/c_{1})(2+c_{1})}{[2(1-c_{2})^{2}+c_{123}(1-c_{2}-c_{123})]} \nonumber\end{aligned}$$ where $c_{ijk..}=c_{i}+c_{j}+c_{k}+..$ and $c_{i}$ are the opposite sign to those considered in [@AE]. Computing the expected degree of gravitational Cherenkov radiation for the above modes and comparing this with observational bounds, it has been found [@ELL] that for the case ${\cal F}({\cal K})={\cal K}$ that extremely stringent constraints could be placed on the $c_{i}$. For instance, they may be satisfied when the $c_{i}$ are mutually related such that the metric and Aether-metric modes propagate at precisely the speed of light. If this is not the case then the magnitudes of the $c_{i}$ must be severely diminished. It is expected that such an analysis could powerfully constrain more general combinations of ${\cal F}$,$c_{i}$, though any nonlinearity is expected to complicate the results when propagation is considered against a curved background spacetime. Note that such an analysis is by no means complete. It has been shown that a very restricted class of Einstein-Aether theories do not have positive Hamiltonians and therefore are inherently unstable at both the classical and quantum levels [@clayton]. Furthermore we are considering non-linear functions of ${\cal K}$ and hence instabilities may arise in non trivial backgrounds. A more detailed analysis of individual cases for ${\cal F}$ is needed yet first indications are that these theories are healthy. Cosmology ========= This class of modified theories are generally covariant. This gives us the possibility of exploring their properties on large scales and in particular we can consider the case of a homogeneous and isotropic universe in which the metric is of the form $ds^2= -{dt}^{2}+a(t)^{2}d{\bf x}^2$ where $t$ is physical time and $a(t)$ is the scale factor. The vector field must respect the spatial homogeneity and isotropy of the system and so will only have a non-vanishing ‘$t$’ component; the constraint fixes $\textbf{A} = (1,0,0,0)$. The energy-momentum tensor of the matter is of the form ${T}^{matter}_{\alpha\beta} = \rho U_{\alpha}U_{\beta} +P(g_{\alpha\beta} + U_\alpha U_\beta)$ where $\rho$ is the energy density, $P$ is pressure and we have introduce a four-vector $\bf{U}$ which satisfies $g_{ab}U^{a}U^{b} = -1$. The equations of motion simplify dramatically with these symmetries and we find that: $$\begin{aligned} \nabla_{e}A^{e}&=& 3H\nonumber \\ {\cal K} &=& 3\frac{\alpha H^{2}}{M^{2}} \nonumber\end{aligned}$$ where $H \equiv \frac{\dot{a}}{a}$, the dot denotes differentiation with respect to $t$, and, following [@CL], we define $\alpha = c_{1}+3c_{2}+c_{3}$. Note that now ${\cal K}$ is negative, unlike the nonrelativistic limit encountered above. This means that the dynamics of static, spherically symmetric systems on the one hand and of relativistic cosmologies on the other probe completely different branches of ${\cal F}$. The modified Einstein’s equations now become: $$\begin{aligned} [1-{\cal F}'\alpha]H^{2}+\frac{1}{6}{\cal F}M^{2} = \frac{8\pi G}{3}\rho \label{00} \nonumber \\ -[1-2{\cal F}'\alpha]H^{2}-2[1-\frac{1}{2}{\cal F}'\alpha]\frac{\ddot{a}}{a}+\dot{{\cal F}'}\alpha H-\frac{1}{2}{\cal F}M^{2}= 8\pi G P\label{trace} \nonumber\end{aligned}$$ These can in fact be rewritten in a more useful form: $$\begin{aligned} \left[1-\alpha{\cal K}^{1/2}\frac{d}{d{\cal K}}\left(\frac{\cal F}{{\cal K}^{1/2}}\right)\right]H^2 &=& \frac{8\pi G}{3}\rho \label{00m} \\ \frac{d}{dt}(-2H+{\cal F}'\alpha H)&=&8 \pi G (\rho+P) \label{summ}\end{aligned}$$ Once again these equations are general but we can see in their structure, interesting possibilities. If we take ${\cal F}=0$, we recover the standard cosmology. We can do this by either setting it to $0$ or choosing ${\cal K}$ to have $c_1=-c_3$ and $c_2=0$ so that it becomes the Maxwell tensor. We then have a theory which will modify gravity on galactic and super-galactic scales but leaves the expansion of the universe unchanged. It is interesting to explore the possibility that the vector field may affect the late time expansion of the Universe. From Einstein’s equations, we can see that for it to behave like a cosmological constant, $\Lambda$ we must have $$\label{eq:c} -{\cal K}{\cal F}'+\frac{1}{2}{\cal F}=\frac{\Lambda}{M^2}$$ which we can solve to find: ${\cal F}=\beta (- {\cal K})^{1/2}-\frac{2\Lambda}{M^2}$. Where $\beta$ is a constant of integration. Now note that for positive arguments of ${\cal F}$ we have been equating the mass scale $M$ with the acceleration scale $a_0\simeq 10^{-10}$ms$^{-2}\simeq cH_0$ where $c$ is the speed of light and $H_0$ is the Hubble constant. If we assume that is also true for negative values, we find that ${\cal F}$ is of order unity. I.e. there is a natural relation between the scale of the cosmological constant and the fundamental mass scale in our theory. A choice of such asymptotic behaviour close to ${\cal K}=0$ does introduce an undesirable discontinuity in ${\cal F}$ at the origin; however, as the behaviour in (\[eq:c\]) will lead to a constant ${\cal K}$ (as $\dot{H}=0$),this form of ${\cal F}$ need only hold in the domain $\|{\cal K}/3\beta\|>\sim H_0$ and does not inform the desirable choice of ${\cal F}$ around ${\cal K}=0$. Accelerate expansion can be reached in a more natural way by positing $$\begin{aligned} {\cal F}=\gamma (-{\cal K})^n \nonumber \nonumber\end{aligned}$$ for a certain range of ${\cal K}$ For arbitrary $n$ Equation (\[00\]) becomes $$\begin{aligned} \left[1+\epsilon\left(\frac{H}{M}\right)^{2(n-1)}\right]H^2 =\frac{8\pi G}{3}\rho \nonumber\end{aligned}$$ where $\epsilon=-(1-2n)\gamma(-3\alpha)^{n}/6$.For an appropriate choice of $\gamma$ and $n$ we have $\epsilon<0$ and we find that $H$ tends to an attractor $$\begin{aligned} H\rightarrow H_{eq}=M(-\epsilon)^{1/2(1-n)} \nonumber\end{aligned}$$ This regime is approached asymptotically with $$\begin{aligned} H-H_{eq}\simeq H_{eq}^{-1}\left(\frac{8\pi G\rho}{3}\right) \nonumber\end{aligned}$$ i.e. as $a^{-3}$ in the matter era. This gives us a particularly elegant, dynamical mechanism for approaching accelerated expansion without invoking a cosmological constant. A small scale is still invoked but it is naturally related to the acceleration scale needed to trigger the onset of the MONDian regime on galactic scales. Discussion ========== The discussion throughout this letter has been general. We have neither chosen a specific form of ${\cal K}$ or ${\cal F}$ although we have constrained the asymptotic form of the latter. It is instructive to pick a simple example. If we choose $c_{1}=-1$ ,$c_{2}=0$ and $c_{3}=1$ we will recover the canonical form ${\cal K}\propto F_{\alpha\beta}F^{\alpha\beta}$ where ${\bf F}$ is the field-strength of the four vector ${\bf A}$. A possible functional form for ${\cal F}$ is then: $$\begin{aligned} {\cal F}({\cal K})=4({\cal K}^{\frac{1}{2}}-\ln({\cal K}^{\frac{1}{2}}+1)) \nonumber\end{aligned}$$ From which we recover the modified Newton-Poisson equation: $$\nabla. ( \frac{\frac{\|\nabla\Phi\|}{M}}{\frac{\|\nabla\Phi\|}{M}+1} \nabla \Phi)=4\pi G \rho$$ With an appropriate choice of the value of M we recover the field equation considered in [@BinneyFamaey], which was found to give a satisfactory fit to the terminal velocity curve of the Milky Way. As noted, the choice $c_{1}=-c_{3},c_{2}=0$ as here forces ${\cal K}$ to vanish in the case of spatial homogeneity and isotropy. Therefore with this choice of ${\cal F}$ the Aether has no influence on cosmological background evolution. The inclusion of a nonzero $c_{2}$ modifies the kinetic term of the metric and may have significant effects both in the static, weak-field limit and the cosmological background. There are a number of theoretical and phenomenological issues that remain to be addressed. For a start, the presence of the non-dynamical Lagrange-multiplier field $\lambda$ in the Lagrangian (\[eq:Lagrangian\]) is perhaps unappealing. Its sole role is to impose the constraint that $A^\mu$ is unit-timelike. The same can be accomplished by replacing the $\lambda$-term in the Lagrangian by a potential term $-V(A^\mu)$, by dint of which $A^\mu$ at low energy acquires a vacuum expectation value such that $A^\mu A_\mu=-1$. For example $V(A^\mu) = \frac{1}{2}\mu^4 \left(A^\mu A_\mu +1 \right)^2$, with $\mu$ a constant with dimensions of mass. At energy scales below $\mu$, one expects that indeed $A^2=-1$, and that “radial” excitation of $A$ (i.e. of $a \equiv \mu A$) will have positive mass-squared $m_a^2\simeq \mu^2$. So long as $\mu^2$ is sufficiently large, the low-energy phenomenology of this model should be identical to that of (\[eq:Lagrangian\]). A more exotic possibility is to construct more complex ${\cal K}$ which have minima for time-like $\textbf{A}$. Indeed the theory proposed in [@Bekenstein2004] is of this form [@ZFS1]. Closely related to the previous point is the fundamental origin of such an Aether. We have kept the discussion general in the hope that a more fundamental theory of fields or strings may pin down the form of ${\cal K}$ and ${\cal F}$. Indeed, the possibility of Lorentz violating vector fields has cropped up in attempts to extend the standard model of particle physics. Most notably it has been argued that such a field may arise in higher dimensional theories as a low energy by product of string theory [@KOSTEL]. It would be interesting to explore the range of current candidates for low energy string theory to find a possible candidate for such an Aether field. What is clear is that a Lagrangian of the form we require will [ not]{} appear from the standard perturbative approach to constructing effective field theories as in [@gripaios]. Non-perturbative effects must come into play. We have endeavoured to explore some of the observational constraints. Preliminary indications are that these theories are compatible with Solar System constraints. 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--- abstract: 'Magnetic excitations in an array of single crystals have been measured using inelastic neutron scattering. Until now, has been thought of as a two-leg antiferromagnetic Heisenberg spin ladder with chains running in the $a$-direction. The present results show unequivocally that is best described as an alternating spin-chain directed along the crystallographic $b$-direction. In addition to the expected magnon with magnetic zone-center energy gap $\Delta = 3.1$ meV, a second excitation is observed at an energy just below $2\Delta$. The higher mode may be a triplet two-magnon bound state. Numerical results in support of bound modes are presented.' address: \| $^1$Oak Ridge National Laboratory, Oak Ridge, TN 37831-6393, USA\ $^2$University of Tennessee, Knoxville, TN 37996-1501, USA\ $^3$University of Florida, Gainesville, FL 32611-0448, USA\ $^4$Insituto de Fisica Rosario, 2000 Rosario, Argentina\ author: - 'D.A. Tennant$^1$, S.E. Nagler$^1$, T. Barnes$^{1,2}, $ A.W. Garrett$^3$, J. Riera$^4$, and B.C. Sales$^1$' title: 'Excitations and Possible Bound States in the $S=1/2$ Alternating Chain Compound ' --- \#1 [Abstract]{} [Keywords: ]{} Bound Magnons, Alternating Heisenberg Chain.\ [Corresponding Author: ]{}\ Dr Alan Tennant\ Bldg 7962 MS 6393, Solid State Division\ Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6393, U.S.A.\ tel.: +1-423-576-7747\ fax.: +1-423-574-6268\ email: [email protected]\ The S=1/2 alternating Heisenberg chain (AHC) is a fascinating quantum system that is currently the subject of much interest. We have established [@gntsb] that the material , previously considered to be a spin ladder, is in fact an excellent realization of the AHC. The physics of the AHC is also very relevant to spin-Peierls materials such as CuGeO3 [@nishi]. Recent theoretical work [@uhrig] on the AHC underscored the potential importance of two-magnon bound modes. In this paper, we review our neutron scattering experiments on the alternating chain material . In addition to the expected one-magnon excitations, we observe an extra mode, which may be a two-magnon bound state. We follow with a discussion of some related theoretical issues. The crystal structure of is nearly orthorhombic, with a slight monoclinic distortion so that the space group is P2$_{1}$[@nhs]. The room temperature lattice parameters are $a$=7.73Å, $b$=16.59Å, $c$=9.58Åand $\beta$=89.98$^{\circ}$. The magnetic properties of arise from $S=1/2$ V$^{4+}$ ions situated within distorted VO$_6$ octahedra. Face-sharing pairs of VO$_6$ octahedra are stacked in two-leg ladder structures oriented along the $a$-axis. The ladders are separated by large, covalently bonded PO$_4$ complexes. The structure is illustrated schematically in figure 1. The susceptibility of powder [@jjgj] can be accurately reproduced by either a spin ladder (with $J_\\| \approx J_\perp$) or by an alternating chain [@jjgj; @br], but the expectation that the PO$_4$ group would provide a weak superexchange path led to a general acceptance of the spin ladder interpretation of . Pulsed inelastic neutron scattering measurements on powders [@ebbj] showed a spin gap of 3.7 meV, which was interpreted as further support for the ladder model. Beltrán-Porter [et al.]{}[@euro_chemists] examined the superexchange pathways in several vanadyl phosphate compounds, and were led to question the spin-ladder interpretation of . Instead they proposed that an alternating V-O-V-PO$_4$-V chain in the $b$-direction was a more likely magnetic system. The observation of a second spin excitation near 6 meV (not predicted by the ladder model) in a recent triple-axis neutron scattering experiment [@gnbs] on powder, and the discovery of strong superexchange through PO$_4$ groups in the precursor compound [@vodpo_paper], also cast doubt on the spin ladder interpretation. For these reasons we undertook studies of the spin dynamics in single crystals. Measurements of the excitations were made using a single crystal array of approx. 200 oriented crystals of typical size 1x1x0.25 mm$^3$. The resulting sample had an effective mosaic spread of $8-10^\circ$ FWHM. Inelastic neutron scattering measurements were carried out using triple-axis spectrometers at the HFIR reactor, Oak Ridge National Laboratory; full experimental details can be found in [@gntsb]. Scans [@gntsb] at $T=10$K showed two modes of roughly equal strength at the antiferromagnetic zone-centre $(0,\pi,0)$ at energies of $\Delta_{l}=3.12(3)$ meV and $\Delta_{u}=5.75(2)$ meV. Full resolution convolutions with the fitted dispersion showed these modes to be resolution limited. The disappearance of both modes at higher temperatures confirmed their magnetic origin. The modes were found to track approximately in $Q$ close to $(0,\pi,0)$ (see Figure (2)). At the zone-boundary $(0,\pi/2,0)$ only a single mode was observed at an energy of $\approx 15$meV. Because of poor instrumental resolution it was not possible to tell whether the modes had coalesced or were simply not resolved. Fig. 2 shows the measured dispersion for both modes along $a^,b^$ and $c^$. The excitation energy is almost independent of $Q_{c}$ (middle panel), implying a very weak coupling along $c$. The dependence of energy on $Q_{a}$ is much weaker than on $Q_{b}$ and is ferromagnetic. The strong $Q_{b}$ dependence implies that the exchange coupling is dominantly one-dimensional along the $b$-direction, confirming the V-O-V-PO$_4$-V alternating chain proposed in [@euro_chemists] and [@vodpo_paper]. For any exchange alternation $-$ as occurs with the two inequivalent exchanges along $b$ $-$ a gap should appear in the dispersion (as observed) and the absence of magnetic ordering in is consistent with a singlet ground state. However the observation of an extra mode requires a more thorough theoretical investigation. Since the high temperature limit of the magnetic susceptibility[@jjgj] is consistent with expectations for simple $S=1/2$, $g=2$ spins, the possibility that the upper mode is an additional low lying single ion excitation can be ruled out. Two other plausible explanations of the second peak are (a) splitting due to an exchange anisotropy, and (b) a physical two-magnon bound state. Although a pseudo-Boson calculation including exchange anisotropy gave an excellent fit to the dispersion [@gntsb] $-$ the solid line in Figure (2) is a fit to this model [@gntsb] $-$ considerable exchange anisotropy ($\approx 15 \%$) was necessary to account for the mode splitting. Recent single crystal magnetic susceptibility measurements [@thompson] were quantitatively consistent with the previous powder results[@jjgj] and found little if any evidence for anisotropy. Also the coupling in the precursor compound was found to be consistent with isotropic exchange [@vodpo_paper] suggesting that one should seek another explanation for the second mode, and because the energy of the upper mode at $(0,\pi,0)$, $\Delta_{u}$, is just below $2\Delta_{l}$, a bound two-magnon mode may provide a good explanation. In support of this explanation, preliminary high-field measurements show splitting of both modes which is consistent with both modes being triplets [@vopo_field]. To gain some insight into the formation of two-magnon modes in we have studied the $S=1/2$ AHC using numerical techniques. The isotropic AHC Hamiltonian is $$H = \sum_{i=1}^{L/2} \ J \; {\vec S_{2i-1}} \cdot {\vec S_{2i}} + \alpha J \; {\vec S_{2i}} \cdot {\vec S_{2i+1}} \ ,$$ where $J>0$ and $1 \ge \alpha \ge 0$. Equation (1) has been studied analytically and numerically over many years, but it had not been appreciated until recently that $S=0$ and $S=1$ bound magnon states may form. Uhrig and Schulz [@uhrig] have used field theory and RPA methods to study these modes at $k=\pi/2$ and $k=0,\pi$. The existence of these bound states depends subtly on the kinetic and potential energies of pair formation, and occur for only certain values of $k$. Perturbation theory in $\alpha$ about the dimer limit ($\alpha=0$) provides a quantitative basis for understanding the excitations for small $\alpha$, and also provides insight into the competition between potential and kinetic energy effects in bound states [@brt]. Figure (3) shows the one- and two-magnon excitation spectra calculated within a simplified approximate first order (one- and two-magnon manifold) treatment of the AHC. At $k=\pi/2$ there is a node in the two-magnon continuum which corresponds to a degeneracy in the total kinetic energy $\omega(k_{1})+\omega(\pi/2-k_{1})$ of two magnons. The $S=0$ and $S=1$ bound states lie well below the continuum lower boundary. However at $k=0$ and $\pi$ only the $S=0$ bound state is seen. The continuum is much broader at $k=0,\pi$ indicating larger mixing effects which disrupt the $S=1$ bound state. Although no $S=1$ bound state forms, the attractive potential still leads to a strongly enhanced scattering cross-section $S(Q,\omega)$ at the continuum lower boundary [@brt], see dashed line in Figure (4). The $S=1$ bound state appears clearly at the $k=\pi/2$ point (solid line in Figure (4)). It should be noted that the neutron scattering cross-section for the $S=0$ mode is zero, however this mode may be visible by light scattering [@brt]. Harris [@abh] used a reciprocal space perturbation theory to calculate the ground state and excited state energy up to $O(\alpha^3)$. This gives a $k=0,\pi$ energy gap of $E_{gap}=J(1-\alpha/2-3 \alpha^2 /8 + \alpha^3 /32)$. However these results can be derived more easily in real space [@brt], and in the case of the ground state energy, we have extended the calculation to $O(\alpha^5)$, $$\begin{aligned} e_0(\alpha)/J=-3/2^3-(3/2^6) \cdot \alpha^2 - (3/2^8) \cdot \alpha^3 \nonumber \\ - (13/2^{12}) \cdot \alpha^4 - (95/3) \cdot (1/2^{14}) \cdot \alpha^5 - O(\alpha^6) .\end{aligned}$$ The perturbation series appears to be rapidly converging for $\alpha \le 0.5$, and may have a radius of convergence of unity. Because has $\alpha \approx 0.8$ [@gntsb], we have used a numerical Lanczos algorithm on finite $L=4n$ lattices of up to $L=28$ and with approximately 14 place accuracy to study the ground states and binding energies up to similar values of $\alpha$. Full details will be given elsewhere [@brt]. Figure (4) shows the calculated binding energies of the $S=0$ bound mode at $k=\pi/2$, and $k=0$, as well as those for the $S=1$ bound mode at $k=\pi/2$. The results show strong binding at $\alpha=0.8$ of the $S=0$ mode at $k=\pi/2$ but the situation is not clear for $k=0$. They also suggest weak binding for the S=1 mode at $\pi/2$ at the alternation for . Unfortunately finite size effects precluded an accurate determination of this binding energy. In order to make a quantitative comparison with $S(Q,\omega)$ is required for the bound modes and continuum. We are currently undertaking calculations to quantify this. The effects of interchain coupling have been neglected and these may enhance the binding. Next-nearest neighour exchange within the chains may have a similar effect. The $\alpha$ perturbation theory provides a useful quantitative guide to such effects, and further theoretical studies are in progress. We also note that similar dynamics are also important in many other low-dimensional Hamiltonians such as spin ladders and we shall present some work on those in the future. In conclusion, we have measured an extra mode in the alternating chain system . The evidence suggests that this is a two-magnon bound state. Perturbation theory and Lanczos calculations give an insight into the formation of bound modes. We thank J.Thompson for sharing his susceptibility results with us prior to publication. Oak Ridge National Laboratory is managed for the U.S. D.O.E. by Lockheed Martin Energy Research Corporation under contract DE-AC05-96OR22464. Work at U.F. is supported by the U.S. D.O.E. under contract DE-FG05-96ER45280. A.W. Garrett, S.E. Nagler, D.A. Tennant, B.C. Sales and T. Barnes, Phys. Rev. Lett. 79, 745 (1997). M. Nishi [et al.]{} Phys. Rev. B 32, 6508 (1994). G.S.Uhrig and H.J.Schulz, Phys. Rev. B 54, R9624 (1996). P.T. Nguyen, [et al.]{} Mat. Res. Bull. 30, 1055 (1995). D.C. Johnston [et al.]{}, Phys. Rev. B 35, 219 (1987). T.Barnes and J.Riera, Phys. Rev. B 50, 6817 (1994). R.S.Eccleston [et al.*]{} Phys. Rev. Lett. 73, 2626 (1994). D. Beltrán-Porter P. Amorós, R. Ibáñez, E. Martínez, A. Beltrán-Porter, A. LeBail, G. Ferey, and G. Villeneuve, Solid State Ionics 32-33, 57 (1989). A.W.Garrett, S.E.Nagler, T.Barnes and B.C.Sales, Phys. Rev. B 55, 3631 (1997). D.A.Tennant, S.E.Nagler, A.W.Garrett, T.Barnes, and C.C.Torardi, Phys. Rev. Lett. 78, 4998 (1997). J.R. Thompson and K.J. Song, private communication S.E. Nagler, D.A. Tennant, A.W. Garrett, B.Sales, and T.Barnes, in preparation. T. Barnes, J. Riera, and D.A. Tennant, in preparation. A.B. Harris, Phys. Rev. B. 7, 3166 (1973). [Figure Captions]{} [Figure 1. Schematic depiction of the structure and magnetic interactions in VOPO. The spin ladder model previously thought to describe VOPO has nearest neighbor exchange constants $J_{\parallel}$ along the $a$ (“ladder”) direction and $J_{\perp}$ along the $b$ (“rung”) direction. In the alternating chain model, nearest neighbor V$^{4+}$ ions are alternately coupled by constants $J_{1}$ and $J_{2}$ along the $b$ (chain) direction. Neighboring spins in adjacent chains are coupled by $J_{a}$. Magnetic coupling in the $c$ direction is negligible. ]{} [Figure 2. Measured dispersion of magnetic excitations in VOPO at T = 10K. When not visible error bars are smaller than the size of the plotted symbols. Filled circles (open diamonds) are points from the lower (upper) energy mode. The solid lines are dispersion curves calculated using parameters obtained by fitting to a pseudo-Boson model [@gntsb]. Wavevectors are plotted in units corresponding to the VOPO reciprocal lattice. ]{} [Figure 3. Schematic depiction of the one- and two-magnon excitation spectra of the $S=1/2$ AHC with an alternation of $\alpha=0.2$. An $S=1$ bound mode appears below the continuum at $k \approx \pi/2$. The more deeply bound $S=0$ mode (dashed line) is not visible to neutrons scattering. ]{} [Figure 4. Calculated $S(Q,\omega)$ for constant-Q scans at $k=\pi/2$ (solid line) and $k=\pi$ (dashed line) using the first order perturbation approach with $\alpha=0.2$. ]{} [Figure 5. Calculated binding energies of the $S=0$ and $S=1$ bound states using a Lanczos method [@brt] at $k=0$ and $\pi/2$. The binding energies are given in units of $J$. ]{}	ArXiv
--- abstract: 'We present observations with the Cosmic Origins Spectrograph onboard the [Hubble Space Telescope]{} of eight compact star-forming galaxies at redshifts $z$ = 0.02811 – 0.06540, with low oxygen abundances 12 + log(O/H) = 7.43 – 7.82 and extremely high emission-line flux ratios O$_{32}$ = \[O [iii]{}\]$\lambda$5007/\[O [ii]{}\]$\lambda$3727 $\sim$ 22 – 39, aiming to study the properties of Ly$\alpha$ emission in such conditions. We find a diversity in Ly$\alpha$ properties. In five galaxies Ly$\alpha$ emission line is strong, with equivalent width (EW) in the range 45 – 190Å. In the remaining galaxies, weak Ly$\alpha$ emission with EW(Ly$\alpha$) $\sim$ 2 – 7Å is superposed on a broad Ly$\alpha$ absorption line, indicating a high neutral hydrogen column density $N$(H [i]{}) $\sim$ (1 – 3)$\times$10$^{21}$ cm$^{-2}$. We examine the relation between the Ly$\alpha$ escape fraction $f_{\rm esc}$(Ly$\alpha$) and the Lyman continuum escape fraction $f_{\rm esc}$(LyC), using direct measures of the latter in eleven low-redshift LyC leakers, to verify whether $f_{\rm esc}$(Ly$\alpha$) can be an indirect measure of escaping LyC radiation. The usefulness of O$_{32}$, of the Ly$\alpha$ equivalent width EW(Ly$\alpha$) and of the Ly$\alpha$ peak separation $V_{\rm sep}$ as indirect indicators of Ly$\alpha$ leakage is also discussed. It is shown that there is no correlation between O$_{32}$ and $f_{\rm esc}$(Ly$\alpha$). We find an increase of $f_{\rm esc}$(Ly$\alpha$) with increasing EW(Ly$\alpha$) for EW(Ly$\alpha$)$<$100Å, but for higher EW(Ly$\alpha$)$\ga$150Å the $f_{\rm esc}$(Ly$\alpha$) is nearly constant attaining the value of $\sim$0.25. We find an anticorrelation between $f_{\rm esc}$(Ly$\alpha$) and $V_{\rm sep}$, though not as tight as the one found earlier between $f_{\rm esc}$(LyC) and $V_{\rm sep}$. This finding makes $V_{\rm sep}$ a promising indirect indicator of both the Ly$\alpha$ and ionizing radiation leakage.' author: - \| Y. I. Izotov$^{1,2}$, D. Schaerer$^{3,4}$, G. Worseck$^{5}$, A. Verhamme$^{3}$, N. G. Guseva$^{1,2}$, T. X. Thuan$^{6}$, I. Orlitová$^{7}$ & K. J. Fricke$^{8,2}$\ $^{1}$Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, 14-b Metrolohichna str., Kyiv, 03143, Ukraine,\ E-mail: [email protected], [email protected]\ $^{2}$Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53121 Bonn, Germany\ $^{3}$Observatoire de Genève, Université de Genève, 51 Ch. des Maillettes, 1290, Versoix, Switzerland,\ E-mail: [email protected], [email protected]\ $^{4}$IRAP/CNRS, 14, Av. E. Belin, 31400 Toulouse, France\ $^{5}$ Institut für Physik und Astronomie, Universität Potsdam, Karl-Liebknecht-Str. 24/25, D-14476 Potsdam, Germany,\ E-mail: [email protected]\ $^{6}$Astronomy Department, University of Virginia, P.O. Box 400325, Charlottesville, VA 22904-4325, USA,\ E-mail: [email protected]\ $^{7}$Astronomical Institute, Czech Academy of Sciences, Bočn[í]{} II 1401, 141 00, Prague, Czech Republic,\ E-mail: [email protected]\ $^{8}$Institut für Astrophysik, Göttingen Universität, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany,\ E-mail: [email protected] date: 'Accepted XXX. Received YYY; in original form ZZZ' title: 'Diverse properties of Ly$\alpha$ emission in low-redshift compact star-forming galaxies with extremely high \[O [iii]{}\]/\[O [ii]{}\] ratios' --- \[firstpage\] (cosmology:) dark ages, reionization, first stars — galaxies: abundances — galaxies: dwarf — galaxies: fundamental parameters — galaxies: ISM — galaxies: starburst Introduction {#intro} ============ The extremely high O$_{32}$ = \[O [iii]{}\]$\lambda$5007/\[O [ii]{}\]$\lambda$3727 ratio in spectra of low-redshift dwarf star-forming galaxies (SFGs) indicates the presence of either an ionization-bounded compact H [ii]{} region with intense ionizing radiation from a very young star-forming region or a density-bounded H [ii]{} region losing its ionizing radiation, or both [e.g. @JO13; @NO14; @S15; @I17]. The ionizing photon production efficiency $\xi_{\rm ion}$ = $N_{\rm LyC}$/$L_\nu$ in these galaxies could be very high, where $N_{\rm LyC}$ is the Lyman continuum photon production rate and $L_\nu$ is the monochromatic UV luminosity at $\lambda$ = 1500Å. Therefore, they are considered as the local counterparts of the dwarf SFGs at high redshifts, which are likely the main source of the last phase transition of the matter in the Universe, namely its reionization, which ends at $z$ $\sim$ 6 [@O09; @WC09; @K11; @J13; @M13; @Y11; @B15a; @Sm18; @N18; @St18; @K19]. An alternative hypothesis is the dominant role of active galactic nuclei (AGN) in the reionization of the Universe, as proposed e.g. by @Madau15. However, in recent years, growing evidence for a small contribution of AGN to reionization has accumulated [e.g. @H18; @M18; @P18; @Mat18; @Ku19]. On the other hand, SFGs can be considered as an important source of ionization only if the product $\xi_{\rm ion}$$\times$$f_{\rm esc}$(LyC) is sufficiently high, where $f_{\rm esc}$(LyC) is the escape fraction of Lyman continuum (LyC), i.e. the fraction of ionizing radiation emitted into the intergalactic medium (IGM). It has been estimated that $f_{\rm esc}$(LyC) should not be less than 10–20 per cent on average in order to accomplish reionization [e.g. @O09; @R13; @D15; @Robertson15; @K16]. Currently, direct observations of the LyC are available for only a limited number of galaxies at high and low redshifts. Searches for and confirmation of LyC emission from high redshift galaxies are difficult due to their faintness, contamination by lower-redshift interlopers and attenuation by residual H [i]{} in the highly ionized IGM. [e.g., @V10; @V12; @Inoue14; @Gr15]. They have resulted in the discovery of approximately three dozens of LyC leakers with reliably high escape fraction: [Ion2]{} [@Va15; @B16], Q1549-C25 [@Sh16], A2218-Flanking [@B17], [Ion3]{} [@Va18], SunBurst Arc [@RT19], 15 SFGs studied by @St18, and a dozen of galaxies at $z$ = 3.1 with strong \[O [iii]{}\] $\lambda$5007 emission [@F19]. At low redshifts, $z$ $\sim$ 0.3 - 0.4, @I16 [@I16b; @I18; @I18b] have detected LyC leaking emission in eleven compact SFGs with high O$_{32}$ $\sim$ 5 - 28 and derived $f_{\rm esc}$(LyC) ranging between 2 and 72 per cent. These values are higher than $f_{\rm esc}$(LyC) of $\la$ 1 per cent derived by @B14 and @C17 in SFGs with lower O$_{32}$, or by @He18 in SFGs with lower equivalent widths of the H$\beta$ emission line. There is hope that the number of low-$z$ LyC leakers will considerably be increased after the completion in 2020 of the [HST]{} Large Program GO 15626 (P.I. A. Jaskot), which includes more than 70 targets. The [Hubble Space Telescope]{} ([HST]{})/Cosmic Origins Spectrograph (COS) sensitivity curve limits LyC observations to $z \ga 0.25$. At lower redshifts, only very bright targets are accessible [e.g. @L16]. Therefore, for low-redshift SFGs, reliable indirect indicators of LyC leakage are needed. @NO14 proposed that high O$_{32}$ can be an indication of density-bounded H [ii]{} region and thus of LyC leakage. However, O$_{32}$ depends on some other parameters, such as ionization parameter and hardness of radiation which in turn depend on metallicity. @F16 has found a correlation between $f_{\rm esc}$(LyC) and O$_{32}$, using a small sample of galaxies which includes low-$z$ LyC leakers with O$_{32}$ $<$ 8. On the other hand, no correlation has been found by @I18b for eleven confirmed LyC leakers with O$_{32}$ in a wide range of $\sim$ 5 - 28. A similar conclusion was obtained by @N18 and @B19 [see also @R17]. However, this conclusion is based mainly on observations of galaxies with O$_{32}$ $\la$ 16. Only one observed galaxy has a higher value of O$_{32}$. It is thus important to observe LyC emission in more galaxies with higher O$_{32}$. The He [i]{} emission-line ratios in optical spectra of galaxies can also be used to estimate the column density of neutral gas and thus to indirectly estimate $f_{\rm esc}$(LyC) [@I17]. However, deep spectra in the optical range are needed since these lines are weak with intensities $\la$10 per cent of the H$\beta$ emission line. Finally, the properties of the Ly$\alpha$ emission line can serve as indirect indicators of escaping LyC emission. Its profile is double-peaked in the spectra of most of LyC leakers, with the separation between the peaks $V_{\rm sep}$ decreasing with decreasing neutral hydrogen column density $N$(H [i]{}) [e.g. @I16; @I16b; @I18; @I18b]. In rare cases, the Ly$\alpha$ profile is more complex and consists of three and more peaks [@RT17; @I18b; @Va18]. @V15 and @I18b have shown that there is a tight relation between the LyC escape fraction $f_{\rm esc}$(LyC) and the separation between Ly$\alpha$ peaks, making $V_{\rm sep}$ the most reliable indicator of escaping ionizing radiation. Furthermore, @V17 have shown for a limited sample of LyC leakers that the Ly$\alpha$ escape fractions $f_{\rm esc}$(Ly$\alpha$) are larger than the LyC escape fractions $f_{\rm esc}$(LyC), in accord with theoretical predictions of @D16. In this paper we present new [HST]{}/COS observations of the Ly$\alpha$ in eight low-redshift ($z$ $<$ 0.07) compact SFGs with the highest O$_{32}$ $\sim$ 22 – 39 found in the Sloan Digital Sky Survey (SDSS), aiming to examine the properties of Ly$\alpha$ emission in these galaxies and to verify possible indirect indicators of escaping Ly$\alpha$ and LyC emission. The studied galaxies have lower stellar masses than previously observed Ly$\alpha$ emitting galaxies such as the “green pea” (GP) galaxies at $z$ $\sim$ 0.1 – 0.3 studied e.g. by @J17, @Y17, @MK19, and the confirmed LyC leakers at $z$ $\sim$ 0.3 – 0.4 [@I16; @I16b; @I18; @I18b]. The selection criteria and the properties of the selected galaxies derived from observations in the optical range are presented in Section \[sec:select\]. The [HST]{} observations and data reduction are described in Section \[sec:obs\]. The surface brightness profiles in the UV range are discussed in Section \[sec:sbp\]. Ly$\alpha$ emission is considered in Section \[sec:lya\]. In Section \[discussion\], our results are compared with the Ly$\alpha$ escape fractions obtained for other galaxies in some recent studies. We summarize our findings in Section \[summary\]. Selection criteria and the integrated properties of selected galaxies {#sec:select} ====================================================================== Our galaxies were selected from the SDSS Data Release 14 (DR14) [@A18]. We adopted the following selection criteria for each galaxy: 1\) it has a very high O$_{32}$ $>$ 20, which is an indication of either a density-bounded H [ii]{} region or a high ionization parameter. 2\) it has a compact structure in the SDSS image in order to observe the total flux of the galaxy with the small 2.5 arcsec COS aperture. 3\) it is isolated from other sources to avoid contamination in the [HST]{}/COS spectra. 4\) it is sufficiently bright in the UV range ([GALEX]{} NUV magnitude brighter than 22 mag) to be acquired with COS. In total, we selected the 8 brightest galaxies satisfying the above conditions. The SDSS spectra of the selected galaxies reveal considerably lower oxygen abundances, in the range 12 + log(O/H) = 7.43 – 7.82 (see below), as compared to the confirmed LyC leakers [@I16; @I16b; @I18; @I18b]. They show comparably high rest-frame equivalent widths EW(H$\beta$) = 258 – 483 Å (Table \[tab1\]), indicating very recent star formation. These galaxies are located at the extreme upper tip of the SFG branch in the Baldwin-Phillips-Terlevich (BPT) diagram (filled circles in Fig. \[fig2\]a) implying the presence of high-excitation H [ii]{} regions. They are considerably more extreme than the confirmed LyC leakers shown by open circles. It is seen in Fig. \[fig2\]b, which shows the O$_{32}$ – R$_{23}$ diagram, that the selected SFGs (filled circles) have the highest O$_{32}$ ratios among all SDSS SFGs, and that they are offset to low R$_{23}$ values compared to the main sequence of SFGs (grey dots) because of their lower metallicities. On the other hand, most of the confirmed LyC leakers with higher metallicities are located in the upper part of the main sequence (open circles). The SDSS spectra were used to derive the interstellar extinction from the hydrogen Balmer decrement and ionized gas metallicity using the prescriptions of @ITL94, @I06 and @G13. The extinction-corrected emission-line flux densities, the extinction coefficients $C$(H$\beta$), the rest-frame equivalent widths of H$\beta$, \[O [iii]{}\]$\lambda$5007 and H$\alpha$ emission lines, and the observed H$\beta$ flux densities are shown in Table \[taba1\]. We note that the H$\alpha$ emission line in the SDSS spectra of J0820$+$5431 and J1032$+$4919 and the \[O [iii]{}\]$\lambda$5007 emission line in the SDSS spectrum of J1032$+$4919 are clipped. Therefore, for J1032$+$4919, $I$(\[O [iii]{}\]$\lambda$5007) = 3$\times$$I$(\[O [iii]{}\]$\lambda$4959) is used to derive O$_{32}$, the electron temperature $T_{\rm e}$(O [iii]{}) and oxygen abundance. Furthermore, the H$\alpha$ emission line in this galaxy and in J0820$+$5431 is excluded in the calculation of extinction. The He [ii]{} $\lambda$4686Å emission line is detected in seven out of eight galaxies implying hard ionizing radiation. The flux densities from Table \[taba1\] and the direct $T_{\rm e}$ method are used to derive the electron temperature, electron number density and the element abundances in the H [ii]{} regions. These quantities are shown in Table \[taba2\]. The oxygen abundances of our galaxies are on average $\sim$ 0.2 - 0.3 dex lower than those in known low-redshift LyC leakers [@I16; @I16b; @I18; @I18b]. Four galaxies from our sample have been observed with the Large Binocular Telescope (LBT) by @I17. Those authors derived oxygen abundances that are in good agreement with the present determinations from SDSS spectra. The neon, sulfur and argon to oxygen abundance ratios (Fig. \[fig3\]) are similar to those in dwarf emission-line galaxies [e.g. @I06; @G13] and in confirmed LyC leakers, with similar small dispersions. On the other hand, the N/O ratios are scattered over a large range (Fig. \[fig3\]a) which can not be explained by N/O errors (Table \[taba2\]). Compared to SDSS SFGs (shown by grey dots), they tend to have systematically higher values both for the galaxies in our sample and for the confirmed LyC leakers. The emission-line luminosities and stellar masses of our galaxies were obtained adopting a luminosity distance [NED, @W06] with the cosmological parameters $H_0$=67.1 km s$^{-1}$Mpc$^{-1}$, $\Omega_\Lambda$=0.682, $\Omega_m$=0.318 [@P14]. Using these parameters and apparent magnitudes, we also derived absolute magnitudes $M_g$ and $M_{\rm FUV}$ in the SDSS $g$ and [GALEX]{} FUV bands (Table \[tab3\]), assuming negligible bandpass corrections and neglecting dust extinction. The H$\beta$ luminosity $L$(H$\beta$) and corresponding star-formation rates SFR shown in Table \[tab3\] were obtained from the extinction-corrected H$\beta$ flux densities using the relation by @K98 and adopting the value of 2.75 for the H$\alpha$/H$\beta$ flux ratio. They are one order of magnitude lower than the respective values for LyC leakers. Specific star formation rates sSFR = SFR/$M_\star$ of $\sim$ 100 Gyr$^{-1}$ are similar to sSFRs for LyC leakers [@I16; @I16b; @I18; @I18b] and are among the highest known for GPs and luminous compact galaxies [@Ca09; @I11], and dwarf SFGs at $z < 1$ [@I16c]. On the other hand, they are several orders of magnitude higher than the sSFRs of $\sim$ 0.01 - 1 Gyr$^{-1}$ of the SDSS main sequence galaxies. The extinction-corrected SDSS spectra of our galaxies were used to fit the spectral energy distribution (SED) of both the stellar and nebular components and to derive their stellar masses, starburst ages and the modelled intrinsic absolute magnitudes $M_{\rm FUV}^{\rm SED}$ in the FUV range. The fitting method, using a two-component model, is described in @I18. We find that the stellar masses and FUV luminosities of our galaxies are $\sim$ 2 orders of magnitude lower compared to the confirmed LyC leakers (Table \[tab3\]), but their extinction-corrected absolute FUV magnitudes are similar to those of the faintest detected galaxies at $z$ = 6 – 8 [e.g. @L17; @Bo17b]. [HST]{}/COS observations and data reduction {#sec:obs} ============================================= [HST]{}/COS spectroscopy of the eight galaxies was obtained in program GO 15136 (PI: Y. I. Izotov) during the period January 2018 – February 2019. The observational details are presented in Table \[tab4\]. The galaxies were acquired by COS near ultraviolet (NUV) imaging. Since our targets are compact but faint, as based on shallow [GALEX]{} imaging, a considerable fraction of one orbit was spent per object for deep NUV imaging and reliable acquisition. The NUV-brightest region of each target was centered in the 2.5arcsec diameter spectroscopic aperture (Fig. \[fig4\]). We note that acquisition observations failed for J1242$+$4851. We do not know the reason for that. The acquisition image has a flag raised in the header: EXPFLAG=‘INVALID CHECK IMAGE’. Possibly, something went wrong during one of the two acquisition exposures. Therefore, the imaging data for J1242$+$4851 are not present in this paper. However, the spectroscopic observations were not affected. For the other galaxies, some structure with an extended low-surface-brightness (LSB) component is seen. However, their sizes are smaller than the central unvignetted $0.8$arcsec diameter region of the spectroscopic aperture. Hence, the galaxy quantities derived from the COS spectra here do not require corrections for vignetting. In fact, most of the NUV continuum in these galaxies is concentrated in much smaller central regions, with full widths at half maximum $\la$ 0.1 arcsec. We obtained spectra of the Ly$\alpha$ emission lines with the COS G130M grating positioned at the central wavelength 1222Å and a small wavelength coverage of $\sim$ 300Å at COS Detector Lifetime Position 4, yielding a spectral resolving power $\lambda/\Delta\lambda\simeq 14,000$ in the wavelength range of interest. All four focal-plane offset positions were employed to correct grid wire shadows and detector blemishes. The individual exposures were reduced with the CALCOS pipeline v3.3.4, followed by accurate background subtraction and co-addition with custom software [@W16]. Apart from adjustments of the rectangular extraction aperture due to the wide spatial profile of the G130M 1222Å setup (57 pixels) and adjustments in the detector pulse-height thresholds to reflect the current detector state, the reduction procedures were analogous to @I16. We verified in the two-dimensional spectra that the extraction aperture contains all visible Ly$\alpha$ flux and preserves the spectrophotometry of the continuum. The estimated background uncertainties of a few per cent do not affect our analysis. The homogeneous reduction enables a fair comparison to our previous results [@I16; @I16b; @I18; @I18b]. Surface brightness distribution in the NUV range {#sec:sbp} ================================================ Using the COS NUV acquisition images and the routine [ellipse]{} in [iraf]{}[^1]/[stsdas]{}[^2] we obtain the surface brightness (SB) profiles of our galaxies in the UV continuum. The [GALEX]{} NUV magnitudes were used to express the profiles in the units mag arcsec$^{-2}$. For this, we have scaled the total NUV flux of the galaxy measured in the acquisition image to the flux corresponding to the [GALEX]{} NUV magnitude. The shape of the SB profiles (Fig. \[fig5\]) in our galaxies is similar to those of confirmed LyC leakers [@I16; @I16b; @I18; @I18b] with a sharp increase in the central part because of the presence of the bright star-forming region(s) and a linear decrease (in magnitudes) in the outer part, reminiscent of a disc structure. However, the scale lengths $\alpha$ of our galaxies in the range $\sim$ 0.09 – 0.48 kpc (Fig. \[fig5\]) are considerably lower than $\alpha$ = 0.6 – 1.8 kpc in LyC leakers [@I16; @I16b; @I18; @I18b], presumably because of their considerably lower stellar masses. The scale lengths of our galaxies are among the lowest found for local blue compact dwarf galaxies [@P02]. On the other hand, the corresponding surface densities of SFR in the studied galaxies $\Sigma$ = SFR/($\pi \alpha^2$) are comparable to those of LyC leakers. Because of the compactness of the bright star-forming regions, the half-light radii $r_{50}$ of our galaxies in the NUV are considerably smaller than $\alpha$ (see Table \[tab3\]). Adopting $r_{50}$ as a measure of the size of these galaxies (Table \[tab3\]), the corresponding $\Sigma$s are typically two orders of magnitude larger, and comparable to those found for low-redshift LyC leakers and SFGs at high redshifts [@CL16; @PA18; @Bo17]. Ly$\alpha$ emission {#sec:lya} =================== The LyC emission can not be directly observed in our galaxies because of their low redshifts. Therefore, we wish to use the Ly$\alpha$ profiles in their spectra to derive information about possible leaking LyC radiation for galaxies with much lower stellar masses than those of confirmed LyC leakers. @D16 and @V17 have proposed that the presence of a double-peaked Ly$\alpha$ profile, with a small peak separation, would be a good indicator of LyC leakage. According to the models of @V15, the peak separation decreases with decreasing column density of the neutral gas. This in turn would result in a higher escape fraction of the LyC radiation. Additionally, the role of the blue Ly$\alpha$ peak in the LyC escape has been discussed by e.g. @H15 and @O18. They found that the escape fraction of the Ly$\alpha$ radiation increases when the blue peak velocity decreases. The Ly$\alpha$ profiles in the medium-resolution spectra of the 8 observed galaxies are shown in Fig. \[fig6\]. In the upper panel for each object, both the flux and the wavelength are scaled so that the continuum behaviour can be seen over a broader wavelength range around the Ly$\alpha$ line, while in the lower panel both the flux and the wavelength scales are adjusted to examine the profile of the Ly$\alpha$ emission line. A strong Ly$\alpha$ $\lambda$1216Å emission-line without any evidence for an absorption profile is detected in five galaxies (Fig. \[fig6\]b,c,f,g,h), similarly to the low-$z$ LyC leakers [fig. 7 in @V17]. In the remaining 3 galaxies, the weak Ly$\alpha$ emission line is superposed on a broad absorption component (Fig. \[fig6\]a,d,e). The fraction of galaxies with a detected Ly$\alpha$ absorption line is similar to that obtained by @MK19. However, the intensity at the bottom of the absorption lines is above zero in these galaxies, indicating that the covering factor of the H [i]{} cloud surrounding the ionizing source is less than unity, allowing for some Ly$\alpha$ emission and some continuum emission in this wavelength range to escape. The full widths at zero intensity of the Ly$\alpha$ emission lines in all our galaxies are FWZI = 3 – 4Å (Fig. \[fig6\]). On the other hand, the widths of the absorption lines at the bottom of their profiles are considerably broader, $\sim$ 10 – 20Å in the three galaxies J0007$+$0226, J0926$+$4504 and J1032$+$4919. Therefore, it is unlikely that the emission above zero at the bottom of the absorption profiles is due to the Ly$\alpha$ emission line. Six out of eight observed galaxies show double-peaked Ly$\alpha$ emission profiles with the peak separations much larger than the nominal COS spectral resolution of $\sim$ 20 km s$^{-1}$ for the G130M grating. The Ly$\alpha$ emission line in J0926$+$4504 is too weak to definitely derive its profile. The two-peak shape is similar to that observed in known LyC leakers [@I16; @I16b; @I18; @I18b; @V17] and in GP galaxies [@JO14; @H15; @Y17; @MK19]. The Ly$\alpha$ emission-line profile of J1032$+$4919 is more complex and consists of three peaks, similar to that of one low-$z$ LyC leaker [@I18b], and two $z$ $>$ 2 LyC leakers [@Va18; @RT17]. However, at variance with J1032$+$4919, all other triple-peaked spectra in the literature do not show clear signs of underlying Ly$\alpha$ absorption. Some parameters of the Ly$\alpha$ emission profile are presented in Table \[tab5\]. The Ly$\alpha$ emission-line fluxes in the spectra of galaxies with detected broad Ly$\alpha$ absorption are measured placing the continuum at the bottom of the absorption profiles. For J1032$+$4919, two separations (blue peak - centre peak and centre peak - red peak) are given. It is seen that the separation between the peaks is in a narrow range $\sim$ 240 – 310 km s$^{-1}$, indicating low neutral hydrogen column densities $N$(H [i]{}) and implying considerable fractions of escaping LyC emission. However, in the spectra of three galaxies, J0007$+$0226, J0926$+$4504 and J1032+4919, strong Ly$\alpha$ absorption is present as well. The flux in the central part of absorption profiles is above zero, although their shape indicates that they are clearly saturated. This appearance is in contrast to expectations for a single source surrounded by an uniform neutral gas cloud with high $N$(H [i]{}). We derive $N$(H [i]{}) using the relation $$I(\lambda)=I_{\rm cont} \exp{[-\tau(\lambda)]}, \label{profile}$$ where $I_{\rm cont}$ and $I(\lambda)$ are the flux of the continuum and the flux in the line at the wavelength $\lambda$, respectively, after subtraction of the flux at the bottom of the Ly$\alpha$ absorption line. The optical depth $\tau$($\lambda$) in the Ly$\alpha$ profile at the wavelength $\lambda$ is defined by the relation of @B75 $$\tau(\lambda) = 4.26\times 10^{-20} \frac{N({\rm H~I})}{6.04\times 10^{-10} + (\lambda - \lambda_0)^2}, \label{Lya}$$ where $N$(H [i]{}) is in cm$^{-2}$, $\lambda$ is in Å, $\lambda_0$ = 1215.67 Å, Using Eqs. \[profile\] and \[Lya\] we find $N$(H [i]{}) = 1.8$\times$10$^{21}$ cm$^{-2}$, 1.0$\times$10$^{21}$ cm$^{-2}$ and 3.0$\times$10$^{21}$ cm$^{-2}$ from the fit of wings of absorption Ly$\alpha$ profiles in the spectra of J0007$+$0226, J0926$+$4504 and J1032$+$4919, respectively (grey lines in the upper panels of Fig. \[fig6\]a, \[fig6\]d, and \[fig6\]e). We note that only the blue damped wing was used for fitting because the red wing is contaminated by the stellar N [v]{} $\lambda$1240Å line with a P Cygni profile. Then the superposition of the Ly$\alpha$ absorption and emission lines could be explained by the emission of two star-forming regions, where the fainter one, with a flux in the continuum corresponding to the one at the bottom of the absorption profile, is surrounded by neutral gas with low $N$(H [i]{}), and the brighter one is surrounded by a neutral optically thick cloud. The flux of the continuum for the brighter region is determined by the difference between the observed flux outside the absorption profile and the flux at its bottom. The presence of Ly$\alpha$ emission in galaxies with broad Ly$\alpha$ absorption could also be explained by low-column-density holes in a thick cloud of neutral hydrogen. Discussion ========== Comparison with other galaxy samples ------------------------------------ We compare our galaxies with those studied in other recent papers. All galaxies from the sample in this paper are dwarf systems with SDSS $g$ absolute magnitudes $M_g$ $\sim$ –16 to –18 mag. Their absolute FUV magnitudes $M_{\rm FUV}$, in the range –14 to –16, compare well with the average $M_{\rm FUV}$ $\sim$ –15 mag of the faintest spectroscopically confirmed LAEs at redshifts 2.9 - 6.7, similar to those of the sources believed to reionize the Universe [@Ma18]. Our galaxies are much more compact in the NUV continuum (Fig. \[fig5\]) and have lower metallicities (Table \[tab1\]) than most Ly$\alpha$ emitters (LAEs) and LyC leakers. They have high ionizing photon production efficiencies $\xi_{\rm ion}$ (Table \[tab3\]), above the canonical value in log scale of $\log\xi_\mathrm{ion}\sim 25.2$ in units of log Hz erg$^{-1}$ adopted in high-$z$ studies [e.g. @R13] and similar to the values in LAEs at $z$ $\sim$ 3 - 7 [@Ha18; @Na18]. These high values are a natural consequence of a very young starburst age (Table \[tab3\]). High $\xi_{\rm ion}$ in our galaxies are also similar to those in confirmed low-$z$ LyC leakers [e.g. @S16]. If applicable to all high-$z$ galaxies, such $\xi_{\rm ion}$ values would suffice to reionize the Universe with a constant escape fraction $f_{\rm esc}$(LyC) $\sim$ 10 per cent [e.g. @O09; @R13; @D15; @Robertson15; @K16]. However, we note that the $\xi_{\rm ion}$ values in Table \[tab3\] should be considered only as lower limits, as including possible escaping LyC radiation, which is not taken into account, would somewhat increase these values. The high-$z$ LAEs show a tight relation between the Ly$\alpha$ luminosity $L$(Ly$\alpha$) and the absolute magnitude $M_{\rm FUV}$ at $\lambda$ = 1500Å(small open grey circles in Fig. \[fig7\]a). The confirmed LyC leakers [open black circles, @I16; @I16b; @I18; @I18b] follow the same relation. On the other hand, GPs at redshifts $\la$ 0.3 (grey asterisks) are offset to lower $L$(Ly$\alpha$) by one order of magnitude. This offset can be caused by aperture effects because GPs are on average closer to us and have larger angular sizes. Furthermore, the EW(Ly$\alpha$)s in GPs are considerably lower than in low-$z$ LyC leakers (Fig. \[fig7\]a), indicating older burst ages and thus lower Ly$\alpha$ luminosities. We note that for the LyC leakers and GPs we do not use the published values for Ly$\alpha$ in @I16 [@I16b; @I18; @I18b], @JO14, @H15, @J17 and @Y17. Instead for the sake of data homogeneity, we measured Ly$\alpha$ fluxes and equivalent widths using the public [HST]{} data, correcting the fluxes for Milky Way extinction and converting them to luminosities. As for the absolute FUV magnitudes, we derived them from the apparent [GALEX]{} FUV magnitudes. Our galaxies with extreme O$_{32}$ extend the range of $M_{\rm FUV}$ to appreciably lower FUV brightesses. Their low Ly$\alpha$ luminosities ($L$(Ly$\alpha$)$\la$10$^{42}$ erg s$^{-1}$) make the $L$(Ly$\alpha$)-$M_{\rm FUV}$ relation bend downwards at low $M_{\rm FUV}$ (Fig. \[fig7\]a)). We note that LAEs with low Ly$\alpha$ luminosities, similar to those of our galaxies, are also detected in the high-$z$ Universe [e.g. @L19]. A notable feature of the four galaxies with the highest EW(Ly$\alpha$) and highest $L$(Ly$\alpha$) in our sample is that their Ly$\alpha$ luminosities are lower by a factor $>$ 2 than the value derived from the extrapolation of the relation for high-$z$ galaxies to fainter absolute FUV magnitudes (dashed line). While they are consistent with the extrapolation of the trend defined by the GPs to fainter absolute UV magnitudes, the Ly$\alpha$ equivalent widths of the galaxies with extreme O$_{32}$ are similar to those of the low-$z$ LyC leakers [@I16; @I16b; @I18; @I18b] and are much higher than those of GPs. It was shown e.g. by @Wi16 [@Wi18] that high-$z$ galaxies at $z$ $>$ 3 are surrounded by extended diffuse Ly$\alpha$-emitting haloes tracing neutral gas up to several tens of kpc and contributing 40 – 90 per cent of the total Ly$\alpha$ flux. These extended haloes are seen even in low-mass SFGs with stellar masses down to 10$^{8}$ – 10$^{9}$ M$_\odot$ [@E18; @E19]. @Y17b have shown that Ly$\alpha$ emission in Ly$\alpha$-emitting galaxies at lower redshifts $z$ $\la$ 0.3 is spatially more extended compared to that of the UV continuum. The difference between our high-EW(Ly$\alpha$) and high-$z$ galaxies may possibly be caused by a low-intensity diffuse Ly$\alpha$ emission in our galaxies extending outside the extraction aperture used to obtain the one-dimensional spectrum. In this case, an aperture correction is needed for comparison of the Ly$\alpha$ flux with fluxes of other hydrogen emission lines. However, we do not expect a large aperture correction for $L$(Ly$\alpha$) ($>$ 2) because the adopted aperture for the extraction of one-dimensional spectra includes nearly all light from our galaxies. Additionally, we have compared Ly$\alpha$ fluxes in one-dimensional spectra obtained with different extraction apertures and reduced with the same custom and pipeline data reduction packages. We find that they differ by not more than 5 per cent. Ly$\alpha$ imaging of the lowest-mass SFGs, with stellar masses $\la$ 10$^{8}$ M$_\odot$, is needed to verify the importance of aperture corrections. Other factors may play a role in reducing $L$(Ly$\alpha$) in our low-mass galaxies, e.g. unaccounted contribution of underlying Ly$\alpha$ absorption, enhanced UV extinction due to a larger fraction of dust and/or a steeper reddening law in the UV range than in LyC leakers and high-$z$ galaxies. But these factors are unlikely to be enough to explain the observed large offset of our galaxies from the extrapolated value (dashed line in Fig. \[fig7\]a). We note that the H$\beta$ luminosities in our galaxies are more than one order of magnitude smaller than in the LyC leakers, corresponding to a smaller number of massive stars by the same factor. Probably, dynamical processes such as large-scale outflows due to the evolution of massive stars in our galaxies are considerably less energetic and may result in a reduced escaping Ly$\alpha$ emission. The distribution of the Ly$\alpha$ rest-frame equivalent widths EW(Ly$\alpha$) for the same samples of galaxies as in Fig. \[fig7\]a is shown in Fig. \[fig7\]b. The range of EW(Ly$\alpha$) for our objects is larger than that for LyC leakers with four galaxies having EW(Ly$\alpha$) smaller than those for LyC leakers, and four galaxies with EW(Ly$\alpha$) greater than the average value for LyC leakers. On the other hand, GPs from the literature have in general considerably lower EW(Ly$\alpha$). Comparing to high-$z$ galaxies we note that the EW(Ly$\alpha$)s of our galaxies are several times lower than of high-$z$ galaxies, due to the same reasons (i.e possible aperture effects) as for Ly$\alpha$ luminosities. @Ha17 found a trend of increasing EW(Ly$\alpha$) with decreasing UV luminosity. They attributed this trend to the lower metallicities of lower-luminosity galaxies. However, the distribution of our galaxies which extend the EW(Ly$\alpha$)-$M_{\rm FUV}$ relation to lower UV luminosities does not support the existence of this trend. Although a dependence of EW(Ly$\alpha$) on metallicity is not excluded, the relation in Fig. \[fig7\]b can also be explained by increasingly higher EW(Ly$\alpha$) errors at their high values for high-$z$ galaxies and by the fact that only galaxies with high EW(Ly$\alpha$) at high redshifts can be selected. The presence of many low-$z$ GPs with low EW(Ly$\alpha$) favours this selection effect. The Ly$\alpha$ escape fraction ------------------------------ An important quantity to consider is the escape fraction $f_{\rm esc}$(Ly$\alpha$). In this paper we define $f_{\rm esc}$(Ly$\alpha$) as the ratio of the extinction-corrected Ly$\alpha$ to H$\beta$ flux ratio to its Case B value of 23.3, corresponding to that at a low electron number density, $N_{\rm e}$ $\sim$ 10$^2$ cm$^{-3}$ [@SH95]: $$f_{\rm esc}({\rm Ly}\alpha) = \frac{1}{23.3}\frac{I({\rm Ly}\alpha)}{I({\rm H}\beta)}, \label{fesc}$$ where $I$(Ly$\alpha$) is the flux density corrected for the Milky Way extinction, and $I$(H$\beta$) is the flux density corrected for both the Milky Way and internal galaxy extinction. At higher electron densities, for example at $N_{\rm e}$ $\sim$ 10$^3$ cm$^{-3}$, the Case B Ly$\alpha$/H$\beta$ ratio is higher by $\sim$ 10 per cent, resulting in a lower $f_{\rm esc}$(Ly$\alpha$). Using Eq. \[fesc\], we calculate $f_{\rm esc}$(Ly$\alpha$) for the eleven LyC leakers from @I16 [@I16b; @I18; @I18b] and compare them with their directly measured $f_{\rm esc}$(LyC) in Fig. \[fig8\]. It is seen that there is a general trend of increasing $f_{\rm esc}$(LyC) with increasing $f_{\rm esc}$(Ly$\alpha$). That trend can be approximated by the maximum likelihood relation (solid line) $$\log f_{\rm esc}({\rm LyC})=(2.67\pm 0.51)\times \log f_{\rm esc}({\rm Ly}\alpha) + (0.40\pm 0.29), \label{LyC-Lya}$$ but with large uncertainties. The distribution of the data points in Fig. \[fig8\] is somewhat different from that of the dusty, clumpy interstellar medium models of @D16 (shaded region in Fig. \[fig8\]). Their models predict $f_{\rm esc}$(Ly$\alpha$) $\ga$ 2$\times$$f_{\rm esc}$(LyC) and a wide range of $f_{\rm esc}$(Ly$\alpha$) at fixed $f_{\rm esc}$(LyC), depending on the covering factor of the neutral gas. The observational data show a much narrower range of $f_{\rm esc}$(Ly$\alpha$) at fixed $f_{\rm esc}$(LyC), with some galaxies lying outside the model prediction regions. A relatively small dispersion of the data points in the observed relation in Fig. \[fig8\] gives hope that $f_{\rm esc}$(Ly$\alpha$) can be used to indirectly derive $f_{\rm esc}$(LyC). However, the number of confirmed LyC leakers is still very small and larger statistics are needed to verify the relation given by Eq. \[LyC-Lya\]. There is also the possibility that there exist sources with significant Ly$\alpha$ escape but with no leaking LyC radiation which would flatten the relation shown by the solid line. The filled circles in Fig. \[fig8\] show the galaxies from our sample with the exclusion of the three galaxies with Ly$\alpha$ absorption. For these galaxies, $f_{\rm esc}$(LyC) is derived using the velocity separation between the peaks of Ly$\alpha$ emission lines. It is seen that, for a given $f_{\rm esc}$(LyC), the galaxies are offset from the confirmed LyC leakers to lower $f_{\rm esc}$(Ly$\alpha$). Possible aperture corrections for Ly$\alpha$ emission would be too small to account for those offsets. On the other hand, corrections for unaccounted Ly$\alpha$ absorption would increase $f_{\rm esc}$(Ly$\alpha$). Overall, these corrections do not appear to be enough to account for the offsets. Alternatively, parts of the offsets could be due to an overestimation of the indirect value of $f_{\rm esc}$(LyC), as determined by eq. 2 of @I18b. It was suggested by @NO14 that the fraction of escaping ionizing radiation from galaxies correlates with the O$_{32}$ ratio. However, @I18b found that the O$_{32}$ – $f_{\rm esc}$(LyC) correlation for the confirmed LyC leakers is rather weak and a high O$_{32}$ does not garantee a high $f_{\rm esc}$(LyC). Adopting the correlation between the Ly$\alpha$ and LyC escape fractions (Fig. \[fig8\]), we investigate whether O$_{32}$ can be used as indirect indicator of escaping ionizing radiation for galaxies with the highest O$_{32}$. We show in Fig. \[fig9\]a the O$_{32}$ – $f_{\rm esc}$(Ly$\alpha$) diagram for the same galaxies as in Fig. \[fig7\], excluding the high-$z$ LAEs. The galaxies discussed in this paper (filled circles) extend the range of O$_{32}$ up to $\sim$ 40, a factor of two higher than in previous studies. However, no correlation is found. Thus, we confirm and strengthen the conclusion of @I18b that O$_{32}$ is not a reliable indicator of escaping Ly$\alpha$ emission and likely also of escaping LyC radiation. This conclusion is in agreement with the models of @B19 who have shown that galaxies with various metallicities and ionization parameters fill all the $f_{\rm esc}$(LyC) – O$_{32}$ space. However, we note that $f_{\rm esc}$(Ly$\alpha$) for our galaxies (filled circles in Fig. \[fig9\]) could be somewhat higher because of unaccounted possible aperture corrections for Ly$\alpha$ emission and underlying absorption. We discuss other possible indicators of $f_{\rm esc}$(Ly$\alpha$). A possible candidate is the rest-frame equivalent width EW(Ly$\alpha$). The diagram EW(Ly$\alpha$) – $f_{\rm esc}$(Ly$\alpha$) is shown in Fig. \[fig9\]b. This diagram is somewhat puzzling. At low EW(Ly$\alpha$), the Ly$\alpha$ escape fraction $f_{\rm esc}$(Ly$\alpha$) increases with increasing EW(Ly$\alpha$). However, no such dependence is found at high EW(Ly$\alpha$) $\ga$ 60Å for four of our galaxies and three confirmed LyC leakers with the highest EW(Ly$\alpha$). One possible explanation for that behavior has been proposed by @SM19. They have shown that the EW(Ly$\alpha$) – $f_{\rm esc}$(Ly$\alpha$) relation flattens considerably in the presence of dust absorption and is nearly independent of EW(Ly$\alpha$), given a varying ionizing photon production efficiency $\xi_{\rm ion}$. In particular, our data, in the flat part of the diagram at EW(Ly$\alpha$) $\ga$ 150Å, can be reproduced adopting $E(B-V)$ $\sim$ 0.2 mag, corresponding to $C$(H$\beta$) $\sim$ 0.3, given that log $\xi_{\rm ion}$ increases from $\sim$ 25.0 to $\sim$ 25.7 with increasing EW(Ly$\alpha$) (solid line in Fig. \[fig9\]b). This value of $C$(H$\beta$) is somewhat higher than the values derived from the SDSS spectra (Table \[tab5\]), but is consistent with the values obtained from the high signal-to-noise ratio LBT spectra for J0159$+$0751, J1032$+$4919, J1205$+$4551 and J1355$+$4651 [@I17]. Another indirect indicator of escaping radiation is the velocity separation between Ly$\alpha$ peaks $V_{\rm sep}$. @I18b have found a tight correlation between $f_{\rm esc}$(LyC) and $V_{\rm sep}$. Fig. \[fig9\]c shows a possible correlation between the peak separation $V_{\rm sep}$ and the Ly$\alpha$ escape fraction $f_{\rm esc}$(Ly$\alpha$), described by the maximum likelihood relation (solid line in Fig. \[fig9\]c): $$f_{\rm esc}({\rm Ly}\alpha)=6.56\times 10^{-7} V_{\rm sep}^2 -1.25\times 10^{-3}V_{\rm sep}+0.607. \label{fesc1}$$ The large scatter of the data around the relation found between an integrated property of Ly$\alpha$ ($f_{\rm esc}$(Ly$\alpha$)) and a characteristic of the Ly$\alpha$ profile ($V_{\rm sep}$) is probably to be expected, since several physical properties, such as dust content, H [i]{} column density, H [i]{} kinematics and geometry affect the Ly$\alpha$ radiative transfer [see e.g. @S11; @V15]. Finally, we consider whether He [i]{} emission lines in the optical range can be indicators of Ly$\alpha$ emission. @I17 have argued that He [i]{} emission line ratios are promising diagnostics because some of these lines can be optically thick, indicating a high column density of the neutral gas. The He [i]{} $\lambda$3889Å emission line is most sensitive to this effect. This line is blended with the hydrogen H8 $\lambda$3889Å emission line, and its intensity can be obtained by subtracting the H8 intensity equal to 0.107$\times$$I$(H$\beta$). The recombination intensity of the He [i]{} $\lambda$3889Å emission line is $\sim$ 0.1$\times$$I$(H$\beta$). This line can be enhanced by collisions with electrons but weakened if its optical depth is high. Therefore, the condition $I$(He [i]{} $\lambda$3889)/$I$(H$\beta$) $<$ 0.1, corresponding to $I$(H8 + He [i]{} $\lambda$3889)/$I$(H$\beta$) $<$ 0.2, would be an indication of high optical depth and high $N$(H [i]{}). It is seen in Table \[taba1\] that $I$(H8 + He [i]{} $\lambda$3889)/$I$(H$\beta$) is considerably lower than 0.2 in only two galaxies, J0926$+$4504 and J1032$+$4919. Both of these show broad Ly$\alpha$ absorption line in their spectra (Fig. \[fig6\]d,e), indicating high $N$(H [i]{}). For the remaining galaxies, this technique predicts lower $N$(H [i]{}) and thus stronger Ly$\alpha$ in emission. The technique fails for only one galaxy, J0007$+$0226, which shows Ly$\alpha$ absorption in its spectrum (Fig. \[fig6\]a), but has $I$(H8 + He [i]{} $\lambda$3889)/$I$(H$\beta$) $\sim$0.2. Thus we conclude that the He [i]{} $\lambda$3889Å emission line is generally a good indicator of $N$(H [i]{}), and hence of Ly$\alpha$ emission. Conclusions {#summary} =========== We present here new [HST]{}/COS observations of eight low-redshift ($z$ $<$ 0.07) compact star-forming galaxies (SFG) with extremely high O$_{32}$ in the range $\sim$ 22 – 39. All studied objects are compact low-mass (log ($M_\star$/M$_\odot$) = 5.8 – 8.6) and low-metallicity (12$+$log(O/H)= 7.43 – 7.82) galaxies with strong nebular emission lines in their spectra (EW(H$\beta$) = 258 - 483Å), indicating very young starburst ages of 0.5 – 3.3 Myr. The Ne/O, S/O and Ar/O abundance ratios in our galaxies are similar to the values for the bulk of compact SDSS SFGs. On the other hand, similarly to confirmed LyC leakers, the N/O abundance ratios in our galaxies are higher than those of the most compact SDSS SFGs. We also study the Ly$\alpha$ emission and indirect indicators of the escaping Ly$\alpha$ and Lyman continuum (LyC) radiation of these SFGs. Our main results are summarized as follows: 1\. A strong Ly$\alpha$ emission line with two peaks was observed in the spectra of five galaxies, while a Ly$\alpha$ emission line with a low equivalent width on top of a broad Ly$\alpha$ absorption profile is present in the remaining three galaxies. Using the damped wings of these absorption profiles, we derive neutral hydrogen column densities $N$(H [i]{}) in the range (1 – 3)$\times$10$^{21}$ cm$^{-2}$. 2\. We discuss various indirect indicators of escaping Ly$\alpha$ and ionizing radiation, such as the O$_{32}$ ratio, EW(Ly$\alpha$) and the velocity separation between Ly$\alpha$ emission line peaks $V_{\rm sep}$, assuming that the strength of the Ly$\alpha$ emission line is a measure of LyC leakage. We found that there is no correlation between O$_{32}$ and the Ly$\alpha$ escape fraction $f_{\rm esc}$(Ly$\alpha$). The dependence of $f_{\rm esc}$(Ly$\alpha$) on EW(Ly$\alpha$) is such that at low EW(Ly$\alpha$) $\la$ 100Å, there is a linear increase of $f_{\rm esc}$(Ly$\alpha$), while at high EW(Ly$\alpha$) $>$ 150Å, $f_{\rm esc}$(Ly$\alpha$) is nearly constant with a value $\sim$ 0.25. This behavior may be explained by dust absorption. We find a trend of increasing of $f_{\rm esc}$(Ly$\alpha$) with decreasing of $V_{\rm sep}$. However, even galaxies showing double-peaked Ly$\alpha$ profiles exhibit a large scatter between the Ly$\alpha$ escape fraction and the separation of their peaks $V_{\rm sep}$. The latter quantity has been shown to correlate more strongly with the LyC escape fraction [@I18b]. 3\. Bright compact star-forming regions are seen in the COS near ultraviolet (NUV) acquisition images. The surface brightness profile at the outskirts can be approximated by an exponential disc profile, with a scale length of $\sim$ 0.09 – 0.48 kpc. These scale lengths are several times lower than those of confirmed LyC leakers and are among the lowest ones found for local blue compact dwarf galaxies. The global properties of the selected galaxies are very similar to those of the lowest-mass high-$z$ galaxies. They are thus ideal nearby laboratories for investigating the mechanisms responsible for the escape of Ly$\alpha$ and ionizing radiation from galaxies during the epoch of the reionization of the Universe. Acknowledgements {#acknowledgements .unnumbered} ================ We thank the anonymous referee for valuable comments. These results are based on observations made with the NASA/ESA [Hubble Space Telescope]{}, obtained from the data archive at the Space Telescope Science Institute. STScI is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS 5-26555. Support for this work was provided by NASA through grant number HST-GO-15136.002-A from the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555. Y.I.I. and N.G.G. acknowledge support from the National Academy of Sciences of Ukraine (Project No. 0116U003191) and by its Program of Fundamental Research of the Department of Physics and Astronomy (Project No. 0117U000240) of the National Academy of Sciences of Ukraine. G.W. has been supported by the Deutsches Zentrum für Luft- und Raumfahrt (DLR) through grant number 50OR1720. I.O. acknowledges grants GACR 14-20666P and 17-06217Y of the Czech National Foundation. Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS web site is www.sdss.org. SDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration. GALEX is a NASA mission managed by the Jet Propulsion Laboratory. This publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space AdministrationGALEX is a NASA mission managed by the Jet Propulsion Laboratory. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. Emission-line fluxes in SDSS spectra and element abundances =========================================================== \[lastpage\] [^1]: [iraf]{} is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation. [^2]: [stsdas]{} is a product of the Space Telescope Science Institute, which is operated by AURA for NASA.	ArXiv
--- abstract: 'Let $W,W''\subseteq G$ be nonempty subsets in an arbitrary group $G$. $W''$ is said to be a complement to $W$ if $WW''=G$ and it is minimal if no proper subset of $W''$ is a complement to $W$. We show that, if $W$ is finite then every complement of $W$ has a minimal complement, answering a problem of Nathanson. We also give necessary and sufficient conditions for the existence of minimal complements of a certain class of infinite subsets $W$ in finitely generated abelian groups, partially answering another problem of Nathanson.' address: - 'Universität Wien, Fakultät für Mathematik, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria.' - 'Department of Mathematics, Indian Institute of Science Education and Research Bhopal, Bhopal Bypass Road, Bhauri, Bhopal 462066, Madhya Pradesh, India' author: - Arindam Biswas - Jyoti Prakash Saha title: On minimal complements in groups --- [^1] [^2] Introduction ============ Motivation ---------- Let $(G,.)$ be a group and $W\subseteq G$ be a nonempty subset. A nonempty set $W'\subseteq G$ is said to be a complement to $W$ if $$WW' = G.$$ Let $\mathcal{W}$ denote the set of all complements of $W$. Then it is clear that $\mathcal{W}\neq \emptyset$ (since $G\in \mathcal{W}$) and also the fact that the elements of $\mathcal{W}$ form a partially ordered set under inclusion. A complement $W'$ to $W$ is minimal if no proper subset of $W'$ is a complement to $W$, i.e., $$WW' = G \,\text{ and }\, W.(W'\setminus \lbrace w'\rbrace)\subsetneq G \,\,\, \forall w'\in W'.$$ Given a minimal complement $W'$ of $W$, we see that the right translation $W'g$ is also a minimal complement of $W$ and $W'$ is a minimal complement of $gW$ for all $g \in G$. Thus, the existence of a minimal complement of a nonempty subset is equivalent to the existence of a minimal complement of any of its left translates. It was shown by Nathanson (see [@NathansonAddNT4 Theorem 8]) that for a non-empty, finite subset $W$ in the additive group $\mathbb{Z}$, any complement to $W$ has a minimal complement. In the same paper he asked the following questions: [@NathansonAddNT4 Problem 11] \[nathansonprob11\] “Let $W$ be an infinite set of integers. Does there exist a minimal complement to $W$? Does there exist a complement to $W$ that does not contain a minimal complement?” [@NathansonAddNT4 Problem 12] \[nathansonprob12\] “Let $G$ be an infinite group, and let $W$ be a finite subset of $G$. Does there exist a minimal complement to $W$? Does there exist a complement to $W$ that does not contain a minimal complement?” [@NathansonAddNT4 Problem 13] \[nathansonprob13\] “Let $G$ be an infinite group, and let $W$ be an infinite subset of $G$. Does there exist a minimal complement to $W$? Does there exist a complement to $W$ that does not contain a minimal complement?” Since then the problems have generated considerable interest. Chen and Yang in 2012 gave examples of two infinite sets $W_1,W_2 \subset \mathbb{Z}$, such that $W_1$ has a complement that does not contain a minimal complement and every complement to $W_2$ contains a minimal complement (see [@ChenYang12]). They also gave certain necessary and certain sufficient conditions on the infinite set $W \subset \mathbb{Z}$ such that $W$ has a minimal complement (see [@ChenYang12 Theorem 1, 2]). Very recently, Kiss, Sándor and Yang [@KissSandorYangJCT19] succeeded in giving necessary and sufficient conditions for the existence of minimal complements of several other class of infinite sets in $\mathbb{Z}$ (which were not covered in the previous work of Chen and Yang). See [@KissSandorYangJCT19 Theorems 1, 2, 3]. Statement of results -------------------- All the aforementioned progresses were in the setting of Question \[nathansonprob11\]. In this article, we deal with the Questions \[nathansonprob12\] and \[nathansonprob13\]. Specifically, we show that - \[Theorem \[theorem1\]\] \[theorem1.1\] Let $G$ be an arbitrary group with $S$ a nonempty finite subset of $G$. Then every complement of $S$ in $G$ has a minimal complement. See Section \[sec2\], Theorem \[theorem1\]. This answers Question \[nathansonprob12\] of Nathanson.\ We turn to Question \[nathansonprob13\]. Before commencing the discussion in detail, we state that it has a simple answer when the subset $W$ is a subgroup. Namely, in that case a minimal complement always exist. See Proposition \[prop4.1\]. For general infinite subsets, the situation is more delicate. To give an answer to Question \[nathansonprob13\], one needs to consider the infinite subsets which have less algebraic structure. Our goal in this case is to establish the existence of minimal complements for a large class of infinite sets in finitely generated abelian groups. This will give the claimed partial solution to this question of Nathanson. In section \[Sec: Minimal complement\], we focus on the minimal complements of certain infinite subsets of free abelian groups of finite rank, which are of the form ${\ensuremath{\mathbb{Z}}}^d$ for some integer $d\geq 1$. It is interesting to consider the subsets $X$ of ${\ensuremath{\mathbb{Z}}}^d$ such that $x + {\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ is contained in $X$ for any $x\in X$, i.e., $$\label{Eqn: preperiodic} X \supseteq X + {\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d,$$ where $u_1, \cdots, u_d$ are element of ${\ensuremath{\mathbb{Z}}}^d$ satisfying no nontrivial ${\ensuremath{\mathbb{Z}}}$-linear relation. However it turns out that such sets do not necessarily have a minimal complement. For instance, ${\ensuremath{\mathbb{Z}}}\times {\ensuremath{\mathbb{N}}}$ is a subset of ${\ensuremath{\mathbb{Z}}}^d$ which satisfies Equation , but do not have any minimal complement. To obtain examples of infinite subsets of ${\ensuremath{\mathbb{Z}}}^d$ having minimal complements, we consider the periodic subsets of ${\ensuremath{\mathbb{Z}}}^d$ (these are subsets of ${\ensuremath{\mathbb{Z}}}^d$ satisfying Equation along with a finiteness condition, see Definition \[Defn: periodic\]). Unfortunately, there exist periodic subsets of ${\ensuremath{\mathbb{Z}}}^d$, which do not admit any minimal complement (see Proposition \[Cor: Nd minimal complement\]). So we consider a more general class of subsets of ${\ensuremath{\mathbb{Z}}}^d$ satisfying a weaker version of Equation along with certain finiteness condition (which we call eventually periodic subsets, see Definition \[Defn: periodic\] - these are $d$-dimensional analogues of the eventually periodic sets in $\mathbb{Z}$ considered by Kiss-Sándor-Yang in [@KissSandorYangJCT19]). Given an eventually periodic subset $W$ of ${\ensuremath{\mathbb{Z}}}^d$ (with periods $u_1, \cdots, u_d$), by Theorem \[Thm: Structure of Eventually Periodic\], there exist subsets ${\ensuremath{\mathcal {W}}}, {\ensuremath{\mathscr{W}}}$ of $W$ such that $$W = {\ensuremath{\mathscr{W}}}\sqcup ({\ensuremath{\mathcal {W}}}+ ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d))$$ holds. It turns out that certain eventually periodic subsets of ${\ensuremath{\mathbb{Z}}}^d$ have minimal complements. The following result provides a necessary condition for an eventually periodic subset of ${\ensuremath{\mathbb{Z}}}^d$ to have a minimal complement. We prove the following results - \[Theorem \[Thm: existence of min complement implies\]\] \[theorem1.2\] Let $W$ be an eventually periodic subset of ${\ensuremath{\mathbb{Z}}}^d$ with periods $u_1, \cdots, u_d$. Let ${\ensuremath{\mathscr{W}}}_1$ be as in Theorem \[Thm: existence of min complement implies\]. Suppose $W$ has a minimal complement in ${\ensuremath{\mathbb{Z}}}^d$. Then ${\ensuremath{\mathscr{W}}}_1$ is nonempty and there exists a nonempty finite subset ${\ensuremath{\mathcal{M}}}$ of ${\ensuremath{\mathbb{Z}}}^d$ such that the following conditions hold. 1. The map $\pi: ({\ensuremath{\mathcal{M}}}+ ({\ensuremath{\mathcal {W}}}\cup {\ensuremath{\mathscr{W}}}_1)) \to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is surjective. 2. For any $m\in {\ensuremath{\mathcal{M}}}$, there exists $w\in {\ensuremath{\mathscr{W}}}_1$ such that $m + w \not\equiv m' + w' {\ensuremath{\mathrm{\;mod\;}}}{\ensuremath{\mathcal {L}}}$ for any $m'\in {\ensuremath{\mathcal{M}}}, w'\in {\ensuremath{\mathcal {W}}}$. This is a simplified version of Theorem \[Thm: existence of min complement implies\]. We refer to Theorem \[Thm: existence of min complement implies\] for a more general statement. As a consequence, we obtain that no periodic subset of ${\ensuremath{\mathbb{Z}}}^d$ has a minimal complement (Corollary \[Cor: No periodic has minimal\]). We also prove the result below, giving a sufficient condition for an eventually periodic subset of ${\ensuremath{\mathbb{Z}}}^d$ to have a minimal complement. \[Theorem \[Thm: Implies existence of min comple\]\] \[theorem1.3\] Let $W$ be an eventually periodic subset of ${\ensuremath{\mathbb{Z}}}^d$ with periods $u_1, \cdots, u_d$. Let ${\ensuremath{\mathscr{W}}}_1$ be as in Theorem \[Thm: existence of min complement implies\]. Suppose there exists a nonempty finite subset ${\ensuremath{\mathcal{M}}}$ of ${\ensuremath{\mathbb{Z}}}^d$ such that the following conditions hold. 1. The set ${\ensuremath{\mathscr{W}}}_1$ is nonempty. 2. The map $\pi: {\ensuremath{\mathcal{M}}}\to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is injective. 3. The map $\pi: ({\ensuremath{\mathcal{M}}}+ ({\ensuremath{\mathcal {W}}}\cup {\ensuremath{\mathscr{W}}}_1)) \to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is surjective. 4. For any $m\in {\ensuremath{\mathcal{M}}}$, there exists $w\in {\ensuremath{\mathscr{W}}}_1$ such that $m + w \not\equiv m' + w' {\ensuremath{\mathrm{\;mod\;}}}{\ensuremath{\mathcal {L}}}$ for any $m'\in {\ensuremath{\mathcal{M}}}\setminus\{m\}, w'\in {\ensuremath{\mathcal {W}}}\cup {\ensuremath{\mathscr{W}}}_1$. Then $W$ has a minimal complement in ${\ensuremath{\mathbb{Z}}}^d$. Using the above, we get a necessary and sufficient condition for certain type of eventually periodic subsets of ${\ensuremath{\mathbb{Z}}}^d$ to have a minimal complement. \[Theorem \[Thm: Even peri 1 has min comp iff\]\] Let $W$ be an eventually periodic subset of ${\ensuremath{\mathbb{Z}}}^d$ with periods $u_1, \cdots, u_d$. Let ${\ensuremath{\mathscr{W}}}_1$ be as in Theorem \[Thm: existence of min complement implies\]. Suppose ${\ensuremath{\mathscr{W}}}_1$ contains only one element. Then $W$ has a minimal complement in ${\ensuremath{\mathbb{Z}}}^d$ if and only if there exists nonempty finite subset ${\ensuremath{\mathcal{M}}}$ of ${\ensuremath{\mathbb{Z}}}^d$ satisfying the following. 1. The map $\pi: {\ensuremath{\mathcal{M}}}\to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is injective. 2. The map $\pi: ({\ensuremath{\mathcal{M}}}+ {\ensuremath{\mathcal {W}}}\cup {\ensuremath{\mathscr{W}}}_1)) \to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is surjective. 3. For any $m\in {\ensuremath{\mathcal{M}}}$, there exists $w\in {\ensuremath{\mathscr{W}}}_1$ such that $m + w \not\equiv m' + w' {\ensuremath{\mathrm{\;mod\;}}}{\ensuremath{\mathcal {L}}}$ for any $m'\in {\ensuremath{\mathcal{M}}}\setminus\{m\}, w'\in {\ensuremath{\mathcal {W}}}\cup {\ensuremath{\mathscr{W}}}_1$. Finally, we conclude a general statement on the existence of minimal complements for certain infinite sets in finitely generated abelian groups. \[Theorem \[sec5Thm\]\] Let $G$ be a finitely generated (infinite) abelian group with the decomposition $G\simeq \mathbb{Z}^{d}\times(\mathbb{Z}/a_{1}\mathbb{Z})\times \cdots \times(\mathbb{Z}/a_{s}\mathbb{Z})$ with $d\geqslant 1$ and where $a_{1}\|a_{2}\|...\|a_{s}$ are positive integers $>1$ (determined uniquely by the isomorphism type of $G$). Suppose $W\subset \mathbb{Z}^{d}$ such that either $W$ satisfies the conditions of Theorem \[theorem1.3\] (equivalently Theorem \[Thm: Implies existence of min comple\]) or $W$ is a product set of the form $W_{1}\times W_{2}\times \cdots \times W_{d}$ where each $W_{i}\subseteq \mathbb{Z}, \forall 1\leqslant i\leqslant d$ has minimal complements in $\mathbb{Z}$, then $W\times H$ will have a minimal complement in $G$ where $H \subseteq (\mathbb{Z}/a_{1}\mathbb{Z})\times \cdots \times(\mathbb{Z}/a_{s}\mathbb{Z})$ is any arbitrary nonempty subset. See also Remark \[Remark:MinCompEg\] for further discussion. Plan of the paper ----------------- The article is divided into $6$ main sections. These are\ 1. Introduction - Here we give the background, motivation and results.\ 2. Existence of minimal complements for finite sets in arbitrary groups - This section deals with the finite subset case in arbitrary groups, answering the Question \[nathansonprob12\] of Nathanson cf Theorem \[theorem1\]. The subsequent sections deal with Question \[nathansonprob13\]. 3. Inexistence of minimal complements for certain infinite sets - This section deals with specific examples of sets which do not have minimal complements. It also provides insight on which sets to avoid in the search for sets having minimal complements in $\mathbb{Z}^{d}$. See Propositions \[prop3.1\], \[prop3.3\].\ 4. Minimal complements of infinite sets in free abelian groups - This section is concerned with the necessary and sufficient conditions for a class of infinite sets to have minimal complements in $\mathbb{Z}^{d}$ cf Theorem \[Thm: existence of min complement implies\], Theorem \[Thm: Implies existence of min comple\] and Theorem \[Thm: Even peri 1 has min comp iff\].\ 5. Minimal complements in finitely generated abelian groups - This section is concerned with the existence of minimal complements of a class of infinite sets in arbitrary finitely generated abelian groups cf Theorem \[sec5Thm\], giving a partial answer to Question \[nathansonprob13\] of Nathanson.\ 6. Conclusion and further remarks - In this concluding section we give specific examples of sets which don’t fall in the purview of Theorem \[Thm: existence of min complement implies\] and Theorem \[Thm: Implies existence of min comple\] but have minimal complements. See Propositions \[prop6.1\], \[prop6.2\], \[prop6.3\] and \[prop6.4\]. Existence of minimal complements for finite sets in arbitrary groups {#sec2} ==================================================================== In this section we answer Question \[nathansonprob12\]. \[theorem1\] Let $G$ be an arbitrary group with $S$ a nonempty finite subset of $G$. Then every complement of $S$ in $G$ has a minimal complement. We divide the proof into two propositions, see Proposition \[prop1\] (for the countable case) and Proposition \[prop2\] (for the uncountable case). \[prop1\] Every complement of a nonempty finite subset of an at most countable group contains a minimal complement. For finite groups, the proof of the proposition is clear. Now, let $S$ be a nonempty finite subset of a countably infinite group $G$ and $C$ be a complement of $S$ in $G$. Since $S$ is finite and $G$ is countably infinite, the complement $C$ is countably infinite. Hence the elements of $C$ can be enumerated using the positive integers. Write $C = \{c_i\}_{i=1}^\infty$. Define $C_1$ to be equal to $C$ and for any integer $i\geq 1$, define $$C_{i+1} := \begin{cases} C_i\setminus\{c_i\} & \text{ if $C_i\setminus\{c_i\}$ is a complement to $S$},\\ C_i & \text{ otherwise}. \end{cases}$$ Since each $C_i$ is a complement to $S$, for each $x\in G$ and any $i\geq 1$, there exist elements $c_{x, i}\in C_i, s_{x, i}\in S$ such that $x = s_{x, i}c_{x, i}$. Since $S$ is finite, by the Pigeonhole principle, for some element $s\in S$, the equality $s = s_{x,i}$ holds for infinitely many $i\geq 1$. Hence for infinitely many $i$, we obtain $s^{{-1}}x = c_{x, i}$, which is an element of $C_i$. Consequently, for each positive integer $k$, there exists an integer $i_k > k$ such that $s^{{-1}}x$ is an element of $C_{i_k}$, which is contained in $C_k$ (as $i_k > k$) and thus $C_k$ contains $s^{{-1}}x$. Define $M$ to be the intersection $\cap_{i\geq 1} C_i$. Then for each $x\in G$, there exists an element $s\in S$ such that $s^{{-1}}x$ belongs to $M$. Hence $M$ is a complement to $S$ in $G$. We claim that $M$ is a minimal complement to $S$. On the contrary, assume that $M\setminus \{c_j\}$ is a complement to $S$ for some element $c_j$ in $M$. Since $C_j$ contains $M$, $C_j\setminus \{c_j\}$ is also a complement to $S$. Then $C_{j+1}$ is equal to $C_j\setminus \{c_j\}$. Hence $c_j$ cannot lie in $M$, which is absurd. So $M$ is a minimal complement of $S$ contained in $C$. \[prop2\] Every complement of a nonempty finite subset of an uncountable group contains a minimal complement. Let $S$ be a finite subset of a group $G$ and $C$ be a complement of $S$ in $G$. Let ${\ensuremath{\mathscr{C}}}$ denote the set of all complements of $S$ in $G$ which are contained in $C$. Note that ${\ensuremath{\mathscr{C}}}$ is partially ordered with respect to strict inclusion $\subsetneq$. Note also that the minimal elements of the partially ordered set $({\ensuremath{\mathscr{C}}}, \subsetneq)$ are the minimal complements of $S$ in $G$ which are contained in $C$. If every chain in ${\ensuremath{\mathscr{C}}}$ has a lower bound in ${\ensuremath{\mathscr{C}}}$, then by Zorn’s lemma, it would follow that ${\ensuremath{\mathscr{C}}}$ has a minimal element. We claim that every chain in ${\ensuremath{\mathscr{C}}}$ has a lower bound in ${\ensuremath{\mathscr{C}}}$. Let ${\ensuremath{\mathcal{C}}}= \{C_\lambda\}_{\lambda\in \Lambda}$ be a chain in ${\ensuremath{\mathscr{C}}}$. If a member of ${\ensuremath{\mathcal{C}}}$ is contained in all other members of ${\ensuremath{\mathcal{C}}}$, then it is a lower bound of ${\ensuremath{\mathcal{C}}}$ in ${\ensuremath{\mathscr{C}}}$. Henceforth we assume that no member of ${\ensuremath{\mathcal{C}}}$ is a lower bound of ${\ensuremath{\mathcal{C}}}$. Note that if ${\ensuremath{\mathcal{C}}}_1, \cdots, {\ensuremath{\mathcal{C}}}_k$ are pairwise disjoint subsets of ${\ensuremath{\mathcal{C}}}$ such that their union is equal to ${\ensuremath{\mathcal{C}}}$, then for some integer $i$ with $1\leq i \leq k$, each member of ${\ensuremath{\mathcal{C}}}$ contains some member of ${\ensuremath{\mathcal{C}}}_i$. Otherwise, for each $i$, some member of $C_{\lambda_i}$ of ${\ensuremath{\mathcal{C}}}$ would be contained in every member of ${\ensuremath{\mathcal{C}}}_i$. Let $j$ be a positive integer $\leq k$ such that $C_{\lambda_j}$ is contained in $C_{\lambda_i}$ for any $i$ satisfying $1\leq i\leq k$. So $C_{\lambda_j}$ is contained in every member of ${\ensuremath{\mathcal{C}}}_i$ for each $i$, which is a contradiction to the assumption that no member of ${\ensuremath{\mathcal{C}}}$ is a lower bound of ${\ensuremath{\mathcal{C}}}$. Since $SC_\lambda = G$, for each $x\in G$ and $\lambda\in \Lambda$, there exist elements $s_{x, \lambda}\in S, c_{x,\lambda}\in C_\lambda$ such that $x=s_{x, \lambda} c_{x, \lambda}$. Let $S_x$ denote the subset of elements of $S$ of the form $s_{x, \lambda}$ for some $\lambda\in \Lambda$. For $s\in S_x$, define ${\ensuremath{\mathcal{C}}}_s$ to be the subchain $\{C_\lambda\,\|\, s_{x,\lambda} =s\}$ of $\{C_\lambda\}_{\lambda\in \Lambda}$. Then the sets ${\ensuremath{\mathcal{C}}}_s$, for $s\in S_x$, form a collection of finitely many pairwise disjoint subsets of ${\ensuremath{\mathcal{C}}}$ and their union is equal to ${\ensuremath{\mathcal{C}}}$. Hence by the observation made in the preceding paragraph, there exists an element $s'\in S$ such that each member of ${\ensuremath{\mathcal{C}}}$ contains some element of ${\ensuremath{\mathcal{C}}}_{s'}$. Since $s'^{{-1}}x$ belongs to each member of ${\ensuremath{\mathcal{C}}}_{s'}$, it also belongs to each member of ${\ensuremath{\mathcal{C}}}$ and hence $s'^{{-1}}x$ belongs to the intersection $\cap_{\lambda\in \Lambda} C_\lambda$. Consequently, $\cap_{\lambda\in \Lambda} C_\lambda$ is a complement of $S$ in $G$ contained in $C$. In other words, it is an element of ${\ensuremath{\mathscr{C}}}$. So each chain in ${\ensuremath{\mathscr{C}}}$ has a lower bound in ${\ensuremath{\mathscr{C}}}$. This proves the claim. So the proposition follows from Zorn’s lemma. In particular, for finitely generated free abelian groups, e.g., $\mathbb{Z}^k$, every (additive) complement of a finite set contains a minimal complement. Later on, we shall give specific examples of infinite subsets (both countably infinite subsets and uncountably infinite subsets) in certain groups which admit minimal complements. See section \[Sec:Conclusion\], Propositions \[prop6.1\], \[prop6.2\], \[prop6.3\] and \[prop6.4\]. Also see section \[Sec: Minimal complement\] for minimal complements of eventually periodic sets. Inexistence of minimal complements for certain infinite sets ============================================================ In the following, $d$ denotes a positive integer and ${\ensuremath{\mathbb{N}}}$ denotes the set of all nonnegative integers. \[prop3.1\] \[Prop: Complement iff\] Let $M$ be a subset of ${\ensuremath{\mathbb{Z}}}^d$. Then the following statements are equivalent. 1. The set $M$ is a complement of ${\ensuremath{\mathbb{N}}}^d$ in ${\ensuremath{\mathbb{Z}}}^d$. 2. $M$ contains a sequence of elements of ${\ensuremath{\mathbb{Z}}}^d$ such that the maximum of their coordinates is arbitrarily small negative number. 3. $\liminf _{ (x_1, \cdots, x_d)\in M} \max\{x_1, \cdots, x_d\}=-\infty.$ Suppose statement (1) holds. We need to show that given any integer $n\in {\ensuremath{\mathbb{Z}}}$, $M$ contains a point all whose coordinates are less than or equal to $n$. On the contrary, suppose this is not true. Then for some $n\in {\ensuremath{\mathbb{Z}}}$, $M$ is contained in ${\ensuremath{\mathbb{Z}}}^d \setminus(n-{\ensuremath{\mathbb{N}}}^d)$. So $M+ {\ensuremath{\mathbb{N}}}^d$ would also be contained in ${\ensuremath{\mathbb{Z}}}^d \setminus(n-{\ensuremath{\mathbb{N}}}^d)$, which is absurd. So the second statement holds. It is clear that statement (3) follows from statement (2). Suppose statement (3) holds. Then for each integer $n$, the set $M$ contains a point $P_n=(x_{n1}, \cdots, x_{nd})$ with each coordinate $\leq n$, i.e., $x_{ni}\leq n$ for any $1\leq i\leq d$. So ${\ensuremath{\mathbb{N}}}^d$ contains the point $(n-x_{n1}, \cdots, n-x_{nd})$. Hence ${\ensuremath{\mathbb{N}}}^d$ contains $(n-x_{n1}, \cdots, n-x_{nd}) + {\ensuremath{\mathbb{N}}}^d$, and thus $$P_n + {\ensuremath{\mathbb{N}}}^d \supseteq P_n + ((n-x_{n1}, \cdots, n-x_{nd}) + {\ensuremath{\mathbb{N}}}^d) = (n, \cdots, n) + {\ensuremath{\mathbb{N}}}^d .$$ Consequently, $M+{\ensuremath{\mathbb{N}}}^d$ contains $\cup_{n\in {\ensuremath{\mathbb{Z}}}} ((n, \cdots, n) + {\ensuremath{\mathbb{N}}}^d) = {\ensuremath{\mathbb{Z}}}^d$. This proves statement (1). \[cor3.2\] \[Cor: Nd minimal complement\] 1. The subset ${\ensuremath{\mathbb{N}}}^d$ has complements in ${\ensuremath{\mathbb{Z}}}^d$. 2. The subset ${\ensuremath{\mathbb{N}}}^d$ has no minimal complement in ${\ensuremath{\mathbb{Z}}}^d$. 3. No complement of ${\ensuremath{\mathbb{N}}}^d$ in ${\ensuremath{\mathbb{Z}}}^d$ contains a minimal complement of ${\ensuremath{\mathbb{N}}}^d$. The first statement follows since $\{(-n,\cdots,-n)\,\|\,n\in {\ensuremath{\mathbb{N}}}\}$ and any of its infinite subsets are complements of ${\ensuremath{\mathbb{N}}}^d$ in ${\ensuremath{\mathbb{Z}}}^d$ (which can be seen by applying Proposition \[Prop: Complement iff\]). For any complement $M$ of ${\ensuremath{\mathbb{N}}}^d$ and for any finite subset $F$ of $M$, it follows from Proposition \[Prop: Complement iff\] that $M\setminus F$ is also a complement of ${\ensuremath{\mathbb{N}}}^d$. Hence the second and the third statement hold. We strengthen Proposition \[Prop: Complement iff\] and Corollary \[Cor: Nd minimal complement\] in Proposition \[Prop: AP Complement iff\] below. \[prop3.3\] \[Prop: AP Complement iff\] Let $u_1, \cdots, u_d$ be elements of ${\ensuremath{\mathbb{Z}}}^d$ which satisfy no nontrivial ${\ensuremath{\mathbb{Z}}}$-linear relation. Denote the subgroup ${\ensuremath{\mathbb{Z}}}u_1 + \cdots + {\ensuremath{\mathbb{Z}}}u_d$ of ${\ensuremath{\mathbb{Z}}}^d$ by ${\ensuremath{\mathcal {L}}}$. 1. A subset $M$ of ${\ensuremath{\mathbb{Z}}}^d$ is a complement of ${\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ in ${\ensuremath{\mathbb{Z}}}^d$ if and only if each fibre of the map $\pi: M \to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ contains a sequence of elements of ${\ensuremath{\mathbb{Z}}}^d$ such that the maximum of their coordinates with respect to $u_1, \cdots, u_d$ is arbitrarily small negative number. 2. The subset ${\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ in ${\ensuremath{\mathbb{Z}}}^d$ has complements in ${\ensuremath{\mathbb{Z}}}^d$. 3. The subset ${\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ in ${\ensuremath{\mathbb{Z}}}^d$ has no minimal complement in ${\ensuremath{\mathbb{Z}}}^d$. 4. No complement of ${\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ in ${\ensuremath{\mathbb{Z}}}^d$ contains a minimal complement of ${\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ in ${\ensuremath{\mathbb{Z}}}^d$. Let ${\ensuremath{\mathfrak{L}}}$ be a subset of ${\ensuremath{\mathbb{Z}}}^d$ such that the map $\pi: {\ensuremath{\mathfrak{L}}}\to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is bijective. For any subset $X$ of ${\ensuremath{\mathbb{Z}}}^d$ and any ${\ensuremath{\mathfrak{l}}}\in {\ensuremath{\mathfrak{L}}}$, let $X_{\ensuremath{\mathfrak{l}}}$ denote the set of all elements of $X$ which are congruent to ${\ensuremath{\mathfrak{l}}}{\ensuremath{\mathrm{\;mod\;}}}{\ensuremath{\mathcal {L}}}$. Since each element of ${\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ is congruent to zero modulo ${\ensuremath{\mathcal {L}}}$, it follows that a subset $M$ of ${\ensuremath{\mathbb{Z}}}^d$ is a complement of ${\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ in ${\ensuremath{\mathbb{Z}}}^d$ if and only if $M_{\ensuremath{\mathfrak{l}}}+ ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d) = {\ensuremath{\mathbb{Z}}}^d_{\ensuremath{\mathfrak{l}}}$ for any ${\ensuremath{\mathfrak{l}}}\in {\ensuremath{\mathfrak{L}}}$. Choose an element ${\ensuremath{\mathfrak{l}}}\in {\ensuremath{\mathfrak{L}}}$. Suppose $M_{\ensuremath{\mathfrak{l}}}+ ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d) = {\ensuremath{\mathbb{Z}}}^d_{\ensuremath{\mathfrak{l}}}$ holds. We claim that $M_{\ensuremath{\mathfrak{l}}}$ contains a sequence of elements of ${\ensuremath{\mathbb{Z}}}^d_{\ensuremath{\mathfrak{l}}}$ such that the maximum of their coordinates with respect to $u_1, \cdots, u_d$ is arbitrarily small negative number. On the contrary, suppose this is false. Then for some $n\in {\ensuremath{\mathbb{Z}}}$, $$M_{\ensuremath{\mathfrak{l}}}\subseteq {\ensuremath{\mathbb{Z}}}^d_{\ensuremath{\mathfrak{l}}}\setminus({\ensuremath{\mathfrak{l}}}+ (n(u_1 + \cdots + u_n))-({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)).$$ So $M_{\ensuremath{\mathfrak{l}}}+ ( {\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$ would also be contained in ${\ensuremath{\mathbb{Z}}}^d_{\ensuremath{\mathfrak{l}}}\setminus({\ensuremath{\mathfrak{l}}}+ (n(u_1 + \cdots + u_n))-({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d))$, which is absurd. So the claim follows. Conversely, assume that $M_{\ensuremath{\mathfrak{l}}}$ contains a sequence of elements of ${\ensuremath{\mathbb{Z}}}^d_{\ensuremath{\mathfrak{l}}}$ such that the maximum of their coordinates with respect to $u_1, \cdots, u_d$ is arbitrarily small negative number. So for any $n\in {\ensuremath{\mathbb{Z}}}$, the set $M_{\ensuremath{\mathfrak{l}}}$ contains a point $P_n={\ensuremath{\mathfrak{l}}}+ x_{n1}u_1 + \cdots + x_{nd}u_d$ with $x_{ni} \leq n$ for any $1\leq i\leq d$. So ${\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ contains the point $(n-x_{n1})u_1 + \cdots + ( n-x_{nd})u_d$. Hence ${\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ contains $((n-x_{n1})u_1 + \cdots + ( n-x_{nd})u_d) + ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$, and thus $$\begin{aligned} P_n + {\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d &\supseteq P_n + (((n-x_{n1})u_1 + \cdots + ( n-x_{nd})u_d)\\ & \quad + ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d))\\ &={\ensuremath{\mathfrak{l}}}+ (n(u_1 + \cdots + u_n)) + ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d) .\end{aligned}$$ Consequently, $M_{\ensuremath{\mathfrak{l}}}+ ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$ contains $\cup_{n\in {\ensuremath{\mathbb{Z}}}} ({\ensuremath{\mathfrak{l}}}+ (n(u_1 + \cdots + u_n)) + ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d) ) = {\ensuremath{\mathbb{Z}}}^d_{\ensuremath{\mathfrak{l}}}$. This proves statement (1). From the first statement, it follows that the set ${\ensuremath{\mathfrak{L}}}+ (-{\ensuremath{\mathbb{N}}}(u_1+\cdots + u_d))$ and any of its subsets $C$ containing infinitely elements of ${\ensuremath{\mathfrak{l}}}+ (-{\ensuremath{\mathbb{N}}}(u_1+\cdots + u_d))$ for any ${\ensuremath{\mathfrak{l}}}\in {\ensuremath{\mathfrak{L}}}$ is a complement to ${\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$. So the second statement holds. For any complement $M$ of ${\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ in ${\ensuremath{\mathbb{Z}}}^d$ and any finite subset $F$ of $M$, it follows from the first statement that $M\setminus F$ is also a complement to ${\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$. This proves the third and the fourth statement. Minimal complements of infinite sets in free abelian groups {#Sec: Minimal complement} =========================================================== We start the section with the statement on existence of minimal complements of subgroups of arbitrary groups as stated in the introduction. \[prop4.1\] Let $G$ be an arbitrary group with $S$ a subgroup of $G$. Then every complement of $S$ in $G$ has a minimal complement. Let $C$ be a complement of $S$ in $G$. Let $M$ be a subset of $C$ such that $m_1m_2^{{-1}}\notin S$ for any two distinct elements $m_1, m_2$ of $M$, and for any $c\in C$, $S$ contains $m_c^{{-1}}c$ for some $m_c\in M$. Then $M$ is a minimal complement to $S$ contained in $C$. With the subgroup case done, we shall study minimal complements of sets which are not subgroups. In this section we will be in free abelian groups of finite rank. For notational convenience, we will consider the free abelian group ${\ensuremath{\mathbb{Z}}}^d$ only. Note that a condition holds for sufficiently large elements of ${\ensuremath{\mathbb{Z}}}^d$ if and only if the condition holds for almost all elements of ${\ensuremath{\mathbb{Z}}}^d$. This enables us to use the terms “sufficiently large” and “almost all” interchangeably without any confusion. However, to refer to sufficiently large elements, the underlying space is certainly required to have a notion of a metric (which in general is not available in an arbitrary free abelian group). We will prove Theorems \[Thm: existence of min complement implies\], \[Thm: Implies existence of min comple\] providing a necessary and a sufficient condition for the existence of minimal complements for eventually periodic sets. We also provide a necessary and sufficient condition for certain eventually periodic sets to have a minimal complement. See Theorem \[Thm: Even peri 1 has min comp iff\]. Periodic and eventually periodic sets ------------------------------------- The union of two subsets $A, B$ of ${\ensuremath{\mathbb{Z}}}^d$ is denoted by $A\cup B$. When $A, B$ are disjoint, we write $A\sqcup B$ to denote the union $A\cup B$ and to indicate that $A, B$ are disjoint. In the following, $u_1, \cdots, u_d$ denote elements of ${\ensuremath{\mathbb{Z}}}^d$ which satisfy no nontrivial ${\ensuremath{\mathbb{Z}}}$-linear relation. Let ${\ensuremath{\mathcal {L}}}$ denote the subgroup ${\ensuremath{\mathbb{Z}}}u_1 + \cdots + {\ensuremath{\mathbb{Z}}}u_d$ of ${\ensuremath{\mathbb{Z}}}^d$. Two elements $x,y\in {\ensuremath{\mathbb{Z}}}^d$ are said to be equivalent modulo ${\ensuremath{\mathcal {L}}}$ if $x-y\in {\ensuremath{\mathcal {L}}}$. Denote by $\pi$ the quotient map $\pi: {\ensuremath{\mathbb{Z}}}^d \twoheadrightarrow {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$. The image of an element $v\in {\ensuremath{\mathbb{Z}}}^d$ under the quotient map $\pi: {\ensuremath{\mathbb{Z}}}^d \twoheadrightarrow {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is denoted by $\bar v$. A typical element of ${\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ would be denoted by $\bar v$ (which is legitimate since the quotient map $\pi: {\ensuremath{\mathbb{Z}}}^d \twoheadrightarrow {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is surjective), and by $v$, we would denote a lift of $\bar v$ to ${\ensuremath{\mathbb{Z}}}^d$. For any subset $X$ of ${\ensuremath{\mathbb{Z}}}^d$ and any $\bar v\in {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$, denote by $X_{\bar v}$ the intersection $X\cap \pi^{{-1}}(\bar v)$. For any subset $X$ of ${\ensuremath{\mathbb{Z}}}^d$, the restriction of the map $\pi$ to $X$ is the composite map $X{\hookrightarrow}{\ensuremath{\mathbb{Z}}}^d {\xrightarrow}{\pi} {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$, which we denote by $\pi_X$ (or, simply by $\pi$ to avoid cumbersome notation). \[Defn: periodic\] A nonempty subset $X$ of ${\ensuremath{\mathbb{Z}}}^d$ is called [periodic with periods]{.nodecor} $u_1,\cdots, u_d$ if $X$ is contained in $F_X + {\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ for some nonempty finite subset $F_X\subset {\ensuremath{\mathbb{Z}}}^d$ and $x + {\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ is contained in $X$ for any $x\in X$. A nonempty subset $X$ of ${\ensuremath{\mathbb{Z}}}^d$ is called [eventually periodic with periods]{.nodecor} $u_1,\cdots, u_d$ if $X$ is contained in $F_X + {\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ for some nonempty finite subset $F_X\subset {\ensuremath{\mathbb{Z}}}^d$ and $x + {\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ is contained in $X$ for any sufficiently large element $x\in X$. For any integer $k\geq 0$, the set $\{(r, -r)\in {\ensuremath{\mathbb{Z}}}^2\,\|\, -k\leq r\leq k\} + {\ensuremath{\mathbb{N}}}^2$ is a periodic subset of ${\ensuremath{\mathbb{Z}}}^2$ with periods $(1,0), (0,1)$. The set ${\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ appearing in Proposition \[Prop: AP Complement iff\] is periodic and also eventually periodic. Note that periodic subsets of ${\ensuremath{\mathbb{Z}}}^d$ are also eventually periodic with the same periods. The set ${\ensuremath{\mathbb{Z}}}^d$ is neither periodic nor eventually periodic. First we prove Proposition \[Prop: Structure of Periodic\] and Theorem \[Thm: Structure of Eventually Periodic\], which describe the structure of periodic sets and eventually periodic sets. \[Prop: Structure of Periodic\] Let $W$ be a periodic subset of ${\ensuremath{\mathbb{Z}}}^d$ with periods $u_1, \cdots, u_d$. Let ${\ensuremath{\mathcal{Q}}}$ denote the image of $W$ under the map $\pi: W\to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$. 1. The nonempty fibres of the map $\pi: W \to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$, i.e., the sets $W_{\bar v}$ for $\bar v\in {\ensuremath{\mathcal{Q}}}$ are periodic with periods $u_1, \cdots, u_d$. 2. For any $\bar v\in {\ensuremath{\mathcal{Q}}}$, denote by ${\ensuremath{\mathcal {W}}}'_{\bar v}$ the set of elements $w\in W_{\bar v}$ such that the intersection of the sets $w - ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$ and $W_{\bar v}$ contains no element other than $w$. Then for any $\bar v\in {\ensuremath{\mathcal{Q}}}$, the set ${\ensuremath{\mathcal {W}}}'_{\bar v}$ is a nonempty finite subset of $W_{\bar v}$ and the equality $$W_{\bar v} = {\ensuremath{\mathcal {W}}}'_{\bar v} + ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$$ holds. 3. Let ${\ensuremath{\mathcal {W}}}$ denote the set of elements $w\in W$ such that the intersection of the sets $w- ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$ and $W$ contains no element other than $w$. Then the equalities $$\begin{aligned} {\ensuremath{\mathcal {W}}}&= \sqcup_{\bar v\in {\ensuremath{\mathcal{Q}}}} {\ensuremath{\mathcal {W}}}'_{\bar v}, \\ {\ensuremath{\mathcal {W}}}_{\bar v}& = {\ensuremath{\mathcal {W}}}'_{\bar v}, \\ W &= {\ensuremath{\mathcal {W}}}+ ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)\\ &= \sqcup _{\bar v\in {\ensuremath{\mathcal{Q}}}} ( {\ensuremath{\mathcal {W}}}_{\bar v} + ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d )) \end{aligned}$$ hold. Since $W$ is periodic with periods $u_1, \cdots, u_d$, and ${\ensuremath{\mathcal {L}}}$ is equal to ${\ensuremath{\mathbb{Z}}}u_1 + \cdots + {\ensuremath{\mathbb{Z}}}u_d$, it follows that the nonempty fibres of the map $\pi: W\to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ are periodic with periods $u_1, \cdots, u_d$. These fibres are precisely the sets $W_{\bar v}$ for $\bar v\in {\ensuremath{\mathcal{Q}}}$. This proves the first statement. Let $\bar v$ be an element of ${\ensuremath{\mathcal{Q}}}$ and $v\in W_{\bar v}$ be a lift of $\bar v$ modulo ${\ensuremath{\mathcal {L}}}$. Note that $-v+ W_{\bar v}$ is periodic with periods $u_1, \cdots, u_d$ and there exists a nonempty finite subset $F$ of ${\ensuremath{\mathbb{Z}}}^d$ such that $-v+ W_{\bar v}$ is contained in $F+ {\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$. Since $-v+ W_{\bar v}$ is contained in ${\ensuremath{\mathbb{Z}}}u_1 + \cdots + {\ensuremath{\mathbb{Z}}}u_d$, note that $F$ can be taken to be a subset of ${\ensuremath{\mathbb{Z}}}u_1 + \cdots + {\ensuremath{\mathbb{Z}}}u_d$. For $1\leq i\leq d$, let $\lambda_i$ denote the minimum of the $i$-th coordinates of the elements of $F$. Then it follows that $F+ {\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ is contained in $(\lambda_1 u_1 + \cdots + \lambda_d u_d)+({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$. So $-v+ W_{\bar v}$ is periodic with periods $u_1, \cdots, u_d$ and it is contained in $(\lambda_1 u_1 + \cdots + \lambda_d u_d)+({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$. Note that $$-v+{\ensuremath{\mathcal {W}}}'_{\bar v} = \{u\in (-v+ W_{\bar v})\,\|\, (u-({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d))\cap (-v+ W_{\bar v}) = \{u\} \}.$$ Then the second statement follows since if a periodic subset $U$ of ${\ensuremath{\mathbb{Z}}}^d$ with periods $u_1, \cdots, u_d$ is contained in $(\mu_1 u_1 + \cdots + \mu_d u_d)+({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$ for some integers $\mu_1, \cdots, \mu_d$, then ${\ensuremath{\mathcal{U}}}:= \{u\in U \,\|\, (u-({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d))\cap U = \{u\} \}$ is a nonempty finite subset of $U$ and $$U = {\ensuremath{\mathcal{U}}}+ ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$$ holds. If $w$ is an element of ${\ensuremath{\mathcal {W}}}$ and $\pi(w)$ is equal to $\bar v$, then the intersection of the sets $w - ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$ and $W_{\bar v}$ contains no element other than $w$. Hence $w$ belongs to ${\ensuremath{\mathcal {W}}}'_{\bar v}$. So ${\ensuremath{\mathcal {W}}}$ is contained in $\sqcup_{\bar v\in {\ensuremath{\mathcal{Q}}}} {\ensuremath{\mathcal {W}}}'_{\bar v}$. Note that for any $\bar v\in {\ensuremath{\mathcal{Q}}}$ and any $w\in W_{\bar v}$, $$(w - ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d) )\cap W = (w - ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d) )\cap W_{\bar v} = \{w\}.$$ This implies that $\sqcup_{\bar v\in {\ensuremath{\mathcal{Q}}}} {\ensuremath{\mathcal {W}}}'_{\bar v}$ is contained in ${\ensuremath{\mathcal {W}}}$, proving the first equality. Then the remaining equalities are immediate. \[Thm: Structure of Eventually Periodic\] Let $W$ be an eventually periodic subset of ${\ensuremath{\mathbb{Z}}}^d$ with periods $u_1, \cdots, u_d$. 1. Let ${\ensuremath{\mathscr{W}}}$ denote the set of all elements $w\in W$ such that $w+ {\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ is not contained in $W$. Then ${\ensuremath{\mathscr{W}}}$ is a finite set. 2. The set $W\setminus {\ensuremath{\mathscr{W}}}$ is periodic with periods $u_1, \cdots,u_d$. 3. The following sets are equal. 1. The set of all elements of ${\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ having infinite fibre under the composite map $W{\hookrightarrow}{\ensuremath{\mathbb{Z}}}^d \twoheadrightarrow {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$. 2. The set of all elements of ${\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ having infinite fibre under the composite map $W\setminus {\ensuremath{\mathscr{W}}}{\hookrightarrow}{\ensuremath{\mathbb{Z}}}^d \twoheadrightarrow {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$. 3. The image $\pi(W\setminus {\ensuremath{\mathscr{W}}})$ of $W\setminus {\ensuremath{\mathscr{W}}}$ under $\pi$, to be denoted by ${\ensuremath{\mathcal{Q}}}$. 4. The nonempty fibres of the map $\pi: W\setminus {\ensuremath{\mathscr{W}}}\to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$, i.e., the sets $(W\setminus {\ensuremath{\mathscr{W}}})_{\bar v}$ for $\bar v\in {\ensuremath{\mathcal{Q}}}$ are periodic with periods $u_1, \cdots, u_d$. 5. For any $\bar v\in {\ensuremath{\mathcal{Q}}}$, denote by ${\ensuremath{\mathcal {W}}}'_{\bar v}$ the set of elements $w\in (W\setminus{\ensuremath{\mathscr{W}}})_{\bar v}$ such that the intersection of the sets $w - ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$ and $(W\setminus{\ensuremath{\mathscr{W}}})_{\bar v}$ contains no element other than $w$. Then for any $\bar v\in {\ensuremath{\mathcal{Q}}}$, the equality $$(W\setminus{\ensuremath{\mathscr{W}}})_{\bar v} = {\ensuremath{\mathcal {W}}}'_{\bar v} + ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$$ holds. Moreover, ${\ensuremath{\mathcal {W}}}'_{\bar v}$ is a finite set. 6. Let ${\ensuremath{\mathcal {W}}}$ denote the set of elements $w\in W\setminus {\ensuremath{\mathscr{W}}}$ such that the intersection of the sets $w- ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$ and $W\setminus {\ensuremath{\mathscr{W}}}$ contains no element other than $w$. Then the equalities $$\begin{aligned} {\ensuremath{\mathcal {W}}}& = \sqcup_{\bar v\in {\ensuremath{\mathcal{Q}}}} {\ensuremath{\mathcal {W}}}'_{\bar v}, \\ {\ensuremath{\mathcal {W}}}_{\bar v} & = {\ensuremath{\mathcal {W}}}'_{\bar v}, \\ W \setminus {\ensuremath{\mathscr{W}}}&= {\ensuremath{\mathcal {W}}}+ ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)\\ &= \sqcup _{\bar v\in {\ensuremath{\mathcal{Q}}}} ( {\ensuremath{\mathcal {W}}}_{\bar v} + ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d )) \end{aligned}$$ hold. 7. The following $$\label{Eqn: Decomposition} W = {\ensuremath{\mathscr{W}}}\sqcup ({\ensuremath{\mathcal {W}}}+ ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)) = \mathscr W \sqcup ( \sqcup _{\bar v\in {\ensuremath{\mathcal{Q}}}} ( {\ensuremath{\mathcal {W}}}_{\bar v} + ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d )) )$$ expresses $W$ as a disjoint union of its subsets. Since $W$ is eventually periodic with periods $u_1, \cdots, u_d$, it contains $w + {\ensuremath{\mathbb{N}}}u_1 + \cdots +{\ensuremath{\mathbb{N}}}u_d$ for almost all $w\in W$. So it follows that ${\ensuremath{\mathscr{W}}}$ is finite. Since $W$ is nonempty, it is infinite. So $W\setminus{\ensuremath{\mathscr{W}}}$ is nonempty. Moreover, since $W$ is eventually periodic, it follows that $W\setminus{\ensuremath{\mathscr{W}}}$ is contained in $F + ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$ for some finite subset $F$ of ${\ensuremath{\mathbb{Z}}}^d$. By the definition of ${\ensuremath{\mathscr{W}}}$, it follows that $(W\setminus {\ensuremath{\mathscr{W}}}) + ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$ is contained in $W$. To prove that $W\setminus {\ensuremath{\mathscr{W}}}$ is periodic with periods $u_1, \cdots, u_d$, it only remains to show that $(W\setminus {\ensuremath{\mathscr{W}}}) + ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$ does not intersect with ${\ensuremath{\mathscr{W}}}$. Suppose ${\ensuremath{\mathscr{W}}}$ contains $w+ \lambda_1 u_1 + \cdots + \lambda_d u_d$ for some $w\in W\setminus {\ensuremath{\mathscr{W}}}$ and for some $\lambda_1, \cdots, \lambda_d\in {\ensuremath{\mathbb{N}}}$. So for some $\mu_1, \cdots, \mu_d\in {\ensuremath{\mathbb{N}}}$, the vector $(w + \lambda_1 u_1 + \cdots + \lambda_d u_d) + (\mu_1 u_1 + \cdots + \mu_d u_d)$ would not be contained in $W$, which is absurd since $w\in W\setminus {\ensuremath{\mathscr{W}}}$. Consequently, $(W\setminus {\ensuremath{\mathscr{W}}}) + ( {\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$ is contained in $W\setminus {\ensuremath{\mathscr{W}}}$. Hence $W\setminus {\ensuremath{\mathscr{W}}}$ is periodic with periods $u_1, \cdots,u_d$. Note $W= {\ensuremath{\mathscr{W}}}\sqcup (W\setminus {\ensuremath{\mathscr{W}}})$ is a decomposition of $W$ into two disjoint subsets where ${\ensuremath{\mathscr{W}}}$ is finite and $W\setminus {\ensuremath{\mathscr{W}}}$ is infinite. Since the set ${\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is finite, it follows that an element $\bar v$ of ${\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ has infinite fibre under $\pi: W \to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ if and only if it has infinite fibre under $\pi: W\setminus {\ensuremath{\mathscr{W}}}\to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$. Moreover, any element in the image of the map $\pi: W\setminus {\ensuremath{\mathscr{W}}}\to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ has infinite fibre because $W\setminus {\ensuremath{\mathscr{W}}}$ contains the $({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$-translate of any of its points. Hence the three sets in part (a), (b), (c) are equal. The statements (4), (5), (6), (7) follow from Proposition \[Prop: Structure of Periodic\]. Note that not all the nonempty subsets $X$ of ${\ensuremath{\mathbb{Z}}}^d$ satisfying $$\label{Eqn: Periodic} X\supseteq X +({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$$ are of the form ${\ensuremath{\mathcal {W}}}+ ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$ for some subset ${\ensuremath{\mathcal {W}}}$ of ${\ensuremath{\mathbb{Z}}}^d$ (for instance, consider the subset ${\ensuremath{\mathbb{Z}}}\times {\ensuremath{\mathbb{N}}}$ of ${\ensuremath{\mathbb{Z}}}^2$). Moreover, there are subsets $X$ of ${\ensuremath{\mathbb{Z}}}^d$ which satisfy Equation and are of the form ${\ensuremath{\mathcal {W}}}+ ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$, but are not periodic, i.e., not contained in the sum of ${\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ and a nonempty finite set (for instance, consider the subset $\{ (n,-n)\,\|\, n\in {\ensuremath{\mathbb{N}}}\} + {\ensuremath{\mathbb{N}}}^2$ of ${\ensuremath{\mathbb{Z}}}^2$). Necessary condition ------------------- \[Thm: existence of min complement implies\] Let $W$ be an eventually periodic subset of ${\ensuremath{\mathbb{Z}}}^d$ with periods $u_1, \cdots, u_d$. Let ${\ensuremath{\mathscr{W}}}, {\ensuremath{\mathcal {W}}}$ be as in Theorem \[Thm: Structure of Eventually Periodic\]. Let $M$ be a minimal complement of $W$ in ${\ensuremath{\mathbb{Z}}}^d$. Let $M_\infty$ denote the union of those fibres of the map $\pi: M \to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ containing a sequence of elements of ${\ensuremath{\mathbb{Z}}}^d$ such that the maximum of their coordinates with respect to $u_1, \cdots, u_d$ is arbitrarily small negative number. Let ${\ensuremath{\mathcal{M}}}$ be a subset of $M_\infty$ such that the composite map ${\ensuremath{\mathcal{M}}}{\hookrightarrow}M_\infty \twoheadrightarrow \pi(M_\infty)$ is a bijection. Let ${\ensuremath{\mathscr{W}}}_0$ (resp. ${\ensuremath{\mathscr{W}}}_1$) denote the set of elements of ${\ensuremath{\mathscr{W}}}$ which are congruent to some element (resp. no element) of ${\ensuremath{\mathcal {W}}}$ modulo ${\ensuremath{\mathcal {L}}}$[^3]. Then the following statements hold. 1. The set $M_\infty$ is an infinite set, ${\ensuremath{\mathcal{M}}}$ is a nonempty finite subset of $M_\infty$, ${\ensuremath{\mathscr{W}}}_1$ is nonempty, and the map $\pi: ({\ensuremath{\mathcal{M}}}+ ({\ensuremath{\mathcal {W}}}\cup {\ensuremath{\mathscr{W}}}_1)) \to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is surjective. 2. For any $m\in {\ensuremath{\mathcal{M}}}$, there exists $w\in {\ensuremath{\mathscr{W}}}_1$ such that $m + w \not\equiv m' + w' {\ensuremath{\mathrm{\;mod\;}}}{\ensuremath{\mathcal {L}}}$ for any $m'\in {\ensuremath{\mathcal{M}}}, w'\in {\ensuremath{\mathcal {W}}}$. Let $M_{\ensuremath{\mathrm{fin}}}$ denote $M\setminus M_\infty$. Let ${\ensuremath{\mathfrak{L}}}$ be a subset of ${\ensuremath{\mathbb{Z}}}^d$ such that the map $\pi: {\ensuremath{\mathfrak{L}}}\to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is bijective. By Theorem \[Thm: Structure of Eventually Periodic\] (7), it follows that for each ${\ensuremath{\mathfrak{l}}}\in {\ensuremath{\mathfrak{L}}}$, there exists a positive integer $\lambda_{\ensuremath{\mathfrak{l}}}$ such that the intersection of $M_{\ensuremath{\mathrm{fin}}}+ W$ and ${\ensuremath{\mathfrak{l}}}+(-\lambda_{\ensuremath{\mathfrak{l}}}(u_1 + \cdots + u_d)) +(-({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d))$ is empty. Hence $M_{\ensuremath{\mathrm{fin}}}+W$ contains no element of the set ${\ensuremath{\mathfrak{L}}}+ (-(\max_{{\ensuremath{\mathfrak{l}}}\in {\ensuremath{\mathfrak{L}}}}\lambda_{\ensuremath{\mathfrak{l}}}) (u_1 + \cdots + u_d)) +(-({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d))$. Since $M$ is a complement of $W$ in ${\ensuremath{\mathbb{Z}}}^d$ and $M$ is equal to the union of $M_{\ensuremath{\mathrm{fin}}}$ and $M_\infty$, it follows that $M_\infty$ is nonempty and $$\label{Eqn: Containment} M_\infty + W \supseteq {\ensuremath{\mathfrak{L}}}+ (-(\max_{{\ensuremath{\mathfrak{l}}}\in {\ensuremath{\mathfrak{L}}}}\lambda_{\ensuremath{\mathfrak{l}}}(u_1 + \cdots + u_d))) +(-({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)).$$ So $M_\infty$ is infinite and ${\ensuremath{\mathcal{M}}}$ is nonempty. If ${\ensuremath{\mathscr{W}}}$ were empty, then Equation would imply $$\label{Eqn: Containment 2} M_\infty + {\ensuremath{\mathcal {W}}}+ ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d) \supseteq {\ensuremath{\mathfrak{L}}}+ (-(\max_{{\ensuremath{\mathfrak{l}}}\in {\ensuremath{\mathfrak{L}}}}\lambda_{\ensuremath{\mathfrak{l}}}(u_1 + \cdots + u_d))) +(-({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)).$$ Then for any $m_0\in M_\infty$, $M\setminus\{m_0\}$ would be a complement to $W$. Indeed, given an element $x\in {\ensuremath{\mathbb{Z}}}^d$, it is equal to $m+w$ for some $m\in M$ and $w\in W$. If $m\neq m_0$, then $x$ belongs to $M\setminus\{m_0\}+W$. If $m=m_0$, then for some positive integers $\mu_1, \cdots, \mu_d$, $M_\infty$ would contain $m_0-(\mu_1 u_1 + \cdots + \mu_d u_d)$. Since $w$ belongs to $W = {\ensuremath{\mathcal {W}}}+ ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$, it follows that $w + (\mu_1 u_1 + \cdots + \mu_d u_d)$ belongs to $W$. Hence $x$ belongs to $M\setminus\{m_0\}+W$. So $M\setminus\{m_0\}+W$ is equal to ${\ensuremath{\mathbb{Z}}}^d$, which is absurd. Hence ${\ensuremath{\mathscr{W}}}$ is nonempty. Now, using Theorem \[Thm: Structure of Eventually Periodic\] (7) again, we obtain from Equation that the map $$\pi: ({\ensuremath{\mathcal{M}}}+ ({\ensuremath{\mathcal {W}}}\cup {\ensuremath{\mathscr{W}}})) \to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$$ is surjective. By the definition ${\ensuremath{\mathscr{W}}}_0$, it follows that the map $\pi: ({\ensuremath{\mathcal{M}}}+ ({\ensuremath{\mathcal {W}}}\cup {\ensuremath{\mathscr{W}}}_1)) \to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is surjective. This proves the first statement. Now we prove that the second statement is true. On the contrary, suppose the second statement is false. So there exists an element $m\in {\ensuremath{\mathcal{M}}}$ such that for each $w\in {\ensuremath{\mathscr{W}}}_1$, there exist elements $m'\in {\ensuremath{\mathcal{M}}}, w'\in {\ensuremath{\mathcal {W}}}$ such that $m+w \equiv m'+w'{\ensuremath{\mathrm{\;mod\;}}}{\ensuremath{\mathcal {L}}}$. We prove that $M\setminus \{m\}$ is a complement to $W$, which would contradict the minimality of $M$, and thereby establish the second statement. Let $x$ be an element of ${\ensuremath{\mathbb{Z}}}^d$. Since $M$ is a complement of $W$, it follows that $x = m_0 + w_0$ for some $m_0\in M, w_0\in W$. If $m_0\neq m$, then $x$ belongs to $M\setminus\{m\} + W$. Suppose $m_0$ is equal to $m$, i.e., $x = m+w_0$. If $w_0$ belongs to ${\ensuremath{\mathscr{W}}}_1$, then there exist elements $m'\in {\ensuremath{\mathcal{M}}}, w'\in {\ensuremath{\mathcal {W}}}$ such that $m+w_0\equiv m'+w'{\ensuremath{\mathrm{\;mod\;}}}{\ensuremath{\mathcal {L}}}$. This gives $x = m'+w'+(n_1u_1 + \cdots + n_du_d)$ for some integers $n_1, \cdots, n_d$. Since $w'$ belongs to ${\ensuremath{\mathcal {W}}}$, for any integers $r_1, \cdots, r_d$ with $r_i\geq - n_i$, $W$ contains $w' + (n_1u_1 + \cdots + n_du_d) + (r_1u_1 + \cdots + r_du_d)$. Since $m'\in {\ensuremath{\mathcal{M}}}\subseteq M_\infty$, it follows that $m'-(\lambda_1 u_1 + \cdots + \lambda_d u_d)\in M_\infty \setminus\{m\}\subseteq M\setminus\{m\}$ for some integers $\lambda_1, \cdots, \lambda_d$ satisfying $\lambda_i\geq \max \{0, -n_i\}$ for $1\leq i\leq d$. Hence $x = (m'-(\lambda_1 u_1 + \cdots + \lambda_d u_d)) + (w'+ (n_1u_1 + \cdots + n_du_d) +( \lambda_1 u_1 + \cdots + \lambda_d u_d) )$ belongs to $M \setminus \{m\} + W$. On the other hand, if $w_0$ does not belong to ${\ensuremath{\mathscr{W}}}_1$, then $w_0$ belongs to ${\ensuremath{\mathscr{W}}}_0 \sqcup ({\ensuremath{\mathcal {W}}}+ ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d))$. Since the image of ${\ensuremath{\mathscr{W}}}_0$ under $\pi: {\ensuremath{\mathscr{W}}}_0\to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is contained in ${\ensuremath{\mathcal{Q}}}$ and the image of ${\ensuremath{\mathcal {W}}}$ under $\pi: {\ensuremath{\mathcal {W}}}\to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is equal to ${\ensuremath{\mathcal{Q}}}$, there exist positive integers $\mu_1, \cdots, \mu_d$ such that $w_0 + \mu_1 u_1 + \cdots + \mu_d u_d$ belongs to ${\ensuremath{\mathcal {W}}}+ ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$. Moreover, since $m\in M\subseteq M_\infty$, there exist integers $\lambda_1, \cdots, \lambda_d$ with $\lambda_i\geq \mu_i$ for any $1\leq i\leq d$ such that $m - (\lambda_1 u_1 + \cdots + \lambda_d u_d)$ belongs to $M_\infty \setminus \{m\} \subseteq M\setminus\{m\}$. Also note that $w_0 + \lambda_1 u_1 + \cdots + \lambda_d u_d$ belongs to $W$ because $w_0 + \mu_1 u_1 + \cdots + \mu_d u_d$ belongs to ${\ensuremath{\mathcal {W}}}+ ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)\subseteq W$ and $\lambda_i\geq \mu_i$ for any $1\leq i\leq r$. So $x = m+w_0$ belongs to $M\setminus\{m\} + W$. Consequently, $x$ belongs to $M\setminus\{m\} + W$ whether $w_0$ belongs to ${\ensuremath{\mathscr{W}}}_1$ or not. Hence $M\setminus\{m\}$ is a complement to $W$, which is a contradiction to the given condition that $M$ is a minimal complement to $W$. So the second statement is true. Note that Theorem \[Thm: existence of min complement implies\] is in accordance with Corollary \[Cor: Nd minimal complement\](2), (3) and Proposition \[Prop: AP Complement iff\](3), (4). \[Cor: No periodic has minimal\] By Theorem \[Thm: existence of min complement implies\](2), periodic subsets of ${\ensuremath{\mathbb{Z}}}^d$ do not have minimal complements in ${\ensuremath{\mathbb{Z}}}^d$. Sufficient condition -------------------- \[Thm: Implies existence of min comple\] Let $W$ be an eventually periodic subset of ${\ensuremath{\mathbb{Z}}}^d$ with periods $u_1, \cdots, u_d$. Let ${\ensuremath{\mathscr{W}}}, {\ensuremath{\mathcal {W}}}$ be as in Theorem \[Thm: Structure of Eventually Periodic\], and ${\ensuremath{\mathscr{W}}}_0, {\ensuremath{\mathscr{W}}}_1$ be as in Theorem \[Thm: existence of min complement implies\]. Suppose there exists a nonempty finite subset ${\ensuremath{\mathcal{M}}}$ of ${\ensuremath{\mathbb{Z}}}^d$ such that the following conditions hold. 1. The set ${\ensuremath{\mathscr{W}}}_1$ is nonempty. 2. The map $\pi: {\ensuremath{\mathcal{M}}}\to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is injective. 3. The map $\pi: ({\ensuremath{\mathcal{M}}}+ ({\ensuremath{\mathcal {W}}}\cup {\ensuremath{\mathscr{W}}}_1)) \to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is surjective. 4. For any $m\in {\ensuremath{\mathcal{M}}}$, there exists $w\in {\ensuremath{\mathscr{W}}}_1$ such that $m + w \not\equiv m' + w' {\ensuremath{\mathrm{\;mod\;}}}{\ensuremath{\mathcal {L}}}$ for any $m'\in {\ensuremath{\mathcal{M}}}\setminus\{m\}, w'\in {\ensuremath{\mathcal {W}}}\cup {\ensuremath{\mathscr{W}}}_1$. Then $W$ has a minimal complement in ${\ensuremath{\mathbb{Z}}}^d$. Let $C$ denote the collection of all elements in ${\ensuremath{\mathbb{Z}}}^d$ which are congruent to some element of ${\ensuremath{\mathcal{M}}}$ modulo ${\ensuremath{\mathcal {L}}}$, and $C'$ denote the collection of all elements in ${\ensuremath{\mathbb{Z}}}^d$ which are congruent to no element of ${\ensuremath{\mathcal{M}}}+ {\ensuremath{\mathcal {W}}}$ modulo ${\ensuremath{\mathcal {L}}}$. Note that $C+{\ensuremath{\mathscr{W}}}_1$ contains all elements of ${\ensuremath{\mathbb{Z}}}^d$ which are congruent to some element of ${\ensuremath{\mathcal{M}}}+{\ensuremath{\mathscr{W}}}_1$. By the second condition, it follows that each element of $C'$ is congruent to some element of ${\ensuremath{\mathcal{M}}}+{\ensuremath{\mathscr{W}}}_1$. So $C+{\ensuremath{\mathscr{W}}}_1$ contains $C'$. We claim that $C$ has a subset $M$ such that $M$ is minimal among the subsets of $C$ with respect to the property that $M+{\ensuremath{\mathscr{W}}}_1$ contains $C'$. Note that $C$ is countably infinite. Hence its elements can be enumerated by the positive integers. Write $C=\{c_i\,\|\, i\geq 1\}$. Define $C_1$ to be equal to $C$ and for any integer $i\geq 1$, define $$C_{i+1}= \begin{cases} C_i\setminus\{c_i\} & \text{ if $C_i\setminus\{c_i\}+{\ensuremath{\mathscr{W}}}_1$ contains }C',\\ C_i & \text{ otherwise}. \end{cases}$$ Let $M$ denote the intersection $\cap_{i\geq 1} C_i$. Note that for each $x\in C'$ and any integer $i\geq 1$, there exist elements $c_{x,i}\in C_i$ and $w_{x,i}\in {\ensuremath{\mathscr{W}}}_1$ such that $x = c_{x,i} + w_{x,i}$ holds. Since ${\ensuremath{\mathscr{W}}}_1$ is a nonempty finite set, by the Pigeonhole principle, for some element $t\in {\ensuremath{\mathscr{W}}}_1$, the equality $t=w_{x,i}$ holds for infinitely many positive integers $i$. Hence for infinitely many $i\geq 1$, we obtain $x-t=c_{x,i}$, which is an element of $C_i$. Consequently, for each positive integer $k$, there exists an integer $i_k > k$ such that $x-t$ is an element of $C_{i_k}$, which is contained in $C_k$ (as $i_k > k$) and thus $C_k$ contains $x-t$. We conclude that for each $x\in C'$, there exists an element $t\in {\ensuremath{\mathscr{W}}}_1$ such that $x-t$ belongs to $C_k$ for any $k\geq 1$, i.e., $x-t\in M$. In other words, $M+{\ensuremath{\mathscr{W}}}_1$ contains $C'$. Moreover, $M$ is minimal with respect to the property that $M+{\ensuremath{\mathscr{W}}}_1$ contains $C'$. On the contrary, assume that for some integer $j\geq 1$, $M$ contains $c_j$ and $M\setminus\{c_j\}+{\ensuremath{\mathscr{W}}}_1$ contains $C'$. Since $M$ is contained in $C_j$, it follows that $C_j\setminus\{c_j\}+{\ensuremath{\mathscr{W}}}_1$ contains $C'$. So $C_{j+1}$ does not contain $c_j$ and hence $c_j$ does not belong to $M= \cap _{i\geq 1} C_i$, which is absurd. This proves the claim. We claim that $M$ is a minimal additive complement to $S$ in ${\ensuremath{\mathbb{Z}}}^d$. For $m\in {\ensuremath{\mathcal{M}}}$, define $M_m$ to be the set of all elements of $M$ congruent to $m$ modulo ${\ensuremath{\mathcal {L}}}$, i.e., $M_m$ is the fibre of the map $M\to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ at the point $\bar m$. Note that for any $m\in {\ensuremath{\mathcal{M}}}$, the set $M_m$ contains a sequence of points such that the maximum of their coordinates with respect to $u_1, \cdots, u_d$ is arbitrary small negative number. Indeed, if we fix $m\in {\ensuremath{\mathcal{M}}}$, then note that by condition (4), there exists $w\in {\ensuremath{\mathscr{W}}}_1$ such that $m+w \not \equiv m' + w'{\ensuremath{\mathrm{\;mod\;}}}{\ensuremath{\mathcal {L}}}$ for any $m'\in {\ensuremath{\mathcal{M}}}\setminus\{m\}$ and $w'\in {\ensuremath{\mathcal {W}}}\cup {\ensuremath{\mathscr{W}}}_1$. Then it follows that $m+w$ is congruent to no element of ${\ensuremath{\mathcal{M}}}+{\ensuremath{\mathcal {W}}}$ modulo ${\ensuremath{\mathcal {L}}}$, and hence $m+w+{\ensuremath{\mathcal {L}}}$ is contained in $$C'\subseteq M+{\ensuremath{\mathscr{W}}}_1=(M_m+{\ensuremath{\mathscr{W}}}_1)\sqcup (\sqcup_{m'\in {\ensuremath{\mathcal{M}}}\setminus\{m\}} (M_{m'}+{\ensuremath{\mathscr{W}}}_1)).$$ Furthermore, condition (4) implies that $\sqcup_{m'\in {\ensuremath{\mathcal{M}}}\setminus\{m\}} (M_{m'}+{\ensuremath{\mathscr{W}}}_1)$ contains no element of $m+w+ {\ensuremath{\mathcal {L}}}$. So $m+w+{\ensuremath{\mathcal {L}}}$ is contained in $M_m+{\ensuremath{\mathscr{W}}}_1$. Thus $$m+w+(-{\ensuremath{\mathbb{N}}}(u_1 + \cdots + u_d))\subseteq M_m+{\ensuremath{\mathscr{W}}}_1=\cup_{w'\in {\ensuremath{\mathscr{W}}}_1}(w'+M_m).$$ Since ${\ensuremath{\mathscr{W}}}_1$ is finite, it follows that for some $w'\in {\ensuremath{\mathscr{W}}}_1$, $w'+M_m$ contains a sequence of points of $m+w+(-{\ensuremath{\mathbb{N}}}(u_1 + \cdots + u_d))$ such that the maximum of their coordinates with respect to $u_1, \cdots, u_d$ is arbitrarily small number. Consequently, $M_m$ contains a sequence of points such that the maximum of their coordinates with respect to $u_1, \cdots, u_d$ is arbitrary small. Let $x$ be an element of ${\ensuremath{\mathbb{Z}}}^d$ which is congruent to some element of ${\ensuremath{\mathcal{M}}}+{\ensuremath{\mathcal {W}}}$ modulo ${\ensuremath{\mathcal {L}}}$, i.e., $x$ is equal to $m+w+\ell$ for some $m\in {\ensuremath{\mathcal{M}}}, w\in {\ensuremath{\mathcal {W}}}, \ell\in {\ensuremath{\mathcal {L}}}$. Since the set $M_m$ contains a sequence of points such that the maximum of their coordinates with respect to $u_1, \cdots, u_d$ is arbitrary small, it follows that for some large enough positive integers $\lambda_1, \cdots, \lambda_d$, the set $M_m$ contains $m-(\lambda_1 u_1 + \cdots + \lambda_d u_d)$ and $w+\ell+(\lambda_1 u_1 + \cdots + \lambda_d u_d) = w + (\ell + \lambda_1 u_1 + \cdots + \lambda_d u_d)$ belongs to $({\ensuremath{\mathbb{N}}}u_1 + \cdots+{\ensuremath{\mathbb{N}}}u_d)+{\ensuremath{\mathcal {W}}}$. Hence $$x = m + w+\ell = (m-(\lambda_1 u_1 + \cdots + \lambda_d u_d)) + (w+\ell+(\lambda_1 u_1 + \cdots + \lambda_d u_d))$$ belongs to $$M_m+ (({\ensuremath{\mathbb{N}}}u_1 + \cdots+{\ensuremath{\mathbb{N}}}u_d)+{\ensuremath{\mathcal {W}}})\subseteq M+W.$$ So $M+W$ contains all elements of ${\ensuremath{\mathbb{Z}}}^d$ which are congruent to some elements of ${\ensuremath{\mathcal{M}}}+ {\ensuremath{\mathcal {W}}}$ modulo ${\ensuremath{\mathcal {L}}}$. Moreover, $M+W$ contains $C'$, i.e., it contains all elements of ${\ensuremath{\mathbb{Z}}}^d$ which are congruent to no element of ${\ensuremath{\mathcal{M}}}+ {\ensuremath{\mathcal {W}}}$ modulo ${\ensuremath{\mathcal {L}}}$. Hence $M+W$ is equal to ${\ensuremath{\mathbb{Z}}}^d$. Let $M'$ be a subset of $M$ such that $M' + W = {\ensuremath{\mathbb{Z}}}^d$. Then each element of $M'+(W\setminus {\ensuremath{\mathscr{W}}}_1)$ is congruent to some element of ${\ensuremath{\mathcal{M}}}+ {\ensuremath{\mathcal {W}}}$ modulo ${\ensuremath{\mathcal {L}}}$. So $M'+(W\setminus {\ensuremath{\mathscr{W}}}_1)$ and $C'$ has no common element. Thus $M'+{\ensuremath{\mathscr{W}}}_1$ contains $C'$. Since $M$ is minimal among the subsets of $C$ with respect to the property that $M+{\ensuremath{\mathscr{W}}}_1$ contains $C'$, we conclude that $M'$ is equal to $M$. Hence $M$ is a minimal additive complement to $W$ in ${\ensuremath{\mathbb{Z}}}^d$. Note that the set $M$ constructed in the proof of Theorem \[Thm: Implies existence of min comple\] satisfies $M= M_\infty$, i.e., $M_{\ensuremath{\mathrm{fin}}}= \emptyset$. Equivalent condition -------------------- \[Thm: Even peri 1 has min comp iff\] Let $W$ be an eventually periodic subset of ${\ensuremath{\mathbb{Z}}}^d$ with periods $u_1, \cdots, u_d$. Let ${\ensuremath{\mathscr{W}}}_1$ be as in Theorem \[Thm: existence of min complement implies\]. Suppose ${\ensuremath{\mathscr{W}}}_1$ contains only one element. Then $W$ has a minimal complement in ${\ensuremath{\mathbb{Z}}}^d$ if and only if there exists nonempty finite subset ${\ensuremath{\mathcal{M}}}$ of ${\ensuremath{\mathbb{Z}}}^d$ satisfying the following. 1. The map $\pi: {\ensuremath{\mathcal{M}}}\to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is injective. 2. The map $\pi: ({\ensuremath{\mathcal{M}}}+ {\ensuremath{\mathcal {W}}}\cup {\ensuremath{\mathscr{W}}}_1)) \to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is surjective. 3. For any $m\in {\ensuremath{\mathcal{M}}}$, there exists $w\in {\ensuremath{\mathscr{W}}}_1$ such that $m + w \not\equiv m' + w' {\ensuremath{\mathrm{\;mod\;}}}{\ensuremath{\mathcal {L}}}$ for any $m'\in {\ensuremath{\mathcal{M}}}\setminus\{m\}, w'\in {\ensuremath{\mathcal {W}}}\cup {\ensuremath{\mathscr{W}}}_1$. If ${\ensuremath{\mathbb{Z}}}^d$ has a subset ${\ensuremath{\mathcal{M}}}$ such that the conditions (1), (2), (3) hold, then by Theorem \[Thm: Implies existence of min comple\], $W$ has a minimal complement in ${\ensuremath{\mathbb{Z}}}^d$. Suppose $W$ has a minimal complement $M$ in ${\ensuremath{\mathbb{Z}}}^d$. Let ${\ensuremath{\mathcal{M}}}$ be as in Theorem \[Thm: existence of min complement implies\]. Then ${\ensuremath{\mathcal{M}}}$ is a nonempty finite set. Since the composite map ${\ensuremath{\mathcal{M}}}{\hookrightarrow}M_\infty \twoheadrightarrow \pi(M_\infty)$ is bijective, it follows that the map $\pi: {\ensuremath{\mathcal{M}}}\to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is injective. By Theorem \[Thm: existence of min complement implies\], condition (2) follows, and for any $m\in {\ensuremath{\mathcal{M}}}$, there exists $w\in {\ensuremath{\mathscr{W}}}_1$ such that $m + w \not\equiv m' + w' {\ensuremath{\mathrm{\;mod\;}}}{\ensuremath{\mathcal {L}}}$ for any $m'\in {\ensuremath{\mathcal{M}}}, w'\in {\ensuremath{\mathcal {W}}}$, which gives $m + w \not\equiv m' + w' {\ensuremath{\mathrm{\;mod\;}}}{\ensuremath{\mathcal {L}}}$ for any $m'\in {\ensuremath{\mathcal{M}}}\setminus\{m\}, w'\in {\ensuremath{\mathcal {W}}}$. Note that for $m'\in {\ensuremath{\mathcal{M}}}\setminus\{m\}, w'\in {\ensuremath{\mathscr{W}}}_1$, we obtain $m+w \not\equiv m' + w' {\ensuremath{\mathrm{\;mod\;}}}{\ensuremath{\mathcal {L}}}$ (since ${\ensuremath{\mathscr{W}}}_1$ contains only one element). This proves condition (3). Minimal complements in finitely generated abelian groups {#Sec: Minimal complement fngen} ======================================================== In this section $G$ is a finitely generated abelian group. We have already seen in Theorem \[theorem1\] that for finite sets $W\subset G$, a minimal complement of $W$ always exists. So we shall consider infinite subsets $W\subset G$ throughout this section. By the structure theorem for finitely generated abelian groups, $$G\simeq \mathbb{Z}^{d}\times(\mathbb{Z}/a_{1}\mathbb{Z})\times \cdots \times(\mathbb{Z}/a_{s}\mathbb{Z})$$ with $d\geqslant 0$ and where $a_{1}\|a_{2}\|...\|a_{s}$ are positive integers $>1$, determined uniquely by the isomorphism type of the group $G$. $\mathbb{Z}^{d}$ is said to be the torsion free part and $F :=(\mathbb{Z}/a_{1}\mathbb{Z})\times \cdots \times(\mathbb{Z}/a_{s}\mathbb{Z})$ is the torsion part. $G$ is finite if and only if $d=0$. Since we are in the realm of infinite subsets $W\subset G$, so $G$ is infinite. Hence, $d>0$ in this section. We first show the following proposition which describes the behaviour of minimal complements of product sets. \[MinCompProd\] Let $n$ be a positive integer. Let $G_1, \cdots, G_n$ be groups. For each $1\leq i\leq n$, let $A_i$ be a subset of $G_i$ with minimal complement $M_i$. Then $\prod_{1\leqslant i\leqslant n}M_{i}$ is a minimal complement of $\prod_{1\leqslant i\leqslant n}A_{i}$ in $\prod_{1\leqslant i\leqslant n}G_{i}$. Let $A_{1}$ and $A_{2}$ be two given sets in the groups $G_{1}$ and $G_{2}$ with minimal complements $M_{1}$ and $M_{2}$ respectively. Then $$\begin{aligned} A_{1}.M_{1} & = G_{1}, \,\, A_{1}.\big\lbrace M_{1}\setminus \lbrace m\rbrace \big\rbrace \subsetneq G_{1} \,\,\forall m\in M_{1} ,\\ A_{2}.M_{2} & = G_{2}, \,\, A_{2}.\big\lbrace M_{2}\setminus \lbrace m\rbrace \big\rbrace \subsetneq G_{2} \,\,\forall m\in M_{2} . \end{aligned}$$ Now $A_{1}\times A_{2}\subset G_{1}\times G_{2}$. First we show that $M_{1}\times M_{2}$ is a complement of $A_{1}\times A_{2}$ in $G_{1}\times G_{2}$. Pick any element $(g_{1},g_{2})\in G_{1}\times G_{2}$. Then $\exists a_{1}\in A_{1},a_{2}\in A_{2}, m_{1}\in M_{1},m_{2}\in M_{2}$ with $a_{1}m_{1} = g_{1}$ and $a_{2}m_{2} = g_{2}$. Thus any $(g_{1},g_{2})\in G_{1}\times G_{2}$ can be represented as $(a_{1}m_{1},a_{2}m_{2})$ for some $a_{1}\in A_{1},a_{2}\in A_{2}, m_{1}\in M_{1},m_{2}\in M_{2}$, i.e., $$(A_{1}\times A_{2}). (M_{1}\times M_{2}) = G_{1}\times G_{2},$$ thus it is a complement of $A_{1}\times A_{2}$. Now, we show that $M_{1}\times M_{2}$ is minimal. Remove an element $(m,n)$ from $M_{1}\times M_{2}$ and look at the set $ M :=M_{1}\times M_{2} \setminus \lbrace (m,n)\rbrace $. We show that $M$ is not a complement of $A_{1}\times A_{2}$ in $G_{1}\times G_{2}$, i.e., $(A_{1}\times A_{2}).M\subsetneq G_{1}\times G_{2}$. Since $M_{1}$ is a minimal complement of $A_{1}$, $\exists a_{1}\in A_{1},g_{1}\in G_{1}$ such that the only way of representing $g_{1}$ in $A_{1}.M_{1}$ is $a_{1}m$. Similarly, $\exists a_{2}\in A_{2},g_{2} \in G_{2}$ with $g_{2} = a_{2}n$. We show that this $(g_{1},g_{2}) \notin (A_{1}\times A_{2}).M$. Indeed, this is clear from the fact that $(g_{1},g_{2}) $ can only be represented in $(A_{1}\times A_{2}). (M_{1}\times M_{2})$ as $(a_{1}m,a_{2}n)$. To prove the general case we use induction. Without loss of generality, let us assume that the statement is true for $k$ groups $G_{1},G_{2},\cdots, G_{k}$ with $k<n$. We show that the statement holds for $(k+1)$-groups. By hypothesis, $$\begin{aligned} A_{1} . M_{1} = G_{1}, \,\, A_{1} . M_{1}\setminus & \lbrace m_{1}\rbrace \subsetneq G_{1} \,\,\forall m_{1}\in M_{1} \\ A_{2} . M_{2} = G_{2}, \,\, A_{2} . M_{2}\setminus & \lbrace m_{2}\rbrace \subsetneq G_{2} \,\,\forall m_{2}\in M_{2} \\ \vdots \\ A_{k} . M_{k} = G_{k}, \,\, A_{k} . M_{k}\setminus & \lbrace m_{k}\rbrace \subsetneq G_{k} \,\,\forall m_{k}\in M_{k}\\ A_{k+1} . M_{k+1} = G_{k+1}, \,\, A_{k+1} . M_{k+1}\setminus & \lbrace m_{k+1}\rbrace \subsetneq G_{k+1} \,\,\forall m_{k+1}\in M_{k+1}. \\ \end{aligned}$$ By the inductive assumption $N :=M_{1}\times \cdots \times M_{k}$ is a minimal complement of $A_{1}\times \cdots \times A_{k}$ in $G_{1}\times \cdots \times G_{k}$. To show that $N\times M_{k+1}$ is a minimal complement of $A_{1}\times \cdots \times A_{k+1}$ in $G_{1}\times \cdots \times G_{k+1}$. It is clear that $N\times M_{k+1}$ is a complement. To show that it is minimal, we remove an arbitrary point $(x,y)$ from $N\times M_{k+1}$ and argue as above to get the required statement. An immediate consequence of the above proposition (combined with previous work of Chen–Yang [@ChenYang12] and Kiss–Sándor–Yang [@KissSandorYangJCT19]) is the construction of infinite sets in $\mathbb{Z}^{d}$ having minimal complements. We first state their results - \[Chen-Yang\] Let $W\subset \mathbb{Z}$. Suppose either of the following holds 1. $\inf W = -\infty, \,\,\sup W = +\infty$. 2. $\limsup_{i\rightarrow \infty} (w_{i+1}-w_{i}) = +\infty$. Then $W$ has a minimal complement in $\mathbb{Z}$. \[Kiss-Sandor-Yang\] Let $W\subset \mathbb{Z}$ be of the following form, $$W = (m\mathbb{N} + X_{m}) \cup Y^{(0)} \cup Y^{(1)},$$ where $X_{m}\subseteq \lbrace 0, 1,\cdots, m-1\rbrace, Y^{(0)}\subseteq \mathbb{Z}^{-}, Y^{(1)} $ are finite sets with $Y^{(0)} \mod m \subseteq X_{m}$ and $(Y^{(1)}\mod m) \cap X_{m} = \emptyset .$ There exists a minimal complement to $W$ if and only if there exists $T\in \mathbb{Z}^{+}, m \|T$, and $C\subseteq \lbrace 0, 1, \cdots, T-1\rbrace $ such that 1. $C+(X_{T}\cup Y^{(1)} ) \mod T = \lbrace 0, 1,\cdots, T-1\rbrace$ where $X_{T}=\cup_{i=0}^{\frac{T}{m}-1}\lbrace im +X_m\rbrace ;$ 2. for any $c\in C$, there exists $y\in Y^{(1)} $ for which $c +y \not\equiv c'+x \,(mod\, T),$ where $c'\in C\setminus{c} $ and $x \in X_{T}$. Using Theorem \[Chen-Yang\], Theorem \[Kiss-Sandor-Yang\] and Proposition \[MinCompProd\] we have the following immediate result. \[MinProdThm\] Fix $d$ a positive integer. Let $W_{i}\subset \mathbb{Z}$ be of the form described in Theorem \[Chen-Yang\] or Theorem \[Kiss-Sandor-Yang\] or finite, for each $1\leqslant i\leqslant d$. Then $W_{1}\times W_{2}\times \cdots \times W_{d}$ has a minimal complement in $\mathbb{Z}^{d}$. Each $W_{i}\subset \mathbb{Z}$ has a minimal complement $M_{i}$ in $\mathbb{Z}$ (using Theorem \[Chen-Yang\] or Theorem \[Kiss-Sandor-Yang\] or Theorem \[theorem1\]). Using Proposition \[MinCompProd\], we see that $M_{1}\times M_{2}\times \cdots \times M_{d}$ is a required minimal complement. However, not all infinite sets in $\mathbb{Z}^{d}$ are product sets. We have seen in the previous section a sufficient condition for the existence of minimal complements for eventually periodic sets in $\mathbb{Z}^{d}$ (which are not product sets, when they are not periodic). We shall now exploit the structure of the finitely generated abelian group $G$. \[lemmasec5\] Let $G\simeq \mathbb{Z}^{d}\times F$ be any finitely generated abelian group ($F$ is the finite torsion part). Let $\emptyset\neq A\subset \mathbb{Z}^{d}$. Suppose minimal complement of $A$ in $\mathbb{Z}^{d}$ exist. Then a minimal complement of $A\times H $ exists $\forall \,\emptyset\neq H\subseteq F$. We have $G\simeq \mathbb{Z}^{d}\times F$, with $F$ a finite group. Suppose $\emptyset\neq A\subset \mathbb{Z}^{d}, \emptyset\neq H\subset F$. Let $B$ be a minimal complement of $A$ in $\mathbb{Z}^{d}$. Since $F$ is finite, $H$ is also finite. By Theorem \[theorem1\], we know that a minimal complement of $H$ exists. Let it be $H'$. Now using the previous proposition, we have that $B\times H'$ is a minimal complement of $A\times H$ in $G$. We come to the main theorem which describes a large class of infinite sets having minimal complements in finitely generated abelian groups. \[sec5Thm\] Let $G\simeq \mathbb{Z}^{d}\times F$ be any finitely generated abelian group. Suppose $W\subseteq \mathbb{Z}^{d}$ be either of the form given in Theorem \[Thm: Implies existence of min comple\] or a product set $W_{1}\times W_{2}\times \cdots \times W_{d}$ as described in Theorem \[MinProdThm\]. Then $W\times H$ will have a minimal complement in $G$ where $H \subseteq F$ is any arbitrary nonempty subset. The form of $W\subseteq \mathbb{Z}^{d}$ ensures that $W$ has a minimal complement in $\mathbb{Z}^{d}$. After this we apply Lemma \[lemmasec5\] to get the desired conclusion. \[Remark:MinCompEg\] The upshot of the discussion in Section \[Sec: Minimal complement fngen\] is to provide examples of subsets of finitely generated abelian groups admitting minimal complements. Some such examples were already given in Section \[Sec: Minimal complement\] (see Theorem \[Thm: Implies existence of min comple\]), where we considered the group ${\ensuremath{\mathbb{Z}}}^d$, i.e., free abelian groups only. The immediate question is look for examples of such subsets in finitely generated abelian groups having nontrivial torsion. By Proposition \[MinCompProd\], a subset $W$ of a group $G$ admits a minimal complement if $G$ is isomorphic to the direct product of groups $G_1\times \cdots \times G_n$ and under such an isomorphism $W$ corresponds to the product of subsets $W_i$ of $G_i$ having minimal complements. Hence, as a consequence of Theorems \[Chen-Yang\], \[Kiss-Sandor-Yang\], \[Thm: Implies existence of min comple\], \[theorem1\] and Propositions \[prop4.1\], \[Prop:Hypersurf\], it follows that a subset $W$ of a finitely abelian group $G$ admits a minimal complement in $G$ if there exists finitely generated free abelian groups $G_1, \cdots, G_n$ and an isomorphism $\psi: G{\xrightarrow}{\sim} G_{\ensuremath{\mathrm{tors}}}\times G_1\times \cdots \times G_n$ (where $G_{\ensuremath{\mathrm{tors}}}$ denotes the torsion part of $G$) such that $\psi(W)$ is equal to the product $W_0 \times W_1 \times \cdots \times W_n$ where $W_0$ is a nonempty subset of $G_{\ensuremath{\mathrm{tors}}}$, $W_1, \cdots, W_n$ are subsets of $G_1, \cdots, G_n$ and for each $1\leq i\leq n$, one of the following conditions hold. 1. $G_i={\ensuremath{\mathbb{Z}}}$ and $W_i$ satisfies the conditions of Theorem \[Chen-Yang\]. 2. $G_i = {\ensuremath{\mathbb{Z}}}$ and $W_i$ satisfies the conditions of Theorem \[Kiss-Sandor-Yang\]. 3. $G_i = {\ensuremath{\mathbb{Z}}}^{d_i}$ for some integer $d_i\geq 1$ and $W_i$ satisfies the conditions of Theorem \[Thm: Implies existence of min comple\]. 4. $G_i = {\ensuremath{\mathbb{Z}}}^{d_i}$ for some integer $d_i\geq 1$ and $W_i$ is a nonempty finite subset of $G_i$. 5. $G_i$ is a free abelian group of finite rank and $W_i$ is a subgroup of $G_i$. 6. $G_i = {\ensuremath{\mathbb{Z}}}^{d_i}$ for some integer $d_i\geq 1$ and $W_i$ is equal to $\{(\pm n,\cdots, \pm n)\in {\ensuremath{\mathbb{Z}}}^{d_i}\,\|\, n\in {\ensuremath{\mathbb{Z}}}\}$ (see Proposition \[Prop:Hypersurf\]). To conclude this section, we see that a combination of Theorem \[sec5Thm\], Theorem \[MinProdThm\] and Proposition \[MinCompProd\] gives us infinite sets in arbitrary finitely generated abelian groups having minimal complements, providing a partial answer to Nathanson’s Question \[nathansonprob13\].\ Conclusion and further remarks {#Sec:Conclusion} ============================== Finally, we conclude the article with a class of examples of infinite sets which are not eventually periodic (i.e., they don’t fall in the class of Theorem \[Thm: existence of min complement implies\] and Theorem \[Thm: Implies existence of min comple\]) but have minimal complements. \[prop6.1\] \[Prop:Hypersurf\] Let $d\geq 2$ be a positive integer. Then the subset $${\ensuremath{\mathcal{D}}}:= \{(\pm n,\cdots, \pm n)\in {\ensuremath{\mathbb{Z}}}^d\,\|\, n\in {\ensuremath{\mathbb{Z}}}\}$$ of ${\ensuremath{\mathbb{Z}}}^d$ is not eventually periodic. For each $1\leq i\leq d$, the hypersurface defined by $x_i=0$, i.e., the set $${\ensuremath{\mathcal{H}}}_i:=\{(x_1, \cdots, x_d)\in {\ensuremath{\mathbb{Z}}}^d\,\|\, x_i=0\}$$ is a minimal complement of ${\ensuremath{\mathcal{D}}}$. Suppose ${\ensuremath{\mathcal{D}}}$ is eventually periodic. So there exist elements $u_1, \cdots, u_d$ in ${\ensuremath{\mathbb{Z}}}^d$ satisfying no nontrivial ${\ensuremath{\mathbb{Z}}}$-linear relation and an integer $n_0\geq 1$ such that for any integer $n$ with $\|n\|> n_0$, $${\ensuremath{\mathcal{D}}}\supseteq (\pm n, \cdots, \pm n) + ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d).$$ Let $n'$ denote the integer $\max \{n_0 , \|\|u_1\|\|, \cdots, \|\|u_d\|\|\}$. For any positive integer $n$ with $n>\sqrt d n'$ and any $1\leq i\leq d$, the vector $(n, \cdots, n)+u_i$ is contained in the open ball $B_{(n, \cdots, n), n/\sqrt d}$ in ${\ensuremath{\mathbb{R}}}^d$ with center at the point $(n,\cdots, n)$ with radius $n/\sqrt d$, and it is also contained in ${\ensuremath{\mathcal{D}}}$. Hence the vectors $u_1, \cdots, u_d$ are scalar multiples of $(1, \cdots, 1)$, which is absurd. So ${\ensuremath{\mathcal{D}}}$ is not eventually periodic. For any element $(x_1, \cdots, x_d)$ of ${\ensuremath{\mathbb{Z}}}^d$, $(x_i, \cdots, x_i)$ belongs to ${\ensuremath{\mathcal{D}}}$ and ${\ensuremath{\mathcal{H}}}_i$ contains $(x_1, \cdots, x_d) - (x_i, \cdots, x_i)$ and hence ${\ensuremath{\mathcal{H}}}_i$ is a complement of ${\ensuremath{\mathcal{D}}}$. Suppose $M_i$ is a subset of ${\ensuremath{\mathcal{H}}}_i$ such that $M_i + {\ensuremath{\mathcal{D}}}= {\ensuremath{\mathbb{Z}}}^d$. Note that the $i$-th coordinate of any element of $M_i + {\ensuremath{\mathcal{D}}}\setminus\{(0,\cdots,0)\}$ is nonzero. So it contains no element of ${\ensuremath{\mathcal{H}}}_i$. Thus ${\ensuremath{\mathcal{H}}}_i$ is contained in $(0,\cdots, 0) + M_i$. So $M_i$ is equal to ${\ensuremath{\mathcal{H}}}_i$. Hence ${\ensuremath{\mathcal{H}}}_i$ is a minimal complement to ${\ensuremath{\mathcal{D}}}$. \[prop6.2\] Let $d\geq 2$ be a positive integer. Let $f_1, \cdots, f_d$ be elements of ${\ensuremath{\mathbb{Z}}}[X_1, \cdots, X_m]$. Define the subset ${\ensuremath{\mathcal{S}}}$ of ${\ensuremath{\mathbb{Z}}}^d$ by $${\ensuremath{\mathcal{S}}}:= \{( f_1(n_1, \cdots, n_m),\cdots, f_d(n_1, \cdots, n_m))\,\|\, (n_1, \cdots, n_m)\in {\ensuremath{\mathbb{Z}}}^m\}.$$ Let $1\leq i\leq d$ be an integer such that the map $f_i:{\ensuremath{\mathbb{Z}}}^m\to {\ensuremath{\mathbb{Z}}}$ is surjective. Then the hypersurface defined by $x_i=0$, i.e., the set $${\ensuremath{\mathcal{H}}}_i:=\{(x_1, \cdots, x_d)\in {\ensuremath{\mathbb{Z}}}^d\,\|\, x_i=0\}$$ is a complement of ${\ensuremath{\mathcal{S}}}$. Moreover, if ${\ensuremath{\mathcal{S}}}\cap {\ensuremath{\mathcal{H}}}_i$ contains only one element, then ${\ensuremath{\mathcal{H}}}_i$ is a minimal complement of ${\ensuremath{\mathcal{S}}}$. For each element $(x_1, \cdots, x_d)$ of ${\ensuremath{\mathbb{Z}}}^d$, there exists an element $(n_1, \cdots, n_m)\in {\ensuremath{\mathbb{Z}}}^m$ such that $f_i(n_1, \cdots, n_m) = x_i$. So ${\ensuremath{\mathcal{S}}}$ contains the point $(f_1(n_1, \cdots, n_m), \cdots,$ $ f_d(n_1, \cdots, n_m))$ and ${\ensuremath{\mathcal{H}}}_i$ contains the difference $(x_1, \cdots, x_d)-(f_1(n_1, \cdots, n_m), \cdots,$ $ f_d(n_1, \cdots, n_m))$. Hence ${\ensuremath{\mathcal{H}}}_i$ is a complement to ${\ensuremath{\mathcal{S}}}$. Suppose ${\ensuremath{\mathcal{S}}}\cap {\ensuremath{\mathcal{H}}}_i$ contains only one element $P$ of ${\ensuremath{\mathbb{Z}}}^d$. Let $M_i$ be a subset of ${\ensuremath{\mathcal{H}}}_i$ such that $M_i + {\ensuremath{\mathcal{S}}}= {\ensuremath{\mathbb{Z}}}^d$. Note that the $i$-th coordinate of any point of $M_i + {\ensuremath{\mathcal{S}}}\setminus\{P\}$ is nonzero. So ${\ensuremath{\mathcal{H}}}_i$ is contained in $P + M_i$. Hence ${\ensuremath{\mathcal{H}}}_i - P = {\ensuremath{\mathcal{H}}}_i$ is contained in $M_i$. Hence ${\ensuremath{\mathcal{H}}}_i$ is a minimal complement of ${\ensuremath{\mathcal{S}}}$. In fact, the ambient group (and even the set, the existence/inexistence of the minimal complement of which we seek to investigate) can be taken to be uncountable. The following propositions shed light on this fact - \[prop6.3\] \[Prop:Hypersurf R\] Let $d\geq 2$ be a positive integer. For each $1\leq i\leq d$, the hypersurface defined by $x_i=0$, i.e., the set $${\ensuremath{\mathcal{H}}}_i:=\{(x_1, \cdots, x_d)\in {\ensuremath{\mathbb{R}}}^d\,\|\, x_i=0\}$$ is a minimal complement of $${\ensuremath{\mathcal{D}}}:= \{(\pm x,\cdots, \pm x)\,\|\, x\in {\ensuremath{\mathbb{R}}}\}$$ in ${\ensuremath{\mathbb{R}}}^d$. The proof is similar to the proof of Proposition \[Prop:Hypersurf\]. \[prop6.4\] \[Prop:Hypersurfaces R\] Let $d\geq 2$ be a positive integer. Let $f_1, \cdots, f_d$ be elements of ${\ensuremath{\mathbb{R}}}[X_1, \cdots, X_m]$. Define the subset ${\ensuremath{\mathcal{S}}}$ of ${\ensuremath{\mathbb{R}}}^d$ by $${\ensuremath{\mathcal{S}}}:= \{( f_1(n_1, \cdots, n_m),\cdots, f_d(n_1, \cdots, n_m))\,\|\, (n_1, \cdots, n_m)\in {\ensuremath{\mathbb{R}}}^m\}.$$ Let $1\leq i\leq d$ be an integer such that the map $f_i:{\ensuremath{\mathbb{R}}}^m\to {\ensuremath{\mathbb{R}}}$ is surjective. Then the hypersurface defined by $x_i=0$, i.e., the set $${\ensuremath{\mathcal{H}}}_i:=\{(x_1, \cdots, x_d)\in {\ensuremath{\mathbb{R}}}^d\,\|\, x_i=0\}$$ is a complement of ${\ensuremath{\mathcal{S}}}$. Moreover, if ${\ensuremath{\mathcal{S}}}\cap {\ensuremath{\mathcal{H}}}_i$ contains only one element, then ${\ensuremath{\mathcal{H}}}_i$ is a minimal complement of ${\ensuremath{\mathcal{S}}}$. The proof is similar to the proof of Proposition \[prop6.2\]. Acknowledgements ================ The first author would like to thank the Fakultät für Mathematik, Universität Wien where a part of the work was carried out. The second author would like to acknowledge the Initiation Grant from the Indian Institute of Science Education and Research Bhopal, and the INSPIRE Faculty Award from the Department of Science and Technology, Government of India. [KSY19]{} Yong-Gao Chen and Quan-Hui Yang, On a problem of [N]{}athanson related to minimal additive complements, SIAM J. Discrete Math. 26 (2012), no. 4, 1532–1536. [MR ]{}[3022150]{} Sándor Z. Kiss, Csaba Sándor, and Quan-Hui Yang, On minimal additive complements of integers, J. Combin. Theory Ser. A 162 (2019), 344–353. [MR ]{}[3875615]{} Melvyn B. Nathanson, Problems in additive number theory, [IV]{}: [N]{}ets in groups and shortest length [$g$]{}-adic representations, Int. J. Number Theory 7 (2011), no. 8, 1999–2017. [MR ]{}[2873139]{} [^1]: [^2]: [^3]: Equivalently, ${\ensuremath{\mathscr{W}}}_0$ is the inverse image of ${\ensuremath{\mathcal{Q}}}$ under the map $\pi: {\ensuremath{\mathscr{W}}}\to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$, and ${\ensuremath{\mathscr{W}}}_1$ is its complement in ${\ensuremath{\mathscr{W}}}$.	ArXiv
--- abstract: 'We present a systematic theory of Coulomb-induced correlation effects in the nonlinear optical processes within the strong-coupling regime. In this paper we shall set a dynamics controlled truncation scheme [@Axt; @Stahl] microscopic treatment of nonlinear parametric processes in SMCs including the electromagnetic field quantization. It represents the starting point for the microscopic approach to quantum optics experiments in the strong coupling regime without any assumption on the quantum statistics of electronic excitations (excitons) involved. We exploit a previous technique, used in the semiclassical context, which, once applied to four-wave mixing in quantum wells, allowed to understand a wide range of observed phenomena [@Sham; @PRL95]. We end up with dynamical equations for exciton and photon operators which extend the usual semiclassical description of Coulomb interaction effects, in terms of a mean-field term plus a genuine non-instantaneous four-particle correlation, to quantum optical effects.' author: - 'S. Portolan$^{1,3,}$[^1], O. Di Stefano$^2$, S. Savasta$^2$, F. Rossi$^3$, and R. Girlanda$^2$' title: 'Dynamics-Controlled Truncation Scheme for Quantum Optics and Nonlinear Dynamics in Semiconductor Microcavities' --- Introduction ============ Since the early Seventies [@Esaki; @Tsu] researchers have been exploring the possible realization of semiconductor-based heterostructures, devised according to the principles of quantum mechanics. The development of sophisticated growth techniques started a revolution in semiconductor physics, determined by the possibility of confining electrons in practical structures. In addition, the increasing ability in controlling fabrication processes has enabled the manipulation of the interaction between light and semiconductors by engineering, in addition to the electronic wave functions, the light modes. Entanglement is one of the key features of quantum information and communication technology [@Nielsen-Chuang] and a hot topic in quantum optics too. Parametric down-conversion is the most frequently used method to generate highly entangled pairs of photons for quantum-optics applications, such as quantum cryptography and quantum teleportation. Rapid development in the field of quantum information requires monolithic, compact sources of nonclassical photon states enabling efficient coupling into optical fibres and possibly electrical injection. Semiconductor-based sources of entangled photons would therefore be advantageous for practical quantum technologies. The strong light-matter interaction in these systems gives rise to cavity polaritons which are hybrid quasiparticles consisting of a superposition of cavity photons and quantum well (QW) excitons [@Weisbuch-Houdre]. Demonstrations of parametric amplification and parametric emission in semiconductor microcavities (SMCs) with embedded QWs[@Baumberg; @Erland; @Langbein; @PRB2004], together with the possibility of ultrafast optical manipulation and ease of integration of these microdevices, have increased the interest on the possible realization of nonclassical cavity-polariton states [@squeezing; @Quattropani; @CiutiBE; @Savasta; @PRL2005; @LosannaCC; @SSC; @Savasta]. In 2004 squeezed light generation in SMCs in the strong coupling regime has been demonstrated [@Giacobino]. In 2005 an experiment probing quantum correlations of (parametrically emitted) cavity polaritons by exploiting quantum complementarity has been proposed and realized [@Savasta; @PRL2005]. Specifically, it has been shown that polaritons in two distinct idler modes interfere if and only if they share the same signal mode so that which-way information cannot be gathered, according to Bohr’s quantum complementarity principle. Laser spectroscopy in semiconductors and in semiconductor quantum structures has been greatly used because exciting with ultrashort optical pulses in general results in the creation of coherent superpositions of many-particle states. Thus it constitutes a very promising powerful tool for the study of correlation and an ideal arena for semiconductor cavity quantum electrodynamics (cavity QED) experiments as well as coherent control, manipulation, creation and measurement of non-classical states [@AxtKuhn; @Sham; @PRL95; @Nature; @CuCl; @Savasta; @PRL2005]. The analysis of nonclassical correlations in semiconductors constitutes a challenging problem, where the physics of interacting electrons must be added to quantum optics and should include properly the effects of noise and dephasing induced by the electron-phonon interaction and the other environment channels [@Kuhn-Rossi; @PRB; @2005]. The nonlinear optical properties of exciton-cavity system play a key role in driving the quantum correlations and the nonclassical optical phenomena. The crucial role of many-particle Coulomb correlations in semiconductors marks a profound difference from the nonlinear optics of dilute atomic systems, where the optical response is well described by independent transitions between atomic levels, and the nonlinear dynamics is governed only by saturation effects mainly due to the balance of populations between different levels. The Dynamics Controlled Truncation Scheme (DCTS) provides a (widely adopted) starting point for the microscopic theory of the light-matter interaction effects beyond mean-field [@AxtKuhn], supplying a consistent and precise way to stop the infinite hierarchy of higher-order correlations which always appears in the microscopic approaches of many-body interacting systems without need to resort to any assumption on the quantum statistics of the quasi-particle arising in due course. By exploting this scheme, it was possible to express nonlinearities originating from the Coulomb interaction as an instantaneous mean-field exciton-exciton interaction plus a noninstantaneous term where four-particle correlation effects beyond menfield are contained entirely in a retarded memory function [@Sham; @PRL95]. In 1996 the DCTS was extended in order to include in the description the quantization of the electromagnetic field and polariton effects [@Savasta; @PRL96]. This extension has been applied to the study of quantum optical phenomena in semiconductors and it was exploited to predict polariton entanglement [@SSC; @Savasta]. The obtained equations showed that quantum optical correlations (as nonlinear optical effects) arise from both saturation effects (phase-space filling) and Coulomb induced correlations due to four-particle states (including both bound and unbound biexciton states). The dynamical equations included explicitly biexciton states. The structure of those equations didn’t allow the useful separation of Coulomb interaction in terms of a mean-field interaction term plus a noninstantaneous correlation term performed in the semiclassical description. In this paper we shall set a DCTS microscopic treatment of nonlinear parametric processes in SMCs including the light-field quantization. It represents the starting point for the microscopic approach to quantum optics experiments in the strong coupling regime. For this purpose we shall exploit a previous technique [@Sham; @PRL95] which, once applied to four-wave mixing in QWs, it allowed to understand a wide range of observed phenomena. Here all the ingredients contributing to the dynamics are introduced and commented. We shall give in great details the manipulations required in order to provide an effective description of the nonlinear parametric contributions beyond mean-field in an exciton-exciton correlation fashion. In particular we derive the coupled equations of motion for the excitonic polarization and the intracavity field. It shows a close analogy to the corresponding equation describing the semiclassical (quantized electron system, classical light field) coherent $\chi^{(3)}$ response in a QW [@Sham; @PRL95], the main difference being that here the (intracavity) light field is regarded not as a driving external source but as a dynamical field [@Savasta; @PRL2003]. This correspondence is a consequence of the linearization of quantum fluctuations in the nonlinear source term here adopted, namely the standard linearization procedure of quantum correlations adopted for large systems [@Walls]. However the present approach includes the light field quantization and can thus be applied to the description of quantum optical phenomena. Indeed, striking differences between the semiclassical and the full quantum descriptions emerge when considering expectation values of exciton and photon numbers or even higher order correlators, key quantities for the investigation of coherence properties of quantum light [@Savasta; @PRL2005]. This is the main motivation for the derivation of fully operatorial dynamical equations, within such lowest order nonlinear coherent response, we address in the last section. The results here presented provide a microscopic theoretical starting point for the description of quantum optical effects in interacting electron systems with the great accuracy accomplished for the description of the nonlinear optical response in such many-body systems, see e.g. [@Sham; @PRL95; @Savasta; @PRL2003; @Savasta; @PRB2001; @Buck; @AxtKuhn] and references therein. The proper inclusion of the detrimental environmental interaction, an important and compelling issue, is left for a detailed analysis in another paper of ours [@nostro; @PRB]. In Section \[1\] the generality of the coupled system taken into account are exposed, here all the ingredients contributing to the dynamics are introduced and commented. The linear and the lowest nonlinear dynamics is the subject of Sec. \[2\], whereas in Sec. \[3\] we shall give in great details the manipulations required in order to provide an effective description of the nonlinear parametric contributions beyond mean-field in an exciton-exciton correlation fashion. In Sec. \[4\] the operatorial equations of motion for exciton and intracavity photon operators are derived. The Coupled System {#1} ================== The system we have in mind is a semiconductor QW grown inside a semiconductor planar Fabry-Perot resonator. In the following we consider a zinc-blende-like semiconductor band structure. The valence band is made from $p$-like ($l=1$) orbital states which, after spin-orbit coupling, give rise to $j = 3/2$ and $j=1/2$ decoupled states. In materials like GaAs, the upper valence band is fourfold degenerate ($j=3/2$), whereas in GaAs-based QWs the valence subbands with $j=3/2$ are energy splitted into two-fold degenerate heavy valence subbands with $j_z=\pm 3/2$ and light lower energy subbands with $j_z= \pm 1/2$. The conduction band, arising from an $s$-like orbital state (l=0), gives rise to $j=1/2$ twofold states. In the following we will consider for the sake of simplicity only twofold states from the upper valence and lowest conduction subbands. As a consequence electrons in a conduction band as well as holes have an additional spin-like degree of freedom as electrons in free space. When necessary both heavy and light hole valence bands or subbands can be included in the present semiconductor model. Only electron-hole ([[eh]{}]{}) pairs with total projection of angular momentum $\sigma = \pm 1$ are dipole active in optical interband transitions. In GaAs QWs photons with circular polarizations $\sigma = -$($+$) excite electrons with $j_z^{\it e}=+1/2$ ($j_z^{\it e}=-1/2$) and holes with $j_z^{\it h}=-3/2$ ($j_z^{\it h}=3/2$). We label optically active [[eh]{}]{} pairs with the same polarization label of light generating them; e.g. $\sigma = +1$ indicates an [[eh]{}]{} pair with $j_z^{\it e}=-1/2$ and $j_z^{\it h}=3/2$. We start from the usual model for the electronic Hamiltonian of semiconductors [@Haugh; @AxtKuhn]. It is obtained from the many-body Hamiltonian of the interacting electron system in a lattice, keeping explicitly only those terms in the Coulomb interaction preserving the number of electrons in a given band, see Appendix \[Npair states\]. The system Hamiltonian can be rewritten as $$\label{Ham electron} \hat{H}_e = \hat{H}_0 + \hat{V}_{\text{Coul}}= \sum_{N \alpha} E_{N \alpha} {\mid{N \alpha}\rangle} {\langle{N \alpha}\mid}\, ,$$ where the eigenstates of $\hat{H}_e$, with energies $E_{N \alpha} =\hbar \omega_{N \alpha}$, have been labelled according to the number N of [eh]{} pairs. The state ${\mid{N=0}\rangle}$ is the electronic ground state, the $N=1$ subspace is the exciton subspace with the additional collective quantum number $\alpha$ denoting the exciton energy level $n$, the in-plane wave vector ${\bf k}$ and the spin index $\sigma$. When needed we will adopt the following notation: $\alpha\equiv (n,k)$ with $k\equiv ({\bf k}, \sigma)$. In QWs, light and heavy holes in valence band are split off in energy. Assuming that this splitting is much larger than kinetic energies of all the involved particles and, as well, much larger than the interaction between them, we shall consider only heavy hole states as occupied. On the contrary to the bulk case, in a QW single particle states experience confinement along the growth direction and subbands appear, anyway in the other two orthogonal directions translational invariance is preserved and the in-plane exciton wave vector remains a good quantum number. Typically, the energy difference between the lowest QW subband level and the first excited one is larger than the Coulomb interaction between particles, and we will consider excitonic states arising from electrons and heavy holes in the lowest subbands. Eigenstates of the model Hamiltonian with N=1 (called excitons) can be created from the ground state by applying the exciton creation operator: $$\label{exciton def} \bigl\| 1 n \sigma {\bf k}\bigr>=\hat{B}^\dagger_{n \sigma {\bf k}} \bigl\| N=0\bigr>\, ,$$ which can be written in terms of electrons and holes operators as $$\hat{B}^\dagger_{n \sigma {\bf k}} = \sum_{{\bf k}'} \Phi^{\bf k}_{n \sigma {\bf k}'} \hat{c}^\dagger_{\sigma, {\bf k}' + \eta_e{\bf k}/2}\hat{d}^\dagger_{\sigma, -{\bf k}' + \eta_h{\bf k}/2}\, , \label{Bdag}$$ here $\Phi^{\bf k}_{n \sigma {\bf k}'}$ is the exciton wave function, being ${\bf k}$ the total wave vector ${\bf k} = {\bf k}_e + {\bf k}_h$, and ${\bf k}' = \eta_e{\bf k}_e - \eta_h{\bf k}_h$ with $\eta_{(e,h)} = m_{(e,h)}/(m_{(e)} + m_{(h)})$ ($m_{e}$ and $m_{h}$ are the electron and hole effective masses). These exciton eigenstates can be obtained by requiring the general one [eh]{} pair states to be eigenstates of $\hat{H}_e$: $$\hat{H}_e \bigl\| 1 n \sigma {\bf k}\bigr> = \hbar \omega_{1 n \sigma {\bf k}} \bigl\| 1 n \sigma {\bf k}\bigr>\, , \label{secular}$$ and projecting this secular equation onto the set of product ([eh]{}) states $\bigl\|k_e,k_h\bigr> = \hat{c}^\dag_{k_e} \hat{d}^\dag_{k_h} {\mid0\rangle}$, (see Appendix \[Npair states\] for details): $$\label{Schrodinger} \sum_{k_e,{k}_h} ( {\langlek'_e,k'_h\mid}\hat{H}_c {\midk_e ,k_h\rangle} - \hbar \omega_{n \sigma {\bf k}} \delta_{k'_e k'_h,k_e k_h} ) \bigl< k_e,k_h \bigl\| 1 n \sigma {\bf k}\bigr> = 0\, .$$ Thus, having expressed the correlated exciton state as a superposition of uncorrelated product states, $$\bigl\| 1 n \sigma {\bf k}\bigr> = \sum_{k_e,k_h} {\Bigg (}\bigl< k_e,k_h \bigl\| 1 n \sigma{\bf k}\bigr> {\Bigg )}\bigl\|k_e,k_h\bigr> \, ,$$ the scalar products, coefficients of this expansion, represent nothing but the envelope function $\Phi^{\bf k}_{n, \sigma, {\bf k}'}$ of the excitonic aggregate being the solution of the corresponding Schrödinger equation (\[Schrodinger\]). It describes the correlated [eh]{} relative motion in k-space. In order to simplify a bit the notation, the spin convention in Eq. (\[Bdag\]) has been changed by using the same label for the exciton spin quantum number and for the spin projections of the electron and hole states forming the exciton. The next relevant subspace ($N=2$) is the biexciton one, spanning all the states with 2 [eh]{} pairs. It seems worth noting that the above description of [eh]{} complexes arises from the properties of quantum states and, once fixed the system Hamiltonian, no approximations have been introduced insofar. Indeed such a property hold for any N [eh]{} pairs aggregate and we will give a full account of it in Appendix \[Npair states\]. The eigenstates of the Hamiltonian $\hat{H}_{c}$ of the cavity modes can be written as ${\midn, \lambda\rangle}$ where $n$ stands for the total number of photons in the state and $\lambda = ({\bf k}_1, \sigma_1; ... ;{\bf k}_n, \sigma_n)$ specifies wave vector and polarization $\sigma$ of each photon. Here we shall neglect the longitudinal- transverse splitting of polaritons [@Kavokin] originating mainly from the corresponding splitting of cavity modes. It is more relevant at quite high in-plane wave vectors and often it results to be smaller than the polariton linewidths. The present description can be easily extended to include it. We shall treat the cavity field in the quasi-mode approximation, that is to say we shall quantize the field as the mirrors were perfect and subsequently we shall couple the cavity with a statistical reservoir of a continuum of external modes. This coupling is able to provide the cavity losses as well as the feeding of the coherent external impinging pump beam. The cavity mode Hamiltonian, thus, reads $$\label{Ham cavity} \hat{H}_{c} = \sum_k \hbar \omega^c_k \hat{a}_k ^{\dag} \hat{a}_k\, ,$$ with the operator $\hat{a}^\dag_k$ which creates a photon state with energy $\hbar \omega^c_k =\hbar (\omega^2_{\text{exc}} + v^2 \|{\bf k}\|^2)^{1/2}$, $v$ being the velocity of light inside the cavity and $k = (\sigma,{\bf k})$. The coupling between the electron system and the cavity modes is given in the usual rotating wave approximation [@Savasta; @PRL96; @HRS; @Savasta] $$\label{Ham inter cav-exc} \hat{H}_{I} = - \sum_{n k} V^_{n k} \hat{a}_k ^{\dag} \hat{B}_{n k} + H.c.\, ,$$ $V_{n,k }$ is the photon-exciton coupling coefficient enhanced by the presence of the cavity [@Savona; @Quattropani; @SSC] set as $V_{n,k} = \tilde V_{\sigma} \sqrt{A} \phi^_{n,\sigma}({\bf x}=0)$, the latter being the real-space exciton envelope function calculated in the origin whereas $A$ is the in-plane quantization surface, $\tilde V_{\sigma}$ is proportional to the interband dipole matrix element. Modeling the loss through the cavity mirrors within the quasi-mode picture means we are dealing with an ensemble of external modes, generally without a particular phase relation among themselves. An input light beam impinging on one of the two cavity mirrors is an external field as well and it must belong to the family of modes of the corresponding side (i.e. left or right). Being coherent, it will be the non zero expectation value of the ensemble. It can be shown [@Savasta; @PRL96; @nostro; @PRB] that for a coherent input beam, the driving of the cavity modes may be described by the model Hamiltonian [@Savasta; @PRL96; @nostro; @PRB] $$\label{H quasi modi} \hat{H}_p = i\, t_c \sum_{\bf k} ({E}_{\bf k} \hat{a}^\dag_{\bf k} - {E}^{}_{\bf k} \hat{a}_{\bf k})\, ,$$ where ${E}_{\bf k}$ (${E}^_{\bf k}$) is a $\mathbb{C}$-number describing the positive (negative) frequency part of the coherent input light field amplitude. Linear and Nonlinear Dynamics {#2} ============================= The idea is not to use a density matrix approach, but to derive directly expectation values of all the quantities at play. The dynamics is described by “transition" operators (known as generalized Hubbard operators): $$\begin{aligned} \label{Hubbard} \hat{X}_{N,\alpha;M,\beta} = {\mid{N,\alpha}\rangle} {\langle{M,\beta}\mid} \nonumber \\ \hat{Y}_{n,\lambda;m,\mu}= {\midn,\lambda\rangle} {\langlem,\mu\mid}\, .\end{aligned}$$ The fundamental point in the whole analysis is that, thanks to the form of the interaction Hamiltonian $\hat{H}_I$ and thanks to the quasiparticle conservation the free Hamiltonians possess, we can use the so-called dynamics controlled truncation scheme, stating that we are facing a rather special model where the correlation have their origin only in the action of the electromagnetic field and thus the general theorem due to Axt and Stahl [@Axt; @Stahl] holds. For our purpose we will need its generalization in order to include the quantization of the electromagnetic field [@Savasta; @PRL96], it reads: $$\begin{aligned} \label{DCTS} \langle \hat{X}_{N,\alpha;M,\beta} \hat{Y}_{n,\lambda;m,\mu} \rangle = \sum_{i=0}^{i_0} \langle \hat{X}_{N,\alpha;M,\beta} \hat{Y}_{n,\lambda;m,\mu} \rangle ^{(N+M+n+m+2i)} \nonumber \\+ \mathcal{O}(E^{(N+M+n+m+2i_0+2)})\, ,\end{aligned}$$ i.e. the expectation value of a zero to N-pair transition is at least of order N in the external electromagnetic field. There are only even powers because of the spatial inversion symmetry which is present. Once a perturbative order in the external coherent fields is chosen, Eq. (\[DCTS\]) limits the expectation values to take into account, thus providing a precise way to truncate the hierarchy of equations of motions. The exciton and photon operators can be expressed as $$\begin{aligned} \label{a,B} \hat{a}_k &=& \hat{Y}_{0;1 k} + \sum_{n \geq 1} \sqrt{n_k+1} \hat{Y}_{n_k k; (n_k+1) k} \nonumber \\ \hat{B}_{n k} &=& \hat{X}_{0;1 n k} + \sum_{N \geq 1, \alpha \beta} {\langleN \alpha\mid}\hat{B}_{n k} {\mid(N+1) \beta\rangle} \hat{X}_{N \alpha;(N+1) \beta}\, , \end{aligned}$$ where in writing the photon expansion we omitted all the states not belonging to the $k$-th mode which add up giving the identity in every Fock sector [@detail]. The equation of motion for the generic quantity of interest $\hat{X}_{N,\alpha;M,\beta} \hat{Y}_{n,\lambda;m,\mu}$ is reported in Appendix \[gen eq\]. In the Heisenberg picture we start considering the equation of motion for the photon and exciton operators, once taken the expectation values we exploit theorem (\[DCTS\]) retaining only the linear terms. With the help of the generalized Hubbard opertors all this procedure may be done by inspection. The linear dynamics for $ \left<\right. \hat{a}_k \left.\right>^{(1)} = \left<\right. \hat{Y}_{0;1 n k} \left.\right>^{(1)}$ and $ \left<\right. \hat{B}_{n k} \left.\right>^{(1)} = \left<\right. \hat{X}_{0;1 n k} \left.\right>^{(1)}$ reads: $$\begin{aligned} \label{lin order} && \frac{d}{dt}\left<\right. \hat{a}_k \left.\right>^{(1)} = -i \bar{\omega}^c_k \left<\right. \hat{a}_k \left.\right>^{(1)} +i \sum_{n} \frac{V^_{n k}}{\hbar} \left<\right. \hat{B}_{n k} \left.\right>^{(1)} + t_c \frac{E_k}{\hbar} \\ && \frac{d}{dt}\left<\right. \hat{B}_{n k} \left.\right>^{(1)} = -i \bar{\omega}_{1 n k} \left<\right. \hat{B}_{n k} \left.\right>^{(1)} + i \frac{V_{n k}}{\hbar} \left<\right. \hat{a}_{k} \left.\right>^{(1)} \, . \end{aligned}$$ In these equations $\bar{\omega}^c_k = \omega^c_k-i\gamma_k$, where $\gamma_k$ is the cavity damping, analogously $\bar{\omega}_{1 n k} = \omega_{1 n k} -i\Gamma_{\text{x}}$ and $\bar{\omega}_{2 \beta} = \omega_{2 \beta}-i\Gamma_{\text{xx}}$. The dynamics up to the third order is a little bit more complex, we shall make extensively use of (\[dt gen op\]) (in the following the suffix $^{+(n)}$ stands for “up to" $n$-th order terms in the external electromagnetic exciting field). With Eq. (\[a,B\]) the exciton and the photon expectation values can be expanded as follows: $$\label{B up to 3} \left<\right. \hat{B}_{n k} \left.\right>^{+(3)} = \left<\right. \hat{X}_{0;1 n k} \left.\right>^{+(3)} + \sum_{\alpha \beta}{\langle1 \alpha\mid} \hat{B}_{n k} {\mid2 \beta\rangle} \left<\right. \hat{X}_{1 \alpha;2 \beta} \left.\right>^{(3)}\, ,$$ $$\label{a up to 3} \left<\right. \hat{a}_{k} \left.\right>^{+(3)} = \left<\right. \hat{Y}_{0;1 k} \left.\right>^{+(3)} + \sqrt{2} \left<\right. \hat{Y}_{1 k;2 k} \left.\right>^{(3)}\, .$$ With a bit of algebra we obtain $$\label{dt a up to 3} \frac{d}{dt}\left<\right. \hat{a}_k \left.\right>^{+(3)} = -i \bar{\omega}^c_k \left<\right. \hat{a}_k \left.\right>^{+(3)} + i \sum_{n} \frac{V^_{n k}}{\hbar} \left<\right. \hat{B}_{n k} \left.\right>^{+(3)} + t_c \frac{E_k}{\hbar}\, ,$$ $$\begin{aligned} \label{dt B up to 3} &&\frac{d}{dt}\left<\right. \hat{B}_{n k} \left.\right>^{+(3)} = -i \bar{\omega}_{1 n k} \left<\right. \hat{B}_{n k} \left.\right>^{+(3)} + i \frac{V_{n k}}{\hbar} \left<\right. \hat{a}_{k} \left.\right>^{+(3)} + \nonumber \\ && \hspace{2.0cm} + \sum_{\tilde{n} \tilde{k}} {\Bigg [} \frac{i}{\hbar} \sum_{n' k', \alpha} V_{n' k'} {\langle1 \tilde{n} \tilde{k}\mid} [\hat{B}_{n k}, \hat{B}^\dag_{n' k'} ] - \delta_{(n' k');(n k)} {\mid1 \alpha\rangle} \langle \hat{X}_{1 \tilde{n} \tilde{k}; 1 \alpha} \hat{Y}_{0; 1 k'} \left.\right>^{(3)} - \nonumber \\ && \hspace{3.0cm} - i \sum_{\beta}(\omega_{2 \beta} - \omega_{1 \tilde n \tilde k} - \omega_{1 n k}) {\langle1 \tilde n \tilde k\mid} \hat{B}_{n k} {\mid2 \beta\rangle} \langle \hat{X}_{1 \tilde n \tilde k; 2 \beta} \hat{Y}_{0;0} \left.\right>^{(3)} {\Bigg ]}\, ,\end{aligned}$$ in analogy with the eqs [@Savasta; @PRL96] (see also Ref. [@Sham; @PRL95]). The resulting equation of motion for the lowest order biexciton amplitude is $$\begin{aligned} \label{X02} && \frac{d}{dt}\left<\right. \hat{X}_{0;2 \beta} \left.\right>^{(2)} = -i \bar{\omega}_{2 \beta} \left<\right. \hat{X}_{0;2 \beta} \left.\right>^{(2)} + \nonumber \\ && \hspace{3.0cm} + \frac{i}{\hbar} \sum_{n' k';n'' k''} V_{n' k'}{\langle2 \beta\mid} \hat{B}^\dag_{n' k'} {\mid1 n'' k''\rangle} \left<\right. \hat{X}_{0,1 n'' k''} \hat{Y}_{0,1 k'} \left.\right>^{(2)}\, .\end{aligned}$$ Coherent Response {#3} ================= Thanks to the fact we want to treat coherent optical processes it is possible to manipulate further the parametric contributions under two assumptions. We are addressing a coherent optical response, thus we may consider that a coherent pumping mainly generates coherent nonlinear processes, as a consequence the dominant contribution of the biexciton sector on the third-order nonlinear response can be calculated considering the system quantum state as a pure state, which means the nonlinear term is regarded as originating mainly from coherent contributions. Moreover nonclassical correlations are taken into account up to the lowest order. The first assumption results in the factorizations $\langle \hat{X}_{1 \tilde n \tilde k; 2 \beta} \hat{Y}_{0;0} \left.\right>^{(3)} \simeq \langle \hat{X}_{1 \tilde n \tilde k; 0} \rangle^{(1)} \langle \hat{X}_{0; 2 \beta} \left.\right>^{(2)}$ and $\langle \hat{X}_{1 \tilde{n} \tilde{k}; 1 \beta} \hat{Y}_{0; 1 k'} \left.\right>^{(3)} \simeq \langle \hat{X}_{1 \tilde{n} \tilde{k};0}\rangle^{(1)} \langle \hat{X}_{0; 1 \beta} \hat{Y}_{0; 1 k'} \left.\right>^{(2)}$. The second implies $\langle \hat{X}_{0; 1 \beta} \hat{Y}_{0; 1 k'} \left.\right>^{(2)} \simeq \langle \hat{X}_{0;1 \beta}\rangle^{(1)} \langle \hat{Y}_{0; 1 k'} \left.\right>^{(1)}$, in the nonlinear source term, namely the standard linearization procedure of quantum correlations adopted for large systems [@Walls]. Of course these two approximations can be avoided at the cost of enlarging the set of coupled equations in order to include the equation of motions for the resulting correlation functions. It neglects higher order quantum optical correlation effects between the electron system and the cavity modes leading to a renormalization of the biexciton dynamics with intriguing physical perspectives. However for extended systems, like QWs in planar microcavities, these are effects in most cases of negligible impact, on the contrary in fully confined geometries such as cavity embedded quantum dots they could give significant contributions. In the end, within such a coherent limit, we are able to describe the biexciton contribution effectively as an exciton-exciton correlation [@Sham; @PRL95]. The resulting equations for the coupled exciton an cavity-field expectation values coincide with those obtained within a semiclassical theory (quantized electron-system and classical cavity field). Nevertheless completely different results can be obtained for exciton or photon number expectation values or for higher order correlation function [@SSC; @Savasta; @HRS; @Savasta]. In the next section we will derive operator equations useful for the calculation of such correlation functions. After the two approximations described above (linearization of quantum fluctuations and coherent limit), Eqs (\[dt B up to 3\]) becomes $$\label{dt B up to 3fin} \frac{d}{dt}\left<\right. \hat{B}_{n k} \left.\right>^{+(3)} = -i \bar{\omega}_{1 n k} \left<\right. \hat{B}_{n k} \left.\right>^{+(3)} +i \frac{V_{n k}}{\hbar} \left<\right. \hat{a}_{k} \left.\right>^{+(3)} -\frac{i}{\hbar} \sum_{\tilde{n} \tilde{k}} \langle \hat{B}_{\tilde n \tilde k} \rangle^{(1)} R^{(2)}_{n k;\tilde n \tilde k}\, ,$$ where \[R nl\] &&R\^[(2)]{}\_[n k;n k]{} = Q\^[(2)]{}\_[n k;n k]{} + Q\^[(2)]{}\_[n k;n k]{}\ &&Q\^[(2)]{}\_[n k;n k]{} = \_[n’ k’, n” k”]{} C\^[n’ k’,n” k”]{}\_[n k,n k]{} \_[n” k”]{} \^[(1)]{} \_[k’]{} \^[(1)]{}\ &&Q\^[(2)]{}\_[n k;n k]{} = \_c\^[(1)]{}\_[n k;n k;]{} \_[0; 2 ]{} \^[(2)]{} , with $$\begin{aligned} \label{coeffs} && C^{n' k',n'' k''}_{\tilde n \tilde k,n k} = V_{n' k'} {\langle1 \tilde{n} \tilde{k}\mid} \delta_{(n' k');(n k)} - [\hat{B}_{n k}, \hat{B}^\dag_{n' k'} ] {\mid1 n'' k''\rangle} \\ && c^{(1)}_{n k;\tilde n \tilde k;\beta} = \hbar (\omega_{2 \beta} - \omega_{1 \tilde n \tilde k} - \omega_{1 n k}) {\langle1 \tilde n \tilde k\mid} \hat{B}_{n k} {\mid2 \beta\rangle}\, .\end{aligned}$$ This equation is analogous to the corresponding equation describing the semiclassical (quantized electron system, classical light field) coherent $\chi^{(3)}$ response in a QW [@Sham; @PRL95], the main difference being that here the (intracavity) light field is regarded not as a driving external source but as a dynamical field [@Savasta; @PRL2003]. This close correspondence for the dynamics of expectation values of the exciton operators, is a consequence of the linearization of quantum fluctuations. However the present approach includes the light field quantization and can thus be applied to the description of quantum optical phenomena. By explicit calculation it is easy to see that the first term in Eq. (\[coeffs\]) is zero unless all the involved polarization labels $\sigma$ coincide. In order to manipulate the last term we follow the procedure of Ref. [@Sham; @PRL95] which succeeded in reformulating the nonlinear term coming from the Coulomb interaction as an exciton-exciton (X-X) mean-field contribution plus a correlation term driven by a two-exciton correlation function. Even if we are about to perform more or less the same steps of Ref. [@Sham; @PRL95] we shall provide a detailed account of all the key points of the present derivation. A clear comprehension of these details will be essential for the extension to operatorial dynamical equations of the next section. In performing this we shall need the two identities: $$\begin{aligned} && c^{(1)}_{n k;\tilde n \tilde k;\beta} = \hbar (\omega_{2 \beta} - \omega_{1 \tilde n \tilde k} - \omega_{1 n k}) {\langle1 \tilde n \tilde k\mid} \hat{B}_{n k} {\mid2 \beta\rangle} = \nonumber \\ && \hspace{2.5cm} = \hbar {\langle1 \tilde n \tilde k\mid} \hat{B}_{n k} \big( \frac{\hat{H}_{c}}{\hbar} - \omega_{1 \tilde n \tilde k} - \omega_{1 n k} \big) {\mid2 \beta\rangle} \end{aligned}$$ and $$\begin{aligned} \label{Sham identity} && \frac{d}{dt} \Bigg( \langle \hat{B}_{n' k'} \left.\right>^{(1)} \langle \hat{B}_{n'' k''} \left.\right>^{(1)} e^{-i \Omega(u-t)} \bigg) = \\ && \hspace{2.5cm} = +\frac{i}{\hbar} \bigg( V_{n' k'} \langle \hat{a}_{k'} \rangle^{(1)} \langle \hat{B}_{n'' k''} \rangle^{(1)} + V_{n'' k''} \langle \hat{a}_{k''} \rangle^{(1)} \langle \hat{B}_{n' k'} \rangle \bigg)e^{-i \Omega(t-t')} \nonumber\, , \end{aligned}$$ or $$\label{Sham identity2} \frac{1}{2}\ \frac{d}{dt} \sum_{n' k';n'' k''} \Bigg( \langle \hat{B}_{n' k'} \left.\right>^{(1)} \langle \hat{B}_{n'' k''} \left.\right>^{(1)} e^{-i \Omega(t-t')} \bigg) = +\frac{i}{\hbar} \sum_{n' k';n'' k''} V_{n' k'} \langle \hat{a}_{k'} \rangle^{(1)} \langle \hat{B}_{n'' k''} \rangle^{(1)} e^{-i \Omega(t-t')}\, ,$$ where $\Omega \doteq \omega_{1 n' k'} + \omega_{1 n'' k''} - 2 i \Gamma_{\text{x}}\, .$ Employing the forma solution of the biexciton amplitude Eq. (\[X02\]) we have: $$\begin{aligned} \label{conti c} && \hspace{-1.0cm} \sum_{\beta}c^{(1)}_{n k;\tilde n \tilde k;\beta} \langle \hat{X}_{0; 2 \beta} \rangle^{(2)} = \hbar \sum_{\beta} {\langle1 \tilde n \tilde k\mid} \hat{B}_{n k} \big( \frac{\hat{H}_{c}}{\hbar} - \omega_{1 \tilde n \tilde k} - \omega_{1 n k} \big) {\mid2 \beta\rangle} \cdot \\ && \hspace{0.0cm} i \sum_{n' k';n'' k''} \frac{V_{n' k'}}{\hbar} {\langle2 \beta\mid} \hat{B}^\dag_{n' k'} {\mid1 n'' k''\rangle} \int_{-\infty}^t dt' e^{-i (\omega_{2 \beta} - i \Gamma_{\text{xx}})(t-t')} \langle \hat{a}_{k'} \rangle^{(1)}(t') \langle \hat{B}_{n'' k''} \rangle^{(1)}(t') \nonumber\, .\end{aligned}$$ We observe that the matrix elements entering the nonlinear source terms are largely independent on the wave vectors for the range of wave vectors of interest in the optical response. Neglecting such dependence we can thus exploit the identity (\[Sham identity2\]), obtaining $$\begin{aligned} \label{conti c 2} && \hspace{-1.0cm} = \hbar \sum_{\beta} {\langle1 \tilde n \tilde k\mid} \hat{B}_{n k} \big( \frac{\hat{H}_{c}}{\hbar} - \omega_{1 \tilde n \tilde k} - \omega_{1 n k} \big) {\mid2 \beta\rangle} \int_{-\infty}^t dt' e^{-i (\omega_{2 \beta} - i \Gamma_{\text{xx}})(t-t')} \nonumber \\ && \hspace{0.0cm} \sum_{n' k';n'' k''} {\langle2 \beta\mid} \hat{B}^\dag_{n' k'} {\mid1 n'' k''\rangle} \frac{1}{2}\ \frac{d}{dt'} \Bigg( \langle \hat{B}_{n' k'} \left.\right>^{(1)}(t') \langle \hat{B}_{n'' k''} \left.\right>^{(1)}(t') e^{-i \Omega(t-t')} \Bigg) e^{+i \Omega(t-t')} = \nonumber\\ && \hspace{-1.0cm} = \hbar \sum_{n' k';n'' k''} \int_{-\infty}^t dt' {\langle1 \tilde n \tilde k\mid} \hat{B}_{n k} \big( \frac{\hat{H}_{c}}{\hbar} - \omega_{1 \tilde n \tilde k} - \omega_{1 n k} \big) e^{-i \frac{\hat{H}_{c}}{\hbar} (t-t')} \hat{B}^\dag_{n' k'} {\mid1 n'' k''\rangle} e^{- \Gamma_{\text{xx}}(t-t')} \nonumber \\ && \hspace{0.0cm} \frac{1}{2}\ \frac{d}{dt'} \Bigg( \langle \hat{B}_{n' k'} \left.\right>^{(1)}(t') \langle \hat{B}_{n'' k''} \left.\right>^{(1)}(t') e^{-i \Omega(t-t')} \Bigg) e^{+i \Omega(t-t')}\, ,\end{aligned}$$ where in the last lines we have resummed all the biexciton subspace by virtue of its completeness. By performing an integration by part, Eq.(\[conti c 2\]) can be rewritten as $$\begin{aligned} \label{conti c 3} && \hspace{-1.0cm} = \frac{1}{2}\hbar \!\!\! \sum_{n' k';n'' k''}\!\! \Bigg[ \Bigg\{ e^{i(\omega_{1 n' k'} + \omega_{1 n'' k''} - 2 i \Gamma_{\text{x}} + i \Gamma_{\text{xx}})(t-t')} {\langle1 \tilde n \tilde k\mid} \hat{B}_{n k} \big( \frac{\hat{H}_{c}}{\hbar} - \omega_{1 \tilde n \tilde k} - \omega_{1 n k} \big) e^{-i \frac{\hat{H}_{c}}{\hbar} (t-t')}\! \hat{B}^\dag_{n' k'}\!\! {\mid1 n'' k''\rangle} \nonumber \\ && \langle \hat{B}_{n' k'} \left.\right>^{(1)}(t') \langle \hat{B}_{n'' k''} \left.\right>^{(1)}(t') e^{-i \Omega(t-t')} \Bigg\}^t_{-\infty}- \nonumber \\ && - \int_{-\infty}^t dt' \langle \hat{B}_{n' k'} \left.\right>^{(1)}(t') \langle \hat{B}_{n'' k''} \left.\right>^{(1)}(t') e^{-i \Omega(t-t')} \frac{d}{dt'} \Bigg\{ e^{i(\omega_{1 n' k'} + \omega_{1 n'' k''} - 2 i \Gamma_{\text{x}} + i \Gamma_{\text{xx}})(t-t')} \nonumber \\ && {\langle1 \tilde n \tilde k\mid} \hat{B}_{n k} \big( \frac{\hat{H}_{c}}{\hbar} - \omega_{1 \tilde n \tilde k} - \omega_{1 n k} \big) e^{-i \frac{\hat{H}_{c}}{\hbar} (t-t')} \hat{B}^\dag_{n' k'} {\mid1 n'' k''\rangle} \Bigg\} \Bigg]=\end{aligned}$$ $$\begin{aligned} \label{conti c 4} && \hspace{-1.0cm} = \frac{1}{2} \hbar \!\!\!\! \sum_{n' k';n'' k''}\!\! \Bigg\{ {\langle1 \tilde n \tilde k\mid} \hat{B}_{n k} \big( \frac{\hat{H}_{c}}{\hbar} - \omega_{1 \tilde n \tilde k} - \omega_{1 n k} \big) \! \hat{B}^\dag_{n' k'}\!\! {\mid1 n'' k''\rangle} \langle \hat{B}_{n' k'} \left.\right>^{(1)}(t) \langle \hat{B}_{n'' k''} \left.\right>^{(1)}(t) - \nonumber \\ && - \int_{-\infty}^t dt' \langle \hat{B}_{n' k'} \left.\right>^{(1)}(t') \langle \hat{B}_{n'' k''} \left.\right>^{(1)}(t') e^{-i \Omega(t-t')} \frac{d}{dt'} \Bigg\{ e^{i(\omega_{1 n' k'} + \omega_{1 n'' k''} - 2 i \Gamma_{\text{x}} + i \Gamma_{\text{xx}})(t-t')} \nonumber \\ && {\langle1 \tilde n \tilde k\mid} \hat{B}_{n k} \big( \frac{\hat{H}_{c}}{\hbar} - \omega_{1 \tilde n \tilde k} - \omega_{1 n k} \big) e^{-i \frac{\hat{H}_{c}}{\hbar} (t-t')} \hat{B}^\dag_{n' k'} {\mid1 n'' k''\rangle} \Bigg\}\, .\end{aligned}$$ The first and the second term can be expressed in terms of a double commutator structure: $$\label{doppio comm 1} {\langle1 \tilde n \tilde k\mid} \hat{B}_{n k} \big( \frac{\hat{H}_{c}}{\hbar} - \omega_{1 \tilde n \tilde k} - \omega_{1 n k} \big) = {\langle0\mid} [\hat{B}_{\tilde n \tilde k},[\hat{B}_{n k},\hat{H}_{c}]] \doteq {\langle0\mid} \hat{D}_{\tilde n \tilde k,n k}\, ,$$ where a force* operator $\hat{D}$ is defined [@Sham; @PRL95] and $$\begin{aligned} \label{doppio comm 2} && \frac{d}{dt'} \Bigg\{ e^{i(\omega_{1 n' k'} + \omega_{1 n'' k''} - 2 i \Gamma_{\text{x}} + i \Gamma_{\text{xx}})(t-t')} \nonumber \\ && {\langle1 \tilde n \tilde k\mid} \hat{B}_{n k} \big( \frac{\hat{H}_{c}}{\hbar} - \omega_{1 \tilde n \tilde k} - \omega_{1 n k} \big) e^{-i \frac{\hat{H}_{c}}{\hbar} (t-t')} \hat{B}^\dag_{n' k'} {\mid1 n'' k''\rangle} \Bigg\} = \nonumber \\ && = \frac{d}{dt'} \Bigg\{ {\langle0\mid} \hat{D}_{\tilde n \tilde k,n k} e^{-i \frac{\hat{H}_{c}}{\hbar} (t-t')} \hat{B}^\dag_{n' k'} \hat{B}^\dag_{n'' k''} {\mid0\rangle} e^{i(\omega_{1 n' k'} + \omega_{1 n'' k''} - 2 i \Gamma_{\text{x}} + i \Gamma_{\text{xx}})(t-t')} \Bigg\} = \nonumber \\ && = {\langle0\mid} \hat{D}_{\tilde n \tilde k,n k} e^{-i \frac{\hat{H}_{c}}{\hbar} (t-t')} i \Big( \frac{\hat{H}_{c}}{\hbar} - \omega_{1 n' k'} - \omega_{1 n'' k''} - i (\Gamma_{\text{xx}} - 2 \Gamma_{\text{x}}) \Big) \nonumber \\ && \hspace{1.0cm} \hat{B}^\dag_{n' k'} \hat{B}^\dag_{n'' k''} {\mid0\rangle} e^{i(\omega_{1 n' k'} + \omega_{1 n'' k''} - 2 i \Gamma_{\text{x}} + i \Gamma_{\text{xx}})(t-t')} = \nonumber\\ && = e^{i(\omega_{1 n' k'} + \omega_{1 n'' k''} - 2 i \Gamma_{\text{x}} + i \Gamma_{\text{xx}})(t-t')} i F^{n'' k'', n' k'}_{\tilde n \tilde k,n k}(t-t') + \\ && + (\Gamma_{\text{xx}} - 2 \Gamma_{\text{x}})e^{i(\omega_{1 n' k'} + \omega_{1 n'' k''} - 2 i \Gamma_{\text{x}} + i \Gamma_{\text{xx}})(t-t')} {\langle0\mid} \hat{D}_{\tilde n \tilde k,n k}(t-t')\hat{B}^\dag_{n' k'} \hat{B}^\dag_{n'' k''} {\mid0\rangle} \nonumber\, , \end{aligned}$$ where the memory kernel reads $$\label{F} F^{n'' k'', n' k'}_{\tilde n \tilde k,n k}(t-t') = {\langle0\mid} \hat{D}_{\tilde n \tilde k,n k}(t-t') \hat{D}^\dag_{n'' k'',n' k'} {\mid0\rangle}\, .$$ The usual time dependence in the Heisenberg picture is given by $ \hat{D}(\tau) = e^{i (\hat{H}_{c}/\hbar) \tau}\hat{D}e^{-i (\hat{H}_{c}/\hbar) \tau}$. Altogether, the nonlinear term originating from Coulomb interaction can be written as $$\begin{aligned} \label{H-F} && \hspace{-1.0cm} Q^{\text{COUL}(2)}_{n k;\tilde n \tilde k} = \sum_{\beta}c^{(1)}_{n k;\tilde n \tilde k;\beta} \langle \hat{X}_{0; 2 \beta} \rangle^{(2)} = \nonumber \\ && \hspace{-1.0cm} \frac{1}{2} \hbar \sum_{n' k';n'' k''} \Bigg\{ {\langle0\mid} \hat{D}_{\tilde n \tilde k,n k} \hat{B}^\dag_{n' k'} \hat{B}^\dag_{n'' k''} {\mid0\rangle} \langle \hat{B}_{n' k'} \left.\right>^{(1)}(t) \langle \hat{B}_{n'' k''} \left.\right>^{(1)}(t) - \nonumber \\ && - i \int_{-\infty}^t dt' F^{n'' k'', n' k'}_{\tilde n \tilde k,n k}(t-t') \langle \hat{B}_{n' k'} \left.\right>^{(1)}(t') \langle \hat{B}_{n'' k''} \left.\right>^{(1)}(t') e^{- \Gamma_{\text{xx}}(t-t')} \Bigg\} - \\ && \hspace{-1.0cm} - \frac{\hbar}{2}(\Gamma_{\text{xx}} - 2 \Gamma_{\text{x}}) \hspace{-0.1cm} \sum_{\substack{ n' k'\\n'' k''}} \hspace{-0.1cm} \int_{-\infty}^t dt' {\langle0\mid} \hat{D}_{\tilde n \tilde k,n k}(t-t')\hat{B}^\dag_{n' k'} \hat{B}^\dag_{n'' k''} {\mid0\rangle} \langle \hat{B}_{n' k'} \left.\right>^{(1)}(t') \langle \hat{B}_{n'' k''} \left.\right>^{(1)}(t') \nonumber \, .\end{aligned}$$ For later purpose we are interested in the optical response dominate by the 1S exciton sector, with $\Gamma_{\text{xx}} \simeq 2 \Gamma_{\text{x}}$ in the cases of counter- and co-circularly polarized waves. Specifying to this case the Coulomb-induced term with Eq. (\[H-F\]) becomes $$\begin{aligned} \label{Coul 1S} && \frac{d}{dt}\left<\right. \hat{B}_{\pm {\bf k}} \left.\right>^{+(3)}\Biggl\|_{\text{COUL}}= -i \bar{\omega}_{\bf k} \left<\right. \hat{B}_{\pm {\bf k}} \left.\right>^{+(3)}-\frac{i}{\hbar} \sum_{\tilde{\sigma} {\bf \tilde{k}}} \langle \hat{B}_{\tilde \sigma {\bf \tilde k}} \rangle^{(1)} Q^{\text{COUL}(2)}_{\pm {\bf k};\tilde \sigma {\bf \tilde k}} = \\ && =-i \bar{\omega}_{\bf k} \left<\right. \hat{B}_{\pm {\bf k}} \left.\right>^{+(3)}-\frac{i}{\hbar} \sum_{\bf k' k'' \tilde{k}} \delta_{\bf k+\tilde k;k'+k''} V_{\text{xx}} \langle \hat{B}_{\pm {\bf \tilde k}} \rangle^{(1)}(t) \langle \hat{B}_{\pm {\bf k'}} \rangle^{(1)}(t) \langle \hat{B}_{\pm {\bf k''}}\rangle^{(1)}(t) + \nonumber \\ && -\frac{1}{\hbar} \sum_{\substack{\sigma' \sigma'' \tilde{\sigma}\\ \bf k' k'' \tilde{k}}} \delta_{\bf k+\tilde k;k'+k''} \delta_{\pm+\tilde \sigma;\sigma'+\sigma''} \langle \hat{B}_{\tilde \sigma {\bf \tilde k}} \rangle^{(1)}(t) \nonumber \\ && \hspace{3.0cm} \int_{-\infty}^t dt' F^{\sigma' \sigma''}(t-t') \langle \hat{B}_{\sigma' {\bf k}'} \left.\right>^{(1)}(t') \langle \hat{B}_{\sigma'' {\bf k}''} \left.\right>^{(1)}(t') e^{- \Gamma_{\text{xx}}(t-t')}\, , \nonumber\end{aligned}$$ where, in order to lighten the notation, we dropped the two spin indexes $\sigma$ and $\tilde \sigma$ in the four-particle kernel function $F$ defined in Eq. (\[F\]) for they are already univocally determined once chosen the others (i.e. $\sigma'$ and $\sigma''$) as soon as their selection rule ($\delta_{\sigma+\tilde \sigma;\sigma'+\sigma''}$) is applied. Moreover, the $\hbar/2$ has been reabsorbed in the Coulomb nonlinear coefficients $V_{\text{xx}}$ and $F^{\sigma' \sigma''}(t-t')$. A detail microscopic account for the mean-field $V_{\text{xx}}$, for the $F$’s and their selection rules are considered in [@Takayama; @EPJ; @Kwong-Binder; @PRB; @2001]. For the range of ${\bf k}$-space of interest, i.e. $\|{\bf k}\| \ll \frac{\pi}{a_{\text{x}}}$ (much lower than the inverse of the exciton Bohr radius) they are largely independent on the center of mass wave vectors. While $V_{\text{xx}}$ and $F^{\pm \pm}(t-t')$ (i.e. co-circularly polarized waves) conserve the polarizations, $F^{\pm \mp}(t-t')$ and $F^{\mp \pm }(t-t')$ (counter-circular polarization) give rise to a mixing between the two circularly polarizations. The physical origin of the three terms in Eq. (\[H-F\]) can be easily understood: the first is the Hartee-Fock or mean-field term representing the first order treatment in the Coulomb interaction between excitons, the second term is a pure biexciton (four-particle correlations) contribution. This coherent memory may be thought as a non-Markovian process involving the two-particle (excitons) states interacting with a bath of four-particle correlations [@Sham; @PRL95]. Equation (\[H-F\]) even if formally similar to that of Ref. [@Sham; @PRL95], represents its extension including polaritonic effects due to the presence of the cavity . It has been possible thanks to the inclusion of the dynamics of the cavity modes whereas in Ref. [@Sham; @PRL95] the electromagnetic field entered as a parameter only. Former analogous extensions have been obtained within a semiclassical model [@Takayama; @EPJ; @Kwong-Binder; @PRB; @2001; @Savasta; @PRL2003]. The strong exciton-photon coupling does not modify the memory kernel because four-particle correlations do not couple directly to cavity photons. As pointed out clearly in Ref. [@Savasta; @PRL2003], cavity effects alter the phase dynamics of two-particle states during collisions, indeed, the phase of two-particle states in SMCs oscillates with a frequency which is modified respect to that of excitons in bare QWs, thus producing a modification of the integral in Eq. (\[H-F\]). In this way the exciton-photon coupling $V_{n k}$ affects the exciton-exciton collisions that govern the polariton amplification process. Ref. [@Savasta; @PRL2003] considers the first (mean-field) and the second (four-particle correlation) terms in the particular case of cocircularly polarized waves, calling them without indexes as $V_{\text{xx}}$ and $F(t)$ respectively. In Fig. 1 they show ${\cal F}(\omega)$, the Fourier transform of $F(t)$ plus the mean-field term $V_{\text{xx}}$, $${\cal F}(\omega) = V_{\text{xx}} -i \int^\infty_{- \infty} dt F(t) e^{i \omega t}\, .$$ Its imaginary part is responsible for the frequency dependent excitation induced dephasing, it reflects the density of the states of two-exciton pair coherences. Towards the negative detuning region the dispersive part Re$({\cal F})$ increases whereas the absorptive part Im$({\cal F})$ goes to zero. The former comprises the mean-field contribution effectively reduced by the four-particle contribution. Indeed, the figure shows the case with a binding energy of 13.5 meV, it gives $V_{\text{xx}} n_{\text{sat}} \simeq 11.39$ meV which clearly is an upper bound for Re$({\cal F})$ for negative detuning. The contribution carried by $F(t)$ determines an effective reduction of the mean-field interaction (through its imaginary part which adds up to $V_{\text{xx}}$) and an excitation induced dephasing. It has been shown [@Savasta; @PRL2003] that both effects depends on the sum of the energies of the scattered polariton pairs. The third term in Eq. (\[H-F\]) can be thought as a reminder of the mismatch between the picture of a biexciton as a composite pair of exciton. In the following we will set $\Gamma_{\text{xx}} \simeq 2 \Gamma_{\text{x}}$. The other nonlinear source term in Eq. (\[R nl\]) depends directly on the exciton wave function and reads $$\sum_{\tilde{n} \tilde{k}} \langle \hat{B}_{\tilde n \tilde k} \rangle^{(1)} \sum_{n' k',n'' k''}C^{n' k',n'' k''}_{\tilde n \tilde k,n k} \langle \hat{a}_{k'} \rangle^{(1)} \langle \hat{B}_{n'' k''} \rangle^{(1)}\, .$$ It represents a phase-space filling (PSF) contribution, due to the Pauli blocking of electrons. It can be developed as follows, $$\begin{aligned} \label{PSFconti} && C^{n' k',n'' k''}_{\tilde n \tilde k,n k} = V_{n' k'} {\langle1 \tilde{n} \tilde \sigma {\bf \tilde{k}}\mid} \delta_{(n' k');(n k)} - [\hat{B}_{n \sigma {\bf k}}, \hat{B}^\dag_{n' \sigma' {\bf k'}} ] {\mid1 n'' \sigma'' {\bf k''}\rangle} = \nonumber \\ && = V_{n' k'} \delta_{\sigma,\sigma'} \Biggl\{ \sum_{\bf q} \Phi^{{\bf k}\, }_{n \sigma {\bf q}} \Phi^{\bf k'}_{n' \sigma' ({\bf q}+\eta_h({\bf k'}-{\bf k}))} {\langle1 \tilde{n} \tilde \sigma {\bf \tilde{k}}\mid} \hat{c}^\dag_{\sigma',{\bf q}+\eta_h({\bf k'}-{\bf k})+\eta_e{\bf k'}} c_{\sigma,{\bf q}+\eta_e{\bf k}} {\mid1 n'' \sigma'' {\bf k''}\rangle} + \nonumber \\ && \sum_{\bf q} \Phi^{{\bf k}\, }_{n \sigma {\bf q}} \Phi^{\bf k'}_{n' \sigma' ({\bf q}-\eta_e({\bf k'}-{\bf k}))} {\langle1 \tilde{n} \tilde \sigma {\bf \tilde{k}}\mid} \hat{d}^\dag_{\sigma',-{\bf q}+\eta_e({\bf k'}-{\bf k})+\eta_h{\bf k'}} d_{\sigma,-{\bf q}+\eta_h{\bf k}} {\mid1 n'' \sigma'' {\bf k''}\rangle} \Biggr\} = \nonumber \\ && = V_{n' k'} \delta_{\sigma,\sigma'} \delta_{\bf k+\tilde k;k'+k''} \Biggl\{ \sum_{\bf q} \Phi^{{\bf k}\, }_{n \sigma {\bf q}} \Phi^{\bf k'}_{n' \sigma' {\bf q}_0} \Phi^{{\bf \tilde k}\, }_{{\tilde{n} \tilde \sigma} {\bf q}_1} \Phi^{\bf k''}_{n'' \sigma'' {\bf q}_2} + \nonumber \\ && \sum_{\bf q} \Phi^{{\bf k}\, }_{n \sigma {\bf q}} \Phi^{\bf k'}_{n' \sigma' {\bf q}_3} \Phi^{{\bf \tilde k}\, }_{{\tilde{n} \tilde \sigma} {\bf q}_4} \Phi^{\bf k''}_{n'' \sigma'' {\bf q}_5} \Biggr\} \, , \end{aligned}$$ the explicit expressions of the ${\bf q}$’s are given in [@q's]. Thus, the nonlinear dynamics of Eq. (\[dt B up to 3fin\]) driven by $\hat{H}_I$ can be written $$\begin{aligned} \label{dt B H_I} &&\frac{d}{dt}\left<\right. \hat{B}_{n \sigma {\bf k}} \left.\right>^{+(3)}\Bigl\|_{\hat{H}_I} = +i \frac{V_{n \sigma {\bf k}}}{\hbar} \left<\right. \hat{a}_{\sigma {\bf k}} \left.\right>^{+(3)} -\frac{i}{\hbar} \sum_{\substack{n' n'' \tilde{n}\\ \bf k' k'' \tilde{k}}} \delta_{\bf k+\tilde k;k'+k''} \langle \hat{B}_{\tilde n \sigma {\bf \tilde k}} \rangle^{(1)} \nonumber \\ && \hspace{1.0cm} \langle \hat{a}_{\sigma {\bf k'}} \rangle^{(1)} \langle \hat{B}_{n'' \sigma {\bf k''}}\rangle^{(1)} \tilde V^_{\sigma} \Bigl[ \sum_{\bf q} \Phi^{{\bf k}\, }_{n \sigma {\bf q}} \Phi^{{\bf \tilde k}\, }_{{\tilde{n} \sigma} {\bf q}_1} \Phi^{\bf k''}_{n'' \sigma {\bf q}_2} + \sum_{\bf q} \Phi^{{\bf k}\, }_{n \sigma {\bf q}} \Phi^{{\bf \tilde k}\, }_{{\tilde{n} \sigma} {\bf q}_4} \Phi^{\bf k''}_{n'' \sigma {\bf q}_5} \Bigr] \, .\end{aligned}$$ We are interested in studying polaritonic effects in SMCs where the optical response involves mainly excitons belonging to the 1S band with wave vectors close to normal incidence, i.e. $\|{\bf k}\| \ll \frac{\pi}{a_{\text{x}}}$ (much lower than the inverse of the exciton Bohr radius). In this case the exciton relative wave functions are independent on spins as well as on the center of mass wave vector. They are such that $\sum_{\bf q=-\infty}^\infty \|\Phi_{\bf q}\|^2 = 1$, i.e. $\Phi_{\bf q} = \frac{1}{\sqrt{A}} \frac{\sqrt{2 \pi} 2 a_{\text{x}}}{(1+(a_{\text{x}}\|{\bf q}\|)^2)^{3/2}}$, $a_{\text{x}}$ is the exciton Bohr radius. From now on whenever no excitonic level is specified the 1S label is understood. It yields $$\begin{aligned} \label{dt B H_I 2} &&\frac{d}{dt}\left<\right. \hat{B}_{\sigma {\bf k}} \left.\right>^{+(3)}\Bigl\|_{\hat{H}_I} = +i \frac{V_{\sigma {\bf k}}}{\hbar} \left<\right. \hat{a}_{\sigma {\bf k}} \left.\right>^{+(3)} -\frac{i}{\hbar} \sum_{\bf k' k'' \tilde{k}} \delta_{\bf k+\tilde k;k'+k''} \nonumber \\ && \hspace{1.0cm} \langle \hat{B}_{\tilde \sigma {\bf \tilde k}} \rangle^{(1)} \langle \hat{a}_{\sigma {\bf k'}} \rangle^{(1)} \langle \hat{B}_{\sigma {\bf k''}}\rangle^{(1)} 2 \tilde V^_{\sigma} O^{\text{PSF}}\, ,\end{aligned}$$ where the overlap $O^{\text{PSF}}$ has been calculated in the case of zero center of mass wave vector, namely $$\label{O^PSF} O^{\text{PSF}} = \sum_{\bf q} \Phi^{}_{{\bf q}} \Phi^{}_{\bf q} \Phi_{\bf q}\nonumber \, .$$ In SMCs a measured parameter is the so-called vacuum Rabi splitting $V_{n \sigma {\bf k}}$ [@Baumberg] of the 1S excitonic resonance, for the range of ${\bf k}$-space of interest essentially constant. Defining $V \doteq V_{\sigma} = \tilde V_{\sigma} \sqrt{A} \phi^(0)$ $$\tilde V^_{\sigma} O^{\text{PSF}} = \frac{V}{\sqrt{A}\phi^(0)} O^{\text{PSF}} = \frac{8}{7} \frac{\pi a^2_{\text{x}}}{A}V = \frac{1}{2} \frac{V}{n_\text{sat}}\, ,$$ where we have set $n_{\text{sat}} \doteq (7/16)\!\! \cdot \!\! (A/\pi a^2_{\text{x}})$, called saturation density. In terms of the two circular polarizations the dynamics induced by $\hat{H}_I$ finally reads $$\label{dt B H_I fin} \frac{d}{dt}\left<\right. \hat{B}_{\pm {\bf k}} \left.\right>^{+(3)}\Bigl\|_{\hat{H}_I} = +i \frac{V}{\hbar} \left<\right. \hat{a}_{\pm {\bf k}} \left.\right>^{+(3)} -\frac{i}{\hbar} \sum_{\bf \tilde{k}} \langle \hat{B}_{\pm {\bf \tilde k}} \rangle^{(1)} Q^{\text{PSF}(2)}_{\pm {\bf k};\tilde \sigma {\bf \tilde k}}\, ,$$ where $$\label{PSF term} \sum_{\bf \tilde{k}} \langle \hat{B}_{\pm {\bf \tilde k}} \rangle^{(1)} Q^{\text{PSF}(2)}_{\pm {\bf k};\tilde \sigma {\bf \tilde k}} = \frac{V}{n_\text{sat}} \sum_{\bf k' k'' \tilde{k}} \delta_{\bf k+\tilde k;k'+k''} \langle \hat{B}_{\pm {\bf \tilde k}} \rangle^{(1)} \langle \hat{a}_{\pm {\bf k'}} \rangle^{(1)} \langle \hat{B}_{\pm {\bf k''}}\rangle^{(1)}\, .$$ The same lines of argument can be followed for computing the Coulomb-induced interactions $Q^{\text{COUL}(2)}$ [@Takayama; @EPJ; @Kwong-Binder; @PRB; @2001]. We are lead to introduce the saturation density for two main reasons. The most obvious is our interest to refer this work to the literature where $n_{\text{sat}}$ is extensively used [@Langbein; @PRB2004; @Savasta; @PRL2003; @Ciuti; @SST; @Savasta; @PRB2001]. The other most interesting reason is that we can directly compute this quantity. Indeed, the equation of motion for the exciton operator reads $$\begin{aligned} \label{dt B PSF } && \frac{d}{dt}\left<\right. \hat{B}_{\pm {\bf k}} \left.\right>^{+(3)}= -i \bar{\omega}_{\bf k} \left<\right. \hat{B}_{\pm {\bf k}} \left.\right>^{+(3)} +i \frac{V}{\hbar} \left<\right. \hat{a}_{\pm {\bf k}} \left.\right>^{+(3)} -\frac{i}{\hbar} \sum_{\tilde{\sigma}=\pm {\bf \tilde{k}}} \langle \hat{B}_{\tilde \sigma {\bf \tilde k}} \rangle^{(1)} Q^{\text{COUL}(2)}_{\pm {\bf k};\tilde \sigma {\bf \tilde k}} \nonumber \\ && \hspace{1.0cm} -\frac{i}{\hbar} 2 \frac{V}{\sqrt{A}\phi^(0)} O^{\text{PSF}} \sum_{\bf k' k'' \tilde{k}} \delta_{\bf k+\tilde k;k'+k''} \langle \hat{B}_{\pm {\bf \tilde k}} \rangle^{(1)} \langle \hat{a}_{\pm {\bf k'}} \rangle^{(1)} \langle \hat{B}_{\pm {\bf k''}}\rangle^{(1)} \nonumber\, . \end{aligned}$$ Leaving apart the discrepancy between the order in the DCTS we can compute the so-called oscillator strength* (OS), defined as what multiplies the photon expectation values $\langle \hat{a}_{\pm {\bf k}=0} \rangle$, $$\begin{aligned} \label{OS} OS =\!\! i\frac{V}{\hbar} {\Bigg (} 1 - \frac{2}{\sqrt{A}\phi^(0)} O^{\text{PSF}}\Big[ \langle \hat{B}_{\pm 0} \rangle^{(1)} \langle \hat{B}_{\pm 0} \rangle^{(1)} \Big] {\Bigg )}\, .\end{aligned}$$ The saturation density may be defined as the exciton density that makes the oscillator strength to be zero. We obtain $$\label{nsat value} n_{\text{sat}} = \Biggl( \frac{2}{\sqrt{A}\phi^(0)} O^{\text{PSF}} \Biggr)^{-1} = \frac{A}{\pi a^2_{\text{x}}} \ \frac{7}{16}\, .$$ Eventually, the lowest order ($\chi^{(3)}$) nonlinear optical response in SMCs are described by the following set of coupled equations: $$\label{dt a} \frac{d}{dt}\left<\right. \hat{a}_{\pm {\bf k}} \left.\right>^{+(3)} = -i \bar{\omega}^c_{\bf k} \left<\right. \hat{a}_{\pm {\bf k}} \left.\right>^{+(3)} + i \frac{V}{\hbar} \left<\right. \hat{B}_{\pm {\bf k}} \left.\right>^{+(3)} + t_c \frac{E_{\pm {\bf k}}}{\hbar}\, ,$$ $$\label{dt B} \frac{d}{dt}\left<\right. \hat{B}_{\pm {\bf k}} \left.\right>^{+(3)} = -i \bar{\omega}_{\bf k} \left<\right. \hat{B}_{\pm {\bf k}} \left.\right>^{+(3)} +i \frac{V}{\hbar} \left<\right. \hat{a}_{\pm {\bf k}} \left.\right>^{+(3)} -\frac{i}{\hbar} \sum_{\tilde{\sigma} {\bf \tilde{k}}} \langle \hat{B}_{\tilde \sigma {\bf \tilde k}} \rangle^{(1)} R^{(2)}_{\pm {\bf k};\tilde \sigma {\bf \tilde k}}\, ,$$ with $\sum_{\tilde{\sigma} {\bf \tilde{k}}} \langle \hat{B}_{\tilde \sigma {\bf \tilde k}} \rangle^{(1)} R^{(2)}_{\pm {\bf k};\tilde \sigma {\bf \tilde k}} = \sum_{\tilde{\sigma} {\bf \tilde{k}}} \langle \hat{B}_{\tilde \sigma {\bf \tilde k}} \rangle^{(1)} Q^{\text{COUL}(2)}_{\pm {\bf k};\tilde \sigma {\bf \tilde k}} + \sum_{\bf \tilde{k}} \langle \hat{B}_{\pm {\bf \tilde k}} \rangle^{(1)} Q^{\text{PSF}(2)}_{\pm {\bf k};\tilde \sigma {\bf \tilde k}}$, with the first of the two addenda originating from Coulomb interaction, Eq. (\[Coul 1S\]), whereas the second represents the phase-space filling contribution written in Eqs. (\[PSF term\]). Starting from here, in the strong coupling case, it might be useful to transform the description into a polariton basis. The proper inclusion of dephasing/relaxation and the application of these equations to parametric processes, in the strong coupling regime, is described in another paper of ours [@nostro; @PRB]. Equations (\[dt a\]) and (\[dt B\]) is exact to the third order in the exciting field. While a systematic treatment of higher-order optical nonlinearities would require an extension of the equations of motions (see e.g. Appendix), a restricted class of higher-order effects can be obtained from solving equations (\[dt a\]) and (\[dt B\]) self-consistently up to arbitrary order as it is usually employed in standard nonlinear optics. This can be simply accomplished by replacing, in the nonlinear sources, the linear excitonic polarization and light fields with the total fields [@Sham; @PRL95; @Savasta; @PRL2003; @Buck]. Multiple-scattering processes are expected to be very effective in cavity-embedded QW’s due to multiple reflections at the Bragg mirrors. Parametric Photoluminescence: Towards Semiconductor Quantum Optics {#4} ================================================================== Entanglement is one of the key features of quantum information and communication technology [@Nielsen-Chuang]. Parametric down-conversion is the most frequently used method to generate highly entangled pairs of photons for quantum-optics applications, such as quantum cryptography and quantum teleportation. This $\chi^{(3)}$ optical nonlinear process consists of the scattering of two polaritons generated by a coherent pump beam into two final polariton modes. The total energy and momentum of the final pairs equal that of pump polariton pairs. The scattering can be spontaneous (parametric emission) or stimulated by a probe beam resonantly exciting one of the two final polariton modes. In 2005 an experiment probing quantum correlations of (parametrically emitted) cavity polaritons by exploiting quantum complementarity has been proposed and realized [@Savasta; @PRL2005]. The most common set-up for parametric emission is the one where a single coherent pump feed resonantly excites the structure at a given energy and wave vector, $\bf{k}_p$. Within the DCTS we shall employ Eqs. (\[dt a up to 3\]), (\[dt B up to 3\]) and Eq. (\[X02\]) in operatorial form, provided all the equations to become fully significant as soon as the expectation value quantities we shall work out would lie within the consistent perturbative DCTS order we set from the beginning [@HRS; @Savasta]. In order to be more specific* we shall derive explicitly the case of input light beams activating only the $1 S$ exciton sector with all the same circularly (e.g. $\sigma^+$) polarization, thus excluding the coherent excitation of bound two-pair coherences (biexciton) mainly responsible for polarization-mixing [@Sham; @PRL95]. Equations involving polariton pairs with opposite polarization can be derived in complete analogy following the same steps. Starting from the Heisenberg equations for the exciton and photon operators and keeping only terms providing lowest order nonlinear response (in the input light field) we obtain, $$\label{dt Y up to 3} \frac{d}{dt} \hat{a}_{k} = -i \omega^c_k \hat{a}_{k} + i \frac{V^_{k}}{\hbar} \hat{B}_{k} + t_c \frac{E_k}{\hbar}\, ,$$ $$\begin{aligned} \label{dt X up to 3} &&\frac{d}{dt} \hat{B}_{k} = - i \omega_{k} \hat{B}_{k} +i \frac{V_{k}}{\hbar}\ \hat{a}_{k} + \nonumber \\ && \hspace{2.0cm} + \frac{i}{\hbar} \sum_{\tilde{k}, k', \alpha} V_{k'} {\langle1 \tilde{k}\mid} [\hat{B}_{k}, \hat{B}^\dag_{k'} ] - \delta_{(k'),(k)} {\mid1 \alpha\rangle} \hat{X}_{1 \tilde{k},0} \hat{X}_{0,1 \alpha} \hat{Y}_{0; 1 k'} - \nonumber \\ && \hspace{3.0cm} - \frac{i}{\hbar} \sum_{\tilde{k} \beta}(\omega_{2 \beta} - \omega_{1 \tilde k} - \omega_{1 k}) {\langle1 \tilde k\mid} \hat{B}_{k} {\mid2 \beta\rangle} \hat{X}_{1 \tilde{k},0} \hat{X}_{0, 2 \beta} \, .\end{aligned}$$ In the following we will assume that the pump polaritons driven by a quite strong coherent input field consists of a classical ($\mathbb{C}$-number) field. This approximation is in close resemblance to the two approximations performed in the previous section (linearization of fluctuations and coherent nonlinear processes). We shall show that under this approximation, we may perform the same manipulations ending up to a set of coupled equations analogous to Eqs. (\[dt a\]) and (\[dt B\]). In addition, having a precise set-up chosen, we will be able to specialize our equations and give an explicit account of the parametric contributions as well as the shifts the lowest order nonlinear dynamics provides. We shall retain only those terms containing the semiclassical pump amplitude at $k_p$ twice, thus focusing on the “direct" pump-induced nonlinear parametric scattering processes. It reads $$\begin{aligned} \label{dt X con shift} &&\frac{d}{dt} \hat{B}_{\pm {\bf k}} = - \omega_{\bf k} \hat{B}_{\pm {\bf k}} + i \frac{V}{\hbar}\ \hat{a}_{\pm {\bf k}} - \\ && \hspace{0.0cm} - \frac{i}{\hbar} \frac{V}{n_{\text{sat}}} \sum_{\bf \tilde{k}, k', k''} \delta_{\bf k+\tilde{k},k'+k''} \hat{X}_{1 \pm {\bf \tilde{k}},0} \hat{X}_{0,1 \pm{\bf k''}} \hat{Y}_{0; 1 \pm {\bf k'}} ( \delta_{{\bf k''},{\bf k}_p} \delta_{{\bf k'},{\bf k}_p} + \delta_{{\bf \tilde{k}},{\bf k}_p} \delta_{{\bf k''},{\bf k}_p} + \delta_{{\bf \tilde{k}},{\bf k}_p} \delta_{{\bf k'},{\bf k}_p} ) - \nonumber \\ && \hspace{0.0cm} - \frac{i}{\hbar} \sum_{\tilde \sigma {\bf \tilde{k}}, \sigma_{\beta} {\bf k}_\beta} (\omega_{2 {\bf k}_\beta} - \omega_{1 {\bf \tilde k}} - \omega_{1 {\bf k}}) {\langle1 {\bf \tilde \sigma \tilde k}\mid} \hat{B}_{\pm {\bf k}} {\mid2 \sigma_{\beta} {\bf k}_\beta\rangle} \hat{X}_{1 \tilde \sigma {\bf \tilde{k}},0} \hat{X}_{0, 2 \sigma_{\beta} {\bf k}_\beta} (\delta_{{k}_\beta,2 {k}_p} + \delta_{{\tilde{k}},{k}_p}\delta_{{k}_\beta,{k}+{k}_p} )\nonumber\, ,\end{aligned}$$ where we have already manipulated the phase-space filling matrix element. Here in brackets the first addendum of each line would be responsible for the parametric contribution, whereas the others will give the shifts. It is understood, from now on, that the pump-driven terms (e.g. the $X$ and $Y$ at $k_p$) are ${\mathbb{C}}$-numbers coherent amplitudes like the semiclassical electromagnetic pump field, we will make such distinction in marking with a “hat" the operators only. We need some care in manipulating the Coulomb-induced terms, the last line. Written explicitly it is $$\begin{aligned} \label{X01 Coulomb} &&\frac{d}{dt} \hat{B}_{\pm {\bf k}}{\Bigg \|}_{\text{Coul}} = \nonumber \\ && \hspace{0.5cm} - \frac{i}{\hbar} \sum_{\tilde \sigma {\bf \tilde{k}}, \sigma_{\beta} {\bf k}_\beta} (\omega_{2 {\bf k}_\beta} - \omega_{1 {\bf \tilde k}} - \omega_{1 {\bf k}}) {\langle1 \tilde \sigma {\bf \tilde k}\mid} \hat{B}_{\pm {\bf k}} {\mid2 \sigma_{\beta} {\bf k}_\beta\rangle} \hat{X}_{1 \tilde \sigma {\bf \tilde{k}},0} {X}_{0, 2 \sigma_p {\bf k}_p} + \nonumber \\ && \hspace{0.5cm} - \frac{i}{\hbar} \sum_{\tilde \sigma {\bf \tilde{k}}, \sigma_{\beta} {\bf k}_\beta} (\omega_{2 {\bf k}_\beta} - \omega_{1 {\bf \tilde k}} - \omega_{1 {\bf k}}) {\langle1 \tilde \sigma {\bf \tilde k}\mid} \hat{B}_{\pm {\bf k}} {\mid2 \sigma_{\beta} {\bf k}_\beta\rangle} X_{1 \sigma_p {\bf k}_p,0} \hat{X}_{0, 2 \sigma_{{\bf k}+{\bf k}_p}({\bf k}+{\bf k}_p)} \,\end{aligned}$$ As for the term containing ${X}_{0, 2 k_p}$, we are facing a ${\mathbb{C}}$-number which gives no problem in performing the very same procedure of the previous chapter. As for the other we would exploit the formal biexciton solution $$\begin{aligned} \label{X02 sol} && \hat{X}_{0;2 (k+k_p)} (t) = \int_{-\infty}^t dt' e^{-i \omega_{2 (k+k_p)}(t-t')} \frac{i}{\hbar} {\Bigg (} V_{k_p} {\langle2 (k+k_p)\mid} \hat{B}^\dag_{k_p} {\mid1 k\rangle} \hat{X}_{0,1 k} Y_{0,1 k_p} + \nonumber \\ && \hspace{0.5cm} V_{k} {\langle2 (k+k_p)\mid} \hat{B}^\dag_{k} {\mid1 k_p\rangle} X_{0,1 k_p} \hat{Y}_{0,1 k} {\Bigg )} \, ,\end{aligned}$$ where, for the sake of consistence, we are neglecting $\hat{X}_{0;2 (k+k_p)} (-\infty)$ because the biexciton, within the present approximations, is always generated by an operator at $k$ times a classical amplitude at $k_p$ which is always zero before the electromagnetic impulse arrived. Moreover, an analogous identity such that of Eq. (\[Sham identity\]) is valid in the present context, namely $$\begin{aligned} \label{Sham id per ops} && \frac{d}{dt} \Bigg( \hat{X}_{0,1 k} {X}_{0, 1 k_p} e^{-i (\omega_{1 k} + \omega_{1 k_p})(t-t')} \bigg) = \\ && \hspace{2.5cm} = \bigg( i \frac{V_k}{\hbar} \hat{Y}_{0,1 k} X_{0,1 k_p} + i \frac{V_{k_p}}{\hbar} Y_{0,1 k_p} \hat{X}_{0,1 k} \bigg)e^{-i (\omega_{1 k} + \omega_{1 k_p})(t-t')} \nonumber\, .\end{aligned}$$ With these tools at hand we are able to perform step by step the manipulations of the previous section for all the quantities at play. The final result reads $$\begin{aligned} \label{dt X completa1} &&\frac{d}{dt} \hat{B}_{\pm {\bf k}} = - \omega_{\bf k} \hat{B}_{\pm {\bf k}} + i \frac{V}{\hbar}\ \hat{a}_{\pm {\bf k}} - \nonumber \\ && \hspace{0.5cm} - \frac{i}{\hbar} \frac{V}{n_{\text{sat}}} \bigg( \hat{X}_{1 \pm {\bf k}_i,0} X_{0,1 \pm {\bf k}_p} Y_{0, 1 \pm {\bf k}_p} + X_{1 \pm {\bf k}_p,0} X_{0,1 \pm {\bf k}_p} \hat{Y}_{0, 1 \pm {\bf k}} + X_{1 \pm {\bf k}_p,0} \hat{X}_{0,1 \pm {\bf k}} Y_{0, 1 \pm {\bf k}_p} \bigg) - \nonumber \\ && \hspace{0.5cm} - \frac{i}{\hbar} \hat{X}_{1 \pm {\bf k}_i,0}(t) \Bigg\{ V_{\text{xx}} X_{0,1 \pm {\bf k}_p}(t) X_{0,1 \pm {\bf k}_p}(t) - i \int_{-\infty}^t dt' F^{\pm \pm}(t-t') X_{0,1 \pm {\bf k}_p}(t')X_{0,1 \pm {\bf k}_p}(t') \Bigg\} - \nonumber \\ && \hspace{0.0cm} - 2 \frac{i}{\hbar} X_{1 \sigma_{{\bf k}_p} {\bf k}_p,0}(t) \Bigg\{ V_{\text{xx}} \hat{X}_{0,1 \pm {\bf k}}(t) X_{0,1 \pm {\bf k}_p}(t) - i \int_{-\infty}^t dt' F^{\pm \pm}(t-t') \hat{X}_{0,1 \pm {\bf k}}(t') X_{0,1 \pm {\bf k}_p}(t') \Bigg\}\, ,\end{aligned}$$ where ${\bf k}_i = 2{\bf k}_p-{\bf k}$, and again $V_{\text{xx}}$ and $F^{\pm \pm}(t-t')$ have reabsorbed the $1/2$ originating from Eq. (\[Sham id per ops\]). In the specific case under analysis we are considering co-circularly polarized waves and the mean field term, $V_{\text{xx}}$ as well as the the kernel function $F(t)$ can be found in Refs. [@Takayama; @EPJ; @Kwong-Binder; @PRB; @2001]. Eventually, the lowest order ($\chi^{(3)}$) nonlinear optical response in SMCs is given by the following set of coupled equations where, in the same spirit of the final remark in the previous section, we account for multiple scattering simply by replacing the linear excitonic polarization and light fields with the total fields: $$\begin{aligned} \label{final sys} &&\frac{d}{dt} \hat{a}_{\pm {\bf k}} = -i \omega^c_{\bf k} \hat{a}_{\pm {\bf k}} + i \frac{V}{\hbar}\ \hat{B}_{\pm {\bf k}} + t_c \frac{E_{\pm {\bf k}}}{\hbar} \nonumber \\ &&\frac{d}{dt} \hat{B}_{\pm k} = -i\omega_{\bf k} \hat{B}_{\pm {\bf k}} + \hat{s}_{\pm {\bf k}} + i \frac{V}{\hbar}\ \hat{a}_{\pm {\bf k}} - \frac{i}{\hbar}{R}^{NL}_{\pm {\bf k}}\, ,\end{aligned}$$ where ${R}^{NL}_{\pm {\bf k}}=(R^{sat}_{\pm {\bf k}}+{R}^{\text{xx}}_{\pm {\bf k}})$ $$\begin{aligned} \label{NN terms} && R^{sat}_{\pm {\bf k}} = \frac{V}{n_{\text{sat}}} B_{\pm {\bf k}_p} a_{\pm {\bf k}_p} \hat{B}^\dag_{\pm {\bf k}_i} \nonumber \\ && R^{\text{xx}}_{\pm {\bf k}} = \hat{B}^\dag_{\pm {\bf k}_i}(t) \bigg( V_{\text{xx}} B_{\pm {\bf k}_p}(t) B_{\pm {\bf k}_p}(t) - \nonumber \\ && - i \int_{-\infty}^t dt' F^{\pm \pm}(t-t') B_{\pm {\bf k}_p}(t') B_{\pm {\bf k}_p}(t') \bigg)\, .\end{aligned}$$ The pump induced renormalization of the exciton dispersion gives a frequency shift $$\begin{aligned} && \hat{s}_{\pm {\bf k}} = -i \bigg( \frac{V}{n_{\text{sat}}} \big(B^_{\pm {\bf k}_p} a_{\pm {\bf k}_p} \hat{B}_{\pm {\bf k}} + B^_{\pm {\bf k}_p} B_{\pm {\bf k}_p} \hat{a}_{\pm {\bf k}} \big) + \nonumber \\ && \hspace{2.5cm} 2 \frac{V_{\text{xx}}}{\hbar} B^_{\pm {\bf k}_p} B_{\pm {\bf k}_p} \hat{B}_{\pm {\bf k}} - \nonumber \\ && \hspace{2.5cm} -2 \frac{i}{\hbar} B^_{\pm {\bf k}_p}(t) \int_{-\infty}^t dt' F^{\pm \pm}(t-t') \hat{B}_{\pm {\bf k}}(t') B_{\pm {\bf k}_p}(t') \bigg)\, .\end{aligned}$$ Equations, (\[final sys\]) are the main result of this paper. They can be considered the starting point for the microscopic description of quantum optical effects in SMCs. These equations extend the usual semiclassical description of Coulomb interaction effects, in terms of a mean-field term plus a genuine non-instantaneous four-particle correlation, to quantum optical effects. Analogous equations can be obtained starting from an effective Hamiltonian describing excitons as interacting bosons [@CiutiBE]. The resulting equations (usually developed in a polariton basis) do not include correlation effects beyond Hartree-Fock. Moreover the interaction terms due to phase space filling differs from those obtaind within the present approach not based on an effective Hamiltonian. Only the many-body electronic Hamiltonian, the intracavity-photon Hamiltonian and the Hamiltonian describing their mutual interaction have been taken into account. Losses through mirrors, decoherence and noise due to environment interactions as well as applications of this theoretical framework will be addressed in another paper of ours [@nostro; @PRB]. Conclusion ========== In this paper we set a dynamics controlled truncation scheme approach to nonlinear optical processes in cavity embedded semiconductor QWs without any assumption on the quantum statistics of the excitons involved. This approach represents the starting point for the microscopic analysis to quantum optics experiments in the strong coupling regime. We presented a systematic theory of Coulomb-induced correlation effects in the nonlinear optical processes in SMCs. We end up with dynamical equations for exciton and photon operators which extend the usual semiclassical description of Coulomb interaction effects, in terms of a mean-field term plus a genuine non-instantaneous four-particle correlation, to quantum optical effects. The proper inclusion of the detrimental environment interactions as well as applications of the present theoretical scheme will be presented in another paper of ours [@nostro; @PRB]. The Equation of Motion At Any Order {#gen eq} =================================== The equation of motion for the operators in (\[Hubbard\]), under the Hamiltonian $\hat{H} = \hat{H}_{e} + \hat{H}_{c} + \hat{H}_{I} + \hat{H}_p$ reads: $$\begin{aligned} \frac{d}{dt} \left(\right. \hspace{-0.4cm} && \hat{X}_{N\alpha;M\beta} \hat{Y}_{n \lambda;m\mu} \left.\right) = -i(\omega_{M \beta} -\omega_{N \alpha} + \sum_{i=1}^m \omega^c_{k_i} - \sum_{j=1}^n \omega^c_{k_j} ) \left(\right. \hat{X}_{N\alpha;M\beta} \hat{Y}_{n \lambda;m\mu} \left.\right) + \nonumber \\ && + \hat{X}_{N\alpha;M\beta} \big( \delta_{m,1} t_c \frac{E_{\mu}}{\hbar} \hat{Y}_{n \lambda;0} + \delta_{n,1} t_c \frac{E^{}_{\lambda}}{\hbar} \hat{Y}_{0;m\mu} \big) - \hat{X}_{N\alpha;M\beta} \sum_{\bar{k}} t_c \big( \delta_{m,0} \frac{E^{}_{\bar{k}}}{\hbar} \hat{Y}_{n \lambda;1 \bar{k}} + \delta_{n,0} \frac{E_{\bar{k}}}{\hbar} \hat{Y}_{1 \bar{k};m\mu} \big) + \nonumber \\ && + \sum_{\bar{k} \nu} t_c \hat{X}_{N\alpha;M\beta} \Bigg[ \Theta(m-2) {\langlem \mu\mid}\hat{a}^\dag_{\bar{k}} {\mid(m-1) \nu\rangle}\frac{E_{\bar{k}}}{\hbar} \hat{Y}_{n \lambda;(m-1)\nu} + \nonumber \\ &&\hspace{5.0cm} \Theta(n-2) {\langle(n-1) \nu\mid}\hat{a}_{\bar{k}} {\midn \lambda\rangle} \frac{E^{}_{\bar{k}}}{\hbar} \hat{Y}_{(n-1) \nu;m \mu} - \nonumber \\ &&\hspace{3.0cm} - \Theta(m-1) {\langlem \mu\mid}\hat{a}_{\bar{k}} {\mid(m+1) \nu\rangle} \frac{E^{}_{\bar{k}}}{\hbar} \hat{Y}_{n \lambda;(m+1)\nu} - \nonumber \\ && \hspace{5.0cm} \Theta(n-1) {\langle(n+1) \nu\mid}\hat{a}^\dag_{\bar{k}} {\midn \lambda\rangle} \frac{E_{\bar{k}}}{\hbar} \hat{Y}_{(n+1) \nu;m \mu} \Bigg] + \nonumber \\ && +\frac{i}{\hbar} \delta_{M,0} \delta_{\beta,0} \delta_{m,1} \sum_{\bar{n}}V^_{\bar{n}\mu} \hat{X}_{N\alpha;1 \bar{n} \mu} \hat{Y}_{n \lambda;0} - \frac{i}{\hbar} \delta_{N,0} \delta_{\alpha,0} \delta_{n,1} \sum_{\bar{n}} V_{\bar{n}\lambda} \hat{X}_{1 \bar{n} \lambda;M \beta} \hat{Y}_{0;m \mu} - \nonumber \\ && -\frac{i}{\hbar} \delta_{N,1} \delta_{n,0}\delta_{\lambda,0} V^_{\alpha} \hat{X}_{0;M\beta} \hat{Y}_{1 k_{\alpha};m \mu} +\frac{i}{\hbar} \delta_{M,1} \delta_{m,0}\delta_{\mu,0} V_{\beta} \hat{X}_{N \alpha;0} \hat{Y}_{n \lambda;1 k_{\beta}} + \nonumber \\ && +\frac{i}{\hbar} \delta_{m,1} \Theta(M-1) \sum_{\bar{n} \delta} V^_{\bar{n} \mu} {\langleM \beta\mid}\hat{B}_{\bar{n}\mu} {\mid(M+1) \delta\rangle} \hat{X}_{N \alpha;(M+1) \delta} \hat{Y}_{n \lambda,0} - \nonumber\\ && -\frac{i}{\hbar}\delta_{n,1} \Theta(N-1) \sum_{\bar{n} \eta} V_{\bar{n} \lambda} {\langle(N+1) \eta\mid}\hat{B}^\dag_{\bar{n}\lambda} {\midN \alpha\rangle} \hat{X}_{(N+1) \eta;M \beta} \hat{Y}_{0;m \mu} - \nonumber\\ && -\frac{i}{\hbar} \delta_{n,0} \delta_{\lambda,0} \Theta(N-2) \sum_{\bar{n} \bar{k} \eta} V^_{\bar{n} \bar{k}} {\langle(N-1) \eta\mid}\hat{B}_{\bar{n}\bar{k}} {\midN \alpha\rangle} \hat{X}_{(N-1) \eta;M \beta} \hat{Y}_{1 \bar{k};m \mu} + \nonumber\\ && +\frac{i}{\hbar} \delta_{m,0} \delta_{\mu,0} \Theta(M-2) \sum_{\bar{n} \bar{k} \delta} V_{\bar{n} \bar{k}} {\langleM \beta\mid}\hat{B}^\dag_{\bar{n}\bar{k}} {\mid(M-1) \delta\rangle} \hat{X}_{N \alpha;(M-1) \delta} \hat{Y}_{n \lambda;1 \bar{k}} + \nonumber\\ && +\frac{i}{\hbar} \delta_{M,0} \delta_{\beta,0} \Theta(m-2) \sum_{\bar{n} \bar{k} \nu} V^_{\bar{n} \bar{k}} {\langlem \mu\mid}\hat{a}^\dag_{\bar{k}} {\mid(m-1) \nu\rangle} \hat{X}_{N \alpha;1 \bar{n} \bar{k}} \hat{Y}_{n \lambda;(m-1) \nu} - \nonumber\\ && -\frac{i}{\hbar} \delta_{N,0} \delta_{\alpha,0} \Theta(n-2) \sum_{\bar{n} \bar{k} \gamma} V_{\bar{n} \bar{k}} {\langle(n-1) \gamma\mid}\hat{a}_{\bar{k}} {\midn \lambda\rangle} \hat{X}_{1 \bar{n} \bar{k};M \beta} \hat{Y}_{(n-1) \gamma;m \mu} - \nonumber\\ && -\frac{i}{\hbar} \delta_{N,1} \Theta(n-1) V_{\alpha} \sum_{\gamma} {\langle(n+1) \gamma\mid}\hat{a}^\dag_{k_{\alpha}} {\midn \lambda\rangle} \hat{X}_{0;M \beta} \hat{Y}_{(n+1) \gamma;m \mu} + \nonumber\\ && +\frac{i}{\hbar} \delta_{M,1} \Theta(m-1) V_{\beta} \sum_{\nu} {\langlem \mu\mid}\hat{a}_{k_{\beta}} {\mid(m+1) \nu\rangle} \hat{X}_{N \alpha;0} \hat{Y}_{n \lambda;(m+1) \nu} + \nonumber\end{aligned}$$ $$\begin{aligned} \label{dt gen op} && + \frac{i}{\hbar} \sum_{\bar{n} \bar{k}} \sum_{\nu \delta} {\Bigg [} V^_{\bar{n} \bar{k}} {\Big (} \Theta(M-1) \Theta(m-2) {\langleM \beta\mid}\hat{B}_{\bar{n} \bar{k}} {\mid(M+1) \delta\rangle} \nonumber \\ && \hspace{3.0cm} {\langlem \mu\mid}\hat{a}^\dag_{\bar{k}} {\mid(m-1) \nu\rangle} \hat{X}_{N \alpha;(M+1)\delta} \hat{Y}_{n \lambda;(m-1) \nu} - \nonumber \\ && \hspace{2.5cm} - \Theta(N-2) \Theta(n-1) {\langle(N-1) \delta\mid}\hat{B}_{\bar{n} \bar{k}} {\midN \alpha\rangle} \nonumber \\ && \hspace{3.0cm} {\langle(n+1)\nu\mid}\hat{a}^\dag_{\bar{k}} {\midn \lambda\rangle} \hat{X}_{(N-1) \delta;M\beta} \hat{Y}_{(n+1) \nu;m \mu} {\Big )} + \nonumber \\ && \hspace{2.0 cm} - \frac{i}{\hbar}V_{\bar{n} \bar{k}} {\Big (} \Theta(N-1) \Theta(n-2) {\langle(N+1) \delta\mid}\hat{B}^\dag_{\bar{n} \bar{k}} {\midN \alpha\rangle} \nonumber \\ && \hspace{3.0cm} {\langle(n-1) \nu\mid}\hat{a}_{\bar{k}} {\midn \lambda\rangle} \hat{X}_{(N+1) \delta;M \beta} \hat{Y}_{(n-1) \nu;m \mu} - \nonumber \\ && \hspace{2.5cm} - \Theta(M-2) \Theta(m-1) {\langleM \beta\mid}\hat{B}^\dag_{\bar{n} \bar{k}} {\mid(M-1) \delta\rangle} \nonumber \\ && \hspace{3.0cm} {\langlem\mu\mid}\hat{a}_{\bar{k}} {\mid(m+1) \nu\rangle} \hat{X}_{N \alpha;(M-1)\delta} \hat{Y}_{n \lambda;(m+1) \nu} {\Big )} {\Bigg ]}\, . \end{aligned}$$ Here $\Theta(x)$ is the Heaviside function equal to 1 for positive argument and zero otherwise. N [eh]{} pair aggregates {#Npair states} ========================== We start from the usual model for the electronic Hamiltonian of a direct two-band semiconductor [@Haugh; @AxtKuhn]. It is obtained from the many-body Hamiltonian of the interacting electron system in a lattice, keeping explicitly only those terms in the Coulomb interaction preserving the number of electrons in a given band and can be expressed as $$\label{Ham electron1} \hat{H}_e = \hat{H}_0 + \hat{V}_{\text{Coul}}\, .$$ It comprises the single-particle Hamiltonian terms for electrons in conduction band and holes in valence band (here $k\equiv ({\bf k}, \sigma)$ and $\hat{c}_{\sigma,{\bf k}}$ ($\hat{d}_{\sigma,{\bf k}}$) annihilates an electron (a hole)) : $$\label{H zero} \hat{H}_0 = \sum_{k} E_{c,{k}} \hat{c}^\dag_{k} \hat{c}_{k} + \sum_{k} E_{h,{k}}\hat{d}^\dag_{k} \hat{d}_{k}\, ,$$ and the Coulomb interaction term of three contributions, the two repulsive electron-electron (e-e) and hole-hole (h-h) terms and the attractive (e-h) one: $$\begin{aligned} \label{V coul} \hat{V}_{\text{Coul}} && = \frac{1}{2} \sum_{{\bf q} \neq 0} V_q \sum_{\sigma,{\bf k},\sigma',{\bf k}'} \hat{c}^\dag_{\sigma,{\bf k}+{\bf q}} \hat{c}^\dag_{\sigma',{\bf k}'-{\bf q}} \hat{c}_{\sigma',{\bf k}'} \hat{c}_{\sigma,{\bf k}} + \frac{1}{2} \sum_{{\bf q} \neq 0} V_q \sum_{\sigma,{\bf k},\sigma',{\bf k}'} \hat{d}^\dag_{\sigma,{\bf k}+{\bf q}} \hat{d}^\dag_{\sigma',{\bf k}'-{\bf q}} \hat{d}_{\sigma',{\bf k}'} \hat{d}_{\sigma,{\bf k}} - \nonumber \\ && - \sum_{{\bf q} \neq 0} V_q \sum_{\sigma,{\bf k},\sigma',{\bf k}'} \hat{c}^\dag_{\sigma,{\bf k}+{\bf q}} \hat{d}^\dag_{\sigma',{\bf k}'-{\bf q}} \hat{d}_{\sigma',{\bf k}'}c_{\sigma,{\bf k}}\, .\end{aligned}$$ A many-body interacting state is usually very different from a product state, however a common way to express the former is by a superposition of uncorrelated product states. The physical picture that arises out of it expresses the dressing* the interaction performs over a set of noninteracting particles. The general many-body Schrödinger equation for this Coulomb-correlated system is $$\label{Schr eq} \hat{H}_e {\mid\Psi\rangle} = (\hat{H}_0 + \hat{V}_{\text{Coul}}) {\mid\Psi\rangle} = E {\mid\Psi\rangle}\, ,$$ with ${\mid\Psi\rangle}$ the global interacting many-body state of the whole Fock space and $E$ its corresponding energy. The system Hamiltonian commutes with the total-number operators for electron and holes, i.e. $\hat{N}_e = \sum_{k} \hat{c}^\dag_{k}\hat{c}_{k}$ and $\hat{N}_h = \sum_{k} \hat{d}^\dag_{k}\hat{d}_{k}$. Therefore the state ${\mid\Psi\rangle}$ may be build up corresponding on a given number of electrons and of holes. Moreover, because we shall consider the case of intrinsic semiconductors materials where $N_e = N_h \doteq N $, the good quantum number for the Schrödinger equation (\[Schr eq\]) is the total number of electron-hole pairs $N$, explicitly $$\label{Schr in N} \hat{H}_e {\midN \alpha\rangle} = E_{N \alpha} {\midN \alpha\rangle}\, ,$$ where $\alpha$ is the whole set of proper quantum numbers needed to specify univocally the many-body state. For any given number $N$ of electron-hole pairs, the product-state set, built up from the single-particle states $\{ {\midN a\rangle} \}$ eigenstates of the noninteracting carrier Hamiltonian $\hat{H}_0$, is a natural complete basis of the N-pair sector of the global Fock space: $$\label{nonint H} \hat{H}_0 {\midN a\rangle} = \epsilon_{N a} {\midN a\rangle}\, ,$$ where N identifies the N-pair subspace and $a$ is a compact form for all the single particle indexes, i.e. $a \equiv {j}_{e1},{j}_{e2}, ...,{j}_{eN};{j}_{h1},{j}_{h2},...,{j}_{hN}$. Indeed $$\label{prod} {\midN,a\rangle} = \otimes_{n=1}^N \hat{c}^\dag_{j_{en}} \hat{d}^\dag_{j_{hn}} {\mid0,0\rangle}\ \ \text{and} \ \ \epsilon_{N a} = \sum_{n=1}^N (\epsilon_{{j}_{en}} + \epsilon_{{j}_{hn}} )\, .$$ Being a complete orthonormal basis for the N-pair subspace we may expand the many-body state ${\midN, \alpha\rangle}$ over it, it yields $$\label{U} {\midN \alpha\rangle} = \sum_{a} U^{N \alpha}_{a} {\midN a\rangle}\, .$$ It is only a matter of calculation to show that $U^{N \alpha}_a$ is nothing but the envelope function of the N-pair aggregate, solution of the corresponding secular equation. Indeed the eigenvalue problem (\[Schr in N\]) is transformed into: $$\label{dressing} \sum_{a'} ( {\langleN a\mid}\hat{H}_e {\midN a'\rangle} - E_{N \alpha} \delta_{a,a'} ) U^{N \alpha}_{a'} = 0\, .$$ Namely N=1 leads to the exciton secular equation, whereas N=2 represents the biexciton (two pairs) Coulomb problem. In order to be clearer we shall propose in details the N=1 exciton calculation. We shall work in the direct lattice ${\bf r} \leftrightarrow {\bf r}_i$ (the former is a continuous variable whereas the latter is a point in the $3D$ lattice). Using the general mapping [@Cohen-Tannoudji; @QED] $\sum_{r_i} \leftrightarrow (1/v_0) \int d^3r$, $\delta({\bf r} - {\bf r}') = (\delta_{{\bf r}_i,{\bf r_j}}/v_0)$, and $ \hat{a}^\dag_{{\bf r}_i} = (\hat{a}^\dag({{\bf r}_i})/{\sqrt{v_0}}) $, here $v_0$ is the unit cell volume and for simplicity the spin selection rules for the optically active states has been already taken into account, (\[dressing\]) reads $$\label{sec eq dir l} \sum_{{\bf r}'_e,{\bf r}'_h} {\Bigg (} {\langle{\bf r}_e,{\bf r}_h\mid}\hat{H}_e {\mid{\bf r}'_e,{\bf r}'_h\rangle} - E_{n \sigma {\bf k}} \delta_{{\bf r}_e {\bf r}_h,{\bf r}'_e {\bf r}'_h} {\Bigg )} U^{\alpha}_a({\bf r}'_e,{\bf r}'_h) = 0\, ,$$ with $$\label{Ham in r} {\langle{\bf r}_e,{\bf r}_h\mid}\hat{H}_e {\mid{\bf r}'_e,{\bf r}'_h\rangle} = {\Bigg (} - \frac{\hbar^2}{2m_e} \nabla^2_{r_e} - \frac{\hbar^2}{2m_h} \nabla^2_{r_h} - \frac{e^2}{\varepsilon_r \|{\bf r}_e - {\bf r}_h\|}+ V({\bf r}_e,{\bf r}_h) {\Bigg )} \delta_{{\bf r}_e {\bf r}_h,{\bf r}'_e {\bf r}'_h}\, ,$$ here $ V({\bf r}_e,{\bf r}_h)$ represents all the additional potential, e.g. those of the heterostructures or those of disorder effects, $ V({\bf r}_e,{\bf r}_h) = V^e(z_e) + V^h(z_h)$. Typically, the energy difference between the lowest QW subband level and the first excited one (at least a few meV) is much larger than the Coulomb interaction between particles (a few meV). As a consequence, at least at low temperatures, particles are confined at the lowest quantization level and the (possible) distorsion of the wave function due to the Coulomb-activated admixture of different subbands can be safely neglected. In some extent, then, the particle wave function dependence along the growth (say $z$) direction can be factorized out and the dynamics becomes essentially two-dimensional. However, a purely $2D$ approximation for excitons would miss important effects of the geometrical QWs parameters on the binding energy and would not be able to account for the interaction with a $3D$ continuum environment of surrounding modes (e.g. acoustic phonon modes in heterostructures with alloy lattice constant in close proximity [@Takagahara]). In addition in QWs, light and heavy holes in valence band are split off in energy. Assuming that this splitting is much larger than kinetic energies of all the involved particles and, as well, much larger than the interaction between them, we shall consider only heavy hole states as occupied. In Eq. (\[sec eq dir l\]) the $3D$ Coulomb interaction prevents form factorizing into (free) in-plane and confined directions. Nevertheless if we assume that the quantization energy along $z$ is much larger than the Coulomb energy, at leading order we can factorize out the $z$-dependence $$\begin{aligned} \label{in z} {\Bigg (} - \frac{\hbar^2}{2m_e} \frac{d^2}{dz^2_e} + V^e(z_e) - \frac{\hbar^2}{2m_h} \frac{d^2}{dz^2_h} &+& V^h(z_h) {\Bigg )} U^{\alpha}({\bf r}_e,{\bf r}_h) = \nonumber \\ && E^{z} U^{\alpha}({\bf r}_e,{\bf r}_h)\, .\end{aligned}$$ It means we are solving our secular equation with solutions built up as linear combination of $F^{\alpha}_{n_c,n_v,a}({\bf r}^\\|_e,{\bf r}^\\|_h)c_{n_c}(z_e)v_{n_v}(z_h)$, with ${\bf r} = ({\bf r}^\\|,z)$. Equation (\[in z\]) expresses the lack of translational symmetry along the growth $z$ direction, thus single particle states experience confinement and two additional QW subband quantum numbers $n_v,n_c$ (for valence and conduction states respectively) appear. We still leave $a$ as a reminder that new possible indexes could still arise in due course. Projecting Eq. (\[sec eq dir l\]) on these confined states we end up with an effective Schrödinger equation in the plane $$\begin{aligned} \label{Wannier prima} {\Bigg (} - \frac{\hbar^2}{2m_e} \nabla^2_{\bf r^\\|_e} - \frac{\hbar^2}{2m_h} \nabla^2_{\bf r^\\|_h} - U_{n_c,n_v;n'_c,n'_v}(\|{\bf r}^\\|_e - {\bf r}^\\|_h\|) {\Bigg )} F^{\alpha}_{n_c,n_v,a}({\bf r}^\\|_e,{\bf r}^\\|_h) = \nonumber \\ && \hspace{-5.0cm} = (E_{\alpha} - E^z_{n_c} - E^z_{n_v}) F^{\alpha}_{n_c,n_v,a}({\bf r}^\\|_e,{\bf r}^\\|_h)\, ,\end{aligned}$$ with $$U_{n_c,n_v;n'_c,n'_v}(\|{\bf r^\\|_e} - {\bf r^\\|_h\|}) = \int dz_e \int dz_h \frac{e^2}{\varepsilon_r \sqrt{\|{\bf r}^\\|_e - {\bf r}^\\|_h\|^2 +(z_e - z_h)^2}} c_{n_c}(z_e) c_{n'_c}(z_e) v_{n_v}(z_h)v_{n'_v}(z_h)\, .$$ For what already stated, we shall consider only the lowest confined subband levels, then the resulting effective in-plane secular equation becomes $$\begin{aligned} \label{Wannier GS} {\Bigg (} - \frac{\hbar^2}{2m_e} \nabla^2_{\bf r^\\|_e} - \frac{\hbar^2}{2m_h} \nabla^2_{\bf r^\\|_h} - \int dz_e \int dz_h \frac{e^2}{\varepsilon_r \sqrt{\|{\bf r}^\\|_e - {\bf r}^\\|_h\|^2 +(z_e - z_h)^2}} \|c_{n_c}(z_e)\|^2 \|v_{n_v}(z_h)\|^2 {\Bigg )} F^{\alpha}_{a}({\bf r}^\\|_e,{\bf r}^\\|_h) = \nonumber \\ && \hspace{-8.0cm} = (E_{\alpha} - E^z_c - E^z_v) F^{\alpha}_{a}({\bf r}^\\|_e,{\bf r}^\\|_h)\, ,\end{aligned}$$with the product exciton envelope function $U^{\alpha}({\bf r}_e,{\bf r}_h)=F^{\alpha}_{a}({\bf r}^\\|_e,{\bf r}^\\|_h)c(z_e)v(z_h)$. Equation (\[Wannier GS\]) is solvable by separation of variables once we employ a coordinate transformation into center of mass (CM) ${\bf R}=(m_e{\bf r}^\\|_e+m_h{\bf r}^\\|_h)/(m_e+m_h)$ and relative ${\bf \rho}=({\bf r}^\\|_e - {\bf r}^\\|_h)$ exciton coordinates. It reads $$\label{Wannier R rho} {\Bigg (} - \frac{\hbar^2}{2M} \nabla^2_{\bf R} - \frac{\hbar^2}{2\mu} \nabla^2_{\bf \rho} - U({\bf \rho}) {\Bigg )} F^{\alpha}_{a}({\bf R}, {\bf \rho}) = E F^{\alpha}_{a}({\bf R}, {\bf \rho})\, ,$$ with a solution we can arrange as $F^{\alpha}_{a}({\bf R}, {\bf \rho}) = \frac{e^{i\, {\bf K} \cdot {\bf R}}}{\sqrt{A}} W^{\alpha}_a({\bf \rho})$ the latter solution of the relative hydrogen-like 2D problem. Eventually, in real-space representation, we have our exciton wave function with total in-plane CM wave vector ${\bf K}$ ($A$ is the in-plane quantization surface in the free directions) which reads $$\label{exciton} {\midn \sigma {\bf K}\rangle}= \frac{v_0}{\sqrt{A}} \sum_{\bf r_e,r_h} e^{i \bf k \cdot R} W_{n \sigma}(\rho) c(z_e) v(z_h) a^\dag _{c,{\bf r}_e} a^\dag_{h,{\bf r}_h } {\mid0\rangle}\, ,$$ being $\hat{a}^\dag _{c/v,{\bf r}} (\hat{a}_{c/v,{\bf r }})$ creation (annihilation) operator of the conduction- or valence- band electron in the Wannier representation and ${\bf r}_{e/h} = ({\bf r}^{\\|}_{e/h},z_{e/h})$ are to be considered coordinates of the direct lattice, ${\mid\!0\rangle}$ is the crystal ground state. When e.g. exploring the exciton-phonon interaction, it is useful to express exciton states in reciprocal space. With the usual transformation to Bloch representation, ($N = v_0 A\,L$ is the number of unit cells and $L$ is the quantization dimension along the confined direction, $\nu = c,v$), $$\label{BlochWannier} \hat{a}_{\nu ({\bf r},z)} = \frac{1}{\sqrt{N}} \sum_{{\bf k},k_z} e^{i {\bf k} \cdot {\bf r}_n} \hat{a}_{\nu ({\bf k},k_z)}\, ,$$ one obtains: $$\begin{aligned} \label{exc in Bloch rep 1} && {\midn \sigma {\bf K}\rangle} = \sum_{\substack{{\bf k}, {\bf k}'\\{k_z,k'_z}}} \delta_{{\bf K},{\bf k}-{\bf k}'} {\Bigg (} \frac{1}{\sqrt{A}\, L} \int d{\rho} \int dz_e \int dz_h W_{n \sigma}(\rho) c(z_e) v(z_h) e^{-i\, {\bf \rho} \cdot (\eta_h {\bf k} + \eta_e {\bf k}')} e^{-i\, k_z z_e} e^{-i\, k'_z z_h} {\Bigg )} \nonumber \\ && \hspace{3.0cm} a^\dag _{c,({\bf k},k_z)}a_{v,({\bf k}',k'_z)} \bigl\| 0\bigr>\, .\end{aligned}$$ In order to end up with a form as much as possible in analogy with its bulk counterpart we shall define CM and relative coordinates even in the reciprocal lattice: $$\begin{aligned} \label{rules} \left\{ \begin{array}{l} {\bf K} = {\bf k} - {\bf k}' \\ {\bf k}_r = \eta_h {\bf k} + \eta_e {\bf k}' \end{array} \right. \Longrightarrow \left\{ \begin{array}{l} {\bf k} = {\bf k}_r + \eta_e {\bf K} \\ {\bf k}' = {\bf k}_r - \eta_h {\bf K} \end{array} \right.\, .\end{aligned}$$ It becomes $$\begin{aligned} \label{exc in Bloch rep 2} && {\midn \sigma {\bf K}\rangle} = \sum_{{\bf K}, {\bf k}_r} \delta_{{\bf K},{\bf k}-{\bf k}'} \sum_{k_z,k'_z} {\Bigg (} \frac{1}{\sqrt{A}} \int d{\rho} W_{n \sigma}(\rho) e^{-i\, {\bf \rho} \cdot {\bf k}_r} {\Bigg )} {\Bigg (} \frac{1}{\sqrt{L}} \int dz_e c(z_e) e^{-i\, k_z z_e} {\Bigg )} \nonumber \\ && \hspace{3.0cm} {\Bigg (} \frac{1}{\sqrt{L}}\int dz_h v(z_h) e^{-i\, k'_z z_h} {\Bigg )} a^\dag _{c,({\bf k}_r + \eta_e {\bf K},k_z)}a_{v,({\bf k}_r - \eta_h {\bf K},k'_z)} \bigl\| 0\bigr>\, .\end{aligned}$$ Thus $$\begin{aligned} \label{exc in Bloch rep final} && {\midn \sigma {\bf K}\rangle} = \sum_{{\bf k}_r} \sum_{k_z,k'_z} \Phi^{\bf K}_{n \sigma,{\bf k}_r} u^c_{k_z} u^{v }_{k'_z} a^\dag _{c,({\bf k}_r + \eta_e {\bf K},k_z)}a_{v,({\bf k}_r - \eta_h {\bf K},k'_z)} \bigl\| 0\bigr>\, ,\end{aligned}$$ or in the electron-hole picture ($\hat{a}_{v,{\bf k}} = \hat{d}^\dag_{-{\bf k}}$ and $-{\bf k}\left.\right\|_{el} = {\bf k}\left.\right\|_{hole}$) $$\begin{aligned} \label{exc in Bloch rep final hole picture} && {\midn \sigma {\bf K}\rangle} = \sum_{{\bf k}_r} \sum_{k_z,k'_z} \Phi^{\bf K}_{n \sigma,{\bf k}_r} u^e_{k_z} u^{h}_{k'_z} c^\dag _{({\bf k}_r + \eta_e {\bf K},k_z)}d^\dag_{(-{\bf k}_r + \eta_h {\bf K},-k'_z)} \bigl\| 0\bigr>\, ,\end{aligned}$$ with the relations in Eq. 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--- abstract: 'We investigate the motion of a thin rigid body in Stokes flow and the corresponding slender body approximation used to model sedimenting fibers. In particular, we derive a rigorous error bound comparing the rigid slender body approximation to the classical PDE for rigid motion in the case of a closed loop with constant radius. Our main tool is the slender body PDE framework established by the authors and D. Spirn in [@closed_loop; @free_ends], which we adapt to the rigid setting.' author: - \| Yoichiro Mori, Laurel Ohm [^1]\ School of Mathematics, University of Minnesota, Minneapolis, MN 55455 bibliography: - 'rigid\_bib.bib' title: 'An error bound for the slender body approximation of a thin, rigid fiber sedimenting in Stokes flow' --- Introduction ============ Determining the motion of a three-dimensional rigid body sedimenting in a Stokesian fluid is an important problem in both theoretical and computational fluid mechanics. This motion is described by a classical PDE [@corona2017integral; @galdi1999steady; @weinberger1972variational], which we write below in the case of a thin rigid body. We use ${\mathcal{E}}({\bm{u}}) = \frac{1}{2}(\nabla{\bm{u}}+(\nabla{\bm{u}})^{\rm T})$ to denote the symmetric gradient, and $\bm{\sigma}=\bm{\sigma}({\bm{u}},p) = 2{\mathcal{E}}({\bm{u}})-p{\bf I}$ to denote the stress tensor. Let $\Sigma_\epsilon$ denote a closed loop slender body of radius $\epsilon>0$ (to be made precise in Section \[geometry\]) and let $\Omega_\epsilon ={\mathbb{R}}^3 \setminus \overline{\Sigma_\epsilon}$ and $\Gamma_\epsilon={\partial}\Sigma_\epsilon$ (see Figure \[fig:coord\_sys\]). The full PDE description of a slender body undergoing a rigid motion in Stokes flow may be written as follows: $$\label{rigid} \begin{aligned} -\Delta {\bm{u}}^{\rm r} +\nabla p^{\rm r} &=0 \hspace{2.6cm} \text{ in } \Omega_\epsilon \\ {{\rm{div}\,}}{\bm{u}}^{\rm r} &= 0 \hspace{2.6cm} \text{ in } \Omega_\epsilon \\ {\bm{u}}^{\rm r}({\bm{x}}) &= {\bm{v}}^{\rm r} + \bm{\omega}^{\rm r}\times {\bm{x}}, \qquad {\bm{x}}\in \Gamma_\epsilon \\ {\bm{u}}^{\rm r}({\bm{x}}) &\to 0 \hspace{2.6cm} \text{as }{\left\lvert {\bm{x}}\right\rvert}\to \infty \end{aligned}$$ and $$\begin{aligned} \int_{\Gamma_\epsilon} \bm{\sigma}^{\rm r}\bm{n} \; dS &= \bm{F}, \quad \int_{\Gamma_\epsilon} {\bm{x}}\times (\bm{\sigma}^{\rm r}\bm{n}) \; dS = \bm{T} .\end{aligned}$$ Here the total force $\bm{F}\in {\mathbb{R}}^3$ and torque $\bm{T}\in {\mathbb{R}}^3$ are given and we aim to solve for the linear velocity ${\bm{v}}^{\rm r}\in {\mathbb{R}}^3$ and angular velocity $\bm{\omega}^{\rm r}\in{\mathbb{R}}^3$ of the body. Note that the boundary value problem is in fact valid for rigid bodies of arbitrary shape, but for the purposes of this paper we specifically consider here a slender closed loop. Using the variational framework of [@galdi1999steady; @gonzalez2004dynamics; @weinberger1972variational], it can be shown that is a well-posed PDE.\ On the computational side, there has been much recent interest in numerical simulations of rigid particle sedimentation [@guazzelli2006sedimentation; @guazzelli2011fluctuations], and various tools have been developed to facilitate these simulations [@corona2017integral; @jung2006periodic; @mitchell2015sedimentation].\ ![The geometry of the rigid fiber may be parameterized with respect to the orthogonal frame ${\bm{e}}_t(s)$, ${\bm{e}}_{n_1}(s)$, ${\bm{e}}_{n_2}(s)$ defined in Section \[geometry\].[]{data-label="fig:coord_sys"}](SB_geometry_rigid "fig:")\ For a thin rigid body, a commonly-used tool for simplifying simulations is slender body theory, which exploits the thin geometry of the body by approximating the filament as a one-dimensional force density distributed along the fiber centerline. Slender body theory is a popular method for modeling sedimentation of thin fibers, both rigid [@butler2002dynamic; @park2010cloud; @saintillan2005smooth; @shin2009structure] and semi-flexible [@li2013sedimentation; @manikantan2014instability]. Here we will specifically consider the slender body theory established by Keller and Rubinow [@keller1976slender] and further developed in [@gotz2000interactions; @johnson1980improved; @tornberg2004simulating].\ Let ${\bm{X}}: {\mathbb{T}}\equiv {\mathbb{R}}/ {\mathbb{Z}}\to {\mathbb{R}}^3$ denote the coordinates of the slender body centerline, parameterized by arclength $s$ and defined more precisely in Section \[geometry\]. Given a line force density $\bm{f}^{\rm s}(s)$, $s\in {\mathbb{T}}$, the slender body approximation yields a direct expression approximating the velocity of the fiber, given by [@shelley2000stokesian]: $$\label{SBT_expr} \begin{aligned} {\bm{u}}^{\rm s}_{\rm C}(s) &= \bm{\Lambda}[\bm{f}^{\rm s}](s) + \bm{K}[\bm{f}^{\rm s}](s),\\ \bm{\Lambda}[\bm{f}](s) &:= \frac{1}{8\pi}\big[({\bf I}- 3{\bm{e}}_t{\bm{e}}_t^{\rm T})-2({\bf I}+{\bm{e}}_t{\bm{e}}_t^{\rm T}) \log(\pi\epsilon/4) \big]{\bm f}(s) \\ \bm{K}[\bm{f}](s) &:= \frac{1}{8\pi}\int_{{\mathbb{T}}} \left[ \left(\frac{{\bf I}}{\|\bm{R}_0\|}+ \frac{\bm{R}_0\bm{R}_0^{\rm T}}{\|\bm{R}_0\|^3}\right){\bm f}(s') - \frac{{\bf I}+{\bm{e}}_t(s){\bm{e}}_t(s)^{\rm T} }{\|\sin (\pi(s-s'))/\pi\|} {\bm f}(s)\right] \, ds'. \end{aligned}$$ Here ${\bm{e}}_t(s)$ is the unit tangent vector to ${\bm{X}}(s)$ and $\bm{R}_0(s,s') = {\bm{X}}(s) - {\bm{X}}(s')$. The slender body approximation generally allows for bending and flexing of the filament along its centerline and requires specifying the one-dimensional force density over the length of the fiber centerline. If the fiber is constrained to be fully rigid, only the total force $\bm{F}$ and torque $\bm{T}$ must be specified, where $$\label{SBT_cond1} \int_{{\mathbb{T}}} \bm{f}^{\rm s}(s) \, ds = \bm{F}, \quad \int_{{\mathbb{T}}} {\bm{X}}(s)\times \bm{f}^{\rm s}(s) \, ds = \bm{T}.$$ Additionally, we constrain the motion of the fiber centerline to be rigid, i.e. $$\label{SBT_cond2} {\bm{u}}^{\rm s}_{\rm C}(s) = {\bm{v}}^{\rm s}+\bm{\omega}^{\rm s}\times {\bm{X}}(s)$$ for constant vectors ${\bm{v}}^{\rm s}$, $\bm{\omega}^{\rm s}$. These constraints give rise to a system of integral equations which must be solved to obtain the line force density along the slender body (see [@gustavsson2009gravity; @tornberg2006numerical]).\ The aim of this paper is to establish a rigorous error bound between the slender body approximation for rigid motion - and the classical PDE describing the sedimentation of a rigid fiber immersed in Stokes flow. We show the following theorem. \[rigid\_theorem\] Let $\Sigma_\epsilon$ be a slender body as defined in Section \[geometry\]. Suppose the total force $\bm{F}\in {\mathbb{R}}^3$ and torque $\bm{T}\in {\mathbb{R}}^3$ are given, and assume that rigid slender body approximation - is satisfied by some $\bm{f}^{\rm s}\in C^1({\mathbb{T}})$. Then the difference ${\bm{v}}^{\rm r}-{\bm{v}}^{\rm s}$, $\bm{\omega}^{\rm r}-\bm{\omega}^{\rm s}$ between the linear and angular velocities of true rigid motion and the slender body approximation - satisfies $$\label{rigid_err} {\left\lvert {\bm{v}}^{\rm r}-{\bm{v}}^{\rm s} \right\rvert} + {\left\lvert \bm{\omega}^{\rm r}-\bm{\omega}^{\rm s} \right\rvert} \le C\sqrt{\epsilon}{\left\lvert \log\epsilon \right\rvert}^{3/2} \big({\left\lvert \log\epsilon \right\rvert}^{1/2}{\left\lVert \bm{f}^{\rm s} \right\rVert}_{C^1({\mathbb{T}})} + {\left\lvert \bm{F} \right\rvert}+{\left\lvert \bm{T} \right\rvert} \big)$$ for $C$ depending on $c_\Gamma$, $\kappa_{\max}$, and $\xi_{\max}$. The constants $c_\Gamma$, $\kappa_{\max}$, and $\xi_{\max}$ have to do only with the shape of the fiber centerline and are defined in Section \[geometry\]. Notice that we must assume that we can find $\bm{f}^{\rm s}\in C^1({\mathbb{T}})$ satisfying -, and that this $\bm{f}^{\rm s}$ then appears in the final error bound. In this sense, our bound should be considered as an [a posteriori]{} error estimate, similar to the type of estimates commonly used in finite element analysis [@ainsworth2011posteriori]. To obtain an [a priori]{} bound, we would need to able to say that such an $\bm{f}^{\rm s}$ is then bounded by the given $\bm{F}$ and $\bm{T}$, but the proof of such a bound is hindered by the likely ill-posedness of the rigid slender body approximation itself. Before discussing this issue in greater detail, we note how it arises in the error analysis.\ In order to compare the classical PDE with the rigid slender body approximation, we introduce an intermediary PDE which we will call the [slender body PDE for rigid motion]{}. The idea follows from the notion of slender body PDE proposed by the authors and D. Spirn in [@closed_loop] and [@free_ends] as a framework for analyzing the error introduced by the Keller-Rubinow slender body approximation for closed-loop and open-ended fibers, respectively. To construct the rigid slender body PDE, we impose that the velocity of the slender body is uniform over each cross section $s$ of the fiber. In particular, we approximate ${\bm{x}}\in \Gamma_\epsilon$ as its $L^2$ projection onto the fiber centerline ${\bm{X}}(s)$, thereby ignoring slight differences in torque across the slender body. Note that the slender body geometry is defined in Section \[geometry\] such that this projection onto the fiber centerline is unique; i.e. the notion of “fiber cross section" is well-defined. We define the slender body PDE for rigid motion as follows: $$\label{SB_PDE} \begin{aligned} -\Delta {\bm{u}}^{\rm p} +\nabla p^{\rm p} &=0 \hspace{3.27cm} \text{in } \Omega_\epsilon \\ {{\rm{div}\,}}{\bm{u}}^{\rm p} &=0 \hspace{3.27cm} \text{in } \Omega_\epsilon \\ {\bm{u}}^{\rm p}({\bm{x}}) &= {\bm{v}}^{\rm p} + \bm{\omega}^{\rm p}\times {\bm{X}}(s) \qquad \text{on } \Gamma_\epsilon \\ {\bm{u}}^{\rm p}({\bm{x}}) &\to 0 \hspace{3.23cm} \text{as }{\left\lvert {\bm{x}}\right\rvert}\to \infty \end{aligned}$$ and $$\begin{aligned} \int_{\Gamma_\epsilon} \bm{\sigma}^{\rm p}\bm{n} \; {\mathcal{J}}_\epsilon(s,\theta) \, d\theta \, ds &= \bm{F}, \qquad \int_{{\mathbb{T}}} {\bm{X}}(s)\times \bigg(\int_0^{2\pi} \bm{\sigma}^{\rm p}\bm{n} \, {\mathcal{J}}_\epsilon(s,\theta) d\theta \bigg) ds = \bm{T}.\end{aligned}$$ Here we have written $dS = {\mathcal{J}}_\epsilon(s,\theta)\, d\theta\, ds$, where ${\mathcal{J}}_\epsilon$ is the Jacobian factor on the slender body surface, which we parameterize as a tube about ${\bm{X}}(s)$ using surface angle $\theta$ (see Section \[geometry\] and expression ). We show that for a closed filament, the rigid slender body PDE is in fact close to the classical PDE for rigid motion [@galdi1999steady; @weinberger1972variational] – in particular, the variation in torque over any cross section of the slender body is higher order in $\epsilon$.\ In the case of a flexible filament with a prescribed force density per unit length along the centerline, the slender body PDE of [@closed_loop; @free_ends] is well-posed, and the difference between the slender body approximation and the PDE solution can be estimated in terms of the slender body radius and the given line force density. We would like to use the existing error analysis in [@closed_loop] to bound the difference between the rigid slender body approximation and the rigid slender body PDE solution. The rigid case is complicated by the fact that the existing error bound relies on knowledge of line force density along the filament, and therefore on the solvability of the integral equation for this force density arising from -.\ This raises the issue of the lack of a general solution theory for this particular type of integral equation. Specifically, a detailed spectral analysis by Götz [@gotz2000interactions] in the case of a straight slender body centerline shows that the slender body operator $(\bm{\Lambda}+\bm{K})$ in is not necessarily invertible for all values of $\epsilon$ and all centerline velocities ${\bm{u}}^{\rm s}_{\rm C}$. A similar result for a perfectly circular, planar centerline was shown by Shelley-Ueda in [@shelley2000stokesian]. For fibers with more general centerline curvature, a spectral analysis of the slender body integral operator is complicated and the invertibility properties remain unclear.\ In rigid slender body theory, this same integral operator arises in the integral equations for the line force density $\bm{f}^{\rm s}$. Therefore, to carry out an error analysis for the rigid slender body problem we must assume from the start that we are considering a slender body approximation that gives rise to $\bm{f}^{\rm s}\in C^1({\mathbb{T}})$ satisfying -. This $\bm{f}^{\rm s}$ must be included in the final error bound along with the total force $\bm{F}$ and torque $\bm{T}$, giving rise to a type of [a posteriori]{} error estimate. Furthermore, the $\sqrt{\epsilon}$ bound is the best we can do using our current techniques, but it is possible that a more complete solution theory for the slender body approximation could improve this bound.\ In practice, various regularizations of the slender body integral operator are used to combat this ill-posedness, although rigorous justification for these regularizations is still needed. Given that the slender body PDE framework of [@closed_loop; @free_ends] [is]{} well-posed, we should be able to use this framework to come up with the best regularization for the ill-posed slender body approximation. However, this is truly a deeper issue that we plan to explore in future work. Outline of the proof of Theorem \[rigid\_theorem\] -------------------------------------------------- The strategy for proving Theorem \[rigid\_theorem\] is to show that, given $\bm{F}$ and $\bm{T}$, the solution to the rigid slender body PDE is close to both the classical rigid PDE solution and the rigid slender body approximation - .\ First, we must show that the rigid slender body PDE is well-posed. Using Definition \[rigid\_weak\_p\] of a weak solution to the rigid slender body PDE , we show the following. \[SB\_PDE\_well\] Let $\Sigma_\epsilon$ be a slender body as defined in Section \[geometry\]. Given $\bm{F}$ and $\bm{T}\in{\mathbb{R}}^3$, there exists a unique weak solution $({\bm{u}}^{\rm p},p^{\rm p})\in {\mathcal{R}}_\epsilon^{{{\rm{div}\,}}} \times L^2(\Omega_\epsilon)$ to the slender body PDE for rigid motion satisfying the estimate $$\label{totalPest} {\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} + {\left\lVert p^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} \le C{\left\lvert \log\epsilon \right\rvert}^{1/2} ( {\left\lvert \bm{F} \right\rvert}+{\left\lvert \bm{T} \right\rvert} )$$ for $C$ depending on $c_\Gamma$ and $\kappa_{\max}$. Theorem \[SB\_PDE\_well\] can be established using many of the same tools from the well-posedness theory in [@closed_loop]. In addition, we will make use of the following bound along the slender body centerline ${\bm{X}}(s)$: \[v\_omega\_bd\] Let ${\bm{X}}$ be as in Section \[geometry\] and consider constant vectors ${\bm{v}}$, $\bm{\omega} \in {\mathbb{R}}^3$. Then $${\left\lvert {\bm{v}}\right\rvert}+{\left\lvert \bm{\omega} \right\rvert} \le C{\left\lVert {\bm{v}}+ \bm{\omega}\times {\bm{X}}\right\rVert}_{L^2({\mathbb{T}})}$$ for $C$ depending only on $c_\Gamma$ and $\kappa_{\max}$. We will first prove Lemma \[v\_omega\_bd\] in Section \[2ndLemma\]; then Theorem \[SB\_PDE\_well\] quickly follows using some of the key inequalities collected in Section \[ineq\].\ With the variational framework for , comparing to is relatively straightforward. Using Lemma \[v\_omega\_bd\], we show that the difference between the true rigid motion and the slender body PDE description satisfies the following lemma. \[true\_vs\_SB\] Let ${\bm{X}}$ be as in Section \[geometry\]. Given $\bm{F}$ and $\bm{T}\in {\mathbb{R}}^3$, let $({\bm{v}}^{\rm r},\bm{\omega}^{\rm r})$ be the corresponding boundary values satisfying and let $({\bm{v}}^{\rm p},\bm{\omega}^{\rm p})$ be the boundary values satisfying . Then $$\label{true_vs_SB_bd} {\left\lvert \bm{\omega}^{\rm r}-\bm{\omega}^{\rm p} \right\rvert}+ {\left\lvert {\bm{v}}^{\rm r}-{\bm{v}}^{\rm p} \right\rvert} \le \epsilon{\left\lvert \log\epsilon \right\rvert}C( {\left\lvert \bm{T} \right\rvert}+ {\left\lvert \bm{F} \right\rvert})$$ where $C$ depends only on $c_\Gamma$ and $\kappa_{\max}$. The main difficulties in proving Theorem \[rigid\_theorem\] arise in comparing to - . As discussed, the solvability issue for the rigid slender body approximation is addressed by assuming that we are only considering a rigid slender body approximation - that gives rise to a force density $\bm{f}^{\rm s}\in C^1({\mathbb{T}})$. An additional difficulty arises in that in order to use the error analysis framework of [@closed_loop], the line force density along the slender body must be the same for both the slender body approximation and the slender body PDE. Therefore we need to define yet another intermediary PDE.\ Given $\bm{f}^{\rm s}\in C^1({\mathbb{T}})$ satisfying - for given $\bm{F}$ and $\bm{T}\in {\mathbb{R}}^3$, we define ${\bm{u}}^{\rm p,s}$ as the solution to the PDE: $$\label{SB_PDE_ps} \begin{aligned} -\Delta {\bm{u}}^{\rm p,s} +\nabla p^{\rm p,s} &=0 \hspace{2.75cm} \text{in } \Omega_\epsilon \\ {{\rm{div}\,}}{\bm{u}}^{\rm p,s} &=0 \hspace{2.75cm} \text{in } \Omega_\epsilon \\ \int_0^{2\pi} (\bm{\sigma}^{\rm p,s}\bm{n}) \; {\mathcal{J}}_\epsilon(s,\theta) \, d\theta &= \bm{f}^{\rm s}(s) \hspace{2.05cm} \text{on }\Gamma_\epsilon \\ {\bm{u}}^{\rm p,s}\big\|_{\Gamma_\epsilon} &= {\rm Tr}({\bm{u}}^{\rm p,s})(s), \qquad \text{ unknown but independent of } \theta \\ {\bm{u}}^{\rm p,s}&\to 0 \hspace{2.65cm} \text{as } {\left\lvert {\bm{x}}\right\rvert}\to\infty. \end{aligned}$$ Here ${\rm Tr}({\bm{u}}^{\rm p,s})(s)$ denotes the trace of ${\bm{u}}^{\rm p,s}$ on $\Gamma_\epsilon$, and $\theta$ refers to the parameterization of $\Gamma_\epsilon$ as a tube about ${\bm{X}}(s)$ (see Section \[geometry\]). By [@closed_loop], we know that a (weak) solution $({\bm{u}}^{\rm p,s},p^{\rm p,s})$ exists and is unique. Now, ${\rm Tr}({\bm{u}}^{\rm p,s})(s)$ may not be precisely a rigid motion, but we can show that it is close. In particular, by Theorem 1.3 in [@closed_loop], we may bound the difference between ${\rm Tr}({\bm{u}}^{\rm p,s})(s)$ and ${\bm{u}}^{\rm s}_{\rm C}(s) = {\bm{v}}^{\rm s}+\bm{\omega}^{\rm s}\times {\bm{X}}(s)$ by $$\label{centerline_err} {\left\lVert {\rm Tr}( {\bm{u}}^{\rm p,s}) - ({\bm{v}}^{\rm s}+\bm{\omega}^{\rm s} \times {\bm{X}}) \right\rVert}_{L^2({\mathbb{T}})} \le C\epsilon {\left\lvert \log\epsilon \right\rvert}^{3/2}{\left\lVert \bm{f}^{\rm s} \right\rVert}_{C^1({\mathbb{T}})}$$ for $C$ depending only on $c_\Gamma$ and $\kappa_{\max}$.\ A further technical issue arises in comparing to . In order to obtain a useful estimate of the difference between $({\bm{u}}^{\rm p,s}-{\bm{u}}^{\rm p},p^{\rm p,s}-p^{\rm p})$ in terms of only $\bm{F}$, $\bm{T}$, and $\bm{f}^{\rm s}(s)$, we will need a careful characterization of the $\epsilon$-dependence in a higher regularity estimate for solutions to (see Lemma \[high\_reg\]). Note that for a (sufficiently smooth) sedimenting rigid body, once well-posedness of the PDE has been established, higher regularity of the solution follows by standard arguments for a Stokes Dirichlet boundary value problem. In our case, the novelty is determining how the higher regularity bound scales with $\epsilon$. Our proof (see Appendix \[reg\_lem\]) makes use of the local coordinate system valid near the slender body. We obtain commutator estimates for the tangential derivatives along the slender body surface and use an integration by parts argument, along with the form of the Stokes equations in local coordinates, to show that the bound for an additional derivative of the rigid slender body PDE solution scales like $1/\epsilon$, up to logarithmic corrections.\ Now, using the variational framework for the slender body PDE along with this higher regularity lemma, we can show the following estimate. \[ups\_up\_err\] Let ${\bm{u}}^{\rm p,s}$ satisfy and let $({\bm{v}}^{\rm p},\bm{\omega}^{\rm p})$ denote the rigid slender body PDE boundary values satisfying . Then $${\left\lVert {\rm Tr}( {\bm{u}}^{\rm p,s}) - ( {\bm{v}}^{\rm p} + \bm{\omega}^{\rm p}\times {\bm{X}}) \right\rVert}_{L^2({\mathbb{T}})} \le C\sqrt{\epsilon}{\left\lvert \log\epsilon \right\rvert}^{3/2} \big({\left\lvert \log\epsilon \right\rvert}^{1/2}{\left\lVert \bm{f}^{\rm s} \right\rVert}_{C^1({\mathbb{T}})} + {\left\lvert \bm{F} \right\rvert}+{\left\lvert \bm{T} \right\rvert} \big)$$ for $C$ depending on $c_\Gamma$, $\kappa_{\max}$, and $\xi_{\max}$. Combining estimate with Lemma \[ups\_up\_err\] and using Lemma \[v\_omega\_bd\] with ${\bm{v}}^{\rm p}-{\bm{v}}^s$ and $\bm{\omega}^{\rm p}-\bm{\omega}^{\rm s}$ in place of ${\bm{v}}$ and $\bm{\omega}$, we obtain the following bound for the difference between the slender body approximation - and the slender body PDE : $$\label{PDE_vs_SBT} {\left\lvert {\bm{v}}^{\rm p}-{\bm{v}}^{\rm s} \right\rvert} + {\left\lvert \bm{\omega}^{\rm p}-\bm{\omega}^{\rm s} \right\rvert} \le C\sqrt{\epsilon}{\left\lvert \log\epsilon \right\rvert}^{3/2} \big({\left\lvert \log\epsilon \right\rvert}^{1/2}{\left\lVert \bm{f}^{\rm s} \right\rVert}_{C^1({\mathbb{T}})} + {\left\lvert \bm{F} \right\rvert}+{\left\lvert \bm{T} \right\rvert} \big).$$ Finally, combining the estimate with Lemma \[true\_vs\_SB\] yields Theorem \[rigid\_err\]. The remainder of this paper is thus devoted to showing Lemmas \[v\_omega\_bd\] - \[ups\_up\_err\]. We will begin by introducing the variational framework for and noting some key inequalities in Section \[variational0\]. In Section \[2ndLemma\], we show Lemma \[v\_omega\_bd\] and use it to derive estimates for $({\bm{u}}^{\rm p},p^{\rm p},{\bm{v}}^{\rm p},\bm{\omega}^{\rm p})$ satisfying . These estimates can then be used to show Theorem \[SB\_PDE\_well\]. In Section \[3rdLemma\], we use the variational framework for the rigid slender body PDE to prove Lemma \[true\_vs\_SB\]. Finally, in Section \[1stLemma\], we prove Lemma \[ups\_up\_err\] to complete the proof of Theorem \[rigid\_err\]. Geometry and variational framework {#variational0} ================================== We begin in Section \[geometry\] with a precise definition of the slender body geometry. In Section \[variational\], we introduce the variational form of the slender body PDE for rigid motion , which, along with the variational form of , will provide the framework for obtaining Theorem \[rigid\_theorem\]. Finally, in Section \[ineq\], we make note of some key inequalities that will be used throughout the remainder of this paper. Slender body geometry {#geometry} --------------------- As in [@closed_loop], we let ${\bm{X}}: {\mathbb{T}}\equiv {\mathbb{R}}/ {\mathbb{Z}}\to {\mathbb{R}}^3$ denote the coordinates of a closed, non-self-intersecting $C^3$ curve in ${\mathbb{R}}^3$, parameterized by arclength $s$. We require that $$\label{cgamma} \inf_{s\neq s'}\frac{{\left\lvert {\bm{X}}(s)-{\bm{X}}(s') \right\rvert}}{{\left\lvert s-s' \right\rvert}} \ge c_\Gamma$$ for some constant $c_\Gamma>0$.\ Along ${\bm{X}}(s)$ we consider the orthonormal frame $({\bm{e}}_t(s),{\bm{e}}_{n_1}(s),{\bm{e}}_{n_2}(s))$ defined in [@closed_loop]. Here ${\bm{e}}_t(s) = \frac{d{\bm{X}}}{ds}$ is the unit tangent vector to ${\bm{X}}(s)$ and $({\bm{e}}_{n_1}(s),{\bm{e}}_{n_2}(s))$ span the plane normal to ${\bm{e}}_t(s)$. The frame satisfies the ODEs $$\label{C2_frame} \frac{d}{ds}{\bm{e}}_t = \kappa_1{\bm{e}}_{n_1}+\kappa_2{\bm{e}}_{n_2}, \quad \frac{d}{ds}{\bm{e}}_{n_1} = -\kappa_1{\bm{e}}_t+\kappa_3{\bm{e}}_{n_2}, \quad \frac{d}{ds}{\bm{e}}_{n_2} = -\kappa_2{\bm{e}}_t-\kappa_3{\bm{e}}_{n_1}$$ where $\kappa_1^2(s) +\kappa_2^2(s) = \kappa^2(s)$, the fiber curvature, and $\kappa_3$ is a constant satisfying ${\left\lvert \kappa_3 \right\rvert}\le \pi$. We require the orthonormal frame to be $C^2$ and denote $$\kappa_{\max} := \max_{s\in{\mathbb{T}}} {\left\lvert \kappa(s) \right\rvert}, \quad \xi_{\max} = \max_{s\in{\mathbb{T}}} {\left\lvert \frac{{\partial}^3{\bm{X}}}{{\partial}s^3} \right\rvert} .$$ Note that ${\left\lvert {\partial}\kappa_1 /{\partial}s \right\rvert} +{\left\lvert {\partial}\kappa_2/{\partial}s \right\rvert} \le \xi_{\max}+2(\kappa_{\max}+\pi)$.\ We define $${\bm{e}}_\rho(s,\theta) := \cos\theta{\bm{e}}_{n_1}(s)+\sin\theta{\bm{e}}_{n_2}(s)$$ and, for some $r_{\max}=r_{\max}(c_\Gamma,\kappa_{\max})\le \frac{1}{2\kappa_{\max}}$, we can uniquely parameterize points ${\bm{x}}$ within a neighborhood ${\rm dist}({\bm{x}},{\bm{X}})< r_{\max}$ of ${\bm{X}}(s)$ as $${\bm{x}}= {\bm{X}}(s) + \rho{\bm{e}}_\rho(s,\theta), \quad 0\le \rho <r_{\max}.$$ For $\epsilon<r_{\max}/4$, we may then define a slender body of uniform radius $\epsilon$ as $$\label{SB_def} \Sigma_\epsilon:= \big\{{\bm{x}}\in {\mathbb{R}}^3 \; : \; {\bm{x}}= {\bm{X}}(s) + \rho{\bm{e}}_\rho(s,\theta), \; \rho <\epsilon \big\}.$$ We parameterize the slender body surface $\Gamma_\epsilon = {\partial}\Sigma_\epsilon$ as $$\Gamma_\epsilon = {\bm{X}}(s) + \epsilon{\bm{e}}_\rho(s,\theta).$$ In addition, we may parameterize the Jacobian factor ${\mathcal{J}}_\epsilon(s,\theta)$ on the slender body surface as $$\label{jac_fac} {\mathcal{J}}_\epsilon(s,\theta) = \epsilon\big(1-\epsilon(\kappa_1(s)\cos\theta+\kappa_2(s)\sin\theta) \big).$$ Variational form of {#variational} -------------------- Letting $\Omega_\epsilon={\mathbb{R}}^3\backslash\overline{\Sigma_\epsilon}$ for $\Sigma_\epsilon$ as in Section \[geometry\], we recall the following function spaces, used in [@closed_loop] to study a slender body PDE of the form . We use $D^{1,2}(\Omega_\epsilon)$ to denote the homogeneous Sobolev space $$\label{D12} D^{1,2}(\Omega_\epsilon) = \big\{ {\bm{u}}\in L^6(\Omega_\epsilon) \; : \; \nabla {\bm{u}}\in L^2(\Omega_\epsilon) \big\},$$ which, due to the Sobolev inequality in $\Omega_\epsilon \subset {\mathbb{R}}^3$ (see Lemma \[sobolev\]), is a Hilbert space with norm ${\left\lVert \nabla {\bm{u}}\right\rVert}_{L^2(\Omega_\epsilon)}$. We define $D^{1,2}_0(\Omega_\epsilon)$ as the closure of $C_0^\infty(\Omega_\epsilon)$ (smooth, compactly supported test functions) in $D^{1,2}(\Omega_\epsilon)$.\ We also recall the space ${\mathcal{A}}_\epsilon$, the subspace of $D^{1,2}(\Omega_\epsilon)$ with $\theta$-independent boundary values: $$\label{Aepsilon} {\mathcal{A}}_\epsilon = \big\{ {\bm{u}}\in D^{1,2}(\Omega_\epsilon) \; : \; {\bm{u}}\big\|_{\Gamma_\epsilon} = {\bm{u}}(s) \big\}.$$ Here the boundary value ${\bm{u}}\big\|_{\Gamma_\epsilon} = {\bm{u}}(s)$ is not directly specified but is required to be independent of the surface angle $\theta$. We define ${\mathcal{A}}_\epsilon^{{\rm{div}\,}}$ to be the divergence-free subspace of ${\mathcal{A}}_\epsilon$.\ We also recall the variational form of , examined in detail in [@closed_loop]. A weak solution ${\bm{u}}^{\rm p,s}\in {\mathcal{A}}_\epsilon^{{{\rm{div}\,}}}$ to satisfies $$\label{weak_no_p} \int_{\Omega_\epsilon} 2 {\mathcal{E}}({\bm{u}}^{\rm p,s}): {\mathcal{E}}({\bm{v}}) \, d{\bm{x}}= \int_{\mathbb{T}}{\bm{v}}(s)\cdot\bm{f}^{\rm s}(s) \, ds$$ for any ${\bm{v}}\in{\mathcal{A}}_\epsilon^{{{\rm{div}\,}}}$. In addition, for ${\bm{u}}^{\rm p,s}$ satisfying , there exists a unique pressure $p^{\rm p,s}\in L^2(\Omega_\epsilon)$ satisfying $$\label{weak_with_p} \int_{\Omega_\epsilon} \big( 2 {\mathcal{E}}({\bm{u}}^{\rm p,s}): {\mathcal{E}}({\bm{v}}) - p^{\rm p,s}\, {{\rm{div}\,}}{\bm{v}}\big)\, d{\bm{x}}= \int_{\mathbb{T}}{\bm{v}}(s)\cdot\bm{f}^{\rm s}(s) \, ds$$ for any ${\bm{v}}\in {\mathcal{A}}_\epsilon$. To study , we define the following subspace of ${\mathcal{A}}_\epsilon$, where we further restrict the boundary value to be a rigid motion: $$\label{Repsilon} {\mathcal{R}}_\epsilon = \big\{ {\bm{u}}\in D^{1,2}(\Omega_\epsilon) \; : \; {\bm{u}}\big\|_{\Gamma_\epsilon} = {\bm{v}}+ \bm{\omega}\times{\bm{X}}(s) \text{ for }{\bm{v}},\, \bm{\omega}\in {\mathbb{R}}^3 \big\}.$$ Again, ${\bm{v}}$ and $\bm{\omega}$ are not directly specified but are required to be constant vectors in ${\mathbb{R}}^3$. We let ${\mathcal{R}}_\epsilon^{{\rm{div}\,}}$ denote the divergence-free subspace of ${\mathcal{R}}_\epsilon$.\ We then define a weak solution to the rigid motion slender body PDE as follows. \[rigid\_weak\] A weak solution ${\bm{u}}^{\rm p}\in {\mathcal{R}}_\epsilon^{{\rm{div}\,}}$ to satisfies $$\int_{\Omega_\epsilon} 2\, {\mathcal{E}}({\bm{u}}^{\rm p}): {\mathcal{E}}(\bm{\varphi}) \, d{\bm{x}}= {\bm{v}}_\varphi \cdot\bm{F} + \bm{\omega}_\varphi \cdot\bm{T}$$ for any $\bm{\varphi}\in {\mathcal{R}}_\epsilon^{{\rm{div}\,}}$, where we denote $\bm{\varphi}\big\|_{\Gamma_\epsilon} = {\bm{v}}_\varphi + \bm{\omega}_\varphi \times {\bm{X}}(s)$. Given the existence and uniqueness of ${\bm{u}}^{\rm p}$ satisfying Definition \[rigid\_weak\], using an essentially identical proof to that in Section 2.2 of [@closed_loop], we can establish an equivalent notion of weak solution that includes a corresponding weak pressure $p^{\rm p}\in L^2(\Omega_\epsilon)$ and removes the divergence-free restriction on test functions $\bm{\varphi}$. \[rigid\_weak\_p\] Given ${\bm{u}}^{\rm p}\in {\mathcal{R}}_\epsilon^{{\rm{div}\,}}$ satisfying Definition \[rigid\_weak\], there exists a unique $p^{\rm p}\in L^2(\Omega_\epsilon)$ satisfying $$\int_{\Omega_\epsilon} \big(2\, {\mathcal{E}}({\bm{u}}^{\rm p}): {\mathcal{E}}(\bm{\varphi}) - p\, {{\rm{div}\,}}\bm{\varphi} \big) \, d{\bm{x}}= {\bm{v}}_\varphi \cdot\bm{F} + \bm{\omega}_\varphi\cdot \bm{T}$$ for any $\bm{\varphi}\in {\mathcal{R}}_\epsilon$. Here we again denote $\bm{\varphi}\big\|_{\Gamma_\epsilon} = {\bm{v}}_\varphi + \bm{\omega}_\varphi \times {\bm{X}}(s)$. Important inequalities {#ineq} ---------------------- In addition to the definitions of Section \[variational\], we collect the statements of various inequalities that are used throughout the paper, keeping track of the $\epsilon$-dependence in any constants that arise. The proofs of these inequalities are mostly contained in [@closed_loop], with the exception of Lemma \[trace2\], which appears in Appendix \[appendix\].\ First, we note the following ${\mathcal{A}}_\epsilon$ trace inequality, which holds for functions ${\bm{u}}\in {\mathcal{A}}_\epsilon$ due to $\theta$-independence on $\Gamma_\epsilon$. As a slight abuse of notation, the trace operator ${\rm Tr}$, when applied to ${\mathcal{A}}_\epsilon$ functions, will be considered as both a function on $\Gamma_\epsilon$ and on ${\mathbb{T}}$. ($L^2({\mathbb{T}})$ trace inequality)\[trace1\] Let $\Omega_{\epsilon}={\mathbb{R}}^3 \backslash \overline{\Sigma_{\epsilon}}$ be as in Section \[geometry\]. Then any ${\bm{u}}\in {\mathcal{A}}_\epsilon$ satisfies $$\\|{\rm Tr}({\bm{u}})\\|_{L^2({\mathbb{T}})} \le C \|\log\epsilon\|^{1/2} \\| \nabla {\bm{u}}\\|_{L^2(\Omega_{\epsilon})},$$ where the constant $C$ depends on $\kappa_{\max}$ and $c_{\Gamma}$ but is independent of $\epsilon$. The proof of this lemma appears in Appendix A.2.1 of [@closed_loop].\ On the other hand, for general $D^{1,2}(\Omega)$ functions, the following trace inequality holds over the surface $\Gamma_\epsilon$: ($L^2(\Gamma_\epsilon)$ trace inequality)\[trace2\] Let $\Omega_{\epsilon}={\mathbb{R}}^3 \backslash \overline{\Sigma_{\epsilon}}$ be as in Section \[geometry\]. Then any ${\bm{u}}\in D^{1,2}(\Omega_\epsilon)$ satisfies $$\\|{\rm Tr}({\bm{u}})\\|_{L^2(\Gamma_\epsilon)} \le C \sqrt{\epsilon}\|\log\epsilon\|^{1/2} \\| \nabla {\bm{u}}\\|_{L^2(\Omega_{\epsilon})},$$ where the constant $C$ depends on $\kappa_{\max}$ and $c_{\Gamma}$ but is independent of $\epsilon$. The proof of Lemma \[trace2\] appears in Appendix \[appendix\].\ We will also need the following Korn inequality. (Korn inequality)\[korn\] Let $\Omega_{\epsilon}={\mathbb{R}}^3 \backslash \overline{\Sigma_{\epsilon}}$ be as in Section \[geometry\]. Then any ${\bm{u}}\in D^{1,2}(\Omega_{\epsilon})$ satisfies $$\\|\nabla {\bm{u}}\\|_{L^2(\Omega_{\epsilon})} \le C\\|{\mathcal{E}}({\bm{u}})\\|_{L^2(\Omega_{\epsilon})},$$ where the constant $C$ depends only on $\kappa_{\max}$ and $c_{\Gamma}$. The proof of $\epsilon$-independence in the Korn constant is given in Appendix A.2.2 - A.2.3 in [@closed_loop].\ Finally, we make use of the following pressure estimate. \[pressure\] For $({\bm{u}},p)$ satisfying the Stokes equations in $\Omega_\epsilon$, we have $${\left\lVert p \right\rVert}_{L^2(\Omega_\epsilon)} \le C {\left\lVert {\mathcal{E}}({\bm{u}}) \right\rVert}_{L^2(\Omega_\epsilon)}$$ for $C$ independent of $\epsilon$. The proof of this lemma exactly follows the proof of estimate (2.17) in [@closed_loop]. Proof of Lemma \[v\_omega\_bd\] and a corollary {#2ndLemma} =============================================== Here we prove Lemma \[v\_omega\_bd\] and make note of a corollary which allows us to obtain a useful bound for functions in $\rm{R}_\epsilon$. This corollary, along with the Korn inequality (Lemma \[korn\]) and pressure estimate (Lemma \[pressure\]), then allows us to prove Theorem \[SB\_PDE\_well\]. Note that Lemma \[v\_omega\_bd\] is obviously true when ${\bm{v}}=\bm{\omega}=0$; thus we can assume that at least one of ${\bm{v}},\bm{\omega}$ is nonzero. Suppose that Lemma \[v\_omega\_bd\] does not hold. Then we may choose a sequence of triples $({\bm{v}}_k,\bm{\omega}_k,{\bm{X}}_k(s))$ such that the following properties hold for each $k=1,2,3,\dots$. First, ${\bm{v}}_k,\bm{\omega}_k\in{\mathbb{R}}^3$ satisfy ${\left\lvert {\bm{v}}_k \right\rvert}^2+{\left\lvert \bm{\omega}_k \right\rvert}^2=1$, and ${\bm{X}}_k(s)$ is a closed curve satisfying the geometric constraints of Section \[geometry\] – in particular, ${\left\lvert {\bm{X}}_k'' \right\rvert}\le \kappa_{\max}$. For simplicity, we also take $\int_{{\mathbb{T}}}{\bm{X}}_k(s) \, ds=0$. In addition, $$1 = {\left\lvert {\bm{v}}_k \right\rvert}^2+{\left\lvert \bm{\omega}_k \right\rvert}^2 > k^2\int_{{\mathbb{T}}}{\left\lvert {\bm{v}}_k+\bm{\omega}_k\times {\bm{X}}_k(s) \right\rvert}^2 \, ds.$$ Then $$\int_{{\mathbb{T}}}{\left\lvert {\bm{v}}_k+\bm{\omega}_k\times {\bm{X}}_k(s) \right\rvert}^2 \, ds < \frac{1}{k^2} \to 0$$ as $k\to\infty$. Since ${\bm{v}}_k,\bm{\omega}_k$ are just vectors in ${\mathbb{R}}^3$, some limit ${\bm{v}}_\infty,\bm{\omega}_\infty$ exists. Furthermore, since each ${\bm{X}}_k$ is controlled in $C^2$ by $\kappa_{\max}$, we have that (passing to a subsequence) ${\bm{X}}_k\to {\bm{X}}_\infty$ in $C^1$ for some closed, unit length curve ${\bm{X}}_\infty(s)$. Thus $$\int_{{\mathbb{T}}}{\left\lvert {\bm{v}}_\infty +\bm{\omega}_\infty \times {\bm{X}}_\infty(s) \right\rvert}^2 \, ds =0,$$ and therefore $\bm{\omega}_\infty\times {\bm{X}}_\infty(s)\equiv -{\bm{v}}_\infty$. But $\bm{\omega}_\infty$ and ${\bm{v}}_\infty$ are both constant vectors with ${\left\lvert {\bm{v}}_\infty \right\rvert}^2+{\left\lvert \bm{\omega}_\infty \right\rvert}^2=1$, while ${\bm{X}}_\infty(s)$ necessarily has nonzero curvature. Thus $\bm{\omega}_\infty\times {\bm{X}}_\infty(s)$ cannot identically equal the constant vector $-{\bm{v}}_\infty$. Furthermore, because ${\bm{X}}_k$ was allowed to vary among curves satisfying the constraints of Section \[geometry\], the constant $C$ arising in Lemma \[v\_omega\_bd\] depends only on $c_\Gamma$ and $\kappa_{\max}$. Given $\bm{F}$ and $\bm{T}$, as an immediate corollary to Lemma \[v\_omega\_bd\] we obtain the following useful bound for any function $\bm{\varphi}\in{\mathcal{R}}_\epsilon$. \[omegaP\] Consider $\bm{\varphi}\in {\mathcal{R}}_\epsilon$ with boundary value denoted by ${\bm{v}}_\varphi+\bm{\omega}_\varphi\times {\bm{X}}(s)$. Then $$\label{curved_est} \|{\bm{v}}_\varphi\|+\|\bm{\omega}_\varphi\| \le C \|\log\epsilon\|^{1/2} {\left\lVert {\mathcal{E}}(\bm{\varphi}) \right\rVert}_{L^2(\Omega_\epsilon)}$$ where $C$ depends only on $c_\Gamma$ and $\kappa_{\max}$. Using Lemma \[v\_omega\_bd\] along with the slender body trace estimate (Lemma \[trace1\]) and Korn inequality (Lemma \[korn\]), we immediately obtain $$\begin{aligned} {\left\lvert {\bm{v}}_\varphi \right\rvert}+ {\left\lvert \bm{\omega}_\varphi \right\rvert} &\le {\left\lVert {\bm{v}}_\varphi+\bm{\omega}_\varphi\times{\bm{X}}\right\rVert}_{L^2({\mathbb{T}})} \le C{\left\lvert \log\epsilon \right\rvert}^{1/2}{\left\lVert \nabla \bm{\varphi} \right\rVert}_{L^2(\Omega_\epsilon)} \\ &\le C{\left\lvert \log\epsilon \right\rvert}^{1/2}{\left\lVert {\mathcal{E}}(\bm{\varphi}) \right\rVert}_{L^2(\Omega_\epsilon)}\end{aligned}$$ for $C$ depending only on $c_\Gamma$ and $\kappa_{\max}$. Using Corollary \[omegaP\] and the variational formulation of , we may now prove Theorem \[SB\_PDE\_well\]. We first show the existence of a weak solution ${\bm{u}}^{\rm p}\in {\mathcal{R}}_\epsilon^{{{\rm{div}\,}}}$ satisfying Definition \[rigid\_weak\]. Note that the bilinear form appearing on the left hand side of Definition \[rigid\_weak\] is bounded on ${\mathcal{R}}_\epsilon^{{{\rm{div}\,}}}$, as $${\left\lvert \int_{\Omega_\epsilon} 2\, {\mathcal{E}}({\bm{u}}^{\rm p}): {\mathcal{E}}(\bm{\varphi}) \, d{\bm{x}}\right\rvert} \le 2{\left\lVert {\mathcal{E}}({\bm{u}}^{\rm p}) \right\rVert}_{L^2(\Omega_\epsilon)}{\left\lVert {\mathcal{E}}(\bm{\varphi}) \right\rVert}_{L^2(\Omega_\epsilon)} \le 2{\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)}{\left\lVert \nabla\bm{\varphi} \right\rVert}_{L^2(\Omega_\epsilon)}.$$ Coercivity of the bilinear form also follows by the Korn inequality (Lemma \[korn\]). Furthermore, using Corollary \[omegaP\], the linear functional on the right hand side of Definition \[rigid\_weak\] is bounded for $\bm{\varphi}\in {\mathcal{R}}_\epsilon^{{{\rm{div}\,}}}$, as $${\left\lvert {\bm{v}}_\varphi \cdot\bm{F} + \bm{\omega}_\varphi\cdot \bm{T} \right\rvert} \le C{\left\lvert \log\epsilon \right\rvert}^{1/2}{\left\lVert {\mathcal{E}}(\bm{\varphi}) \right\rVert}_{L^2(\Omega_\epsilon)}({\left\lvert \bm{F} \right\rvert} +{\left\lvert \bm{T} \right\rvert}) \le C{\left\lvert \log\epsilon \right\rvert}^{1/2}{\left\lVert \nabla\bm{\varphi} \right\rVert}_{L^2(\Omega_\epsilon)}({\left\lvert \bm{F} \right\rvert} +{\left\lvert \bm{T} \right\rvert}).$$ Then, by the Lax-Milgram theorem, there exists a unique weak solution ${\bm{u}}^{\rm p}\in {\mathcal{R}}_\epsilon^{{{\rm{div}\,}}}$ to .\ In addition, using the variational form of along with Corollary \[omegaP\], we have that ${\bm{u}}^{\rm p}$ satisfies $$\begin{aligned} \int_{\Omega_\epsilon} {\left\lvert {\mathcal{E}}({\bm{u}}^{\rm p}) \right\rvert}^2 \, d{\bm{x}}&= \int_{\Gamma_\epsilon} ({\bm{v}}^{\rm p}+\bm{\omega}^{\rm p}\times {\bm{X}}(s))\cdot(\bm{\sigma}^{\rm p}\bm{n}) \, dS \\ &= {\bm{v}}^{\rm p}\cdot\int_{\Gamma_\epsilon} \bm{\sigma}^{\rm p}\bm{n} \, dS + \bm{\omega}^{\rm p}\cdot \int_{{\mathbb{T}}} {\bm{X}}(s)\times \bigg(\int_0^{2\pi} (\bm{\sigma}^{\rm p}\bm{n}) \, {\mathcal{J}}_\epsilon(s,\theta) d \theta \bigg) ds \\ &\le {\left\lvert {\bm{v}}^{\rm p} \right\rvert}{\left\lvert \bm{F} \right\rvert} + {\left\lvert \bm{\omega}^{\rm p} \right\rvert}{\left\lvert \bm{T} \right\rvert} \le C{\left\lvert \log\epsilon \right\rvert}^{1/2}{\left\lVert {\mathcal{E}}({\bm{u}}^{\rm p}) \right\rVert}_{L^2(\Omega_\epsilon)}({\left\lvert \bm{F} \right\rvert} +{\left\lvert \bm{T} \right\rvert}) \\ &\le \frac{1}{2}{\left\lVert {\mathcal{E}}({\bm{u}}^{\rm p}) \right\rVert}_{L^2(\Omega_\epsilon)}^2 + C{\left\lvert \log\epsilon \right\rvert}({\left\lvert \bm{F} \right\rvert}^2 +{\left\lvert \bm{T} \right\rvert}^2),\end{aligned}$$ where we have used Young’s inequality in the last line. We thus obtain the estimate $$\label{Eup_est} {\left\lVert {\mathcal{E}}({\bm{u}}^{\rm p}) \right\rVert}_{L^2(\Omega_\epsilon)} \le C{\left\lvert \log\epsilon \right\rvert}^{1/2} ( {\left\lvert \bm{F} \right\rvert}+{\left\lvert \bm{T} \right\rvert} ).$$ As noted after Definition \[rigid\_weak\], the existence of a unique corresponding weak pressure $p^{\rm p}\in L^2(\Omega_\epsilon)$ satisfying Definition \[rigid\_weak\_p\] as well as Lemma \[pressure\] follows by an essentially identical proof to that appearing in Section 2.2 of [@closed_loop].\ Combining with Lemmas \[korn\] and \[pressure\] then yields the bound . Classical versus slender body PDE description of rigid motion {#3rdLemma} ============================================================= Using the variational framework of Section \[variational0\] along with Lemma \[v\_omega\_bd\], we prove Lemma \[true\_vs\_SB\] comparing the classical PDE and slender body PDE descriptions of rigid slender body motion. The difference $\overline{\bm{u}}= {\bm{u}}^{\rm r}- {\bm{u}}^{\rm p}$, $\overline p= p^{\rm r}- p^{\rm p}$, $\overline{\bm{\sigma}}=\bm{\sigma}^{\rm r}- \bm{\sigma}^{\rm p}$, $\overline{\bm{\omega}}= \bm{\omega}^{\rm r}- \bm{\omega}^{\rm p}$, $\overline{\bm{v}}= {\bm{v}}^{\rm r}- {\bm{v}}^{\rm p}$ satisfies the PDE $$\label{diff_PDE} \begin{aligned} -\Delta \overline{\bm{u}}+\nabla \overline p &=0, \quad {{\rm{div}\,}}\overline{\bm{u}}=0 \hspace{2.3cm} \text{in } \Omega_\epsilon \\ \overline{\bm{u}}({\bm{x}}) &= \overline{\bm{v}}+ \overline{\bm{\omega}}\times {\bm{x}}+ \epsilon \bm{\omega}^{\rm p} \times {\bm{e}}_\rho, \qquad {\bm{x}}\in \Gamma_\epsilon \\ \overline{\bm{u}}({\bm{x}}) &\to 0 \hspace{4.3cm} \text{as }{\left\lvert {\bm{x}}\right\rvert}\to \infty \\ \int_{\Gamma_\epsilon} \overline{\bm{\sigma}}\bm{n} \, dS &=0, \quad \int_{\Gamma_\epsilon} {\bm{x}}\times (\overline{\bm{\sigma}}\bm{n}) \, dS = - \epsilon \int_{\Gamma_\epsilon} {\bm{e}}_\rho \times (\bm{\sigma}^{\rm p}\bm{n}) \; dS. \end{aligned}$$ Then, (formally) multiplying by $\overline{\bm{u}}$ and integrating by parts, we have that $\overline{\bm{u}}$ satisfies $$\label{diff_bound0} \begin{aligned} \int_{\Omega_\epsilon} 2\|{\mathcal{E}}(\overline{\bm{u}})\|^2 \, d{\bm{x}}&= \int_{\Gamma_\epsilon} \big( \overline{\bm{v}}+ \overline{\bm{\omega}}\times {\bm{x}}+ \epsilon\bm{\omega}^{\rm p}\times{\bm{e}}_\rho \big) \cdot(\overline{\bm{\sigma}}\bm{n}) \, dS \\ &= \overline{\bm{v}}\cdot\int_{\Gamma_\epsilon}\overline{\bm{\sigma}}\bm{n} \, dS + \overline{\bm{\omega}}\cdot\int_{\Gamma_\epsilon} {\bm{x}}\times(\overline{\bm{\sigma}}\bm{n}) \, dS + \epsilon\bm{\omega}^{\rm p}\cdot\int_{\Gamma_\epsilon} {\bm{e}}_\rho\times(\overline{\bm{\sigma}}\bm{n}) \, dS \\ &= -\epsilon \overline{\bm{\omega}}\cdot\int_{\Gamma_\epsilon} {\bm{e}}_\rho \times (\bm{\sigma}^{\rm p}\bm{n}) \, dS + \epsilon \bm{\omega}^{\rm p}\cdot\int_{\Gamma_\epsilon} {\bm{e}}_\rho \times(\overline{\bm{\sigma}}\bm{n}) \, dS. \end{aligned}$$ To estimate the right hand side of , we first need to define a smooth cutoff function $\phi(\rho)$ satisfying $$\label{cutoff_def} \phi(\rho) = \begin{cases} 1, & \rho < 2 \\ 0, & \rho >4 \end{cases}$$ with smooth decay satisfying $$\label{phi_decay} {\left\lvert \frac{d\phi}{d\rho} \right\rvert} \le c_\phi.$$ Then for ${\bm{x}}={\bm{X}}(s)+\rho{\bm{e}}_\rho(\theta,s)$ in a neighborhood of $\Gamma_\epsilon$, we define $\phi_\epsilon(\rho):= \phi(\rho/\epsilon)$.\ We estimate the second term on the right hand side first, noting that the estimation of the first term will be essentially identical. Using index notation (the subscript $\cdot_{,j}$ signifies $\frac{{\partial}\cdot}{{\partial}x_j}$; sum over repeated indices) along with the divergence theorem, we may write $$\label{cross_est0} \begin{aligned} \int_{\Gamma_\epsilon} \big({\bm{e}}_\rho \times(\overline{\bm{\sigma}}\bm{n}) \big)_i \, dS &= \int_{\Gamma_\epsilon} \varepsilon_{ijk} (e_\rho)_j \overline\sigma_{k\ell} n_\ell \, dS = \int_{\Omega_\epsilon} (\phi \, \varepsilon_{ijk}(e_\rho)_j \overline\sigma_{k\ell})_{,\ell} \, d{\bm{x}}\\ &= \int_{\Omega_\epsilon} \varepsilon_{ijk}\big( \phi_{,\ell}(e_\rho)_j \overline\sigma_{k\ell} + \phi \,(e_\rho)_{j,\ell} \overline\sigma_{k\ell} \big)\, d{\bm{x}}. \end{aligned}$$ Here $\varepsilon_{ijk}$ is the alternating symbol $$\varepsilon_{ijk} = \begin{cases} 1, & \text{ for even permutations of }i,j,k \\ -1, & \text{ for odd permutations of }i,j,k \\ 0, & \text{ if } i=j,j=k, \text{ or }k=i, \end{cases}$$ and we have used that $\overline{\bm{\sigma}}$ is divergence-free. Although $\overline{\bm{\sigma}}$ only belongs to $L^2(\Omega_\epsilon)$ [a priori]{}, we may justify equation because ${{\rm{div}\,}}\overline{\bm{\sigma}}=0\in L^2(\Omega_\epsilon)$, and therefore $\overline{\bm{\sigma}}$ belongs to the space $H_{{{\rm{div}\,}}}(\Omega_\epsilon):= \{\bm{\varphi}\in L^2(\Omega_\epsilon) \, : \, {{\rm{div}\,}}\bm{\varphi}\in L^2(\Omega_\epsilon) \}$. Then we may use the divergence theorem for $H_{{{\rm{div}\,}}}$ functions given in Lemma IV.3.3 of [@boyer2012mathematical] to obtain . Now, due to the cutoff $\phi_\epsilon$, the integrand on the right hand side of is supported only within the region $${\mathcal{O}}_\epsilon := \big\{{\bm{X}}(s)+\rho {\bm{e}}_\rho(s,\theta) \; : \; s\in{\mathbb{T}}, \; \epsilon \le \rho\le4\epsilon, \; 0\le \theta<2\pi \big\}$$ with ${\left\lvert {\mathcal{O}}_\epsilon \right\rvert}= C\epsilon^2$ for some $C$ depending only on $c_\Gamma$ and $\kappa_{\max}$.\ Within ${\mathcal{O}}_\epsilon$, defining ${\widehat{\kappa}}(s,\theta):= \kappa_1(s)\cos\theta+\kappa_2(s)\sin\theta$, we have $$\label{grad_rho} \begin{aligned} {\left\lvert \nabla {\bm{e}}_\rho(s,\theta) \right\rvert} &= {\left\lvert \frac{1}{\rho}\frac{{\partial}{\bm{e}}_\rho}{{\partial}\theta}{\bm{e}}_\theta^{\rm T}+ \frac{1}{1-\rho{\widehat{\kappa}}}\bigg(\frac{{\partial}{\bm{e}}_\rho}{{\partial}s} - \kappa_3\frac{{\partial}{\bm{e}}_\rho}{{\partial}\theta} \bigg){\bm{e}}_t^{\rm T} \right\rvert} \\ &= {\left\lvert \frac{1}{\rho}{\bm{e}}_\theta{\bm{e}}_\theta^{\rm T} - \frac{1}{1-\rho{\widehat{\kappa}}}(\kappa_1\cos\theta+\kappa_2\sin\theta) {\bm{e}}_t{\bm{e}}_t^{\rm T} \right\rvert} \le \frac{1}{\epsilon}+ 4\kappa_{\max}, \end{aligned}$$ where the final $\kappa_{\max}$ bound is shown in Appendix \[reg\_lem\].\ Using , , and Cauchy-Schwarz, we may estimate as $$\label{cross_bound} \begin{aligned} {\left\lvert \int_{\Gamma_\epsilon} {\bm{e}}_\rho \times(\overline{\bm{\sigma}}\bm{n}) \, dS \right\rvert} &\le \int_{{\mathcal{O}}_\epsilon} \big({\left\lvert \phi_\epsilon\nabla{\bm{e}}_\rho \right\rvert}+{\left\lvert \nabla\phi_\epsilon \right\rvert}){\left\lvert \overline{\bm{\sigma}} \right\rvert} \, d{\bm{x}}\\ &\le {\left\lvert {\mathcal{O}}_\epsilon \right\rvert}^{1/2} \bigg(\frac{1}{\epsilon} + 4\kappa_{\max} + \frac{c_\phi}{\epsilon} \bigg)\bigg(\int_{\Omega_\epsilon} {\left\lvert \overline{\bm{\sigma}} \right\rvert}^2 \, d{\bm{x}}\bigg)^{1/2} \\ & \le C \bigg(\int_{\Omega_\epsilon} {\left\lvert \overline{\bm{\sigma}} \right\rvert}^2 \, d{\bm{x}}\bigg)^{1/2} \end{aligned}$$ where $C$ depends only on the shape of ${\bm{X}}$ – in particular, $c_\Gamma$ and $\kappa_{\max}$. Finally, using Lemma \[pressure\], we obtain $$\label{cross_bound2} {\left\lvert \int_{\Gamma_\epsilon} {\bm{e}}_\rho \times(\overline{\bm{\sigma}}\bm{n}) \, dS \right\rvert} \le C \bigg(\int_{\Omega_\epsilon} \big({\left\lvert {\mathcal{E}}(\overline{\bm{u}}) \right\rvert}^2 + \overline p^2 \big) \, d{\bm{x}}\bigg)^{1/2} \le C \bigg(\int_{\Omega_\epsilon} {\left\lvert {\mathcal{E}}(\overline{\bm{u}}) \right\rvert}^2 \, d{\bm{x}}\bigg)^{1/2}.$$ Following exactly the same procedure, we can also show $$\label{cross_bound3} {\left\lvert \int_{\Gamma_\epsilon} {\bm{e}}_\rho \times(\bm{\sigma}^{\rm p}\bm{n}) \, dS \right\rvert} \le C \bigg(\int_{\Omega_\epsilon} {\left\lvert {\mathcal{E}}({\bm{u}}^{\rm p}) \right\rvert}^2 \, d{\bm{x}}\bigg)^{1/2}.$$ Furthermore, in the same way as in Lemma \[omegaP\], it can be shown that $$\label{omegav_bound} {\left\lvert \overline{\bm{\omega}} \right\rvert}+ {\left\lvert \overline{\bm{v}}\right\rvert} \le C{\left\lVert \overline{\bm{v}}+\overline{\bm{\omega}}\times{\bm{x}}+ \epsilon\bm{\omega}^{\rm p}\times {\bm{e}}_\rho \right\rVert}_{L^2(\Gamma_\epsilon)} \le C\sqrt{\epsilon}{\left\lvert \log\epsilon \right\rvert}^{1/2}{\left\lVert {\mathcal{E}}(\overline{\bm{u}}) \right\rVert}_{L^2(\Omega_\epsilon)},$$ where we have used the $L^2(\Gamma_\epsilon)$ trace estimate (Lemma \[trace2\]).\ Then, using and in along with Lemma \[omegaP\] and , we have $$\label{diff_bound1} \begin{aligned} \int_{\Omega_\epsilon} 2\|{\mathcal{E}}(\overline{\bm{u}})\|^2 \, d{\bm{x}}&\le \epsilon C{\left\lvert \overline{\bm{\omega}} \right\rvert}\bigg(\int_{\Omega_\epsilon}{\left\lvert {\mathcal{E}}({\bm{u}}^{\rm p}) \right\rvert}^2 \, d{\bm{x}}\bigg)^{1/2} + \epsilon C{\left\lvert \bm{\omega}^{\rm p} \right\rvert} \bigg(\int_{\Omega_\epsilon}{\left\lvert {\mathcal{E}}(\overline{\bm{u}}) \right\rvert}^2 \, d{\bm{x}}\bigg)^{1/2} \\ &\le \epsilon{\left\lvert \log\epsilon \right\rvert}^{1/2}C\bigg(\int_{\Omega_\epsilon}{\left\lvert {\mathcal{E}}({\bm{u}}^{\rm p}) \right\rvert}^2 \, d{\bm{x}}\bigg)^{1/2}\bigg(\int_{\Omega_\epsilon}{\left\lvert {\mathcal{E}}(\overline{\bm{u}}) \right\rvert}^2 \, d{\bm{x}}\bigg)^{1/2} \\ &\le \epsilon^2{\left\lvert \log\epsilon \right\rvert}C\int_{\Omega_\epsilon}{\left\lvert {\mathcal{E}}({\bm{u}}^{\rm p}) \right\rvert}^2 \, d{\bm{x}}+ \int_{\Omega_\epsilon}{\left\lvert {\mathcal{E}}(\overline{\bm{u}}) \right\rvert}^2 \, d{\bm{x}}, \end{aligned}$$ where we have used Young’s inequality in the last line. Then, using , we obtain $${\left\lVert {\mathcal{E}}(\overline{\bm{u}}) \right\rVert}_{L^2(\Omega_\epsilon)} \le \epsilon{\left\lvert \log\epsilon \right\rvert}C( {\left\lvert \bm{T} \right\rvert}+ {\left\lvert \bm{F} \right\rvert}).$$ Finally, using again, we obtain Lemma \[true\_vs\_SB\]. Proof of Lemma \[ups\_up\_err\] {#1stLemma} ================================ Finally, we prove Lemma \[ups\_up\_err\] comparing the rigid slender body PDE to the intermediary slender body PDE .\ We begin by defining $$\label{fp_def} \bm{f}^{\rm p}(s):=\int_0^{2\pi} (\bm{\sigma}^{\rm p}\bm{n}) \, {\mathcal{J}}_\epsilon(s,\theta) d\theta$$ for $\bm{\sigma}^{\rm p}$ as in , and establish the following: \[fp\_FT\] Suppose the slender body $\Sigma_\epsilon$ is as in Section \[geometry\] – in particular, ${\bm{X}}\in C^3({\mathbb{T}})$. Let the total force $\bm{F}$ and torque $\bm{T}$ be given, and let $\bm{f}^{\rm p}$ be as defined in . Then $${\left\lVert \bm{f}^{\rm p} \right\rVert}_{L^2({\mathbb{T}})} \le C{\left\lvert \log\epsilon \right\rvert}^{3/2}({\left\lvert \bm{F} \right\rvert}+{\left\lvert \bm{T} \right\rvert})$$ for $C$ depending only on $c_\Gamma$, $\kappa_{\max}$, and $\xi_{\max}$. The proof of this lemma relies on a higher regularity estimate for $\bm{\sigma}^{\rm p}$. Note that once Theorem \[SB\_PDE\_well\] has been established, we immediately obtain that ${\bm{u}}^{\rm p}\big\|_{\Gamma_\epsilon} = {\bm{v}}^{\rm p}+\bm{\omega}^{\rm p}\times {\bm{X}}(s)$ is in $C^3(\Omega_\epsilon)$, since ${\bm{v}}^{\rm p}$ and $\bm{\omega}^{\rm p}$ are just constants in ${\mathbb{R}}^3$ and the fiber centerline ${\bm{X}}$ is in $C^3({\mathbb{T}})$. Given this $C^3$ Dirichlet data, $\bm{\sigma}^{\rm p}\in H^1(\Omega_\epsilon)$ follows by standard higher regularity arguments for the exterior Stokes Dirichlet boundary value problem (see the proof of Lemma V.4.3 in [@galdi2011introduction] or Theorem IV.5.8 in [@boyer2012mathematical]). Note that since ${\bm{X}}\in C^3({\mathbb{T}})$, $\bm{\sigma}^{\rm p}$ should in fact be even more regular, but the method we use to show Lemma \[high\_reg\] only allows us to quantify the $\epsilon$-dependence in the estimate for ${\left\lVert \nabla\bm{\sigma}^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)}$. In particular, we can show the following bound on $\nabla \bm{\sigma}^{\rm p}$. \[high\_reg\] Given $\Omega_\epsilon$ as in Section \[geometry\], the solution $\bm{\sigma}^{\rm p}$ to belongs to $H^1(\Omega_\epsilon)$ and satisfies $$\label{high_reg_eqn} {\left\lVert \nabla\bm{\sigma}^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} \le {\left\lVert \nabla^2{\bm{u}}^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} + {\left\lVert \nabla p^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} \le \frac{C}{\epsilon}{\left\lvert \log\epsilon \right\rvert}^{1/2}\big({\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} + {\left\lVert p^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} \big),$$ where $C$ depends on $c_\Gamma$, $\kappa_{\max}$, and $\xi_{\max}$. The proof of the $\epsilon$-dependence in Lemma \[high\_reg\] is given in Appendix \[reg\_lem\].\ Using Lemma \[high\_reg\] and Corollary \[omegaP\], we have the higher regularity estimate $$\label{high_reg2} \begin{aligned} {\left\lVert \nabla\bm{\sigma}^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} &\le \frac{C}{\epsilon}{\left\lvert \log\epsilon \right\rvert}^{1/2}\bigg({\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} + {\left\lVert p^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} \bigg) \le \frac{C}{\epsilon}{\left\lvert \log\epsilon \right\rvert}({\left\lvert \bm{F} \right\rvert}+{\left\lvert \bm{T} \right\rvert}). \end{aligned}$$ Now, using that ${\mathcal{J}}_\epsilon(s,\theta)>0$ for each $(s,\theta)\in \Gamma_\epsilon$ and the surface measure ${\left\lvert \Gamma_\epsilon \right\rvert} = \int_{\mathbb{T}}\int_0^{2\pi} {\mathcal{J}}_\epsilon(s,\theta) d\theta ds = \epsilon$, we have $$\begin{aligned} {\left\lVert \bm{f}^{\rm p} \right\rVert}_{L^2({\mathbb{T}})}^2 &= \int_{\mathbb{T}}{\left\lvert \int_0^{2\pi} \bm{\sigma}^{\rm p}\bm{n} \, {\mathcal{J}}_\epsilon(s,\theta) d\theta \right\rvert}^2 ds \le {\left\lvert \Gamma_\epsilon \right\rvert} \int_{\mathbb{T}}\int_0^{2\pi} {\left\lvert {\rm Tr}(\bm{\sigma}^{\rm p}) \right\rvert}^2 \, {\mathcal{J}}_\epsilon(s,\theta) d\theta \, ds \\ &\le C\epsilon^2{\left\lvert \log\epsilon \right\rvert}{\left\lVert \nabla\bm{\sigma}^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)}^2 \le C{\left\lvert \log\epsilon \right\rvert}^3({\left\lvert \bm{F} \right\rvert}+{\left\lvert \bm{T} \right\rvert})^2.\end{aligned}$$ Here we have applied both the $L^2(\Gamma_\epsilon)$ trace inequality (Lemma \[trace2\]) and the higher regularity estimate in the last line. With Lemma \[fp\_FT\], we are now equipped to show Lemma \[ups\_up\_err\]. The proof relies on estimates for the PDE satisfied by the difference between solutions to and . Letting ${\widetilde{{\bm{u}}}} = {\bm{u}}^{\rm p,s}-{\bm{u}}^{\rm p}$, ${\widetilde{p}} = p^{\rm p,s}- p^{\rm p} $, ${\widetilde{{\bm{v}}}} = {\bm{v}}^{\rm s}-{\bm{v}}^{\rm p}$, ${\widetilde{\bm{\omega}}} = \bm{\omega}^{\rm s}-\bm{\omega}^{\rm p}$, ${\widetilde{\bm{\sigma}}}=\bm{\sigma}^{\rm p,s} - \bm{\sigma}^{\rm p}$, we consider the following boundary value problem: $$\label{wtbu_eqn} \begin{aligned} -\Delta {\widetilde{{\bm{u}}}} +\nabla {\widetilde{p}} &=0, \quad {{\rm{div}\,}}{\widetilde{{\bm{u}}}} =0 \hspace{2.25cm} \text{in } \Omega_\epsilon \\ {\widetilde{{\bm{u}}}}({\bm{x}}) &= {\widetilde{{\bm{v}}}} + {\widetilde{\bm{\omega}}}\times {\bm{X}}(s) + \bm{R}(s), \qquad {\bm{x}}\in \Gamma_\epsilon \\ {\widetilde{{\bm{u}}}}({\bm{x}}) &\to 0 \hspace{4.3cm} \text{as }{\left\lvert {\bm{x}}\right\rvert}\to \infty \\ \int_{\Gamma_\epsilon} {\widetilde{\bm{\sigma}}}\bm{n} \; dS &= 0, \quad \int_{{\mathbb{T}}} {\bm{X}}(s)\times \bigg(\int_0^{2\pi} {\widetilde{\bm{\sigma}}}\bm{n} \, {\mathcal{J}}_\epsilon(s,\theta) d\theta \bigg) ds = 0, \end{aligned}$$ where $\bm{R}(s):={\rm Tr}({\bm{u}}^{\rm p,s})(s) - \big( {\bm{v}}^{\rm s} + \bm{\omega}^{\rm s}\times {\bm{X}}(s)\big)$ satisfies $$\label{Rest} {\left\lVert \bm{R} \right\rVert}_{L^2({\mathbb{T}})} \le C\epsilon{\left\lvert \log\epsilon \right\rvert}^{3/2}{\left\lVert \bm{f}^{\rm s} \right\rVert}_{C^1({\mathbb{T}})},$$ by . We consider the variational form of : formally, multiplying by by ${\widetilde{{\bm{u}}}}$ and integrating by parts, we have $$\begin{aligned} \int_{\Omega_\epsilon} 2\|{\mathcal{E}}({\widetilde{{\bm{u}}}})\|^2 \, d{\bm{x}}&= \int_{\Gamma_\epsilon} \big({\widetilde{{\bm{v}}}} + {\widetilde{\bm{\omega}}}\times {\bm{X}}(s) +\bm{R}(s) \big) \cdot({\widetilde{\bm{\sigma}}}\bm{n}) \; dS \\ &= {\widetilde{{\bm{v}}}} \cdot \int_{\Gamma_\epsilon}({\widetilde{\bm{\sigma}}}\bm{n}) \; dS + {\widetilde{\bm{\omega}}}\cdot\int_{{\mathbb{T}}} {\bm{X}}(s)\times \bigg(\int_0^{2\pi} {\widetilde{\bm{\sigma}}}\bm{n}\, {\mathcal{J}}_\epsilon(s,\theta) d\theta \bigg) ds \\ &\qquad + \int_{\Gamma_\epsilon} \big({\widetilde{{\bm{v}}}} + {\widetilde{\bm{\omega}}}\times {\bm{X}}(s) +\bm{R}(s) \big) \cdot({\widetilde{\bm{\sigma}}}\bm{n}) \; dS \\ &=\int_{\mathbb{T}}\bm{R}(s) \cdot \big( \bm{f}^{\rm s} -\bm{f}^{\rm p} \big) \; ds \le {\left\lVert \bm{R} \right\rVert}_{L^2({\mathbb{T}})} \big( {\left\lVert \bm{f}^{\rm s} \right\rVert}_{L^2({\mathbb{T}})} + {\left\lVert \bm{f}^{\rm p} \right\rVert}_{L^2({\mathbb{T}})} \big) \\ &\le C\epsilon{\left\lvert \log\epsilon \right\rvert}^{3/2}{\left\lVert \bm{f}^{\rm s} \right\rVert}_{C^1({\mathbb{T}})}\big({\left\lVert \bm{f}^{\rm s} \right\rVert}_{L^2({\mathbb{T}})} + {\left\lvert \log\epsilon \right\rvert}^{3/2}({\left\lvert \bm{F} \right\rvert}+{\left\lvert \bm{T} \right\rvert}) \big). \end{aligned}$$ Here we have used and Lemma \[fp\_FT\] in the final line. It would seem to make sense to try to bound the difference $\bm{f}^{\rm s} -\bm{f}^{\rm p}$ appearing in the second-to-last line by ${\left\lVert {\mathcal{E}}({\widetilde{{\bm{u}}}}) \right\rVert}_{L^2(\Omega_\epsilon)}$, or try to use an extension $\overline{\bm{R}}({\bm{x}})\in D^{1,2}(\Omega_\epsilon)$ with $\overline{\bm{R}}\big\|_{\Gamma_\epsilon} = \bm{R}(s)$ and instead take ${\widetilde{{\bm{u}}}} - \overline{\bm{R}}$ as a test function in the above variational estimate to get rid of the boundary term. In either case, we run into difficulties in that we only have an $L^2({\mathbb{T}})$ estimate for $\bm{R}(s)$, when at least an $H^{1/2}({\mathbb{T}})$ estimate would be needed. However, as noted in Lemma \[high\_reg\], bounding the gradient of a function on $\Omega_\epsilon$ incurs an additional factor of $1/\epsilon$. By scaling, an $H^{1/2}({\mathbb{T}})$ estimate for $\bm{R}(s)$ would likely yield the same $\sqrt{\epsilon}$ factor appearing in Lemma \[ups\_up\_err\]. Now, using the $L^2({\mathbb{T}})$ trace inequality (Lemma \[trace1\]), the Korn inequality (Lemma \[korn\]), and Young’s inequality, along the with above ${\left\lVert {\mathcal{E}}({\widetilde{{\bm{u}}}}) \right\rVert}_{L^2(\Omega_\epsilon)}$ estimate, we have $$\begin{aligned} {\left\lVert {\rm Tr}({\bm{u}}^{\rm p,s}) - ({\bm{v}}^{\rm p}+\bm{\omega}^{\rm p}\times{\bm{X}}) \right\rVert}_{L^2({\mathbb{T}})} &\le C{\left\lvert \log\epsilon \right\rvert}^{1/2}{\left\lVert \nabla{\widetilde{{\bm{u}}}} \right\rVert}_{L^2(\Omega_\epsilon)} \le C{\left\lvert \log\epsilon \right\rvert}^{1/2}{\left\lVert {\mathcal{E}}({\widetilde{{\bm{u}}}}) \right\rVert}_{L^2(\Omega_\epsilon)} \\ &\le C\sqrt{\epsilon}{\left\lvert \log\epsilon \right\rvert}^{3/2} \big({\left\lvert \log\epsilon \right\rvert}^{1/2}{\left\lVert \bm{f}^{\rm s} \right\rVert}_{C^1({\mathbb{T}})} + {\left\lvert \bm{F} \right\rvert}+{\left\lvert \bm{T} \right\rvert} \big),\end{aligned}$$ yielding Lemma \[ups\_up\_err\]. Appendix ======== Here we provide proofs for the $L^2(\Gamma_\epsilon)$ trace inequality (Lemma \[trace2\]) and the higher regularity estimate (Lemma \[high\_reg\]).\ We first recall the following lemma, which will be used throughout the appendix. (Sobolev inequality)\[sobolev\] Let $\Omega_{\epsilon}={\mathbb{R}}^3\backslash\overline{\Sigma_{\epsilon}}$ be as in Section \[geometry\]. For any ${\bm{u}}\in D^{1,2}(\Omega_{\epsilon})$, we have $$\label{sobolev_const} \\| {\bm{u}}\\|_{L^6(\Omega_{\epsilon})} \le C\\|\nabla{\bm{u}}\\|_{L^2(\Omega_{\epsilon})}$$ where $C$ depends only on $c_\Gamma$ and $\kappa_{\max}$. The proof of $\epsilon$-independence of $C$ appears in Appendix A.2.4 of [@closed_loop]. Proof of Lemma \[trace2\] ------------------------- The proof of the $L^2(\Gamma_\epsilon)$ trace inequality follows the same outline as the proof of Lemma \[trace1\], contained in Appendix A.2.1 of [@closed_loop]. In particular, using the $\epsilon$-independent $C^2$-diffeomorphisms $\psi_j$ (defined in Appendix A.2.1, [@closed_loop]) which map segments of the curved slender body $\Sigma_\epsilon$ to a straight cylinder, it suffices to show the $\sqrt{\epsilon}{\left\lvert \log\epsilon \right\rvert}$ dependence of the trace constant for a straight cylinder.\ Accordingly, let ${\mathcal{D}}_\rho\subset{\mathbb{R}}^2$ denote the open disk of radius $\rho$ in ${\mathbb{R}}^2$, centered at the origin, and, for some $a<\infty$, define the cylindrical surface $\Gamma_{\epsilon,a}={\partial}{\mathcal{D}}_\epsilon\times [-a,a]$ and the cylindrical shell ${\mathcal{C}}_{\epsilon,a}= ({\mathcal{D}}_1\backslash\overline{{\mathcal{D}}_\epsilon}) \times [-a,a]$. Consider the function space $$D^{1,2}_\Gamma({\mathcal{C}}_{\epsilon,a})= \big\{ {\bm{u}}\in D^{1,2}({\mathcal{C}}_{\epsilon,a}) \; : \; {\bm{u}}\big\|_{{\partial}{\mathcal{C}}_{\epsilon,a}\backslash\Gamma_{\epsilon,a}} = 0 \big\}.$$ As in the proof of Lemma \[trace1\], it suffices to show the $\sqrt{\epsilon}{\left\lvert \log\epsilon \right\rvert}$ dependence of the $L^2(\Gamma_{\epsilon,a})$ trace constant for functions belonging to $D^{1,2}_\Gamma({\mathcal{C}}_{\epsilon,a})$.\ By estimate (A.4) in [@closed_loop], any ${\bm{u}}\in C^1({\mathcal{C}}_{\epsilon,a}) \cap C^0(\overline{{\mathcal{C}}_{\epsilon,a}})\cap D^{1,2}_\Gamma({\mathcal{C}}_{\epsilon,a})$ satisfies $$\begin{aligned} {\left\lvert {\rm Tr}({\bm{u}}) \right\rvert}^2 \le {\left\lvert \log\epsilon \right\rvert} \int_\epsilon^1 {\left\lvert \frac{{\partial}{\bm{u}}}{{\partial}\rho} \right\rvert}^2 \rho \, d\rho.\end{aligned}$$ Then, noting that the surface element on $\Gamma_{\epsilon,a}$ is simply $\epsilon$, we have $$\begin{aligned} {\left\lVert {\rm Tr}({\bm{u}}) \right\rVert}_{L^2(\Gamma_{\epsilon,a})}^2 &= \int_{-a}^a \int_0^{2\pi} {\left\lvert {\rm Tr}({\bm{u}}) \right\rvert}^2 \epsilon \, d\theta\, ds \\ &\le \epsilon{\left\lvert \log\epsilon \right\rvert} \int_{-a}^a \int_0^{2\pi}\int_\epsilon^1 {\left\lvert \frac{{\partial}{\bm{u}}}{{\partial}\rho} \right\rvert}^2 \rho \, d\rho\, d\theta\, ds \le \epsilon{\left\lvert \log\epsilon \right\rvert}{\left\lVert \nabla{\bm{u}}\right\rVert}_{L^2({\mathcal{C}}_{\epsilon,a})}^2.\end{aligned}$$ The same result for ${\bm{u}}\in D^{1,2}_\Gamma({\mathcal{C}}_{\epsilon,a})$ follows by density. Proof of Lemma \[high\_reg\] {#reg_lem} ---------------------------- To determine the $\epsilon$-dependence of the constant in , it suffices to work locally near the slender body surface and show that Lemma \[high\_reg\] holds within an $\epsilon$-independent region about the slender body centerline. We define the region $$\label{mcO} {\mathcal{O}} = \big\{{\bm{x}}\in \Omega_{\epsilon} \; : \; {\bm{x}}= {\bm{X}}(s) + \rho {\bm{e}}_{\rho}(s,\theta), \quad \epsilon < \rho<r_{\max} \big\},$$ where $r_{\max}$ is as in Section \[geometry\]. Within ${\mathcal{O}}$, we can use the orthonormal frame . We will use the notation ${\partial}_s,{\partial}_\theta,{\partial}_\rho$ to denote derivatives ${\partial}/{\partial}s$, ${\partial}/{\partial}\theta$, ${\partial}/{\partial}\rho$ with respect to the variables $s,\theta,\rho$, defined with respect to the orthonormal frame. We verify the $\epsilon$-dependence in the bound for $\nabla^2{\bm{u}}^{\rm p}$ and $\nabla p^{\rm p}$ in two parts: we first show an $L^2$ bound for derivatives $\nabla({\partial}_s{\bm{u}}^{\rm p})$, $\nabla({\partial}_\theta{\bm{u}}^{\rm p})$, ${\partial}_s p^{\rm p}$, and ${\partial}_\theta p^{\rm p}$ in directions tangent to the slender body surface $\Gamma_\epsilon$, and then use these bounds to estimate the derivatives $\nabla({\partial}_\rho{\bm{u}}^{\rm p})$, ${\partial}_\rho p^{\rm p}$ normal to $\Gamma_\epsilon$.\ We begin by estimating the tangential derivatives $\nabla({\partial}_s{\bm{u}}^{\rm p})$ and $\nabla({\partial}_\theta{\bm{u}}^{\rm p})$. Since the derivatives ${\partial}_s$ and ${\partial}_\theta$ with respect to the orthonormal frame do not commute with the “straight” differential operators $\nabla$ and ${{\rm{div}\,}}$, we will need to make use of the following commutator bounds. \[comm\_ests\] For any function ${\bm{u}}\in D^{1,2}_0({\mathcal{O}})$ and for each of the differential operators $D={{\rm{div}\,}},\, \nabla,\, {\mathcal{E}}(\cdot)$, the following commutator estimates hold: $$\begin{aligned} {\left\lVert [D,{\partial}_\theta]{\bm{u}}\right\rVert}_{L^2({\mathcal{O}})} &\le C{\left\lVert \nabla{\bm{u}}\right\rVert}_{L^2({\mathcal{O}})}, \quad {\left\lVert [D,{\partial}_s]{\bm{u}}\right\rVert}_{L^2({\mathcal{O}})} \le C{\left\lVert \nabla{\bm{u}}\right\rVert}_{L^2({\mathcal{O}})}, \end{aligned}$$ where the constant $C$ depends only on $c_\Gamma$, $\kappa_{\max}$, and $\xi_{\max}$. We begin by denoting $$\begin{aligned} {\bm{e}}_\theta(s,\theta) &= -\sin\theta{\bm{e}}_{n_1}(s) + \cos\theta {\bm{e}}_{n_2}(s),\\ u_\rho &= {\bm{u}}\cdot{\bm{e}}_\rho, \; u_\theta={\bm{u}}\cdot{\bm{e}}_\theta, \; u_s = {\bm{u}}\cdot{\bm{e}}_t. \end{aligned}$$ Then, with respect to the orthonormal frame , the divergence and gradient are given by $$\begin{aligned} {{\rm{div}\,}}{\bm{u}}&= \frac{1}{1-\rho{\widehat{\kappa}}} \bigg( \frac{1}{\rho} \frac{{\partial}(\rho(1-\rho{\widehat{\kappa}})u_\rho)}{{\partial}\rho} + \frac{1}{\rho}\frac{{\partial}((1-\rho{\widehat{\kappa}})u_\theta)}{{\partial}\theta} + \frac{{\partial}u_s}{{\partial}s} \bigg) \\ \nabla {\bm{u}}&= {\bm{e}}_\rho(s,\theta)\frac{{\partial}{\bm{u}}}{{\partial}\rho}^{\rm T} + {\bm{e}}_\theta(s,\theta)\frac{1}{\rho}\frac{{\partial}{\bm{u}}}{{\partial}\theta}^{\rm T} + {\bm{e}}_t(s)\frac{1}{1-\rho{\widehat{\kappa}}} \bigg(\frac{{\partial}{\bm{u}}}{{\partial}s} -\kappa_3 \frac{{\partial}{\bm{u}}}{{\partial}\theta} \bigg)^{\rm T},\end{aligned}$$ where $$\label{kappahat} {\widehat{\kappa}}(s,\theta) = \kappa_1(s)\cos\theta + \kappa_2(s)\sin\theta.$$ Direct computation of the commutators yields $$\begin{aligned} [{{\rm{div}\,}},{\partial}_\theta]{\bm{u}}&= \frac{({\partial}_\theta{\widehat{\kappa}})}{1-\rho{\widehat{\kappa}}}\bigg( \rho\,{{\rm{div}\,}}{\bm{u}}- \frac{1}{\rho}\frac{{\partial}}{{\partial}\rho} \big(\rho^2 u_\rho \big) - \frac{{\partial}u_\theta}{{\partial}\theta} \bigg) - \frac{({\partial}_\theta^2{\widehat{\kappa}})}{1-\rho{\widehat{\kappa}}}u_\theta \\ [{{\rm{div}\,}},{\partial}_s]{\bm{u}}&= \frac{({\partial}_s{\widehat{\kappa}})}{1-\rho{\widehat{\kappa}}}\bigg( \rho\,{{\rm{div}\,}}{\bm{u}}- \frac{1}{\rho}\frac{{\partial}}{{\partial}\rho} \big(\rho^2 u_\rho \big) - \frac{{\partial}u_\theta}{{\partial}\theta} \bigg) - \frac{({\partial}_\theta{\partial}_s{\widehat{\kappa}})}{1-\rho{\widehat{\kappa}}}u_\theta \\ [\nabla,{\partial}_\theta]{\bm{u}}&= {\bm{e}}_\theta\frac{{\partial}{\bm{u}}}{{\partial}\rho}^{\rm T} - {\bm{e}}_\rho\frac{1}{\rho}\frac{{\partial}{\bm{u}}}{{\partial}\theta}^{\rm T} + {\bm{e}}_t\frac{\rho({\partial}_\theta{\widehat{\kappa}})}{(1-\rho{\widehat{\kappa}})^2} \bigg(\frac{{\partial}{\bm{u}}}{{\partial}s} -\kappa_3 \frac{{\partial}{\bm{u}}}{{\partial}\theta} \bigg)^{\rm T} \\ [\nabla,{\partial}_s]{\bm{u}}&= ({\partial}_s{\bm{e}}_\rho)\frac{{\partial}{\bm{u}}}{{\partial}\rho}^{\rm T} + ({\partial}_s{\bm{e}}_\theta)\frac{1}{\rho}\frac{{\partial}{\bm{u}}}{{\partial}\theta}^{\rm T} + \bigg({\bm{e}}_t \frac{\rho({\partial}_s{\widehat{\kappa}})}{1-\rho{\widehat{\kappa}}}+({\partial}_s{\bm{e}}_t) \bigg)\frac{1}{1-\rho{\widehat{\kappa}}} \bigg(\frac{{\partial}{\bm{u}}}{{\partial}s} -\kappa_3 \frac{{\partial}{\bm{u}}}{{\partial}\theta} \bigg)^{\rm T} \end{aligned}$$ Using and the orthonormal frame ODEs , we have $$\begin{aligned} {\left\lvert {\partial}_\theta {\widehat{\kappa}} \right\rvert} &= {\left\lvert -\kappa_1\sin\theta+ \kappa_2\cos\theta \right\rvert}\le \kappa_{\max}, \quad {\left\lvert {\partial}_s {\widehat{\kappa}} \right\rvert} = {\left\lvert \kappa_1'\cos\theta+\kappa_2'\sin\theta \right\rvert}\le \xi_{\max} + 2(\kappa_{\max}+\pi), \\ {\left\lvert {\partial}_\theta^2{\widehat{\kappa}} \right\rvert} &= {\left\lvert -{\widehat{\kappa}} \right\rvert}\le \kappa_{\max}, \quad {\left\lvert {\partial}_\theta{\partial}_s {\widehat{\kappa}} \right\rvert} = {\left\lvert -\kappa_1'\sin\theta+ \kappa_2'\cos\theta \right\rvert}\le \xi_{\max}+ 2(\kappa_{\max}+\pi), \\ {\left\lvert {\partial}_s {\bm{e}}_\rho \right\rvert} &= {\left\lvert -{\widehat{\kappa}}{\bm{e}}_t + \kappa_3{\bm{e}}_\theta \right\rvert}\le \kappa_{\max} +\pi, \quad {\left\lvert {\partial}_s{\bm{e}}_\theta \right\rvert} = {\left\lvert -({\partial}_\theta {\widehat{\kappa}}){\bm{e}}_t -\kappa_3{\bm{e}}_t \right\rvert}\le \kappa_{\max} +\pi, \\ {\left\lvert \frac{1}{1-\rho{\widehat{\kappa}}} \right\rvert} &\le \frac{1}{1- r_{\max}\kappa_{\max}(\cos\theta+\sin\theta) } \le \frac{1}{1- \frac{1}{2\kappa_{\max}}\kappa_{\max}\sqrt{2} } \le 4 .\end{aligned}$$ Finally, noting that, by Lemma \[sobolev\], $${\left\lVert u_\theta \right\rVert}_{L^2({\mathcal{O}})} \le {\left\lvert {\mathcal{O}} \right\rvert}^{1/3} {\left\lVert {\bm{u}}\right\rVert}_{L^6({\mathcal{O}})} \le C{\left\lVert \nabla{\bm{u}}\right\rVert}_{L^2({\mathcal{O}})},$$ the desired $L^2(\Omega)$ bounds follow for each of $D={{\rm{div}\,}},\nabla$. The estimate for the symmetric gradient ${\mathcal{E}}({\bm{u}})$ then follows from the gradient commutator bound. Now, to derive an estimate for $\nabla({\partial}_s{\bm{u}}^{\rm p})$, we will make use of Definition \[rigid\_weak\_p\] with a particular test function $\bm{\varphi}$, which we will construct here. First, we want our test function to be supported only within ${\mathcal{O}}$. We define a smooth cutoff function $$\label{Ocutoff} \psi(\rho) = \begin{cases} 1, & \rho < r_{\max}/4 \\ 0, & \rho > r_{\max}/2, \end{cases} \quad {\left\lvert \frac{{\partial}\psi}{{\partial}\rho} \right\rvert} \le C,$$ where $C$ depends only on $r_{\max}$. Note that $\psi(\rho)$ commutes with both ${\partial}_\theta$ and ${\partial}_s$.\ We would like to use ${\partial}_s^2(\psi{\bm{u}}^{\rm p})$ as a test function in Definition \[rigid\_weak\_p\], but it will be more convenient to work with a function which vanishes on $\Gamma_\epsilon$. We therefore construct a correction ${\bm{g}}\in C^2(\Omega_\epsilon)$ supported only in ${\mathcal{O}}$ and satisfying $$\label{g_correction} {\bm{g}}\big\|_{\Gamma_\epsilon} = ({\partial}_s {\bm{u}}^{\rm p})\big\|_{\Gamma_\epsilon} = \bm{\omega}^{\rm p}\times{\bm{e}}_t(s), \quad {\left\lVert \nabla {\bm{g}}\right\rVert}_{L^2({\mathcal{O}})} \le C{\left\lvert \bm{\omega} \right\rvert},$$ where $C$ depends on $c_\Gamma$ and $\kappa_{\max}$. To build ${\bm{g}}$, we follow a similar construction used in Section 4.1 of [@closed_loop]. We define $${\bm{g}}_0(\rho,\theta,s) = \begin{cases} \bm{\omega}^{\rm p}\times {\bm{e}}_t(s) & \text{if } \rho<4\epsilon \\ 0 & \text{otherwise} \end{cases}$$ and take $${\bm{g}}(\rho,\theta,s):= \phi_\epsilon(\rho){\bm{g}}_0(\rho,\theta,s),$$ where $\phi_\epsilon(\rho)$ is the smooth cutoff defined in -. Note that ${\bm{g}}\in C^2$ and is supported within the region $${\mathcal{O}}_\epsilon := \big\{ {\bm{X}}(s) + \rho {\bm{e}}_\rho(s,\theta) \; : \; s\in {\mathbb{T}}, \, \epsilon \le \rho \le 4\epsilon, \, 0\le \theta <2\pi \big\},$$ where ${\left\lvert {\mathcal{O}}_\epsilon \right\rvert} \le C\epsilon^2$. Then, using and , we have $$\begin{aligned} {\left\lVert \nabla {\bm{g}}\right\rVert}_{L^2({\mathcal{O}})} &\le \sqrt{{\left\lvert {\mathcal{O}}_\epsilon \right\rvert}}{\left\lVert \nabla {\bm{g}}\right\rVert}_{C({\mathcal{O}}_\epsilon)} \\ & \le \sqrt{{\left\lvert {\mathcal{O}}_\epsilon \right\rvert}}\bigg( {\left\lVert \frac{{\partial}\phi_\epsilon}{{\partial}\rho} \right\rVert}_{C({\mathcal{O}}_\epsilon)}{\left\lVert {\bm{g}}_0 \right\rVert}_{C({\mathcal{O}}_\epsilon)}+ {\left\lVert \frac{1}{1-\rho{\widehat{\kappa}}}\frac{{\partial}{\bm{g}}_0}{{\partial}s} \right\rVert}_{C({\mathcal{O}}_\epsilon)} \bigg) \le C{\left\lvert \bm{\omega}^{\rm p} \right\rvert}.\end{aligned}$$ Now, we could just use ${\partial}_s({\partial}_s(\psi{\bm{u}}^{\rm p}) -{\bm{g}})$ as a test function in Definition \[rigid\_weak\_p\], but it will actually be useful to include a second correction term in the following way. We consider ${\bm{z}}\in D^{1,2}_0({\mathcal{O}})$ satisfying $$\label{zee_def} \begin{aligned} {{\rm{div}\,}}{\bm{z}}&= {{\rm{div}\,}}(\psi{\partial}_s{\bm{u}}^{\rm p} -{\bm{g}}) \quad \text{in } {\mathcal{O}} \\ {\left\lVert \nabla {\bm{z}}\right\rVert}_{L^2({\mathcal{O}})} &\le C{\left\lVert {{\rm{div}\,}}(\psi{\partial}_s{\bm{u}}^{\rm p} -{\bm{g}}) \right\rVert}_{L^2({\mathcal{O}})} \end{aligned}$$ for $C$ depending only on $c_\Gamma$ and $\kappa_{\max}$. We know that such a ${\bm{z}}$ exists due to [@galdi2011introduction], Section III.3, and the constant $C$ is independence of $\epsilon$ due to Appendix A.2.5 of [@closed_loop]. Furthermore, since ${{\rm{div}\,}}{\bm{u}}^{\rm p}=0$, by Proposition \[comm\_ests\] we have $$\begin{aligned} {\left\lVert {{\rm{div}\,}}(\psi{\partial}_s{\bm{u}}^{\rm p}-{\bm{g}}) \right\rVert}_{L^2({\mathcal{O}})} &\le {\left\lVert {{\rm{div}\,}}({\partial}_\theta{\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})}+ C{\left\lVert {\partial}_\theta{\bm{u}}^{\rm p} \right\rVert}_{L^2({\mathcal{O}})}+{\left\lVert \nabla {\bm{g}}\right\rVert}_{L^2({\mathcal{O}})} \\ &\le {\left\lVert [{{\rm{div}\,}},{\partial}_\theta]{\bm{u}}^{\rm p} \right\rVert}_{L^2({\mathcal{O}})} + C{\left\lVert {\partial}_\theta{\bm{u}}^{\rm p} \right\rVert}_{L^2({\mathcal{O}})} + C{\left\lvert \bm{\omega}^{\rm p} \right\rvert} \\ &\le C{\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2({\mathcal{O}})} + C{\left\lvert \bm{\omega}^{\rm p} \right\rvert}.\end{aligned}$$ Here we have also used that ${\left\lVert {\partial}_\theta{\bm{u}}^{\rm p} \right\rVert}_{L^2({\mathcal{O}})} \le {\left\lVert \rho \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2({\mathcal{O}})} \le r_{\max}{\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2({\mathcal{O}})}$. In particular, ${\bm{z}}$ satisfying also satisfies $$\label{zestimate} {\left\lVert \nabla{\bm{z}}\right\rVert}_{L^2({\mathcal{O}})} \le C{\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2({\mathcal{O}})} + C{\left\lvert \bm{\omega}^{\rm p} \right\rvert}.$$ Using extension by zero to consider ${\bm{z}}$ as a function over all $\Omega_\epsilon$, we can now construct our desired test function for use in Definition \[rigid\_weak\_p\]. In particular, we will use the function ${\partial}_s({\partial}_s(\psi{\bm{u}}^{\rm p})-{\bm{g}}-{\bm{z}})$ in place of $\bm{\varphi}$ in Definition \[rigid\_weak\_p\]. Note that by definition of ${\bm{z}}$, this function may only belong to $L^2(\Omega_\epsilon)$. In this case, we can make sense of the following integration-by-parts argument using finite differences rather than full derivatives (see [@boyer2012mathematical], Section III.2.7 for construction of finite difference operators along a curved boundary). Thus we really only need ${\partial}_s(\psi{\bm{u}}^{\rm p})-{\bm{g}}-{\bm{z}}\in D^{1,2}(\Omega_\epsilon)$ to make sense of the following result. Note that in integrating by parts, we will also need to make use of the fact that, for $i=s,\theta$, $$\label{jacfac_i} {\partial}_i (d{\bm{x}}) = -\frac{\rho{\partial}_i{\widehat{\kappa}}}{1-\rho{\widehat{\kappa}}} d{\bm{x}}:= {\mathcal{J}}_i \, d{\bm{x}}, \quad {\left\lvert {\mathcal{J}}_i \right\rvert} \le C; \quad i=s,\theta ,$$ where $C$ depends on $c_\Gamma$, $\kappa_{\max}$, and $\xi_{\max}$.\ Then, using ${\partial}_s({\partial}_s(\psi{\bm{u}}^{\rm p})-{\bm{g}}-{\bm{z}})$ in Definition \[rigid\_weak\_p\], we have $$\begin{aligned} 0 &= \int_{{\mathcal{O}}} \bigg(2{\mathcal{E}}({\bm{u}}^{\rm p}): {\mathcal{E}}\big({\partial}_s({\partial}_s(\psi{\bm{u}}^{\rm p})-{\bm{g}}-{\bm{z}}) \big) - p^{\rm p} \,{{\rm{div}\,}}({\partial}_s({\partial}_s(\psi{\bm{u}}^{\rm p})-{\bm{g}}-{\bm{z}})) \bigg) \, d{\bm{x}}\\ &= \int_{{\mathcal{O}}} 2{\mathcal{E}}({\bm{u}}^{\rm p}): {\partial}_s{\mathcal{E}}({\partial}_s(\psi{\bm{u}}^{\rm p})-{\bm{g}}-{\bm{z}}) \, d{\bm{x}}+ \int_{{\mathcal{O}}} 2{\mathcal{E}}({\bm{u}}^{\rm p}): [{\mathcal{E}}(\cdot),{\partial}_s]({\partial}_s(\psi{\bm{u}}^{\rm p})-{\bm{g}}-{\bm{z}}) \, d{\bm{x}}\\ &\qquad - \int_{{\mathcal{O}}} p^{\rm p} \,{\partial}_s({{\rm{div}\,}}({\partial}_s(\psi{\bm{u}}^{\rm p})-{\bm{g}}-{\bm{z}})) \, d{\bm{x}}- \int_{{\mathcal{O}}} p^{\rm p} \,[{{\rm{div}\,}},{\partial}_s]({\partial}_s(\psi{\bm{u}}^{\rm p})- {\bm{g}}-{\bm{z}}) \, d{\bm{x}}\\ &= -\int_{{\mathcal{O}}} 2{\partial}_s{\mathcal{E}}({\bm{u}}^{\rm p}): {\mathcal{E}}({\partial}_s(\psi{\bm{u}}^{\rm p})-{\bm{g}}-{\bm{z}}) \, d{\bm{x}}-\int_{{\mathcal{O}}} 2{\mathcal{E}}({\bm{u}}^{\rm p}): {\mathcal{E}}({\partial}_s(\psi{\bm{u}}^{\rm p})-{\bm{g}}-{\bm{z}}) \, {\mathcal{J}}_s \, d{\bm{x}}\\ &\qquad + \int_{{\mathcal{O}}} 2{\mathcal{E}}({\bm{u}}^{\rm p}): [{\mathcal{E}}(\cdot),{\partial}_s]({\partial}_s(\psi{\bm{u}}^{\rm p})-{\bm{g}}-{\bm{z}}) \, d{\bm{x}}- \int_{{\mathcal{O}}} p^{\rm p} \,[{{\rm{div}\,}},{\partial}_s]({\partial}_s(\psi{\bm{u}}^{\rm p})- {\bm{g}}-{\bm{z}}) \, d{\bm{x}}\\ &= -\int_{{\mathcal{O}}} 2{\mathcal{E}}({\partial}_s{\bm{u}}^{\rm p}):{\mathcal{E}}({\partial}_s(\psi{\bm{u}}^{\rm p})-{\bm{g}}-{\bm{z}}) \, d{\bm{x}}- \int_{{\mathcal{O}}} 2{\mathcal{E}}({\bm{u}}^{\rm p}): {\mathcal{E}}({\partial}_s(\psi{\bm{u}}^{\rm p})-{\bm{g}}-{\bm{z}}) \, {\mathcal{J}}_s \, d{\bm{x}}\\ &\qquad +\int_{{\mathcal{O}}} 2[{\mathcal{E}}(\cdot),{\partial}_s]{\bm{u}}^{\rm p}: {\mathcal{E}}({\partial}_s(\psi{\bm{u}}^{\rm p})- {\bm{g}}-{\bm{z}}) \, d{\bm{x}}+ \int_{{\mathcal{O}}} 2{\mathcal{E}}({\bm{u}}^{\rm p}): [{\mathcal{E}}(\cdot),{\partial}_s]({\partial}_s(\psi{\bm{u}}^{\rm p})-{\bm{g}}-{\bm{z}}) \, d{\bm{x}}\\ &\qquad - \int_{{\mathcal{O}}} p^{\rm p} \,[{{\rm{div}\,}},{\partial}_s]({\partial}_s(\psi{\bm{u}}^{\rm p})- {\bm{g}}-{\bm{z}}) \, d{\bm{x}}.\end{aligned}$$ Note that the first integral in the third line vanishes due to the definition of ${\bm{z}}$. In this way we we can avoid having to deal with a ${\partial}_s p^{\rm p}$ term in the resulting estimate.\ Then, using Proposition \[comm\_ests\], estimates and , and Lemma \[korn\], we have $$\begin{aligned} {\left\lVert {\mathcal{E}}({\partial}_s{\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})}^2 &\le C{\left\lVert {\mathcal{E}}({\partial}_s{\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})} \big( {\left\lVert {\partial}_s {\bm{u}}^{\rm p} \right\rVert}_{L^2({\mathcal{O}})} + {\left\lVert {\mathcal{E}}({\bm{z}}) \right\rVert}_{L^2({\mathcal{O}})} + {\left\lVert {\mathcal{E}}({\bm{g}}) \right\rVert}_{L^2({\mathcal{O}})} \big) \\ &\quad + C{\left\lVert {\mathcal{E}}({\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})} \big({\left\lVert {\mathcal{E}}(\psi{\partial}_s{\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})} +{\left\lVert {\mathcal{E}}({\bm{g}}) \right\rVert}_{L^2({\mathcal{O}})} +{\left\lVert {\mathcal{E}}({\bm{z}}) \right\rVert}_{L^2({\mathcal{O}})} \big) \\ &\quad + 2{\left\lVert [{\mathcal{E}}(\cdot),{\partial}_s]{\bm{u}}^{\rm p} \right\rVert}_{L^2({\mathcal{O}})}\big( {\left\lVert {\mathcal{E}}(\psi{\partial}_s{\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})} + {\left\lVert {\mathcal{E}}({\bm{z}}) \right\rVert}_{L^2({\mathcal{O}})} + {\left\lVert {\mathcal{E}}({\bm{g}}) \right\rVert}_{L^2({\mathcal{O}})} \big) \\ &\quad +2{\left\lVert {\mathcal{E}}({\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})} \big( {\left\lVert [{\mathcal{E}}(\cdot),{\partial}_s](\psi{\partial}_s{\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})} + {\left\lVert [{\mathcal{E}}(\cdot),{\partial}_s]({\bm{z}}) \right\rVert}_{L^2({\mathcal{O}})} + {\left\lVert [{\mathcal{E}}(\cdot),{\partial}_s]({\bm{g}}) \right\rVert}_{L^2({\mathcal{O}})} \big) \\ &\quad + {\left\lVert p^{\rm p} \right\rVert}_{L^2({\mathcal{O}})} \big( {\left\lVert [{{\rm{div}\,}},{\partial}_s](\psi{\partial}_s{\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})} + {\left\lVert [{{\rm{div}\,}},{\partial}_s]({\bm{z}}) \right\rVert}_{L^2({\mathcal{O}})} +{\left\lVert [{{\rm{div}\,}},{\partial}_s]({\bm{g}}) \right\rVert}_{L^2({\mathcal{O}})} \big) \\ &\le C({\left\lVert {\mathcal{E}}({\partial}_s{\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})} + {\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2({\mathcal{O}})} +{\left\lvert \bm{\omega} \right\rvert})\big({\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2({\mathcal{O}})}+{\left\lVert p^{\rm p} \right\rVert}_{L^2({\mathcal{O}})} +{\left\lvert \bm{\omega}^{\rm p} \right\rvert} \big) \\ &\le \delta{\left\lVert {\mathcal{E}}({\partial}_s{\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})}^2 + C(\delta) \big({\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2({\mathcal{O}})}^2+{\left\lVert p^{\rm p} \right\rVert}_{L^2({\mathcal{O}})}^2 + {\left\lvert \bm{\omega}^{\rm p} \right\rvert}^2\big)\end{aligned}$$ for any $0<\delta\in{\mathbb{R}}$, by Young’s inequality. Taking $\delta=\frac{1}{2}$ and using Lemma \[korn\], we obtain $$\label{ps_est} \begin{aligned} {\left\lVert \nabla({\partial}_s{\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})} &\le {\left\lVert {\mathcal{E}}({\partial}_s{\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})} \le C \big({\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2({\mathcal{O}})}+{\left\lVert p^{\rm p} \right\rVert}_{L^2({\mathcal{O}})} + {\left\lvert \bm{\omega}^{\rm p} \right\rvert} \big) \\ & \le C{\left\lvert \log\epsilon \right\rvert}^{1/2} \big({\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)}+{\left\lVert p^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} \big), \end{aligned}$$ where we have used Corollary \[omegaP\] to bound ${\left\lvert \bm{\omega}^{\rm p} \right\rvert}$. Here $C$ depends only on $c_\Gamma$, $\kappa_{\max}$, and $\xi_{\max}$.\ We may estimate ${\partial}_\theta{\bm{u}}^{\rm p}$ in a similar way. In fact, the construction of the analogous test function is simpler since $({\partial}_\theta{\bm{u}}^{\rm p})\big\|_{\Gamma_\epsilon} = {\partial}_\theta({\bm{v}}+\bm{\omega}\times {\bm{X}}(s))=0$ and thus we do not need to correct for a nonzero boundary value. Following the same steps used to estimate ${\partial}_s{\bm{u}}^{\rm p}$, we obtain $$\label{ptheta_est} {\left\lVert \nabla({\partial}_\theta{\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})} \le C \big({\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)}+{\left\lVert p^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} \big),$$ where $C$ depends only on $c_\Gamma$, $\kappa_{\max}$, and $\xi_{\max}$.\ In addition to the estimates and , we need bounds for the tangential derivatives ${\partial}_s p^{\rm p}$ and ${\partial}_\theta p^{\rm p}$ of the pressure. We begin by estimating ${\partial}_s p^{\rm p}$; the bound for ${\partial}_\theta p^{\rm p}$ is similar. Since we already know that ${\partial}_s p^{\rm p}\in L^2(\Omega_\epsilon)$, we may consider ${\widetilde{{\bm{z}}}} \in D^{1,2}_0({\mathcal{O}})$ satisfying $$\label{wtzee_def} \begin{aligned} {{\rm{div}\,}}{\widetilde{{\bm{z}}}} &= \psi {\partial}_s p^{\rm p} \quad \text{in }{\mathcal{O}}, \\ {\left\lVert \nabla{\widetilde{{\bm{z}}}} \right\rVert}_{L^2({\mathcal{O}})} &\le C{\left\lVert \psi {\partial}_s p^{\rm p} \right\rVert}_{L^2({\mathcal{O}})}, \end{aligned}$$ where $\psi$ is as in . Again, we know that such a ${\widetilde{{\bm{z}}}}$ exists due to [@galdi2011introduction], Section III.3 and [@closed_loop], Appendix A.2.5.\ Using ${\partial}_s{\widetilde{{\bm{z}}}}$ as a test function in Definition \[rigid\_weak\_p\] (again, we can make sense of the following computation using finite differences, and thus only require ${\widetilde{{\bm{z}}}}\in D^{1,2}({\mathcal{O}})$), we have $$\begin{aligned} 0&= \int_{{\mathcal{O}}} \bigg(2{\mathcal{E}}({\bm{u}}^{\rm p}): {\mathcal{E}}({\partial}_s {\widetilde{{\bm{z}}}}) - p^{\rm p}\, {{\rm{div}\,}}({\partial}_s{\widetilde{{\bm{z}}}})\bigg) \, d{\bm{x}}= \int_{{\mathcal{O}}}2{\mathcal{E}}({\bm{u}}^{\rm p}): {\partial}_s{\mathcal{E}}({\widetilde{{\bm{z}}}}) \, d{\bm{x}}\\ &\quad + \int_{{\mathcal{O}}}2{\mathcal{E}}({\bm{u}}^{\rm p}): [{\mathcal{E}}(\cdot),{\partial}_s]{\widetilde{{\bm{z}}}} \, d{\bm{x}}- \int_{{\mathcal{O}}} p^{\rm p} \,{\partial}_s{{\rm{div}\,}}{\widetilde{{\bm{z}}}} \, d{\bm{x}}- \int_{{\mathcal{O}}} p^{\rm p} \, [{{\rm{div}\,}},{\partial}_s]{\widetilde{{\bm{z}}}} \, d{\bm{x}}\\ &= -\int_{{\mathcal{O}}}2{\partial}_s{\mathcal{E}}({\bm{u}}^{\rm p}): {\mathcal{E}}({\widetilde{{\bm{z}}}}) \, d{\bm{x}}- \int_{{\mathcal{O}}}2{\mathcal{E}}({\bm{u}}^{\rm p}): {\mathcal{E}}({\widetilde{{\bm{z}}}}) \, {\mathcal{J}}_s \, d{\bm{x}}+ \int_{{\mathcal{O}}}2{\mathcal{E}}({\bm{u}}^{\rm p}): [{\mathcal{E}}(\cdot),{\partial}_s]{\widetilde{{\bm{z}}}} \, d{\bm{x}}\\ &\quad - \int_{{\mathcal{O}}} p^{\rm p} \, [{{\rm{div}\,}},{\partial}_s]{\widetilde{{\bm{z}}}} \, d{\bm{x}}+ \int_{{\mathcal{O}}}({\partial}_s p){{\rm{div}\,}}{\widetilde{{\bm{z}}}} \, d{\bm{x}}+ \int_{{\mathcal{O}}} p^{\rm p} \,{{\rm{div}\,}}{\widetilde{{\bm{z}}}} \, {\mathcal{J}}_s \, d{\bm{x}}\\ &= \int_{{\mathcal{O}}}\psi({\partial}_s p)^2 \, d{\bm{x}}-\int_{{\mathcal{O}}}2{\partial}_s{\mathcal{E}}({\bm{u}}^{\rm p}): {\mathcal{E}}({\widetilde{{\bm{z}}}}) \, d{\bm{x}}- \int_{{\mathcal{O}}}2{\mathcal{E}}({\bm{u}}^{\rm p}): {\mathcal{E}}({\widetilde{{\bm{z}}}}) \, {\mathcal{J}}_s \, d{\bm{x}}\\ &\quad + \int_{{\mathcal{O}}}2{\mathcal{E}}({\bm{u}}^{\rm p}): [{\mathcal{E}}(\cdot),{\partial}_s]{\widetilde{{\bm{z}}}} \, d{\bm{x}}- \int_{{\mathcal{O}}} p^{\rm p} \, [{{\rm{div}\,}},{\partial}_s]{\widetilde{{\bm{z}}}} \, d{\bm{x}}+ \int_{{\mathcal{O}}} p^{\rm p} \,{{\rm{div}\,}}{\widetilde{{\bm{z}}}} \, {\mathcal{J}}_s \, d{\bm{x}},\end{aligned}$$ where ${\mathcal{J}}_s \, d{\bm{x}}$ is as in and we have used . Then, using that $\psi^2\le \psi$, we have $$\begin{aligned} {\left\lVert \psi{\partial}_s p^{\rm p} \right\rVert}_{L^2({\mathcal{O}})}^2 &\le 2{\left\lVert {\mathcal{E}}({\partial}_s{\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})} {\left\lVert {\mathcal{E}}({\widetilde{{\bm{z}}}}) \right\rVert}_{L^2({\mathcal{O}})} + 2{\left\lVert [{\mathcal{E}}(\cdot),{\partial}_s]{\bm{u}}^{\rm p} \right\rVert}_{L^2({\mathcal{O}})} {\left\lVert {\mathcal{E}}({\widetilde{{\bm{z}}}}) \right\rVert}_{L^2({\mathcal{O}})} \\ &\quad + C{\left\lVert {\mathcal{E}}({\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})} {\left\lVert {\mathcal{E}}({\widetilde{{\bm{z}}}}) \right\rVert}_{L^2({\mathcal{O}})} + 2{\left\lVert {\mathcal{E}}({\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})}{\left\lVert [{\mathcal{E}}(\cdot),{\partial}_s]{\widetilde{{\bm{z}}}} \right\rVert}_{L^2({\mathcal{O}})} \\ &\quad +{\left\lVert p^{\rm p} \right\rVert}_{L^2({\mathcal{O}})} {\left\lVert [{{\rm{div}\,}},{\partial}_s]{\widetilde{{\bm{z}}}} \right\rVert}_{L^2({\mathcal{O}})} + C{\left\lVert p^{\rm p} \right\rVert}_{L^2({\mathcal{O}})} {\left\lVert {{\rm{div}\,}}{\widetilde{{\bm{z}}}} \right\rVert}_{L^2({\mathcal{O}})}\\ &\le C\big({\left\lVert \nabla({\partial}_s{\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})} + {\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2({\mathcal{O}})} +{\left\lVert p^{\rm p} \right\rVert}_{L^2({\mathcal{O}})} \big) {\left\lVert \psi{\partial}_s p^{\rm p} \right\rVert}_{L^2({\mathcal{O}})} \\ &\le \delta {\left\lVert \psi{\partial}_s p^{\rm p} \right\rVert}_{L^2({\mathcal{O}})}^2 + C(\delta)\big({\left\lVert \nabla({\partial}_s{\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})}^2 + {\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2({\mathcal{O}})}^2 +{\left\lVert p^{\rm p} \right\rVert}_{L^2({\mathcal{O}})}^2 \big)\end{aligned}$$ for $0<\delta\in {\mathbb{R}}$. Here we have used , , Proposition \[comm\_ests\], and Young’s inequality. Taking $\delta=\frac{1}{2}$ and using , we obtain $${\left\lVert \psi{\partial}_s p^{\rm p} \right\rVert}_{L^2({\mathcal{O}})} \le C{\left\lvert \log\epsilon \right\rvert}^{1/2}\big( {\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} +{\left\lVert p^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} \big).$$ Then, using , within the region $${\mathcal{O}}' = \bigg\{{\bm{x}}\in \Omega_{\epsilon} \; : \; {\bm{x}}= {\bm{X}}(s) + \rho {\bm{e}}_{\rho}(s,\theta), \quad \epsilon < \rho<\frac{r_{\max}}{4} \bigg\},$$ we have $$\label{psp_est} {\left\lVert {\partial}_s p^{\rm p} \right\rVert}_{L^2({\mathcal{O}}')} \le C{\left\lvert \log\epsilon \right\rvert}^{1/2}\big( {\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} +{\left\lVert p^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} \big)$$ for $C$ depending only on $c_\Gamma$, $\kappa_{\max}$, and $\xi_{\max}$.\ We can similarly use to show $$\label{pthetap_est} {\left\lVert {\partial}_\theta p^{\rm p} \right\rVert}_{L^2({\mathcal{O}}')} \le C\big( {\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} +{\left\lVert p^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} \big).$$ Now we can use the tangential bounds , , , and to obtain an estimate for derivatives $\nabla({\partial}_\rho{\bm{u}}^{\rm p})$ normal to $\Gamma_\epsilon$. For this, we will use the full Stokes equations , written with respect to the orthonormal frame ${\bm{e}}_t$, ${\bm{e}}_\rho$, ${\bm{e}}_\theta$ in ${\mathcal{O}}$ as $$\begin{aligned} - \Delta {\bm{u}}^{\rm p} +\nabla p^{\rm p} &= -\Delta {\bm{u}}^{\rm p} + \frac{{\partial}p^{\rm p}}{{\partial}\rho}{\bm{e}}_{\rho} + \frac{1}{\rho}\frac{{\partial}p^{\rm p}}{{\partial}\theta}{\bm{e}}_{\theta} + \frac{1}{1-\rho{\widehat{\kappa}}}\bigg(\frac{{\partial}p^{\rm p}}{{\partial}s}-\kappa_3\frac{{\partial}p^{\rm p}}{{\partial}\theta}\bigg){\bm{e}}_t =0 \\ {{\rm{div}\,}}{\bm{u}}^{\rm p} &= \frac{1}{1-\rho{\widehat{\kappa}}}\bigg(\frac{1}{\rho}\frac{{\partial}(\rho (1-\rho{\widehat{\kappa}}) u_{\rho})}{{\partial}\rho}+\frac{1}{\rho}\frac{{\partial}((1-\rho{\widehat{\kappa}}) u_{\theta})}{{\partial}\theta} + \frac{{\partial}u_s}{{\partial}s} \bigg) = 0.\end{aligned}$$ Here ${\widehat{\kappa}}$ is as in and we recall the notation $u_\rho = {\bm{u}}^{\rm p}\cdot{\bm{e}}_\rho$, $u_\theta={\bm{u}}^{\rm p}\cdot{\bm{e}}_\theta$, $u_s={\bm{u}}^{\rm p}\cdot{\bm{e}}_t$.\ From the divergence-free condition on ${\bm{u}}^{\rm p}$, after multiplying through by $\rho (1-\rho{\widehat{\kappa}})$ and differentiating once with respect to $\rho$, we obtain $$\begin{aligned} {\left\lVert \frac{{\partial}^2 u_{\rho}}{{\partial}^2 \rho} \right\rVert}_{L^2(\mathcal{O})} &\le C\bigg({\left\lVert \frac{1}{\rho} \nabla {\bm{u}}^{\rm p} \right\rVert}_{L^2(\mathcal{O})} + \bigg\\|\frac{1}{\rho}\bigg\\|_{L^{\infty}(\mathcal{O})}\|\mathcal{O}\|^{1/3}\big\\|{\bm{u}}^{\rm p} \big\\|_{L^6(\mathcal{O})} \\ &\hspace{2cm}+ \bigg\\|\frac{1}{\rho}\frac{{\partial}}{{\partial}\rho}\bigg(\frac{{\partial}u_{\theta}}{{\partial}\theta}\bigg)\bigg\\|_{L^2(\mathcal{O})} +\bigg\\|\frac{{\partial}}{{\partial}\rho}\bigg(\frac{{\partial}u_s}{{\partial}s}\bigg)\bigg\\|_{L^2(\mathcal{O})} \bigg)\\ &\le \frac{C}{\epsilon}{\left\lvert \log\epsilon \right\rvert}^{1/2}\big( \\|\nabla {\bm{u}}^{\rm p}\\|_{L^2(\Omega_\epsilon)} +{\left\lVert p^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} \big),\end{aligned}$$ where we have used and along with the Sobolev inequality on $\Omega_\epsilon$.\ Furthermore, using the ${\bm{e}}_\rho$ component of $-\Delta {\bm{u}}^{\rm p} +\nabla p=0$, we have $$\begin{aligned} \frac{{\partial}p^{\rm p}}{{\partial}\rho} &= (\Delta{\bm{u}}^{\rm p}) \cdot{\bm{e}}_{\rho} \\ &= \frac{1}{\rho (1-\rho{\widehat{\kappa}})}\frac{{\partial}}{{\partial}\rho}\left(\rho (1-\rho{\widehat{\kappa}})\frac{{\partial}{\bm{u}}^{\rm p}}{{\partial}\rho}\right)\cdot{\bm{e}}_{\rho} +\frac{1}{\rho^2(1-\rho{\widehat{\kappa}})}\frac{{\partial}}{{\partial}\theta}\bigg((1-\rho{\widehat{\kappa}})\frac{{\partial}{\bm{u}}^{\rm p}}{{\partial}\theta}\bigg)\cdot{\bm{e}}_{\rho} \\ &\hspace{4cm} +\frac{1}{1-\rho{\widehat{\kappa}}} \frac{{\partial}}{{\partial}s}\bigg( \frac{1}{1-\rho{\widehat{\kappa}}}\bigg[ \frac{{\partial}{\bm{u}}^{\rm p}}{{\partial}s}- \kappa_3\frac{{\partial}{\bm{u}}^{\rm p}}{{\partial}\theta} \bigg] \bigg)\cdot{\bm{e}}_{\rho} \\ &= \frac{1}{\rho (1-\rho{\widehat{\kappa}})}\frac{{\partial}}{{\partial}\rho}\left(\rho (1-\rho{\widehat{\kappa}})\frac{{\partial}u_{\rho}}{{\partial}\rho}\right) +\frac{1}{\rho^2(1-\rho{\widehat{\kappa}})}\frac{{\partial}}{{\partial}\theta}\bigg((1-\rho{\widehat{\kappa}})\frac{{\partial}{\bm{u}}^{\rm p}}{{\partial}\theta}\bigg)\cdot{\bm{e}}_{\rho} \\ &\hspace{4cm} +\frac{1}{1-\rho{\widehat{\kappa}}} \frac{{\partial}}{{\partial}s}\bigg( \frac{1}{1-\rho{\widehat{\kappa}}}\bigg[ \frac{{\partial}{\bm{u}}^{\rm p}}{{\partial}s}- \kappa_3\frac{{\partial}{\bm{u}}^{\rm p}}{{\partial}\theta} \bigg] \bigg)\cdot{\bm{e}}_{\rho}, \end{aligned}$$ since ${\bm{e}}_{\rho}(s,\theta)$ does not vary with $\rho$. Therefore, using , , , and , along with the the bound on $\frac{{\partial}^2 u_{\rho}}{{\partial}\rho^2}$, we have $$\\|\nabla p^{\rm p}\\|_{L^2(\mathcal{O}')} \le \frac{C}{\epsilon}{\left\lvert \log\epsilon \right\rvert}^{1/2}\big( \\|\nabla {\bm{u}}^{\rm p}\\|_{L^2(\Omega_\epsilon)} +{\left\lVert p^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} \big).$$ Finally, to estimate $\frac{{\partial}^2 u_j}{{\partial}\rho^2}$, $j=\theta,s$, we again use that $$\begin{aligned} \nabla p^{\rm p}\cdot{\bm{e}}_j &= (\Delta {\bm{u}}^{\rm p})\cdot {\bm{e}}_j(s,\theta) \\ &= \frac{1}{\rho (1-\rho{\widehat{\kappa}})}\frac{{\partial}}{{\partial}\rho}\bigg(\rho (1-\rho{\widehat{\kappa}})\frac{{\partial}u_j}{{\partial}\rho}\bigg) +\frac{1}{\rho^2(1-\rho{\widehat{\kappa}})}\frac{{\partial}}{{\partial}\theta}\bigg((1-\rho{\widehat{\kappa}})\frac{{\partial}{\bm{u}}^{\rm p}}{{\partial}\theta}\bigg)\cdot{\bm{e}}_j \\ &\hspace{4cm}+\frac{1}{1-\rho{\widehat{\kappa}}} \frac{{\partial}}{{\partial}s}\bigg( \frac{1}{1-\rho{\widehat{\kappa}}}\bigg[ \frac{{\partial}{\bm{u}}^{\rm p}}{{\partial}s}- \kappa_3\frac{{\partial}{\bm{u}}^{\rm p}}{{\partial}\theta} \bigg] \bigg)\cdot{\bm{e}}_j , \quad j=\theta,s,\end{aligned}$$ since each of ${\bm{e}}_t(s)$, ${\bm{e}}_\rho(s,\theta)$ and ${\bm{e}}_\theta(s,\theta)$ are independent of $\rho$. Then we have $$\begin{aligned} \bigg\\|\frac{{\partial}^2 u_j}{{\partial}\rho^2}\bigg\\|_{L^2(\mathcal{O}')} &\le C \bigg( \bigg\\|\frac{1}{\rho}\bigg\\|_{L^{\infty}(\mathcal{O}')} \\| \nabla {\bm{u}}^{\rm p}\\|_{L^2(\mathcal{O}')} + \bigg\\|\frac{{\partial}^2{\bm{u}}^{\rm p}}{{\partial}s^2}\bigg\\|_{L^2(\mathcal{O}')} \\ &\hspace{2cm} + \bigg\\|\frac{{\partial}^2{\bm{u}}^{\rm p}}{{\partial}s{\partial}\theta}\bigg\\|_{L^2(\mathcal{O}')} +\bigg\\|\frac{{\partial}^2{\bm{u}}^{\rm p}}{{\partial}\theta^2}\bigg\\|_{L^2(\mathcal{O}')}+\\|\nabla p^{\rm p}\\|_{L^2(\mathcal{O}')} \bigg) \\ &\le \frac{C}{\epsilon}{\left\lvert \log\epsilon \right\rvert}^{1/2} \big({\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} + {\left\lVert p^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} \big), \quad j=\theta, s, \end{aligned}$$ where $C$ depends only on $c_\Gamma$, $\kappa_{\max}$, and $\xi_{\max}$. Altogether, we obtain Lemma \[high\_reg\]. $\square$ We note that the factor of $\frac{1}{\epsilon}$ in Lemma \[high\_reg\] is necessary. As a heuristic, we consider an infinite straight cylinder of radius $\epsilon$ and take ${\bm{u}}= (\frac{1}{\rho}-\frac{1}{\epsilon}) {\bm{e}}_{\theta}$, where ${\bm{e}}_{\theta}$ is now the (constant) angular vector in straight cylindrical coordinates, and $p\equiv$ constant. Ignoring decay conditions toward infinity along the cylinder, $({\bm{u}},p)$ solves the Stokes equations with ${\bm{u}}=0$ on the cylinder surface. Then $$\begin{aligned} \|\nabla^2{\bm{u}}\| =\bigg\| \frac{{\partial}^2}{{\partial}\rho^2} \frac{1}{\rho}\bigg\| = \bigg\|\frac{2}{\rho^3}\bigg\| = \frac{2}{\rho}\big\| \nabla {\bm{u}}\big\|,\end{aligned}$$ and within the region $\epsilon < \rho \le 2 \epsilon$, we have $\|\nabla^2{\bm{u}}\| \ge \frac{1}{\epsilon}\|\nabla {\bm{u}}\|$. [^1]: This research was supported in part by NSF grant DMS-1620316 and DMS-1516978, awarded to Y.M., and a Torske Klubben Fellowship, awarded to L.O. L.O. also thanks Dallas Albritton for helpful discussion.	ArXiv
--- abstract: 'We derive the momentum space dynamic equations and state functions for one dimensional quantum walks by using linear systems and Lie group theory. The momentum space provides an analytic capability similar to that contributed by the z transform in discrete systems theory. The state functions at each time step are expressed as a simple sum of three Chebyshev polynomials. The functions provide an analytic expression for the development of the walks with time.' author: - Ian Fuss - 'Langford B. White and Sanjeev Naguleswaran' - 'Peter J. Sherman' bibliography: - 'D:/DPOLP/documents/bib/quant.bib' title: Momentum Dynamics of One Dimensional Quantum Walks --- Introduction ============ The study of quantum walks has received considerable attention since the introductory papers on the subject, such as [@aharanov00; @Kempe03] and references therein. In this paper, we develop an analytic approach to study the properties of these walks based on a momentum space representation. This paper is structured such that in Section 2 of the paper the momentum space dynamic equations for one dimensional quantum walks are derived via the Z transform of the position space dynamic equations and its representation of the discrete Fourier transform when Z lies on the unit circle. An exponential form of of the momentum space time operator is derived in section 3 by using the group theory of $SU(2)$ and a matrix inner product space. The exponential form allows a simple analytic calculation of the time evolution operator for arbitrary time intervals. This is used in Section 4 to obtain analytic expressions for the momentum space wave functions of quantum walks at arbitrary times. These wave functions are expressed quite simply in terms of Chebyshev Polynomials of the second kind. Some plots of the momentum space probability densities for different parameter values and times are provided in section 5. The conclusions are summarised in Section 6. Momentum Space Dynamic Equations ================================ For a given $\psi(0,0)$ we consider the evolution of a quantum state $\psi(t,x)\in C^{2}$ for discrete times $t\ge0$ on a line $x\in Z.$ The dynamics of the state then evolve according to the difference equations,$$\begin{aligned} & \psi_{0}(t,x)=e^{i\alpha}[a\psi_{0}(t-1,x-1)+b\psi_{1}(t-1,x-1)],\nonumber \\ & \psi_{1}(t,x)=e^{i\alpha}[-b^{}\psi_{0}(t-1,x+1)+a^{}\psi_{1}(t-1,x+1)],\label{eq:momdy}\end{aligned}$$ where $\|a\|^{2}+\|b\|^{2}=1$ and $\alpha\in R$. Taking two-dimensional $Z$ transforms of these equations yields$$\begin{aligned} & \psi_{0}(z_{1},z_{2})=e^{i\alpha}z_{1}^{-1}z_{2}^{-1}[a\psi_{0}(z_{1},z_{2})+b\psi_{1}(z_{1},z_{2})\nonumber \\ & \psi_{1}(z_{1},z_{2})=e^{i\alpha}z_{1}^{-1}z_{2}^{-1}[-b^{}\psi_{0}(z_{1},z_{2})+a^{}\psi_{1}(z_{1},z_{2}).\end{aligned}$$ Thus the transfer matrix for the system is$$B(z_{1},z_{2})=e^{i\alpha}z_{1}^{-1}\left[\begin{array}{cc} az_{2}^{-1} & bz_{2}^{-1}\\ -b^{\ast}z_{2} & a^{\ast}z_{2}\end{array}\right]$$ therefore, for any iteration (time) index $n$, the quantum walk state $\ \Psi(n,x)$ has transform $x\leftrightarrow z$ $$\Psi(n,x)\leftrightarrow e^{in\alpha}C^{n}(z)\Psi(0,0),$$ where $C(z)$ is the matrix polynomial$$C(z)=\left[\begin{array}{cc} az^{-1} & bz^{-1}\\ -bz & az\end{array}\right].$$ It should be noted that $C$ is paraunitary, that is $C^{-1}(z)=C^{T}(1/z).$ In particular this implies that $C(z)$ is unitary on $\|z\|=1.$ Further we note that $detC(e^{ip)})=1$ and hence the matrix $$S(p)=C(e^{ip})\label{eq:6}$$ is unimodular. The Fourier transform $x\leftrightarrow p$ is$$\Psi(n,x)\leftrightarrow e^{in\alpha}S^{n}(p)\Psi(0,0).$$ Thus by choosing Planck’s constant $\hbar=1,$ the momentum space representation of the quantum walk state vector $\phi(n,p)$ evolves as$$\phi(n,p)=e^{in\alpha}S^{n}(p)\phi(0,p),\label{eq:8}$$ where$$\phi(0,p)=\psi(0,0)=\left[\begin{array}{c} \psi_{0}(0,0)\\ \psi_{1}(0,0)\end{array}\right].\label{eq:9}$$ Thus the time evolution operator in the momentum space is a $2\times2$ matrix polynomial. Hence, the momentum space equations are much more amenable to analysis than those in position space. Exponentiation of the Time Evolution Operator ============================================= The unimodular matrix $S(p)$ can be written in exponential form as$$S(p)=Exp(i\theta(p)\overrightarrow{c}(p).\overrightarrow{\sigma})\label{eq10}$$ where $\theta$ and $\overrightarrow{c}$ are real functions of $p$ and the matrix vector $\overrightarrow{\sigma}$ has Pauli matrix components [@merz]$$\sigma_{1}=\left[\begin{array}{cc} 0 & 1\\ 1 & 0\end{array}\right],$$ $$\sigma_{2}=\left[\begin{array}{cc} 0 & -i\\ i & 0\end{array}\right]$$ and$$\sigma_{3}=\left[\begin{array}{cc} 1 & 0\\ 0 & -1\end{array}\right].\label{eq:11}$$ The inner product$$(A,B)=\frac{1}{2}Tr(AB)$$ defined on the vector space of $2\times2$ unitary matrices gives an inner product space. The set of matrices $\{ I,\sigma_{1},\sigma_{2,}\sigma_{3}\},$ provide an ortho-normal basis for this space. The coefficients of the matrices can be evaluated by taking the inner product of both sides of (\[eq10\]) $$(\sigma_{i},S(p))=(\sigma_{i},Exp(i\theta(p)\overrightarrow{c}(p).\overrightarrow{\sigma})$$ with each of the matrices $\sigma_{i}.$ In doing this we note that a generalised de-Moivre principle gives$$Exp(i\theta\overrightarrow{c}.\overrightarrow{\sigma})=Icos(\theta)+i\overrightarrow{c}.\overrightarrow{\sigma}sin(\theta),$$ where the $p$ dpendence has been suppressed for simplicity. Hence,$$(I,Exp(i\theta\overrightarrow{c}.\overrightarrow{\sigma}))=cos(\theta)\label{eq13}$$ and$$(\sigma_{j},Exp(i\theta\overrightarrow{c}.\overrightarrow{\sigma}))=ic_{j}sin(\theta).\label{eq14}$$ The equivalent coefficients for $S(p)$ can be obtained by defining $$a=cos(\beta)e^{-i\gamma},$$ $$b=sin(\beta)e^{-i\delta}.\label{eq:define_ab}$$ Substituting in (\[eq:6\]) gives$$S(p)=\left[\begin{array}{cc} cos(\beta)e^{-i(p+\gamma)} & sin(\beta)e^{-i(p+\delta)}\\ -sin(\beta)e^{i(p+\delta)} & cos(\beta)e^{i(p+\gamma)}\end{array}\right].$$ These expressions can be simplified by setting $p'=p+\gamma$ and $p''=p+\delta$. Using de Moivre’s principle once again we obtain the transition matrix coefficients$$\begin{aligned} & (I,S(p))=cos(\beta)cos(p'),\nonumber \\ & (\sigma_{1},S(p))=-isin(\beta)sin(p''),\nonumber \\ & (\sigma_{2},S(p))=isin(\beta)cos(p''),\nonumber \\ & (\sigma_{3},S(p))=-icos(\beta)sin(p').\label{eq17}\end{aligned}$$ Comparing coefficients in equations (\[eq13\]) and (\[eq14\]) with those of (\[eq17\]) we obtain$$cos(\theta)=cos(\beta)cos(p'),$$ $$c_{1}sin(\theta)=-sin(\beta)sin(p''),$$ $$c_{2}sin(\theta)=sin(\beta)cos(p''),$$ $$c_{3}sin(\theta)=-cos(\beta)sin(p').\label{eq20}$$ Momentum Space State Functions ============================== A dynamic equation for momentum space state functions was given in (\[eq:8\]). The exponentiation of the operator in (\[eq10\]) enables us to write the powers of the evolution operator as$$S^{n}(p)=Exp(in\theta\overrightarrow{c}.\overrightarrow{\sigma})=Icos(n\theta)+i\overrightarrow{c}.\overrightarrow{\sigma}sin(n\theta).$$ The trigonometric expressions in the above equation can be expressed in terms of the Chebyshev polynomials $T_{n}$ and $U_{n}$ as [@Arf]$$cos(n\theta)=T_{n}(cos(\theta))$$ and$$sin(n\theta)=U_{n-1}(cos(\theta))sin(\theta).$$ Using these expressions and writing the dot product as a sum of components (\[eq:11\]) becomes$$S^{n}(p)=T_{n}(cos(\theta))I+iU_{n-1}(cos(\theta))\sum_{i=1}^{3}c_{i}sin(\theta)\sigma_{i}.$$ The equalities of (\[eq20\]) enable us to rewrite this as$$S^{n}(p)=T_{n}(cos(\beta)cos(p'))I-iU_{n-1}(cos(\beta)cos(p'))[sin(\beta)sin(p'')\sigma_{1}-sin(\beta)cos(p'')\sigma_{2}+cos(\beta)sin(p')\sigma_{3}]$$ Using the Pauli matrices the matrix polynomial$$\begin{aligned} & S^{n}(p)=\left[\begin{array}{cc} T_{n}(cos(\beta)cos(p') & U_{n-1}(cos(\beta)cos(p'))sin(\beta)cos(p'')\\ -U_{n-1}(cos(\beta)cos(p'))sin(\beta)cos(p'') & T_{n}(cos(\beta)cos(p'))\end{array}\right]\nonumber \\ - & i\left[\begin{array}{cc} U_{n-1}(cos(\beta)cos(p'))cos(\beta)sin(p') & U_{n-1}(cos(\beta)cos(p'))sin(\beta)sin(p'')\\ U_{n-1}(cos(\beta)cos(p'))sin(\beta)sin(p'') & -U_{n-1}(cos(\beta)cos(p'))cos(\beta)sin(p')\end{array}\right]\label{eq25}\end{aligned}$$ is obtained. The evolution of the quantum walk in momentum space representation given in (\[eq:8\] )can also be expressed as $$\phi(n,p)e^{-in\alpha}=S^{n}(p)\phi(0,p).\label{eq:21}$$ (\[eq25\]) and (\[eq:9\]) enable this expression to be written as $$\begin{aligned} & \phi_{0}(n,p)e^{-in\alpha}=[T_{n}(cos(\beta)cos(p'))-iU_{n-1}(cos(\beta)cos(p'))cos(\beta)sin(p')]\Psi_{0}(0,0)\nonumber \\ & +[U_{n-1}(cos(\beta)cos(p'))sin(\beta)cos(p'')-iU_{n-1}(cos(\beta)cos(p'))sin(\beta)sin(p'')]\Psi_{1}(0,0)\end{aligned}$$ $$\begin{aligned} & \phi_{1}(n,p)e^{-in\alpha}=-[U_{n-1}(cos(\beta)cos(p'))sin(\beta)cos(p'')+iU_{n-1}(cos(\beta)cos(p'))sin(\beta)sin(p'')]\Psi_{0}(0,0)\nonumber \\ & +T_{n}(cos(\beta)cos(p'))+iU_{n-1}(cos(\beta)cos(p'))cos(\beta)sin(p')]\Psi_{1}(0,0)\end{aligned}$$ By using the relation$$T_{n}(x)=U_{n}(x)-xU_{n-1}(x)$$ this can be written as $$\begin{aligned} & \phi_{0}(n,p)e^{-in\alpha}=[U_{n}(cos(\beta)cos(p'))-U_{n-1}(cos(\beta)cos(p'))cos(\beta)[cos(p')+isin(p')]]\Psi_{0}(0,0)\nonumber \\ & +[[U_{n-1}(cos(\beta)cos(p'))sin(\beta)[cos(p'')-sin(p'')]]\Psi_{1}(0,0)\end{aligned}$$ $$\begin{aligned} & \phi_{1}(n,p)e^{-in\alpha}=-[U_{n-1}(cos(\beta)cos(p'))sin(\beta)[cos(p'')+isin(p'')]]\Psi_{0}(0,0)\nonumber \\ & +[U_{n}(cos(\beta)cos(p'))-U_{n-1}(cos(\beta)cos(p'))cos(\beta)[cos(p')+isin(p')]]\Psi_{1}(0,0).\end{aligned}$$ Inverting the de Moivre formula and moving the global phase term to the right hand side gives the analytic expressions$$\begin{aligned} & \phi_{0}(n,p)=e^{in\alpha}[U_{n}(cos(\beta)cos(p'))-U_{n-1}(cos(\beta)cos(p'))cos(\beta)e^{ip}]\Psi_{0}(0,0)\nonumber \\ & +e^{in\alpha}[U_{n-1}(cos(\beta)cos(p'))sin(\beta)e^{-ip}]\Psi_{1}(0,0)\end{aligned}$$ $$\begin{aligned} & \phi_{1}(n,p)=-e^{in\alpha}[U_{n-1}(cos(\beta)cos(p'))e^{ip}]\Psi_{0}(0,0)\nonumber \\ & +e^{in\alpha}[U_{n}(cos(\beta)cos(p'))-U_{n-1}(cos(\beta)cos(p'))cos(\beta)e^{-ip}]\Psi_{1}(0,0)\end{aligned}$$ for the general momentum space state functions for a one dimensional quantum walk at time n. Momentum Space Densities ======================== The denisity $\|\phi_{0}(p:t)\|^{2}$ for $\alpha=\gamma=\delta=0,$ $\Psi_{0}(0,0)=1$ and $\Psi_{1}(0,0)=0$ is plotted in figures \[cap:Momentum-Space-Density1\], \[cap:Momentum-Space-Density2\], \[cap:Momentum-Space-Density3\] for $\beta=\frac{\pi}{8},\frac{\pi}{4}$ and $\frac{3\pi}{8}$ and for times $t=10,30,50,70.$ When $\beta$ is fixed the dominant feature of the time series is an increase in oscillation frequency with time. This corresponds to the increase in support of the position space densities with time. The effect of increasing $\beta$ is to trade a decrease in the constant component of the density function for an increase in the oscillatory component. This corresponds to a shift in the position space of probability density from the zero region of the walk to the outer edges of the walk. ![\[cap:Momentum-Space-Density1\]Momentum Space Density functions for $\beta=\frac{\pi}{8}$](paperfig1) ![\[cap:Momentum-Space-Density2\]Momentum Space Density functions for $\beta=\frac{\pi}{4}$](paperfig2) ![\[cap:Momentum-Space-Density3\]Momentum Space Density functions for $\beta=\frac{3\pi}{8}$](paperfig3) The sequences shows that the densities converge to a limit as time increases. They also illustrate the fact that the momentum space is an attractive representation in which to derive this limit because the domain of the wave functions is constant, $p\in[-\pi,\pi].$ This is in contrast to the real space where the domain expands with time. Conclusions =========== It has been shown that the momentum space dynamic equations for a quantum walk can be derived using a z transform of the position space equations for the dynamic walk. An exponential representation of the momentum space time evolution operator was derived by using Lie group theory. This enabled the calculation of general momentum space wave functions in terms of Chebyshev polynomials. Some simple calculations of the momentum space probability densities illustrate the convergence of the momentum wave functions to a limit as time increases.	ArXiv
--- abstract: 'The magnetic structures and the magnetic phase transitions in the Mn-doped orthoferrite TbFeO$_3$ studied using neutron powder diffraction are reported. Magnetic phase transitions are identified at $T^\mathrm{Fe/Mn}_N \approx$ 295 K where a paramagnetic-to-antiferromagnetic transition occurs in the Fe/Mn sublattice, $T^\mathrm{Fe/Mn}_{SR} \approx$ 26 K where a spin-reorientation transition occurs in the Fe/Mn sublattice and $T^\mathrm{R}_N \approx$ 2 K where Tb-ordering starts to manifest. At 295 K, the magnetic structure of the Fe/Mn sublattice in TbFe$_{0.5}$Mn$_{0.5}$O$_3$ belongs to the irreducible representation $\Gamma_4$ ($G_xA_yF_z$ or $Pb''n''m$). A mixed-domain structure of ($\Gamma_1 + \Gamma_4$) is found at 250 K which remains stable down to the spin re-orientation transition at $T^\mathrm{Fe/Mn}_{SR}\approx$ 26 K. Below 26 K and above 250 K, the majority phase ($> 80\%$) is that of $\Gamma_4$. Below 10 K the high-temperature phase $\Gamma_4$ remains stable till 2 K. At 2 K, Tb develops a magnetic moment value of 0.6(2) $\mu_\mathrm{B}/$f.u. and orders long-range in $F_z$ compatible with the $\Gamma_4$ representation. Our study confirms the magnetic phase transitions reported already in a single crystal of TbFe$_{0.5}$Mn$_{0.5}$O$_3$ and, in addition, reveals the presence of mixed magnetic domains. The ratio of these magnetic domains as a function of temperature is estimated from Rietveld refinement of neutron diffraction data. Indications of short-range magnetic correlations are present in the low-$Q$ region of the neutron diffraction patterns at $T < T^\mathrm{Fe/Mn}_{SR}$. These results should motivate further experimental work devoted to measure electric polarization and magnetocapacitance of TbFe$_{0.5}$Mn$_{0.5}$O$_3$.' author: - 'Harikrishnan S. Nair' - Tapan Chatterji - 'C. M. N. Kumar' - Thomas Hansen - Hariharan Nhalil - Suja Elizabeth - 'André M. Strydom' title: 'Magnetic structures and magnetic phase transitions in the Mn-doped orthoferrite TbFeO$_3$ studied by neutron powder diffraction' --- \[INTRO\]Introduction ===================== The orthoferrite \[$R$FeO$_3$; $R$ = rare earth\] oxides have been recently re-investigated experimentally and theoretically from the fascinating perspective of multiferroicity. [@mandal2011spin; @shang2013multiferroic; @deng2015magnetic; @pavlov2012optical; @zhao2014creating] Pursuing the recent line of multiferroics research, theoretical work on $R$FeO$_3$ thinfilms has identified that strain can convert paraelectric phase of orthoferrites in to ferroelectrics thus rendering them multiferroic[@zhao2014creating]. It has been found theoretically that for large values of strain on $R$FeO$_3$ with large rare earth ion, giant polarization is realized. In fact, with increasing strain, a new ferroelectric phase, not observed in any perovskite before, is realized. Multifunctional properties like large magnetoelectric coupling and ultrafast optical control of spins have been observed in the orthoferrites [@tokunaga2008magnetic; @yamaguchi2013terahertz; @mikhaylovskiy2014terahertz]. The $R$FeO$_3$ realize high Néel temperature, $T_N \approx$ 623 -740 K[@marezio1970crystal; @eibschutz1967mossbauer] however, in bulk form they are paraelectric rather than ferroelectric suggesting weak multiferroic effects. Weak ferroelectric polarization has been recently reported in Gd and Sm orthoferrites [@tokunaga2008magnetic; @lee2011spin] which are thought to have “improper” origin induced by magnetic order. In TbFeO$_3$, an unusual incommensurate magnetic phase was discovered through neutron diffraction [@artyukhin2012solitonic] – it was shown that the exchange of spin waves between extended topological defects could result in novel magnetic phases drawing parallels with the Yukawa forces that mediate between protons and neutrons in a nucleus. The Fe$^{3+}$ moments in TbFeO$_3$ exhibit $G_xA_yF_z$ ($Pb'n'm$) spin configuration at room temperature [@bouree1975mise; @bertaut1967structures; @tejada1995quantum] which is accompanied by a spin-reorientation to $F_xC_yG_z$ ($Pbn'm'$). At 3 K, another spin re-orientation occurs to revert to the $G_xA_yF_z$ ($Pb'n'm$) structure. It is considered that the Tb$^{3+}$ spins order in $F_xC_y$ structure in 10 - 3 K interval and in the $A_xG_y$ structure below 3 K. Doping the $R$-site in $R$FeO$_3$ with another rare earth is found to be profitable to realize electric field induced generation and reversal of ferromagnetic moments [@tokunaga2012electric; @tokunaga2014magnetic]. Chemical substitution at the Fe-site in $R$FeO$_3$ also brings about interesting multiferroic effects. For example, in the case of Mn-substituted YFeO$_3$, magnetoelectric and magnetodielectric effects at different temperatures were reported[@mandal2011spin]. First-order spin-reorientation effects were observed as a result of Mn-substitution however, the magnetodielectric effects were observed at lower temperatures than $T_N$ or $T_{SR}$. Giant magnetodielectric coupling is also observed in another doped-orthoferrite, DyMn$_{0.33}$Fe$_{0.67}$O$_3$ [@hong2012temperature]. Spin-reorientation effects and magnetic sublattice effects were also observed in doped-orthoferrites with large $R$[@nagata2001magnetic; @mihalik2013magnetic]. $G$-type magnetic ordering of Mn$^{3+}$ and Cr$^{3+}$ spins were observed below $T_N \approx$ 84 K in the case of TbMn$_{0.5}$Cr$_{0.5}$O$_3$[@staruch2014magnetic], in addition to signatures of short-range magnetic correlations observed below 40 K which was attributed to the ferromagnetic component from canting of magnetic moments along the $c$-axis. In the case of Mn-substituted compound TbFe$_{0.75}$Mn$_{0.25}$O$_3$, the $T_N$ was determined to be 550 K and the $T_{SR}$ as 180 K through magnetic studies and Mößbauer spectroscopy[@kim2011spin].\ In our previous investigation using magnetometry it was inferred that TbFe$_{0.5}$Mn$_{0.5}$O$_3$ orders in $A_xG_yC_z$ ($\Gamma_1$) structure at $T^\mathrm{Fe/Mn}_N \approx$ 286 K followed by a spin re-orientation at $T^\mathrm{Fe/Mn}_{SR} \approx$ 28 K to the structure $G_xA_yF_z$ ($\Gamma_4$)[@hariharan2015reorientation]. No signature of Tb ordering was obtained in the previous study. In the present manuscript, we make a detailed investigation of the magnetic structures and spin re-orientation transitions in TbFe$_{0.5}$Mn$_{0.5}$O$_3$ using neutron powder diffraction in order to confirm the magnetic structure arrived at through macroscopic magnetization earlier. We update the magnetic structures as a function of temperature and observe that they evolve between $\Gamma_1$ and $\Gamma_4$ through mixed-domains of ($\Gamma_1$ + $\Gamma_4$). \[EXP\]Experimental details =========================== Polycrystalline samples of TbFe$_{0.5}$Mn$_{0.5}$O$_3$ were prepared by conventional solid state reaction methods employing the oxides Tb$_2$O$_3$, FeO, MnO$_2$ (all from Sigma Aldrich, 99.9$\%$) as precursors. The thoroughly-mixed powder was heated at 1300$^{\circ}$ C for 4 days with intermediate grinding. The phase-purity of the black powder that resulted was checked using x ray diffraction employing a Philips X’pert diffractometer with Cu-$K\alpha$ radiation. The chemical composition of the prepared sample was determined using the Inductively Coupled Plasma emission Spectroscopy (ICPAES) method. Magnetization measurements were performed on a sintered pellet of the sample in a magnetic property measurement system (MPMS, Quantum Design, San Diego). Neutron powder diffraction experiments were performed on 8 g of TbFe$_{0.5}$Mn$_{0.5}$O$_3$ powder at the instrument D1B in ILL, Grenoble. A wavelength of 2.52 Å was used for the experiment. Diffractograms were recorded at 2 K, 5 K, 10 K, 26 K, 50 K to 100 K in 10 K interval and 100 K to 300 K in 50 K interval. The diffraction data was analyzed using FullProf suite of programs[@fullprof] employing the Rietveld method[@rietveld]. Magnetic structure was determined using the software SARA$h$[@sarah_wills] and was refined using FullProf. \[RESULTS\] Results =================== \[mag\] Magnetization --------------------- The experimentally measured magnetization curves, $M(T)$, in zero field-cooled and field-cooled protocols at 0.02 T are plotted in Fig \[fig\_mag\] (a) and the isothermal magnetization curves of TbFe$_{0.5}$Mn$_{0.5}$O$_3$ at 2 K and 40 K are presented in (b). Three magnetic phase transitions are identified in (a) [viz.,]{} $T^\mathrm{Fe/Mn}_{N} \approx$ 295 K, $T^\mathrm{Fe/Mn}_{SR} \approx$ 26 K and $T^\mathrm{R}_{N} \approx$ 2 K. The magnetic phase transition at $T^\mathrm{Fe/Mn}_{N}$ marks the paramagnetic (PM) to antiferromagnetic (AFM) phase transition. TbFe$_{0.5}$Mn$_{0.5}$O$_3$ adopts $G_xA_yF_z$ magnetic structure below this temperature [@hariharan2015reorientation]. The apparent difference seen in the transition temperatures of the single crystal and the polycrystalline sample might stem from differences in sample quality. As determined by the ICPAES method, the chemical composition of the sample is Tb$_{1.97}$Fe$_{0.51}$Mn$_{0.49}$O$_3$ which is very close to the nominal value. The second transition occurring at $T^\mathrm{Fe/Mn}_{SR} \approx$ 26 K corresponds to the spin-reorientation transition where the structure transforms from $G_xA_yF_z$ to $A_xG_yC_z$. At temperatures close to 2 K, signatures of Tb-order are observed which is reflected in the magnetization measurements at 0.02 T as a irreversibility between the ZFC and FC plots at $\approx$ 5 K. The panel (c) in Fig \[fig\_mag\] magnifies the $M(T)$ curves around $T^\mathrm{Fe/Mn}_{SR}$ where a “loop”-like feature is observed. It can be observed that the “loop”-like feature begins at $\approx$ 36 K and extends till about 18 K. The $M(T)$ at a higher applied field of 1 T was measured where the signs of magnetic phase transitions or bifurcation between ZFC and FC were absent (data not shown). The panel (d) shows the presence of irreversibility extends over a wide temperature range from 36 K to $T^\mathrm{Fe/Mn}_N$. In the panel (e), the derivative $dM/dT$ versus $T$ is plotted to show the $T^\mathrm{Fe/Mn}_{N}$ transition more clearly. It is seen that the $T^\mathrm{Fe/Mn}_{N}$ is a very broad transition with a spread in temperature from about 280 K extending to 303 K. The transition temperature of 295 K is estimated approximately at the point of steepest slope of $dM/dT$.\ 300 K 150 K 26 K 2 K ----------- ------------ ------------ ------------ ------------ -- $a (\AA)$ 5.3055(5) 5.3046(5) 5.3124(4) 5.3138(1) $b (\AA)$ 5.6877(8) 5.6772(6) 5.6676(8) 5.6692(6) $c (\AA)$ 7.5391(9) 7.5270(8) 7.5221(2) 7.5242(4) Tb: $x$ -0.0173(7) -0.0193(7) -0.0223(6) -0.0202(6) $y$ 0.0715(6) 0.0714(6) 0.0732(5) 0.0712(5) $z$ 0.25 0.25 0.25 0.25 O1: $x$ 0.1089(8) 0.1084(9) 0.1091(9) 0.1094(8) $y$ 0.4669(7) 0.4659(8) 0.4665(7) 0.4668(7) $z$ 0.25 0.25 0.25 0.25 O2: $x$ -0.2999(6) -0.3007(6) -0.3007(5) -0.3000(5) $y$ 0.3152(6) 0.3149(5) 0.3145(5) 0.31420(5) $z$ 0.0509(6) 0.0513(6) 0.0549(6) 0.0520(6) : \[tab1\] The refined lattice parameters and fractional coordinates of TbFe$_{0.5}$Mn$_{0.5}$O$_3$ at 300 K, 150 K, 26 K and at 2 K. These parameters are obtained through Rietveld refinement of the neutron powder diffraction data obtained from the instrument D1B, ILL, Grenoble. The nuclear structure model used was $Pbnm$ with Fe/Mn at $4b$ ($\frac{1}{2}$, 0, 0) and Tb at $4c$ ($x$,$y$,$z$). At 2 K, the maximum magnetization attained with application of 5 T is about $4~\mu_\mathrm{B}/$f.u. This value is lower than the value obtained for ferromagnetic alignment of Fe$^{3+}$, Mn$^{3+}$ and Tb$^{3+}$ moments. The observed maximum moment at 2 K, 5 T is comparable to the moment value obtained on the single crystal of TbFe$_{0.5}$Mn$_{0.5}$O$_3$ in the case of $H \parallel b$ at 25 K[@hariharan2015reorientation]. The first-order spin-flip-like transition observed at $H_c \pm$ 26 kOe is not clear in the present measurement on polycrystalline sample. Though a weak hysteresis is observed at 2 K in Fig \[fig\_mag\] (b), prominent hysteresis loops as observed for $H \parallel c$ in single crystals are absent. These features underline the magnetic anisotropy in the crystal sample of TbFe$_{0.5}$Mn$_{0.5}$O$_3$ [@hariharan2015reorientation]. \[npd\] Neutron diffraction --------------------------- The neutron diffraction patterns at different temperature points were refined using FullProf. The values of refined lattice parameters and fractional atomic coordinates of TbFe$_{0.5}$Mn$_{0.5}$O$_3$ at 300 K, 150 K, 26 K and 2 K are collected in Table \[tab1\]. The temperature-dependent variation of the lattice parameters $a$, $b$ and $c$ are shown in Fig \[fig\_latt\]. No significant anomalies are observed for the unit cell parameters as a function of temperature except for a change-of-slope at 150 K and 250 K. However, lack of enough data points makes any inference unreliable here. The trend of thermal evolution of $b$ and $c$ are comparable while that of $a$ is opposite to the other two. Jahn-Teller (JT) or pseudo-JT distortion has been correlated with the observation of multiferroicity in perovskites[@bersuker2012pseudo] including TbMnO$_3$ which are known to show significant JT effect[@zhou2007evidence]. In order to investigate the presence of JT effect in TbFe$_{0.5}$Mn$_{0.5}$O$_3$, estimates of the distortion parameters and bond angles and bond distances were obtained from the refined structural data, Table \[tab2\]. However, with the substitution of 50$\%$ Fe at the Mn-site, the effects of JT-distortion are found to have diminished in TbFe$_{0.5}$Mn$_{0.5}$O$_3$ however, effects of distortion of the perovskite structure are clearly seen. To facilitate comparison, in Table \[tab2\], the values of the JT-parameters of TbMnO$_3$ at 300 K taken from Ref.\[28\] [@zhou2007evidence] are given in parenthesis.\ 300 K 150 K 26 K 2 K ----------------------------------- ------------------------- ------------------------- ------------------------ ------------------------ -- Mn–O(2) $l$=2.123(4) $\times$ 2 $l$=2.112(5) $\times$2 $l$=2.114(6) $\times$2 $l$=2.111(5) $\times$2 $s$=1.947(6) $\times$ 2 $s$=1.949(6) $\times$2 $s$=1.956(7) $\times$2 $s$=1.951(7) $\times$2 Mn–O(1) $m$=1.977(8) $\times$ 2 $m$=1.977(7) $\times$ 2 $m$=1.977(8) $\times$2 $m$=1.978(6) $\times$2 Mn–O(2)–Mn 145.81(8) 146.06(9) 145.14(7) 145.81(11) Mn–O(1)–Mn 144.9(12) 144.2(10) 144.04(8) 144.9(9) O(1)–Mn–O(1) 180 180 180 180 O(2)–Mn–O(2) 90.57(12) $\times$ 2 90.60(8) $\times$ 2 90.16(11) $\times$ 2 90.41(8) $\times$ 2 89.53(9) $\times$ 2 89.40(6) $\times$ 2 89.83(9) $\times$ 2 89.59(7) $\times$ 2 $Q_2 = 2(l -s)/\sqrt(2)$ 0.2489 (0.45) 0.2305 0.2234 0.2263 $Q_3 = 2(2m-l -s)/\sqrt(6)$ -0.0947 (-0.2) -0.0874 -0.0947 -0.0865 $\phi$ = tan($Q_3/Q_2$) -20.84$^{\circ}$ (-24) -20.77$^{\circ}$ -22.98$^{\circ}$ -20.92$^{\circ}$ $\rho_0$ = $\sqrt{Q_3^2 + Q_2^2}$ 0.266 (0.5) 0.2465 0.2426 0.2423 A symmetry analysis of $R$FeO$_3$ in $Pbnm$ space group with Fe$^{3+}$ in $4b$ and $R^{3+}$ in $4c$ Wyckoff positions leads to eight irreducible representations, $\Gamma_1$ through $\Gamma_8$, for magnetic structure. For the $4b$ position, the configurations $\Gamma_5$ to $\Gamma_8$ are not allowed and hence $\Gamma_1$, $\Gamma_2$, $\Gamma_3$ and $\Gamma_4$ are selected as the final possibilities. Table \[tab3\] lists these irreducible representations, the Shubnikov space groups and the magnetic structure notations used for $R$FeO$_3$ in general. The neutron diffraction pattern of TbFe$_{0.5}$Mn$_{0.5}$O$_3$ at 300 K is presented in Fig \[fig\_npd\_300\] as black plus signs. The nuclear structure at 300 K is refined in $Pbnm$ space group. In Fig \[fig\_npd\_300\], the calculated pattern is shown as red solid line, difference curve as gray dotted line and the allowed Bragg peaks for $Pbnm$ space group as vertical bars. From the magnetization measurements presented in Fig \[fig\_mag\] (a, e), it is clear that TbFe$_{0.5}$Mn$_{0.5}$O$_3$ undergoes a magnetic phase transition very close to 300 K (notice from Fig \[fig\_mag\] (e) that the transition extends over a wide range from 280 K to 303 K). Hence, the diffraction data at 300 K is refined with additional magnetic phase. Irreps Space group $4b$ $4c$ ------------ ------------- ------------- ---------- -- -- $\Gamma_1$ $Pbnm$ $A_xG_yC_z$ $C_z$ $\Gamma_2$ $Pbn'm'$ $F_xC_yG_z$ $F_xC_y$ $\Gamma_3$ $Pb'nm'$ $C_xF_yA_z$ $C_xF_y$ $\Gamma_4$ $Pb'n'm$ $G_xA_yF_z$ $F_z$ : \[tab3\] The possible magnetic structures of $R$FeO$_3$ allowed by symmetry. The space group is chosen as $Pbnm$ and $x$, $y$ and $z$ denote orientations parallel to the crystallographic directions $a$, $b$ and $c$. $R$ occupies $4c$ and Fe, $4b$ Wyckoff positions in this structure. In order to solve the magnetic structure at 300 K, the peaks below 2$\Theta \approx$ 40$^{\circ}$ were used to perform a $\bf k$-search to find the propagation vector. The utility called $\bf k$-search within FullProf suite of programs was used for this purpose. Thus, $\bf k$=(000) was identified as the propagation vector. Representation analysis using $\bf k$(000) and $Pbnm$ nuclear cell lead to the listing of four possibilities – $\Gamma_1$ ($Pbnm$), $\Gamma_2$ ($Pbn'm'$), $\Gamma_3$ ($Pb'nm'$) and $\Gamma_4$ ($Pb'n'm$) matching with the selection in Table \[tab3\]. They also match with the magnetic structures of orthoferrites already reported in the literature [@deng2015magnetic]. From the refinement trials it was noted that the representation $\Gamma_3$ contributes zero intensity to the Bragg peaks at (101) and (011) and hence can be excluded. A better visual fit to the experimental data and reasonable agreement factors were obtained for $\Gamma_4$ (the $R_\mathrm{mag}$ factors were, $\Gamma_1\approx$ 70; $\Gamma_2\approx$ 21; $\Gamma_3\approx$ 20 and $\Gamma_4\approx$ 5) and hence was accepted as the solution to the magnetic structure at 300 K. In Fig \[fig\_npd\_300\], the lower set of vertical tick marks correspond to the magnetic Bragg positions.\ After confirming the room-temperature crystal structure to be $Pbnm$ and the magnetic structure as $\Gamma_4$ ($Pb'n'm$), we now discuss the low temperature diffraction data. In Fig \[fig\_npd\], an expanded view of the reflections in the 2$\Theta$-range 31–35$^{\circ}$ is given for 2 K, 90 K and 300 K. At 2 K, the intensity of the (011) reflection is observed to increase, compared to the value at 300 K. The relative intensity, $I_{(011)}/I_{(101)} > 1$ at 2 K and 300 K where as the opposite is true for 90 K. Macroscopic magnetic characterization of TbFe$_{0.5}$Mn$_{0.5}$O$_3$ [@hariharan2015reorientation] clearly suggested spin-reorientations and magnetic phase transformations as a function of temperature and magnetic fields. At 250 K, the nuclear structure was refined in $Pbnm$ and the magnetic structure in $\Gamma_4$ ($Pb'n'm$) similar to the case of 300 K-data. However, at 200 K mixed magnetic domains consisting of ($\Gamma_1$ + $\Gamma_4$) is found to reproduce the experimental data faithfully. Neither of the representations $\Gamma_1$ or $\Gamma_4$ alone could give a satisfactory fit, while the mixed-domain model significantly returned lower values of reliability factors ($R\sim$ 5.3 for mixed-domains whereas $\sim$ 6 for pure phases). A better fit using the mixed-domain model could result from utilizing a bigger parameter space for least-squares however, we also notice the presence of clear irreversibility in the magnetization profiles that are in support of the claim of presence of mixed-domains. The mixed-domains of ($\Gamma_1$ + $\Gamma_4$) were found to exist down till $T^\mathrm{Fe/Mn}_{SR}$ at 26 K. Fig \[fig\_mag\] shows that “loop-like” anomalies are present in the $M(T)$ which correspond to the spin-reorientation transition at $T^\mathrm{Fe/Mn}_{SR}$. In fact, from the panel (c) of Fig \[fig\_mag\] it can be seen that the irreversibility in ZFC and FC curves commences at 36 K itself. Finally at 10 K, the $\Gamma_4$ representation observed at 300 K re-emerges. It was observed that the inclusion of Tb moments in the refinement at 10 K and 5 K did not lead to any appreciable values of refined magnetic moment on the Tb site. At 2 K, the magnetic structure remains in the $\Gamma_4$ representation however, Tb develops a magnetic moment value of 0.6(2) $\mu_\mathrm{B}/$f.u. indicating that Tb is magnetically ordered at this temperature. The irreversibility in $M (T)$ suggesting magnetic ordering in the rare earth sublattice occurs $\approx$ 5 K. However, attempts to refine the 5 K-data assuming magnetic contribution from Tb did not yield better agreement factors. Hence magnetic contributions from Mn/Fe were only considered. The refinement of the magnetic structure of Tb was performed in $F_z$ representation following the symmetry analysis[@deng2015magnetic]. In Fig \[fig\_npd\_refine\] (a)–(e), the refined diffraction patterns of TbFe$_{0.5}$Mn$_{0.5}$O$_3$ at 200, 50, 26, 10 K and 2 K are shown. Though no structural distortions are observed in the entire temperature range of the study, the magnetic structure is seen to evolve between the $\Gamma_4$ and $\Gamma_1$ representation through mixed-domain regions. A schematic of the magnetic structures $\Gamma_1$ and $\Gamma_4$ are shown in Fig \[fig\_magstr\]. In the case of $\Gamma_4$, the $F_z$ type ordering of Tb ions are also depicted (Tb moments are shown in blue). Next, we make an attempt to quantify the volume fraction of the two magnetic domains $\Gamma_1$ and $\Gamma_4$. For this calculation we assume that the total Fe/Mn ordered moment at a particular temperature is equal for both $\Gamma_1$ and $\Gamma_4$ domains. With this assumption and by setting the scale factors of both $\Gamma_1$ and $\Gamma_4$ domains equal to that of the nuclear phase, the ratio of refined ordered moments of $\Gamma_1$ and $\Gamma_4$ should give the percentage of each domains. The result of this calculation is presented as Fig \[fig\_domains\]. As clear from the figure, mixed domain structure is present in the wide temperature range of 250 K to 26 K. Outside this temperature window, the major magnetic phase is that of $\Gamma_4$. \[DISCUSSION\]Discussion ======================== Our investigation of the magnetic structure of TbFe$_{0.5}$Mn$_{0.5}$O$_3$ shows that the predominant structures are $\Gamma_4$ and $\Gamma_1$ however, a competition between these two magnetic phases is evident from the presence of a mixed-domains in certain temperature range. At 295 K, the high temperature paramagnetic phase transforms to the magnetic structure $\Gamma_4$ ($G_xA_yF_z$ or $Pb'n'm$). It is seen that down till $\approx$ 250 K, the $\Gamma_4$ structure remains stable. In the intermediate temperature range close to 200 K a mixed-domains of ($\Gamma_1$ + $\Gamma_4$) is observed which remains down till 26 K. A precise phase space study to determine the boundaries of the mixed phase would demand many more temperature points, but such a detailed study was outside the design and scope of our present experiments. As evidenced by the $M(T)$ curve (Fig \[fig\_mag\] (a)), the $T^\mathrm{Fe/Mn}_{SR}$ is associated with a clear bifurcation of ZFC and FC forming a “loop-like” region between 18 K and 36 K. We deduce that the mixed-domains extend over this temperature window. Further at 10 K, the $\Gamma_4$ structure reemerges and remains till 2 K. In the temperature range 2 - 10 K, the magnetic representation $\Gamma_1$ leads to a high $R_\mathrm{mag}\approx$ 60. It did not adequately account for the magnetic peaks and hence, $\Gamma_4$ ($R_\mathrm{mag}\approx$ 3.5) was chosen as the solution. Note that in both $\Gamma_1$ and $\Gamma_4$, Fe/Mn are constrained to have magnetic moments in $x$, $y$ and $z$ whereas the $R$-moment is constrained to the $z$-direction. We found that in whole temperature range in $\Gamma_1$ setting the Fe/Mn magnetic moments had negligible $x$ and $z$ components (practically zero) and aligned along y-axis (Fig. 6). Also the whole temperature range in $\Gamma_4$ setting Fe/Mn moments have negligible y-component and aligned along $x$-axis with small canting along $y$-direction.\ Different rare earths in $R$FeO$_3$ are observed to have different magnetic structures at low temperatures. For example, DyFeO$_3$ has a $Pb'n'm'$ space group for Dy and $Pnma$ for Fe whereas TbFeO$_3$ has $Pbnm'$ for Tb. In the present case of TbFe$_{0.5}$Mn$_{0.5}$O$_3$, it is found that in the low temperature region significant diffuse scattered intensity is present arising from the short-range magnetic order of Tb. Supporting this feature is the fact that no enhancement on quality of the fit was obtained in refining the 5 K or 10 K diffraction data by introducing a magnetic moment for Tb. Only at 2 K does Tb begin to order as $F_z$ as evident by the development of 0.6(2) $\mu_\mathrm{B}$ for Tb moment. A notable difference of the magnetic structure of Fe/Mn in TbFe$_{0.5}$Mn$_{0.5}$O$_3$ compared to that of the parent compound TbFeO$_3$ is that the latter compound transforms to $\Gamma_2$ ($F_xC_yG_z$) structure at the spin-reorientation transition. In both the cases, the high temperature structure pertaining to Fe/Mn is $\Gamma_4$. In TbFeO$_3$, the Tb moment undergoes two types of ordering below 10 K in to $F_xC_y$ and to $A_xG_y$ whereas in TbFe$_{0.5}$Mn$_{0.5}$O$_3$, Tb develops no significant magnetic moment until 2 K where it orders $F_z$. \[CONCLUSION\]Conclusions ========================= In conclusion, the Mn-doped orthoferrite compound TbFe$_{0.5}$Mn$_{0.5}$O$_3$ orders antiferromagnetically at $T^\mathrm{Fe/Mn}_{N}\approx$ 300 K and undergoes spin-reorientation transition at $T^\mathrm{Fe/Mn}_{SR}$ 26 K. Further, at 2 K the rare earth Tb is found to be magnetically ordered, however, the exact ordering temperature is no confirmed through our study. It is found that in the intermediate temperatures 250 K to 26 K, a mixed-domain model best describe the neutron diffraction data. At 2 K and at 300 K $\Gamma_4$ representation is stable while at the spin-reorientation transition and around 200 K mixed-domains of ($\Gamma_1$ + $\Gamma_4$) exist. An estimate of the domain concentration as a function of temperature is made. Clear indication of diffuse magnetic scattering from Tb is present especially below the $T^\mathrm{Fe/Mn}_{SR}$ while long-range order emerges at 2 K. Acknowledgements {#acknowledgements .unnumbered} ================ H. S. N. acknowledges FRC/URC of UJ for a postdoctoral fellowship. A. M. S. thanks the SA-NRF (93549) and the FRC/URC of UJ for financial assistance. 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\ <span style="font-variant:small-caps;">[W[ł]{}adys[ł]{}aw A. Majewski]{}\ Institute of Theoretical Physics and Astrophysics\ Gda[ń]{}sk University\ Wita Stwosza 57\ 80-952 Gda[ń]{}sk, Poland</span>\ E-mail address: `[email protected]`\ <span style="font-variant:small-caps;">Abstract.</span> We outline the scheme for quantization of classical Banach space results associated with some prototypes of dynamical maps and describe the quantization of correlations as well. A relation between these two areas is discussed. : Primary: 46L53, 46L60: Secondary: 46L45, 46L30 Key words and phrases: $C^$-algebras, positive maps, separable states, entanglement, quantum stochastic dynamics, quantum correlations. INTRODUCTION ============ The aim of this paper is to bring together two areas, theory of positive maps on [[[$\hbox{\bf C}^$]{}]{}]{}-algebras and theory of entanglement considered as a peculiar feature of non-commutative Radon measures. Both topics are at the heart of quantum theory, thus in particular, in the foundations of quantum information theory. It will be shown that such pure quantum features as: peculiar behaviour of positive maps, quantum correlations, entanglement, and quantum stochastic dynamics, can be easily obtained within the framework of the [[[$\hbox{\bf C}^$]{}]{}]{}-algebraic approach to Quantum Mechanics. This approach sheds new light on entanglement and quantum features of correlations of non-commutative systems. In particular, some emphasis will be put on evolution of entanglement. The paper is organized as follows. Section 2 provides sufficient preparation for the concept of “quantization” of classical results related to prototypes of dynamical maps. Section 3 is concerned with entanglement and the coefficient of quantum correlations. The latter is again an example of “quantization” of a classical concept. The last section contains a brief discussion of applications of the presented results to the description of quantum dynamical systems. We will discuss the evolution of entanglement for some selected models as well as relations between classification of positive maps and measures of entanglement. POSITIVE MAPS ============= In this section we compile some basic facts on the theory of positive maps on [[[$\hbox{\bf C}^$]{}]{}]{}-algebras. To begin with, let ${{\mathcal A}}$ and ${{\mathcal B}}$ be [[[$\hbox{\bf C}^$]{}]{}]{}-algebras (with unit), ${{\mathcal A}}_h = \{ a \in {{\mathcal A}}; a = a^ \}$, ${{\mathcal A}}^+ = \{ a \in {{\mathcal A}}_h; a \ge 0 \}$ - the set of all positive elements in ${{\mathcal A}}$, and ${{\mathcal S}}({{\mathcal A}})$ the set of all states on ${{\mathcal A}}$. In particular $$({{\mathcal A}}_h, {{\mathcal A}}^+)\quad is \quad an \quad ordered \quad Banach \quad space.$$ We say that a linear map $\alpha : {{\mathcal A}}\to {{\mathcal B}}$ is positive if $\alpha({{\mathcal A}}^+) \subset {{\mathcal B}}^+$. The theory of positive maps on non-commutative algebras can be viewed as a jig-saw-puzzle with pieces whose exact form is not well known. Therefore, as we address this paper to a readership interested in quantum mechanics and quantum information theory, we will focus our attention on “quantization” procedure of some classical (Banach space) results in order to facilitate access to some main problems of that theory. We begin with the classical Banach-Stone result ([@B], [@S]) [if a unital linear map $T: C(X) \to C(Y)$, where $X,Y$ are compact Hausdorff spaces, is either isometric or an order-isomorphism then it is also an algebraic isomorphism.]{} Thus, even in the Banach space setting, the order and algebraic structures are strongly related. The Banach-Stone theorem has the following non-commutative generalization (Kadison, [@K]): [a unital isometric or order isomorphic linear map $\alpha : {{\mathcal A}}_h \to {{\mathcal B}}_h$ must preserve the Jordan product ( $(a,b) \mapsto {1/2}(ab + ba)$)]{}. In other words, this result indicates the role of a specific algebraic structure - the Jordan structure - in operator algebras and that remark will be frequently used throughout the paper. Moreover, such a result makes it legitimate to study and to classify [[[$\hbox{\bf C}^$]{}]{}]{}-algebras ${{\mathcal A}}$ by a detailed analysis of ordered Banach spaces ${{\mathcal A}}_h$. However, this is [a very difficult task]{}. In particular, it was soon realized that one of the basic problems is the answer to the following question: which compact convex sets can arise as the state spaces of unital $C^$-algebras (again a very difficult task!). To describe the next result we need some preliminaries. Let $(\Omega, \mu)$ be a measure space. Here and subsequently, $\mu$ stands for a probability measure. The triple (semigroup $\{S_t\}$, $\Omega$, $\mu$) will denote the classical dynamical system where $ S_t : \Omega \to \Omega$ is a one parameter family of measure preserving maps. The phase functions $f : \Omega \to {{\mathbb{C}}}$ evolve according to the Koopman operators $$V_tf(\omega)=f(S_t\omega) \ , \ \ \omega\in\Omega \,.$$ It is known that the Koopman operators $V_t$ are isometries on the Banach space $L^p=L^p(\Omega, \mu)$, $p\ge 1$ of $p$-integrable functions and unitary operators when restricted to the Hilbert space $L^2$ and the transformations $S_t$ are automorphisms. The relation of the point dynamics with the Koopman operators is clarified by asking the question: [what types of isometries on $L^p$ spaces are implementable by point transformations?]{} For $L^p$ spaces $p\not=2$, all isometries induce underlying point transformations, i.e. if $\|\|Vf\|\| = \|\|f\|\|$ for all $f \in L^p$, then $V$ is given by an underlying measurable point transformation $S$ and a certain function $h$ according to $(Vf)(x) = h(x)f(Sx)$. Such theorems on the implementability of isometries on $L^p$ spaces, $p\neq 2$, are known as [Banach-Lamperti theorems ]{} [@B], [@La]. They are of great importance for the Misra-Prigogine-Courbage theory [@MPC] which is trying [to reconcile irreversible phenomena with the basic dynamical laws]{}. Again, one may “quantize” Banach-Lamperti theorems [@Ye], see also [@AMS]. To this end one should use the so called non-commutative (quantum) $L_p$-spaces. Namely, using the “quantized” measure theory, let $\{ {{\mathcal A}}, \varphi \}$ be a von Neumann algebra with faithful normal trace and let $L_p({{\mathcal A}}, \varphi)$, $p \ge 1$, be the corresponding quantum $L_p$-space, i.e. a Banach space of operators which is closed under an appropriate norm. Assume that $T: L_p({{\mathcal A}}, \varphi) \to L_p({{\mathcal A}}, \varphi)$ is a linear map. [Then $T$ is an $L_p$-isometry if and only if ]{} $$T(x) = W B J(x), \quad x \in L_p({{\mathcal A}}, \varphi) \cap {{\mathcal A}}$$ where $W \in {{\mathcal A}}$ is a partial isometry, $B$ a selfadjoint operator affiliated with ${{\mathcal A}}$, $J$ a normal Jordan isomorphism mapping ${{\mathcal A}}$ into a weakly closed $^$-subalgebra of ${{\mathcal A}}$ such that $ W^W = J({{\mathbb{I}}}) = supp(B)$ and $B$ commutes strongly with $ J({{\mathcal A}})$. Again, we can see the importance of the Jordan structure. The third example we wish to recall is associated with a very strong notion of positivity: the so called complete positivity (CP). Namely, a linear map $\tau : {{\mathcal A}}\to {{\mathcal B}}$ is CP iff $$\tau_n : M_n({{\mathcal A}}) \to M_n({{\mathcal B}}); [a_{ij} \mapsto [\tau(a_{ij})]$$ is positive for all n. To explain the basic motivation for that concept we need the following notion: [an operator state of ${{\mathcal A}}$ on a Hilbert space ${{\mathcal K}}$ is a CP map $\tau : {{\mathcal A}}\to {{\mathcal B}}({{\mathcal K}})$]{}. Having that concept we can recall the Stinespring result, [@Sti], which is the generalization of GNS construction and which was the starting point for a general interest in the concept of complete positivity. [For operator state $\tau$ there is a Hilbert space ${{\mathcal H}}$, a $^$-representation $\pi : {{\mathcal A}}\to {{\mathcal B}}({{\mathcal H}})$ and a partial isometry $V : {{\mathcal K}}\to {{\mathcal H}}$ for which]{} $$\tau(a) = V^ \pi(a) V.$$ Following the quantization “route”, it was shown - (Choi, [@Ch]) if $\tau : {{\mathcal A}}\to {{\mathcal B}}$ is a CP order isomorphism then it is a $^$-isomorphism. This can be considered as a final “quantization” of the Banach-Stone theorem. - (Arverson, [@Ar]) the Hahn-Banach theorem and its order-theoretical version (due to Krein) has a nice generalization for non-commutative structures in terms of CP maps: [Let ${\mathcal N}$ be a closed self-adjoint subspace of [[[$\hbox{\bf C}^$]{}]{}]{}algebra ${{\mathcal A}}$ containing the identity and let $\tau: {\mathcal N} \to {{\mathcal B}}({{\mathcal H}})$ be a CP map. Then $\tau$ possesses an extension to a CP map $\tilde{\tau} : {{\mathcal A}}\to {{\mathcal B}}({{\mathcal H}})$.]{} It is worth pointing out that plain positivity is not enough for these generalizations. Moreover, Arverson’s extension theorem is the basis of the CP ideology in open system theory. Up to now we considered linear positive maps on an algebra without entering into the (possible) complexity of the underlying algebra. The situation changes when one is dealing with composed systems (for example in the framework of open system theory). Namely, there is a need to use the tensor product structure. In particular, again, we wish to consider positive maps but now defined on the tensor product of two [[[$\hbox{\bf C}^$]{}]{}]{}-algebras, $\tau : {{\mathcal A}}\otimes {{\mathcal B}}\to {{\mathcal A}}\otimes {{\mathcal B}}$. But now the question of order is much more complicated. Namely, there are various cones determining the order structure in the tensor product of algebras (cf. [@W]) $${{\mathcal C}}_{inj} \equiv ({{\mathcal A}}\otimes {{\mathcal B}})^+ \supseteq, ..., \supseteq {{\mathcal C}}_{\beta} \supseteq,..., \supseteq {{\mathcal C}}_{pro}\equiv conv({{\mathcal A}}^+\otimes{{\mathcal B}}^+)$$ and correspondingly in terms of states (cf [@MM]) $${{\mathcal S}}({{\mathcal A}}\otimes {{\mathcal B}}) \supseteq,..., \supseteq {{\mathcal S}}_{\beta} \supseteq, ..., \supseteq conv({{\mathcal S}}({{\mathcal A}})\otimes{{\mathcal S}}({{\mathcal B}})).$$ Here, ${{\mathcal C}}_{inj}$ stands for the injective cone, ${{\mathcal C}}_{\beta}$ for a tensor cone, while ${{\mathcal C}}_{pro}$ for the projective cone. The tensor cone ${{\mathcal C}}_{\beta}$ is defined by the property: the canonical bilinear mappings $\omega :{{\mathcal A}}_h \times {{\mathcal B}}_h \to ({{\mathcal A}}_h \otimes {{\mathcal B}}_h, {{\mathcal C}}_{\beta})$ and $\omega^ : {{\mathcal A}}^_h \times {{\mathcal B}}^_h \to ({{\mathcal A}}^_h \otimes {{\mathcal B}}^_h, {{\mathcal C}}_{\beta}^)$ are positive. The connes ${{\mathcal C}}_{inj}, C_{\beta}, C_{pro}$ are different unless either ${{\mathcal A}}$, or ${{\mathcal B}}$, or both ${{\mathcal A}}$ and ${{\mathcal B}}$ are abelian. This feature is the origin of various positivity concepts for non-commutative composed systems and it was Stinespring who used the partial transposition (transposition tensored with identity map) for showing the difference among $C_{\beta}$ and ${{\mathcal C}}_{inj}$ and ${{\mathcal C}}_{pro}$. Clearly, in dual terms, the mentioned property corresponds to the fact that separable states are different from the set of all states and that there are various special subsets of states if both subsystems are truly quantum. To summarize one can say that contrary to the plain positivity, CP property plays a dominant role in the programme of quantization of classical results for composed systems. However, as it will be discussed in the final section, other types of positivity are helpful for better understanding the relations between various subsets of states, the algebraic and the order structure. QUANTUM CORRELATIONS ==================== Now we wish to discuss problems associated with the partial order structure of tensor product of [[[$\hbox{\bf C}^$]{}]{}]{}-algebras which are related to quantum information theory. Our first remark is the observation that quantum information theory relies on the fact that the restriction of a pure state of a composed system to a subsystem is, in general, not pure. Moreover, if the restriction of a pure state is pure then the state of the composed system is of the product form. This leads to the observation that the coupling of observables of a composed system by an entangled state offers additional possibilities for information exchange as well as a chance to reproduce states. All that follows from the fact that entangled states exhibit non-classical correlations. To be more precise, let ${{\mathcal A}}_1 \subseteq {{\mathcal B}}({{\mathcal H}}_1)$ and ${{\mathcal A}}_2 \subseteq {{\mathcal B}}({{\mathcal H}}_2)$ be two concrete [[[$\hbox{\bf C}^$]{}]{}]{}-algebras and define, for a state $\omega$ on ${{\mathcal A}}_1 \otimes {{\mathcal A}}_2$, the following map: $$(r_1 \omega)(A) \equiv \omega(A \otimes {\bf 1}) \quad ( (r_2\omega(B) = \omega( {\bf 1}\otimes B) )$$ where $A \in {{\mathcal A}}_1$ ($B \in {{\mathcal A}}_2$). $r_{1(2)} \omega$ is a state on ${{\mathcal A}}_{1(2)}$ . Moreover: [Let $(r_1 \omega)$ be a pure state on ${{\mathcal A}}_1$. Then $\omega$ can be written as a product state on ${{\mathcal A}}_1 \otimes {{\mathcal A}}_2$.]{} Let $\omega$ be a state on ${\mathcal A}_1 \otimes {{\mathcal A}}_2$. [The entanglement of formation]{}, EoF, of $\omega$ can be defined as ( [@Mjp], see also [@Ben]) $${E}(\omega) = \inf_{\mu \in M_{\omega}({{\mathcal S}})} \int_{{{\mathcal S}}} d\mu(\varphi) S(r\varphi)$$ where $S(\cdot)$ stands for the von Neumann entropy, i.e. $S(\varphi) = - Tr \varrho_{\varphi} \log \varrho_{\varphi}$ where $\varrho_{\varphi}$ is the density matrix determining the state $\varphi$. We want to stress that other entropy-functions can be used! The given definition of EoF is based on the decomposition theory and in particular $M_{\omega}({{\mathcal S}}) \equiv \{ \mu: \omega = \int_{{{\mathcal S}}}\nu d\mu(\nu)\}$. We recall that the separable states are those which are in the closure of the convex hull of simple tensors (so tensor products of subsystem states) while an entangled state stands for a non-separable one. One can prove [@Mjp] A state $\omega \in {{\mathcal S}}$ is separable if and only if $E(\omega)$ is equal to 0. Let us denote the set of all states on ${{\mathcal A}}\equiv {{\mathcal A}}_1 \otimes {{\mathcal A}}_2$ (${{\mathcal A}}_1$, ${{\mathcal A}}_2$) by ${{\mathcal S}}({{\mathcal A}})$ (${{\mathcal S}}({{\mathcal A}}_1)$, ${{\mathcal S}}({{\mathcal A}}_2)$ respectively). Obviously, $r_i \omega$ is in ${{\mathcal S}}({{\mathcal A}}_i)$, $i=1,2$. Next, take a measure $\mu$ on ${{\mathcal S}}({{\mathcal A}})$. Then, using the restriction maps $r_i$ one can define measures $\mu_i$ on ${{\mathcal S}}({{\mathcal A}}_i)$ in the following way: for a Borel subset $F_i \subset {{\mathcal S}}({{\mathcal A}}_i)$ we put $$\mu_i(F_i) = \mu(r_i^{-1}(F_i)), \quad i =1,2.$$ Having measures $\mu_1$ and $\mu_2$, both coming from the given measure $\mu$ on ${{\mathcal S}}({{\mathcal A}})$, one can define new measure $\boxtimes \mu$ on ${{\mathcal S}}({{\mathcal A}}_1) \times {{\mathcal S}}({{\mathcal A}}_2)$ which encodes classical correlations between the two subsystems described by ${{\mathcal A}}_1$ and ${{\mathcal A}}_2$ respectively (see [@Ma2] for details). The measure $\boxtimes \mu$ leads to the concept of coefficient of local (quantum) correlations for $\phi \in {{\mathcal S}}({{\mathcal A}}), a_1 \in {{\mathcal A}}_1, a_2 \in {{\mathcal A}}_2$, which is defined as $$\begin{aligned} d(\phi, a_1, a_2)& = & \inf_{\mu \in M_{\phi}({{\mathcal S}}({{\mathcal A}}))} \|\phi(a_1 \otimes a_2) \nonumber \\ && - (\int \xi d(\boxtimes \mu)(\xi))(a_1 \otimes a_2)\| \nonumber\end{aligned}$$ The crucial property of the coefficient of quantum correlations is that $d(\phi, \cdot \cdot)$ is equal to $0$ if and only if the state $\phi$ is separable ([@Ma2], [@Ma3]). The advantage of using $d(\cdot)$ lies in the fact that that concept looks more operational and that it does not use an entropy function. Moreover, $d(\cdot)$ is nothing else but the “quantization” of the classical concept of coefficient of independence. Hence, we got a strong indication that entangled states contain new type of correlations which are called quantum. SOME APPLICATIONS ================= QUANTUM STOCHASTIC DYNAMICS ([@MZ1]-[@MZ5], [@MOZ]) --------------------------------------------------- It is well known that in the theory of classical particle systems one of the basic objectives is to produce, describe and analyze dynamical systems with an evolution originated from stochastic processes in such a way that their equilibrium states are given Gibbs states (cf. [@Ligget]). A well known illustration of such an approach are systems with the so called Glauber dynamics [@Rx]. To carry out the analysis of dynamical systems with evolution originated from stochastic processes, it is convenient to use the theory of Markov processes in the framework of $L_p$-spaces. In particular, for the Markov-Feller processes, using the unique correspondence between the process and the corresponding dynamical semigroup, one can give a recipe for the construction of Markov generators (see [@Ligget]). The correspondence uses the concept of conditional expectation which can be nicely characterized within the (classical) $L_p$-space framework (cf. [@Moya]). Furthermore, (classical) $L_p$ spaces are extremely useful in a detailed analysis of the ergodic properties of the evolution. However, as contemporary science is based on [quantum mechanics]{}, it is again legitimate to look for a quantization of the above approach. That task was carried out in the setting of quantum mechanics and the main ingredient of the quantization was the concept of generalized conditional expectation and Dirichlet forms defined in terms of non-commutative (quantum) $L_p$-spaces. We already met these spaces in the description of quantized Banach-Lamperti theorems. The advantage of using quantum $L_p$-spaces for the quantization of stochastic dynamics lies in the fact that we can follow the traditional “route” of analysis of dynamical systems and that it is possible to have a single scheme for the quantum counterparts of stochastic dynamics of both jump and diffusive type. Turning to concrete dynamical systems, for example to jump type evolutions, we recall that one of the essential ingredients of the $L_p$-space approach to the analysis of such evolutions, is the usage of local knowledge. To illustrate that idea let us consider a region $\Lambda_I$ (usually finite) and its environment $\Lambda_{II}$. Then, performing an operation over $\Lambda_I$ (e.g. a block-spin flip or a symmetry transformation) one is changing locally the reference state. Such a change can be expressed in terms of generalized conditional expectations. Guided by the classical theory, one can define, now in terms of generalized conditional expectations, the infinitesimal generator of quantum dynamics. It is important to note that such a dynamics is the result of local operations (associated with the mentioned local knowledge about the system).\ Then, having defined the dynamics, we should pose the natural question of its nontriviality. By this we understand, first of all, that the infinitesimal generator of the dynamics is [not]{} a function of the hamiltonian defining the reference Gibbs state. This requirement arises in a natural way from the methodology of constructing stochastic dynamics as sketched in the preceding paragraph. In fact, it has been shown [@MOZ] that generators defined within the $L_p$-space setting satisfy the above requirement. On the other hand, [in order to confirm that the constructed dynamics are interesting, and the genuine quantum counterparts of classical dynamical maps it is necessary to study the evolution of entanglement and correlations as measures of coupling between two subsystems]{} caused by local (e.g. block-spin flip) operations. Going in that direction, an analysis of stochastic quantum models based on reference systems determined by Ising type and XXZ hamiltonians ([@KM2], see also [@KM1]) was done. It has shown the tendency of enhancement of quantum correlations. In the first example, based on one dimensional Ising model with nearest neighbor interactions, the lack of production of quantum correlations was shown. This is to be expected because the Ising model illustrates a behaviour typical of classical interactions (cf [@BR]). The second example, based on the quantum XXZ model with more interesting and complicated features of propagation, provides [clear signatures of production of quantum correlations.]{} POSITIVE MAPS VERSUS ENTANGLEMENT --------------------------------- The analysis of evolution of entanglement which was described in the previous subsection indicates that there is a need for an operational measure of entanglement. This demand is strenghtened by the observation that the amount of states that can be used for quantum information is measured by the entanglement. On the other hand, the programme of classification of entanglement seems to be a very difficult task. In particular, it was realized that the first step must presumably take the full classification of all positive maps. To see this let us take a positive map $\alpha_{1,t}: {{\mathcal A}}_1 \to {{\mathcal A}}_1$, $t$ being the time, and consider the evolution of a density matrix $\varrho$ ($\varrho$ determines the state $\phi \in {{\mathcal S}}({{\mathcal A}}\otimes {{\mathcal B}})$), i.e. we wish to study $(\alpha_{1,t} \otimes id_2)^d\varrho$. Here $(\alpha_{1,t} \otimes id_2)^d$ stands for the dual map, i.e. for the dynamical map in the Schrödinger picture. Then, if $\varrho$ is an entangled state, $(\alpha_{1,t} \otimes id_2)^d\varrho$ may develop negative eigenvalues and thus lose consistency as a physical state. That observation was the origin of rediscovery, now in the physical context, of Stinespring’s argument saying that the tensor product of transposition with the identity map can distinguish various cones in the tensor product structure (see Section 2). This led to the criterion of separability ([@P], [@H]) saying that only separable states are globally invariant with respect to the familly of all positive maps. However, criterions of that type are not operational. Even worse, they are strongly related to a classification of positive maps. In particular, the old open problem concerning the description of non-decomposable maps was revived. To describe that problem we need (cf [@St]): Let $\tau : {{\mathcal A}}\to {{\mathcal B}}({{\mathcal H}})$ be a linear, positive map. $\tau$ is called decomposable if there exists a Hilbert space ${{\mathcal K}}$, a bounded linear map $V: {{\mathcal H}}\to {{\mathcal K}}$ and a [[[$\hbox{\bf C}^$]{}]{}]{}-homomorphism $\pi : {{\mathcal A}}\to {{\mathcal B}}({{\mathcal K}})$ such that $\tau = V^* \pi V$. [[[$\hbox{\bf C}^$]{}]{}]{}-homomorphism means that $\pi (\{a,b\}) = \{\pi(a), \pi(b)\}$ where $\{ \cdot, \cdot \}$ stands for anticommutator, i.e. $\pi$ preserves the Jordan structure! A more subtle notion is the following: $\tau$ is locally decomposable if for $0 \ne x \in {{\mathcal H}}$, there exists a Hilbert space ${{\mathcal K}}_x$, $V_x:{{\mathcal K}}_x \to {{\mathcal H}}$ and a [[[$\hbox{\bf C}^$]{}]{}]{}-hommomorphism $\pi_x$ of ${{\mathcal A}}$ to ${{\mathcal B}}({{\mathcal K}}_x)$ such that $$V_x \pi_x(a) V^_x x = \tau (a)x$$ for all $a \in {{\mathcal A}}$. It is known ([@Wor], [@Choi]) that for the case $M_k({{\mathbb{C}}}) \otimes M_l({{\mathbb{C}}})$ with $k=2=l$ and $k=2$, $l=3$ all positive maps are decomposable. Then, the criterion for separability simplifies significantly. Namely, to verify separability it is enough to analyse $(\tau \otimes id)^d$, with $\tau$ being the transposition, as other positive maps are just convex combinations of CP maps (they always map states into states) and the composition of CP map with $\tau \otimes id$. The situation changes dramatically when both $k$ and $l$ are larger than 2. In that case there are plenty of non-decomposable maps (see [@Kos] and the references given there) and to analyse entanglement one cannot restrict oneself to study $\tau \otimes id$. Thus, a full description of positive maps is needed. Furthermore, one can constuct examples of entangled states using concrete non-decomposable maps (see [@HKP]). However, the classification of non-decomposable maps is a difficult task which is still not completed ([@St3], [@LMM]). We want to close the section with an important remark. Namely, if $d(\phi, A)= 0$ for any $A \in {{\mathcal A}}_1 \otimes {{\mathcal A}}_2$ then, using the description of locally decomposable maps, one can show that the state $\phi$ is separable [@Ma3]. This result shows how strong the interplay between separability and certain subtle features of positive maps is. 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--- abstract: 'In recent years, a number of experimental studies have been conducted to investigate the mechanical behavior and damage mechanisms of articular cartilage under impact loading. Some experimentally observed results have been explained using a non-linear viscoelastic impact model. At the same time, there is the need of simple mathematical models, which allow comparing experimental results obtained in drop impact testing with impact loads of different weights and incident velocities. The objective of this study was to investigate theoretically whether the main features of articular impact could be qualitatively predicted using a linear viscoelastic theory or the linear biphasic theory. In the present paper, exact analytical solutions are obtained for the main parameters of the Kelvin–Voigt and Maxwell impact models. Perturbation analysis of the impact process according to the standard viscoelastic solid model is performed. Asymptotic solutions are obtained for the drop weight impact test. The dependence of the coefficient of restitution on the impactor parameters has been studied in detail.' address: 'Institute of Mathematics and Physics, Aberystwyth University, Ceredigion SY23 3BZ, Wales, UK' author: - 'I.I. Argatov' title: ' Mathematical modeling of linear viscoelastic impact: Application to drop impact testing of articular cartilage' --- Impact contact problem ,blunt indenter ,asymptotic model ,coefficient of restitution Nomenclature {#nomenclature .unnumbered} ============ --------------- ------------------------------------------------ $b$ damper coefficient $D$ discriminant of the characteristic equation $E_{\rm dyn}$ incremental dynamic modulus $E_{\rm max}$ maximum incremental dynamic modulus $E_{10}$ modulus at stresses of 10 MPa $e_$ coefficient of restitution $F$ contact force $F_M$ maximum contact force $g$ gravitational acceleration $h$ cartilage layer thickness $h_0$ drop height of the impactor $H_A$ aggregate modulus $k$ stiffness coefficient $k_1$, $k_2$ spring stiffnesses in the standard solid model $k_0$ instantaneous stiffness $k_\infty$ long-term stiffness $m$ impactor mass $t$ time variable $t_c$ impact duration $t_m$ time to maximum displacement $t_M$ time to maximum contact force $v_0$ initial impact velocity $x$ displacement $\dot{x}$ velocity $\ddot{x}$ acceleration $x_m$ maximum displacement --------------- ------------------------------------------------ $${}$$ ------------------------------ ------------------------------------------------------------------- $\beta$ damping coefficient in the Kelvin–Voigt model $\beta_1$ real part of complex roots of the characteristic equation $\Delta m$ percentage increase in mass of cartilage sample $\epsilon$ strain $\varepsilon_0$ non-dimensional parameter accounting for the gravitational effect $\zeta$ loss factor in the Maxwell model $\zeta_1$ imaginary part of complex roots of the characteristic equation $\eta$ loss factor in the Kelvin–Voigt model $\kappa$ cartilage permeability $\varkappa_1$, $\varkappa_2$ spring stiffnesses in the standard solid model $\lambda$ Lamé coefficient $\lambda_1$ root of the characteristic equation $\Lambda$ non-dimensional parameter in the standard solid model $\mu$ Lamé coefficient $\xi$ non-dimensional displacement $\rho$ ratio of the long-term and instantaneous stiffnesses $\sigma$ stress $\tau$ non-dimensional time $\tau_D$ typical diffusion time $\tau_R$ relaxation time $\Psi(\tau)$ dimensionless relaxation function $\omega$ angular frequency of damped oscillations $\omega_0$ angular frequency of undamped oscillations ------------------------------ ------------------------------------------------------------------- Introduction {#1dsSectionI} ============ Articular cartilage is a soft hydrated tissue covering the end of each bone at the joints. Cartilage has no known function other than maintaining mechanical competence of joints, allowing bones to move against one another without friction. But there is no need to underline its significance to health of a human body, since almost all the load transmitted by a human joint goes through the articular cartilage, and it prevents biomechanical damage caused by severe loading including impact loading. It is believed that severe articular impact can initiate post-traumatic arthritis [@JeffreyGregoryAspden1995; @QuinnAllen2001]. An impact loading of the joint constitutes the action of extremely high non-physiological loads applied very rapidly (for instance, due to a car accident, sports injury, or a fall from a height). In recent years, a number of experimental studies have been conducted to investigate the mechanical behavior and damage mechanisms of articular cartilage under impact loading [@AtkinsonHautAltiero1998; @VerteramoSeedhom2007; @BurginAspden2008]. In particular, the experimental data on relative dissipation of the impact energy $\Delta E/E_0$ versus overall impactor energy $E_0$ obtained in [@Varga2007] were fitted with quadratic curves. Here, $E_0=mv_0^2/2$, $\Delta E=m(v_1^2-v_0^2)/2$, $v_0$ and $v_1$ are the initial impact and rebound velocities, respectively, $m$ is the impactor mass. Since, $v_1=-e_ v_0$, where $e_$ is the coefficient of restitution, we easily get $\Delta E/E_0=1-e_^2$. Thus, the experimental data and fitting curves for dissipation of the impact energy [@Varga2007] can be recalculated in terms of the coefficient of restitution as presented in Fig. \[Varga2007.pdf\], which shows a non-monotonic dependence of $e_$ on $v_0$. Some experimentally observed results have been explained using a non-linear viscoelastic impact model [@Edelsten2010]. At the same time, there is the need of a simple mathematical model, which allows comparing experimental results obtained in drop impact testing with impact loads of different weights and incident velocities. ![Coefficient of restitution $e_$ versus the impact velocity $v_0$ for articular cartilage samples of different thicknesses. Based on the experimental data and fitting curves obtained in [@Varga2007]. []{data-label="Varga2007.pdf"}](Varga2007.pdf) A variety of mathematical models were suggested to describe the stress-strain response of articular cartilage that represents a multiphasic, structurally complex material possessing viscoelastic properties. It is long known that articular cartilage possesses viscoelastic properties [@HayesMockros1971; @Lau_et_al_2008], though there is no direct correspondence between viscoelastic parameters and parameters of the biphasic/poroelastic models of cartilage. The biphasic theory [@MowKueiLaiArmstrong1980], which models the tissue as a mixture of a solid phase and a fluid phase, has demonstrated very good agreement with experimental results in the creep and stress relaxation tests [@SoltzAteshian2000]. The objective of this study was to investigate theoretically whether the main features of articular impact observed in [@Varga2007; @Edelsten2010] could be qualitatively predicted using a linear viscoelastic theory or the linear biphasic theory. The rest of the paper is organized as follows. In Sections \[1dsSection1\] and \[1dsSection2\], we consider in detail the viscoelastic Kelvin–Voigt and Maxwell impact models, respectively. Since some elements of the presented solutions are known in the literature, we pay a particular attention to the evaluation of the contact force, $F(t)$, and impactor displacement, $x(t)$, at the time moments $t_M$ and $t_m$, when the force and displacement reach their maxima, $F_M$ and $x_m$, respectively. In Section \[1dsSection3\], we outline a closed form solution of the impact equation in the case of standard solid model. In order to get analytical approximations, we consider the standard solid model as a perturbation of the Kelvin–Voigt (Section \[1dsSection4\]) or the Maxwell model (Section \[1dsSection5\]). In particular, simple analytical approximations are derived for the impact duration, $t_c$, and for the coefficient of restitution, $e_$. In Sections \[1dsSection05\] and \[1dsSection06\], we consider the influence of the gravity effect on these parameters in the framework of the Kelvin–Voigt and Maxwell models for drop weight impact. In Section \[1dsSection07\], we develop an asymptotic model for the force-displacement relationship in the indentation problem for a thin biphasic layer corresponding to the conditions of the so-called blunt impact, when the specimen thickness is much smaller than the radius of a flat-ended cylindrical impactor. An example of application of the developed linear theory of viscoelastic impact for analyzing experimental data is given in Section \[1dsSection10\]. Finally, in Sections \[1dsSectionD\] and \[1dsSectionC\], we outline a discussion of the results obtained and formulate our conclusions. Viscoelastic Kelvin–Voigt impact model {#1dsSection1} ====================================== In this section, the deformation of articular cartilage layer is modeled schematically as a parallel combination of linear spring $k$ and dashpot $b$ (Fig. \[Kelvin-Voigt\_Impact.pdf\]). Dynamic balance between the force of cartilage reaction $$F=kx+b\dot{x} \label{1vI(1.0)}$$ and the force of body inertia $m\ddot{x}$ governs the development of collision. According to Newton’s second law, the differential equation of the impact has the form $$m\ddot{x}+b\dot{x}+kx=0, \quad t\in[0,t_c], \label{1vI(1.1)}$$ where $t_c$ is the contact duration, that is $t_c$ denotes the instant, when the cartilage reaction force changes its sign, or the indenter acceleration vanishes. ![Impact viscoelastic Kelvin–Voigt model.[]{data-label="Kelvin-Voigt_Impact.pdf"}](Kelvin-Voigt_Impact.pdf) The initial conditions for Eq. (\[1vI(1.1)\]) are as follows: $$x(0)=0, \quad \dot{x}(0)=v_0. \label{1vI(1.2)}$$ The impact duration is determined by the condition $$kx+b\dot{x}\bigr\vert_{t=t_c}=0, \label{1vI(1.3)}$$ or, in view of Eq. (\[1vI(1.1)\]), by the condition $$\ddot{x}\bigr\vert_{t=t_c}=0. \label{1vI(1.4)}$$ The impact problem (\[1vI(1.1)\]), (\[1vI(1.2)\]) has the following well-known solution [@WinemanRajagopal2000]: $$x(t)=\frac{v_0}{\omega}e^{-\beta t}\sin\omega t, \quad t\in[0,t_c]. \label{1vI(1.5)}$$ Here we used the notation $$\omega_0^2=\frac{k}{m},\quad \omega^2=\omega_0^2-\beta^2, \quad \beta=\frac{b}{2m}. \label{1vI(1.6)}$$ We assume that $\omega_0>\beta$. Fig. \[Kelvin-Voigt\_exact.pdf\] shows the behavior of the dimensionless quantities $\omega_0 x/v_0$, $F/(mv_0\omega_0)$, and $\dot{x}/v_0$ with respect to time. Observe that the time moment $t_M$, when the contact force reaches its maximum, approaches the initial moment of impact as the damping ratio $\eta$ increases. ![Viscoelastic Kelvin–Voigt impact model. Behavior of the main impact variables with time for the following values of the damping ratio: $\eta=0{.}1$ (a), $\eta=0{.}3$ (b), $\eta=0{.}5$ (c).[]{data-label="Kelvin-Voigt_exact.pdf"}](Kelvin-Voigt_exact_01.pdf "fig:") ![Viscoelastic Kelvin–Voigt impact model. Behavior of the main impact variables with time for the following values of the damping ratio: $\eta=0{.}1$ (a), $\eta=0{.}3$ (b), $\eta=0{.}5$ (c).[]{data-label="Kelvin-Voigt_exact.pdf"}](Kelvin-Voigt_exact_03.pdf "fig:") ![Viscoelastic Kelvin–Voigt impact model. Behavior of the main impact variables with time for the following values of the damping ratio: $\eta=0{.}1$ (a), $\eta=0{.}3$ (b), $\eta=0{.}5$ (c).[]{data-label="Kelvin-Voigt_exact.pdf"}](Kelvin-Voigt_exact_05.pdf "fig:") By differentiating the both sides of Eq. (\[1vI(1.5)\]) with respect to $t$, we get $$\dot{x}(t)=\frac{v_0}{\omega}e^{-\beta t}\bigl( \omega\cos\omega t-\beta\sin\omega t\bigr), \label{1vI(1.7a)}$$ $$\ddot{x}(t)=-\frac{v_0}{\omega}e^{-\beta t}\bigl[ (\omega^2-\beta^2)\sin\omega t+2\beta\omega\cos\omega t\bigr]. \label{1vI(1.7b)}$$ After solving Eq. (\[1vI(1.4)\]) for $t_c$ in view of (\[1vI(1.7b)\]), the following expression for the impact duration can be obtained [@ButcherSegalman2000]: $$t_c=\frac{1}{\omega}{\rm Atan\,}\frac{-2\beta\omega}{\omega^2-\beta^2}, \label{1vI(1.8)}$$ where the first positive value of the many-valued ${\rm Atan}$ function should be taken. Using properties of the ${\rm Atan}$ function, we rewrite Eq. (\[1vI(1.8)\]) as follows [@Popov2010]: $$t_c=\frac{1}{\omega} \left\{ \begin{array}{l} \displaystyle \pi-{\,\rm atan\,}\frac{2\beta\omega}{\omega^2-\beta^2},\quad \beta<\omega, \\ \displaystyle {\,\rm atan\,}\frac{2\beta\omega}{\omega^2-\beta^2}, \quad \omega>\beta. \end{array} \right. \label{1vI(1.9)}$$ Here, ${\,\rm atan\,}(z)$ is the principal branch of the arctangent function ${\,\rm Atan\,}(z)$. Finally, using properties of the ${\,\rm atan\,}$ function, we can rewrite formula (\[1vI(1.9)\]) in a more simple form as $$t_c=\frac{2}{\omega}{\,\rm atan\,}\frac{\omega}{\beta}. \label{1vI(1.9a)}$$ Let $\eta$ denote the loss factor, i.e., $$\eta=\frac{\beta}{\omega_0}. \label{1vI(1.10a)}$$ Then, Eq. (\[1vI(1.9a)\]) can be rewritten as $$t_c=\frac{2}{\omega_0\sqrt{1-\eta^2}}{\,\rm atan\,}\frac{\sqrt{1-\eta^2}}{\eta}. \label{1vI(1.10)}$$ Recall that we assume that $\eta\in[0,1]$. Also, note that in view of the notation (\[1vI(1.6)\]), we have $$\eta=\frac{b}{2\sqrt{km}}. \label{1vI(1.10b)}$$ The velocity of the indenter at separation can be obtained by the substitution of (\[1vI(1.10)\]) into (\[1vI(1.7a)\]) in the following form: $$\begin{aligned} \dot{x}(t_c) & = & -v_0\exp(-\beta t_c) \label{1vI(1.11)} \\ {} & = & -v_0\exp\biggl\{- \frac{2\eta}{\sqrt{1-\eta^2}}{\,\rm atan\,}\frac{\sqrt{1-\eta^2}}{\eta}\biggr\}. \label{1vI(1.12)}\end{aligned}$$ From Eq. (\[1vI(1.11)\]), it follows that the coefficient of restitution, $e_$, which is defined as the ratio of the velocity at separation $\vert \dot{x}(t_c)\vert$ to the velocity of the indenter at incidence $\vert \dot{x}(0)\vert=v_0$, is given by $$\begin{aligned} e_* & = & \exp\Bigl\{- \frac{2\beta}{\omega}{\,\rm atan\,}\frac{\omega}{\beta}\Bigr\} \label{1vI(1.13)} \\ {} & = & \exp\biggl\{- \frac{2\eta}{\sqrt{1-\eta^2}}{\,\rm atan\,}\frac{\sqrt{1-\eta^2}}{\eta}\biggr\}. \label{1vI(1.14)}\end{aligned}$$ The peak value of the indenter penetration occurs at the instant $t=t_m$, when $\dot{x}(t_m)=0$. In view of (\[1vI(1.7a)\]), we have $$\begin{aligned} t_m & = & \frac{1}{\omega}{\,\rm atan\,}\frac{\omega}{\beta} \label{1vI(1.15)} \\ {} & = & \frac{1}{\omega_0\sqrt{1-\eta^2}}{\,\rm arcsin\,}\sqrt{1-\eta^2}. \label{1vI(1.16)}\end{aligned}$$ Substituting the expression (\[1vI(1.15)\]) into Eq. (\[1vI(1.5)\]), we obtain the maximum penetration $x_m=x(t_m)$ in the form $$\begin{aligned} x_m & = & \frac{v_0}{\omega_0}\exp\Bigl(-\frac{\beta}{\omega}{\,\rm atan\,}\frac{\omega}{\beta}\Bigr) \label{1vI(1.17)} \\ {} & = & \frac{v_0}{\omega_0}\exp\Bigl(-\frac{\eta}{\sqrt{1-\eta^2}}{\,\rm arcsin\,}\sqrt{1-\eta^2} \Bigr). \label{1vI(1.18)}\end{aligned}$$ Note that from (\[1vI(1.10)\]) and (\[1vI(1.16)\]), it is readily seen that $t_m=t_c/2$. The peak value of the contact force $F$ occurs at the instant $t=t_M$, when $\dot{F}(t_M)=0$. According to Eqs. (\[1vI(1.5)\]), (\[1vI(1.7a)\]), we obtain $$\begin{aligned} F(t) & = & \frac{mv_0}{\omega}\exp(-\beta t)\bigl[(\omega^2-\beta^2)\sin\omega t+2\beta\omega\cos\omega t\bigr] \nonumber \\ {} & = & m v_0\omega_0 \exp(-\eta\omega_0 t)\biggl(\frac{(1-2\eta^2)}{\sqrt{1-\eta^2}} \sin\omega_0\sqrt{1-\eta^2}t +2\eta\cos\omega_0\sqrt{1-\eta^2}t \biggr). \label{1vI(1.19)}\end{aligned}$$ Differentiating the previous expression, we can reduce the equation $\dot{F}(t_M)=0$ to the following one: $$\sqrt{1-\eta^2}(1-4\eta^2)\cos(t_M\omega_0\sqrt{1-\eta^2})+ \eta(4\eta^2-3)\sin(t_M\omega_0\sqrt{1-\eta^2})=0.$$ Thus, for $\eta\in(0,0{.}5)$, we obtain $$\begin{aligned} t_M & = & \frac{1}{\omega}{\,\rm atan\,}\frac{\omega(\omega^2-3\beta^2)}{\beta (3\omega^2-\beta^2)} \nonumber \\ {} & = & \frac{1}{\omega_0\sqrt{1-\eta^2}}{\,\rm atan\,}\frac{\sqrt{1-\eta^2}(1-4\eta^2)}{\eta(3-4\eta^2)}. \label{1vI(1.20)}\end{aligned}$$ For $\eta\in(0{.}5,1)$, the maximum value of the contact force $F_M=F(t_M)$ takes place at the initial instant $t=0$. Substituting (\[1vI(1.20)\]) into Eq. (\[1vI(1.19)\]), we get $$\begin{aligned} F_M & = & m v_0\omega_0 \exp\biggl(-\frac{\eta}{\sqrt{1-\eta^2}} {\,\rm atan\,}\frac{\sqrt{1-\eta^2}(1-4\eta^2)}{\eta(3-4\eta^2)}\biggr),\quad \eta\in(0,0{.}5), \\ \label{1vI(1.21)} F_M & = & m v_0\omega_0 2\eta,\quad \eta\in(0{.}5,1). \label{1vI(1.22)}\end{aligned}$$ Note that the function $F_M(\eta)$ defined by Eqs. (\[1vI(1.21)\]) and (\[1vI(1.22)\]) is continuously differentiable. ![Viscoelastic Kelvin–Voigt impact model. Behavior of the main impact parameters $t_m$, $t_c$, $t_M$ (a) and $x_m$, $F_M$ (b) with the damping ratio.[]{data-label="Kelvin-Voigt_tm+tM.pdf"}](Kelvin-Voigt_tm+tM.pdf "fig:") ![Viscoelastic Kelvin–Voigt impact model. Behavior of the main impact parameters $t_m$, $t_c$, $t_M$ (a) and $x_m$, $F_M$ (b) with the damping ratio.[]{data-label="Kelvin-Voigt_tm+tM.pdf"}](Kelvin-Voigt_xm+FM.pdf "fig:") Fig. \[Kelvin-Voigt\_tm+tM.pdf\]a shows the monotonic behavior of the dimensionless characteristic time moments $t_m/\omega_0$, $t_c/(2\omega_0)$, $t_M/\omega_0$ with the damping ratio $\eta$. Recall that $t_m=t_c/2$. The variations of the relative maximum contact force $F_M/(mv_0\omega_0)$ and displacement $\omega_0 x_m/v_0$ are presented in Fig. \[Kelvin-Voigt\_tm+tM.pdf\]b. It is interesting to observe the non monotonic behavior of $F_M$ with the minimum at $\eta\approx 0{.}26493$. Viscoelastic Maxwell impact model {#1dsSection2} ================================= Assuming that the cartilage layer’s response to impact loading is modeled schematically as a serial combination of linear spring $k$ and dashpot $b$ (Fig. \[Maxwell\_Impact.pdf\]). The force-displacement relation is given by the following differential equation [@WinemanRajagopal2000]: $$\frac{\dot{F}}{k}+\frac{F}{b}=\dot{x}. \label{1vI(2.1)}$$ ![Impact viscoelastic Maxwell model.[]{data-label="Maxwell_Impact.pdf"}](Maxwell_Impact.pdf) From (\[1vI(2.1)\]), it follows that $$F=k\int\limits_0^t\exp\Bigl\{-\frac{k}{b}(t-\tau)\Bigr\}\frac{dx}{d\tau}(\tau)\,d\tau. \label{1vI(2.2)}$$ The differential equation of the impact $m\ddot{x}+F=0$ in view of (\[1vI(2.1)\]) results in the third-order equation $$\dddot{x}+\frac{k}{b}\ddot{x}+\frac{k}{m}\dot{x}=0 \label{1vI(2.3)}$$ with the initial conditions $$x(0)=0, \quad \dot{x}(0)=v_0,\quad \ddot{x}(0)=0. \label{1vI(2.4)}$$ The impact problem (\[1vI(2.3)\]), (\[1vI(2.4)\]) has the following solution [@ButcherSegalman2000; @Stronge2000]: $$x(t)=\frac{v_0}{\omega_0}\exp(-\zeta\omega_0 t)\Bigl\{\frac{\omega_0(1-2\zeta^2)}{\omega}\sin\omega t -2\zeta\cos\omega t\Bigr\}+\frac{2\zeta v_0}{\omega_0}, \label{1vI(2.5)}$$ $$\dot{x}(t)=v_0\exp(-\zeta\omega_0 t)\Bigl\{ \cos\omega t+\frac{\zeta\omega_0}{\omega}\sin\omega t\Bigr\}. \label{1vI(2.5a)}$$ Here we used the notation $$\omega_0^2=\frac{k}{m},\quad \omega=\omega_0\sqrt{1-\zeta^2}, \quad \zeta=\frac{k}{2\omega_0 b}. \label{1vI(2.6)}$$ The variation of the contact force during the impact interaction is $$F=\frac{kv_0}{\omega}\exp(-\zeta\omega_0 t)\sin\omega t. \label{1vI(2.7)}$$ The impact duration $t_c$ is determined by the condition $F\bigr\vert_{t=t_c}=0$. Thus, according to (\[1vI(2.7)\]), the following relation takes place [@ButcherSegalman2000; @Stronge2000]: $$t_c=\frac{\pi}{\omega}. \label{1vI(2.8)}$$ Substituting the value (\[1vI(2.8)\]) into Eq. (\[1vI(2.5a)\]), one gets the coefficient of restitution in the form $$e_=\exp\Bigl(-\frac{\pi\zeta}{\sqrt{1-\zeta^2}}\Bigr). \label{1vI(2.9)}$$ ![Viscoelastic Maxwell impact model. Behavior of the main impact variables with time for the following values of the damping ratio: $\zeta=0{.}1$ (a), $\zeta=0{.}3$ (b), $\zeta=0{.}5$ (c).[]{data-label="Maxwell_exact.pdf"}](Maxwell_exact_01.pdf "fig:") ![Viscoelastic Maxwell impact model. Behavior of the main impact variables with time for the following values of the damping ratio: $\zeta=0{.}1$ (a), $\zeta=0{.}3$ (b), $\zeta=0{.}5$ (c).[]{data-label="Maxwell_exact.pdf"}](Maxwell_exact_03.pdf "fig:") ![Viscoelastic Maxwell impact model. Behavior of the main impact variables with time for the following values of the damping ratio: $\zeta=0{.}1$ (a), $\zeta=0{.}3$ (b), $\zeta=0{.}5$ (c).[]{data-label="Maxwell_exact.pdf"}](Maxwell_exact_05.pdf "fig:") Fig. \[Maxwell\_exact.pdf\] shows the behavior of the dimensionless quantities $\omega_0 x/v_0$, $F/(mv_0\omega_0)$, and $\dot{x}/v_0$ with respect to time. Observe that the time moment $t_m$, when the indenter displacement reaches its maximum, approaches the final moment of impact as the damping ratio $\zeta$ increases. According to Eq. (\[1vI(2.5a)\]), the peak value of the indenter penetration occurs at the instant $$t_m=\frac{\pi}{2\omega}\Bigl(1+\frac{2}{\pi}{\,\rm arcsin\,}\zeta\Bigr). \label{1vI(2.10)}$$ The substitution of the value (\[1vI(2.10)\]) into Eqs. (\[1vI(2.5)\]) and (\[1vI(2.7)\]) gives the maximum penetration $$x_m=\frac{v_0}{\omega_0}\biggl( 2\zeta+\exp\biggl\{-\frac{\pi\zeta}{2\sqrt{1-\zeta^2}}\Bigl(1+\frac{2}{\pi}{\,\rm arcsin\,}\zeta\Bigr) \biggr\}\biggr) \label{1vI(2.11)}$$ and the corresponding force $$F_m=\frac{kv_0}{\omega_0} \exp\biggl\{-\frac{\pi\zeta}{2\sqrt{1-\zeta^2}}\Bigl(1+\frac{2}{\pi}{\,\rm arcsin\,}\zeta\Bigr) \biggr\} \label{1vI(2.11a)}$$ From Eq. (\[1vI(2.7)\]), it follows that the peak value $F_M$ of the contact force occurs at the instant $$t_M=\frac{1}{\omega}{\,\rm atan\,}\frac{\sqrt{1-\zeta^2}}{\zeta}. \label{1vI(2.12)}$$ Substituting (\[1vI(2.12)\]) into Eqs. (\[1vI(2.7)\]) and (\[1vI(2.5)\]), we obtain the maximum contact force $$F_M=\frac{kv_0}{\omega_0} \exp\biggl\{-\frac{\zeta}{\sqrt{1-\zeta^2}}{\,\rm atan\,}\Bigl( \frac{\sqrt{1-\zeta^2}}{\zeta}\Bigr)\biggr\} \label{1vI(2.13)}$$ and the corresponding displacement $$x_M=\frac{v_0}{\omega_0}\biggl(2\zeta+ (1-4\zeta^2) \exp\biggl\{-\frac{\zeta}{\sqrt{1-\zeta^2}}{\,\rm atan\,} \frac{\sqrt{1-\zeta^2}}{\zeta}\biggr\}\biggr). \label{1vI(2.14)}$$ ![Viscoelastic Maxwell impact model. Behavior of the main impact parameters $t_m$, $t_c$, $t_M$ (a) and $x_m$, $F_M$ (b) with the damping ratio.[]{data-label="Maxwell_tm+tM.pdf"}](Maxwell_tm+tM.pdf "fig:") ![Viscoelastic Maxwell impact model. Behavior of the main impact parameters $t_m$, $t_c$, $t_M$ (a) and $x_m$, $F_M$ (b) with the damping ratio.[]{data-label="Maxwell_tm+tM.pdf"}](Maxwell_xm+FM.pdf "fig:") Fig. \[Maxwell\_tm+tM.pdf\] shows the monotonic behavior of the dimensionless characteristic time moments $t_m/\omega_0$, $t_c/(2\omega_0)$, $t_M/\omega_0$ with the damping ratio $\zeta$. The variations of the relative maximum contact force $F_M/(mv_0\omega_0)$ and displacement $\omega_0 x_m/v_0$ are presented in Fig. \[Maxwell\_tm+tM.pdf\]b. Finally, as it was observed [@ButcherSegalman2000], although certain quantities of the Maxwell impact model are equivalent to the so-called half-period Kelvin–Voigt impact model, the inherent physics of these models are completely different. Standard solid model {#1dsSection3} ==================== There are two schematic representations of the standard linear solid model (Figs. \[SSM\_model-1.pdf\] and \[SSM\_model-2.pdf\]). The force-displacement relationship is given by the following two equations: $$(k_1+k_2)F+b\dot{F}=k_1 k_2 x+k_1 b\dot{x}, \label{1vI(3.1)}$$ $$\varkappa_1 F+\beta\dot{F}=\varkappa_1 \varkappa_2 x+\beta(\varkappa_1+\varkappa_2)\dot{x}. \label{1vI(3.2)}$$ ![Standard solid model. Configuration based on the Kelvin–Voigt model.[]{data-label="SSM_model-1.pdf"}](SSM_model-1.pdf) The instantaneous and long-term moduli are $$k_0=k_1=\varkappa_1+\varkappa_2,\quad k_\infty=\varkappa_1=\frac{k_1 k_2}{k_1+k_2}. \label{1vI(3.3)}$$ ![Standard solid model. Configuration based on the Maxwell model.[]{data-label="SSM_model-2.pdf"}](SSM_model-2.pdf) The relaxation time is equal to $$\tau_R=\frac{b}{k_1+k_2}=\frac{\beta}{\varkappa_2}. \label{1vI(3.4)}$$ The differential equations (\[1vI(3.1)\]) and (\[1vI(3.2)\]) are equivalent to the force-displacement relationship $$F=\int\limits_0^t k(t-\tau)\frac{dx}{d\tau}(\tau)\,d\tau \label{1vI(3.5)}$$ with the relaxation stiffness $$k(t)=k_\infty+(k_0-k_\infty)\exp\Bigl(-\frac{t}{\tau_R}\Bigr). \label{1vI(3.6)}$$ Let us also introduce the notation $$\rho=\frac{k_\infty}{k_0}. \label{1vI(3.7)}$$ Note that $\rho\in(0,1)$ in view of (\[1vI(3.3)\]). The differential equation of impact $$m\ddot{x}+F=0, \quad t\in[0,t_c], \label{1vI(3.8)}$$ where the contact force $F$ is determined by Eq. (\[1vI(3.1)\]), can be written as $$\dddot{x}+\frac{(k_1+k_2)}{b}\ddot{x}+\frac{k_1}{m}\dot{x}+\frac{k_1 k_2}{mb}x=0. \label{1vI(3.9)}$$ By introducing the non-dimensional time $$\tau=\frac{t}{\tau_R}, \label{1vI(3.10)}$$ Eq. (\[1vI(3.9)\]) can be reduced to the following equation: $$x^{\prime\prime\prime}+x^{\prime\prime}+\Lambda x^\prime+\Lambda\rho x=0. \label{1vI(3.11)}$$ Here we introduced the notation $$\Lambda=\frac{k_0}{m}\tau_R^2. \label{1vI(3.12)}$$ The initial conditions for Eq. (\[1vI(3.9)\]) are as follows: $$x(0)=0, \quad \dot{x}(0)=v_0,\quad \ddot{x}(0)=0. \label{1vI(3.13)}$$ The solution to the problem (\[1vI(3.9)\]), (\[1vI(3.13)\]) is given by the following formula: $$\begin{aligned} x(t) & = & \frac{\tau_R v_0}{\zeta_1[(\beta_1-\lambda_1)^2+\zeta_1^2]} \biggl\{[(1-\beta_1)(\lambda_1-\beta_1)+\zeta_1^2]\sin\frac{\zeta_1 t}{\tau_R} \nonumber \\ {} & { } & {}-\zeta_1(1-\lambda_1)\cos\frac{\zeta_1 t}{\tau_R}\biggr\} \exp\Bigl(-\frac{\beta_1 t}{\tau_R}\Bigr) %\nonumber \\ %{} & { } & {} +\frac{(1-\lambda_1)\tau_R v_0}{(\beta_1-\lambda_1)^2+\zeta_1^2} \exp\Bigl(-\frac{\lambda_1 t}{\tau_R}\Bigr). \label{1vI(3.14)}\end{aligned}$$ Here, $-\lambda_1$ and $-(\beta_1\pm{\rm i}\zeta_1)$ are the roots of the characteristic equation corresponding to Eq. (\[1vI(3.11)\]). In other words, the following factorization takes place: $$z^3+z^2+\Lambda z+\Lambda\rho =(z+\lambda_1) (z^2+2\beta_1 z +\beta_1^2+\zeta_1^2). \label{1vI(3.15)}$$ The discriminant of the characteristic equation is $$D=4\Lambda(\Lambda^2+\rho)-\Lambda^2(1+18\rho-27\rho^2). \label{1vI(3.16)}$$ We underline that formula (\[1vI(3.14)\]) holds true when $D>0$. In this case, we have $$\begin{aligned} \lambda_1 & = & \frac{1}{3}+\frac{C_1}{3}+\frac{(1-3\Lambda)}{3C_1}, \nonumber \\ \beta_1 & = & \frac{1}{3}-\frac{C_1}{6}-\frac{(1-3\Lambda)}{6C_1}, \nonumber \\ \zeta_1 & = & \frac{\sqrt{3}C_1}{6}-\frac{\sqrt{3}(1-3\Lambda)}{6C_1}, \nonumber\end{aligned}$$ where $$C_1=\sqrt[3]{\frac{1}{2}(Q_1+2-9\Lambda+27\Lambda\rho)},\quad Q_1=\sqrt{(2-9\Lambda+27\Lambda\rho)^2-4(1-3\Lambda)^3}.$$ Perturbation of the Kelvin–Voigt model {#1dsSection4} ====================================== Taking into account (\[1vI(3.3)\]) and (\[1vI(3.7)\]), we rewrite Eq. (\[1vI(3.1)\]) in the following form: $$k_\infty F+\rho(1-\rho)b\dot{F}=k_\infty^2 x+(1-\rho)k_\infty b\dot{x}. \label{1vI(4.1)}$$ Now, letting $\rho\to 0$, we arrive at the equation $$F=k_\infty x+b\dot{x}, \label{1vI(4.2)}$$ which coincides with Eq. (\[1vI(1.0)\]). Thus, for small values of $\rho$, the standard solid model (\[1vI(4.1)\]) is a perturbation of the Kelvin–Voigt model (\[1vI(4.2)\]). Let us introduce the notation $$\omega_0^2=\frac{k_\infty}{m},\quad \beta=\frac{b}{2m},\quad \eta=\frac{\beta}{\omega_0}. \label{1vI(4.3)}$$ Then, the parameters (\[1vI(3.4)\]) and (\[1vI(3.12)\]) can be evaluated as $$\tau_R=2\eta\rho(1-\rho)\frac{1}{\omega_0},\quad \Lambda=4\eta^2\rho(1-\rho)^2. \label{1vI(4.4)}$$ In view of (\[1vI(4.4)\]), the discriminant (\[1vI(3.16)\]) and the roots of the characteristic equation (\[1vI(3.15)\]) can be asymptotically evaluated as follows: $$\begin{aligned} D & = & 16\eta^2(1-\eta^2)\rho^2+O(\rho^3),\quad \rho\to 0, \nonumber \\ \lambda_1 & = & 1-4\eta^2\rho+O(\rho^2), \nonumber \\ \beta_1 & = & 2\eta^2\rho+O(\rho^2), \quad \zeta_1 = 2\eta\sqrt{1-\eta^2}\rho+O(\rho^2). \nonumber\end{aligned}$$ Consequently, we obtain the following asymptotic formulas for the impact duration, $t_c$, and the coefficient of restitution, $e_$: $$\omega_0 t_c \simeq \frac{2}{\sqrt{1-\eta^2}}{\,\rm atan\,}\frac{\sqrt{1-\eta^2}}{\eta}+ \rho\biggl\{4\eta-\frac{8\eta^2}{\sqrt{1-\eta^2}}{\,\rm atan\,}\frac{\sqrt{1-\eta^2}}{\eta} \biggr\}, \label{1vI(4.5)}$$ $$e_* \simeq \exp\biggl(-\frac{2\eta}{\sqrt{1-\eta^2}}{\,\rm atan\,}\frac{\sqrt{1-\eta^2}}{\eta} \biggr)\biggl\{1+\frac{4\rho\eta}{\sqrt{1-\eta^2}}{\,\rm atan\,}\frac{\sqrt{1-\eta^2}}{\eta} \biggr\}. \label{1vI(4.6)}$$ ![Perturbation of the Kelvin–Voigt model. Relative errors of the asymptotic formulas (\[1vI(4.5)\]) and (\[1vI(4.6)\]) for the duration of impact (a) and the coefficient of restitution (b).[]{data-label="SSM_Kelvin-Voigt.pdf"}](SSM_Kelvin-Voigt_e.pdf "fig:") ![Perturbation of the Kelvin–Voigt model. Relative errors of the asymptotic formulas (\[1vI(4.5)\]) and (\[1vI(4.6)\]) for the duration of impact (a) and the coefficient of restitution (b).[]{data-label="SSM_Kelvin-Voigt.pdf"}](SSM_Kelvin-Voigt_time.pdf "fig:") The accuracy of the asymptotic approximations (\[1vI(4.5)\]) and (\[1vI(4.6)\]) is presented in Fig. \[SSM\_Kelvin-Voigt.pdf\]. Note that the asymptotic formulas (\[1vI(4.5)\]) and (\[1vI(4.6)\]) are not uniformly valid as $\eta\to 1$. Perturbation of the Maxwell model {#1dsSection5} ================================= Now, taking into account (\[1vI(3.3)\]) and (\[1vI(3.7)\]), we rewrite Eq. (\[1vI(3.1)\]) as follows: $$k_0 F+(1-\rho)b\dot{F}=\rho k_0^2 x+(1-\rho)k_0 b\dot{x}. \label{1vI(5.1)}$$ Again, by letting $\rho\to 0$, we obtain the limit equation $$\frac{F}{b}+\frac{\dot{F}}{k_0}=\dot{x}, \label{1vI(5.2)}$$ which coincides with Eq. (\[1vI(2.1)\]). Thus, for small values of $\rho$, the standard solid model (\[1vI(5.1)\]) can be regarded as a perturbation of the Maxwell model (\[1vI(5.2)\]). Let us introduce the notation $$\omega_0^2=\frac{k_0}{m},\quad \zeta=\frac{k_0}{2\omega_0 b}. \label{1vI(5.3)}$$ In view of (\[1vI(5.3)\]), the parameters (\[1vI(3.4)\]) and (\[1vI(3.12)\]) can be evaluated as $$\tau_R=\frac{1-\rho}{2\zeta\omega_0},\quad \Lambda=\frac{(1-\rho)^2}{4\zeta^2}. \label{1vI(5.4)}$$ Now, taking into account (\[1vI(5.4)\]), we expand the discriminant (\[1vI(3.16)\]) and the roots of the characteristic equation (\[1vI(3.15)\]) as follows: $$\begin{aligned} D & = & \frac{(1-\zeta^2)}{16\zeta^6}-\frac{\rho(7\zeta^2-8\zeta^4+3)}{8\zeta^6}+O(\rho^2),\quad \rho\to 0, \nonumber \\ \lambda_1 & = & \rho+O(\rho^2),\quad \beta_1 = \frac{1}{2}-\frac{\rho}{2}+O(\rho^2), \nonumber \\ \zeta_1 & = & \frac{\sqrt{1-\zeta^2}}{2\zeta}- \rho\frac{\sqrt{1-\zeta^2}(2+\zeta^2)}{2\zeta(1-\zeta^2)}+O(\rho^2). \nonumber\end{aligned}$$ Consequently, we obtain the following asymptotic approximations for the impact duration, $t_c$, and the coefficient of restitution, $e_$: $$\omega_0 t_c \simeq \frac{\pi}{\sqrt{1-\zeta^2}}+ \frac{2\pi\rho\zeta^2}{(1-\zeta^2)^{3/2}}, \label{1vI(5.5)}$$ $$e_ \simeq \exp\Bigl(-\frac{\pi\zeta}{\sqrt{1-\zeta^2}}\Bigr) +4\rho\zeta^2 \biggl\{1+\biggl(1-\frac{\pi\zeta}{2(1-\zeta^2)^{3/2}}\biggr) \exp\Bigl(-\frac{\pi\zeta}{\sqrt{1-\zeta^2}}\Bigr)\biggr\}. \label{1vI(5.6)}$$ ![Perturbation of the Maxwell model. Relative errors of the asymptotic formulas (\[1vI(5.5)\]) and (\[1vI(5.6)\]) for the duration of impact (a) and the coefficient of restitution (b).[]{data-label="SSM_Maxwell.pdf"}](SSM_Maxwell_e.pdf "fig:") ![Perturbation of the Maxwell model. Relative errors of the asymptotic formulas (\[1vI(5.5)\]) and (\[1vI(5.6)\]) for the duration of impact (a) and the coefficient of restitution (b).[]{data-label="SSM_Maxwell.pdf"}](SSM_Maxwell_time.pdf "fig:") The accuracy of the asymptotic approximations (\[1vI(5.5)\]) and (\[1vI(5.6)\]) is presented in Fig. \[SSM\_Maxwell.pdf\]. Note that the asymptotic formulas (\[1vI(5.5)\]) and (\[1vI(5.6)\]) are not uniformly valid as $\eta\to 1$. Drop weight impact. Viscoelastic Kelvin–Voigt model {#1dsSection05} =================================================== Due to Newton’s second law, the differential equation of the drop weight impact has the form $$m\ddot{x}+b\dot{x}+kx=mg, \quad t\in[0,t_c], \label{1vI(05.1)}$$ where $g$ is the gravitational acceleration. The initial conditions for Eq. (\[1vI(05.1)\]) are $$x(0)=0, \quad \dot{x}(0)=v_0. \label{1vI(05.2)}$$ The drop weight impact problem (\[1vI(05.1)\]), (\[1vI(05.2)\]) has the following solution: $$x(t)=\frac{g}{\omega_0^2}\bigl(1-e^{-\beta t}\cos\omega t\bigr)+ \frac{1}{\omega}\Bigl(v_0-\frac{g\beta}{\omega_0^2}\Bigr)e^{-\beta t}\sin\omega t, \label{1vI(05.3)}$$ $$\dot{x}(t)=v_0 e^{-\beta t}\cos\omega t+ \frac{(g-\beta v_0)}{\omega}e^{-\beta t}\sin\omega t. \label{1vI(05.4)}$$ Here we used the notation (\[1vI(1.6)\]). According to Eqs. (\[1vI(05.3)\]), (\[1vI(05.4)\]), the reaction force $F(x,\dot{x})=kx+b\dot{x}$ is given by $$\begin{aligned} \frac{F}{mv_0\omega_0} & = & \varepsilon_0\biggl\{1+e^{-\beta t}\biggl( \frac{\eta}{\sqrt{1-\eta^2}}\sin\omega t-\cos\omega t\biggr)\biggr\} \nonumber \\ {} & {} & {}+e^{-\beta t}\biggl( \frac{1-2\eta^2}{\sqrt{1-\eta^2}}\sin\omega t+\frac{2\eta}{\sqrt{1-\eta^2}}\cos\omega t\biggr), \label{1vI(05.5)}\end{aligned}$$ where we introduced the notation $$\varepsilon_0=\frac{g}{\omega_0 v_0}. \label{1vI(05.5e)}$$ The problem (\[1vI(05.1)\]), (\[1vI(05.2)\]) was studied in [@Ivanov1997], where the existence of the parameter domain of “plastic impact” was established. This means that for any $\eta>0$, there exists a unique value of $\varepsilon_0^$ such that for all $\varepsilon_0>\varepsilon_0^$ we have $F(x,\dot{x})>0$ in the time interval $t\in(0,+\infty)$. The critical value $\varepsilon_0^$ of the parameter $\varepsilon_0$ determines the critical value $v_0^$ of the initial velocity $v_0$ below which there is no rebound effect. With the aim of application to the drop weight impact testing, we consider the problem (\[1vI(05.1)\]), (\[1vI(05.2)\]) for small values of the dimensionless parameter $\varepsilon_0$ and construct an asymptotic solution for the coefficient of restitution. Let $t_c^0$ and $e_^0$ be the impact duration and the coefficient of restitution for the Kelvin–Voigt impact model, correspondingly. According to Eqs. (\[1vI(1.9a)\]) and (\[1vI(1.14)\]), we have $$t_c^0=\frac{2}{\omega_0\sqrt{1-\eta^2}}{\,\rm atan\,}\frac{\sqrt{1-\eta^2}}{\eta},\quad e_^0=\exp\biggl\{- \frac{2\eta}{\sqrt{1-\eta^2}}{\,\rm atan\,}\frac{\sqrt{1-\eta^2}}{\eta}\biggr\}. \label{1vI(05.6)}$$ Now, solving the transcendental equation $F(x,\dot{x})\bigr\vert_{t=t_c}=0$ by a perturbation method to terms of the first order inclusive, we obtain $$t_c\simeq t_c^0+\varepsilon_0\frac{(1+e_^0)}{e_^0\omega_0}, \label{1vI(05.7)}$$ $$e_\simeq e_^0(1-2\varepsilon_0\eta), \label{1vI(05.8)}$$ where $t_c^0$ and $e_^0$ are given by Eqs. (\[1vI(05.6)\]). From the asymptotic formulas (\[1vI(05.7)\]) and (\[1vI(05.8)\]), it is clearly seen that the gravitational effect increases the duration of the impact process and decreases the coefficient of restitution. But it is more interesting to observe that the coefficient of restitution $e_$ increases with velocity $v_0$, since the parameter $\varepsilon_0$ is inversely proportional to $v_0$ . That is why the effect of decrease in the coefficient of restitution in the drop weight impact test experimentally observed in [@Edelsten2010] for the velocity range $v_0\in(0{.}7,1{.}4)$ m/s and extrapolated for the low velocity region by means of the nonlinear Kelvin–Voigt model $F(x,\dot{x})=kx+c\vert x\vert\dot{x}$ with no account for the impactor weight cannot be explained by the linear viscoelastic Kelvin–Voigt model considered in this section. Drop weight impact. Viscoelastic Maxwell model {#1dsSection06} ============================================== By applying the approach [@Stronge2000], the differential equation of motion $m\ddot{x}+F=mg$ with the initial conditions $x(0)=0$ and $\dot{x}(0)=v_0$ in view of the constitutive relationship (\[1vI(2.1)\]) can be reduced to the following problem: $$\dddot{x}+\frac{k}{b}\ddot{x}+\frac{k}{m}\dot{x}-\frac{kg}{b}=0, \label{1vI(06.1)}$$ $$x(0)=0, \quad \dot{x}(0)=v_0,\quad \ddot{x}(0)=g. \label{1vI(06.2)}$$ The drop weight impact problem (\[1vI(06.1)\]), (\[1vI(06.2)\]) has the following exact solution: $$\begin{aligned} x(t) & = & \frac{v_0}{\omega_0}e^{-\zeta\omega_0 t}\biggl\{ \frac{\omega_0}{\omega}(1-2\zeta^2-\varepsilon_0\zeta)\sin\omega t -(2\zeta+\varepsilon_0)\cos\omega t\biggr\} \nonumber \\ {} & {} & {}+\frac{v_0}{\omega_0}\bigl(2\zeta+\varepsilon_0(1-2\zeta\omega_0 t)\bigr), \label{1vI(06.3)}\end{aligned}$$ $$\begin{aligned} \dot{x}(t) & = & v_0 e^{-\zeta\omega_0 t}\biggl\{ \frac{\omega_0}{\omega}(\zeta+\varepsilon_0)\sin\omega t+\cos\omega t\biggr\} \nonumber \\ {} & {} & {}+2\zeta\varepsilon_0 v_0. \label{1vI(06.4)}\end{aligned}$$ Here we used the notation (\[1vI(2.6)\]), (\[1vI(05.5e)\]). According to Eqs. (\[1vI(06.3)\]) and (\[1vI(06.4)\]), the reaction force $F=mg-\ddot{x}$ is given by $$\frac{F}{mv_0\omega_0}=\frac{\omega_0}{\omega} e^{-\zeta\omega_0 t}\biggl\{ (1+\varepsilon_0\zeta)\sin\omega t-\frac{\omega}{\omega_0}\varepsilon_0\cos\omega t\biggr\} +\varepsilon_0 \label{1vI(06.5)}$$ Now, let $t_c^0$ and $e_^0$ be the impact duration and the coefficient of restitution for the Maxwell impact model, correspondingly. According to Eqs. (\[1vI(2.8)\]) and (\[1vI(2.9)\]), we have $$t_c^0=\frac{\pi}{\omega_0\sqrt{1-\zeta^2}},\quad e_^0=\exp\biggl(- \frac{\pi\zeta}{\sqrt{1-\zeta^2}}\biggr). \label{1vI(06.6)}$$ Applying a perturbation method, we find $$t_c\simeq t_c^0+\varepsilon_0\frac{(1+e_^0)}{e_^0\omega_0}, \label{1vI(06.7)}$$ $$e_\simeq e_^0-2\zeta\varepsilon_0, \label{1vI(06.8)}$$ where $t_c^0$ and $e_^0$ are given by Eqs. (\[1vI(06.6)\]). It is interesting to note that the asymptotic formulas (\[1vI(05.7)\]) and (\[1vI(06.7)\]) coincide. Clearly, the same conclusions can be drawn about the influence of the gravitational effect in the framework of the viscoelastic Maxwell drop weight impact model as those that were formulated in Section \[1dsSection05\]. Short-time asymptotic solution of the indentation problem for a thin biphasic layer {#1dsSection07} =================================================================================== We assume that the deformational behavior of articular cartilage is modeled in the framework of linear biphasic theory [@MowKueiLaiArmstrong1980], which represents the biological tissue as a mixture consisting of a porous solid phase and a fluid phase (mobile interstitial water). The constitution equations for the solid and fluid phase stresses, $\mbox{$\boldsymbol\sigma$}^s$ and $\mbox{$\boldsymbol\sigma$}^f$, are given by $$\mbox{$\boldsymbol\sigma$}^s=-\phi^s p{\bf I}+\lambda^s {\rm tr}(\mbox{$\boldsymbol\varepsilon$}){\bf I} +2\mu^s \mbox{$\boldsymbol\varepsilon$},\quad \mbox{$\boldsymbol\sigma$}^f=-\phi^f p{\bf I}.$$ Here, $\phi^f$ is the fluid volume fraction (porosity), $\phi^s=1-\phi^f$ is the solid volume fraction, $p$ is the true pressure of the fluid, $\lambda^s$ and $\mu^s$ are the Lamé constants, which together define the aggregate modulus $H_A=\lambda^s+2\mu^s$, $\mbox{$\boldsymbol\varepsilon$}$ is the strain tensor of the solid phase, and $\bf I$ is the identity tensor. Note that the fluid phase is assumed to be intrinsically incompressible and inviscid. The continuity equation for the mixture and the momentum equations for each phase are given by $${\rm div}(\phi^s{\bf v}^s+\phi^f{\bf v}^f)=0,$$ $${\rm div}\mbox{$\boldsymbol\sigma$}^s+\frac{(\phi^f)^2}{\kappa}({\bf v}^f-{\bf v}^s)={\bf 0}, \quad {\rm div}\mbox{$\boldsymbol\sigma$}^f-\frac{(\phi^f)^2}{\kappa}({\bf v}^f-{\bf v}^s)={\bf 0},$$ where ${\bf v}^s$ and ${\bf v}^f$ are the solid and fluid velocities, respectively, and $\kappa$ is the permeability of the solid phase. Let us consider an axisymmetric contact problem for a thin biphasic layer indented without friction by a rigid impermeable cylindrical indenter. It is assumed that the contact radius $a$ is much larger than the cartilage layer thickness $h$ (i.e., $h/a\ll 1$). In this case, according to [@Ateshian1994], the vertical displacements $w(r,t)$ of the boundary points of the articular cartilage tissue at the contact zone can be approximated by the following asymptotic formula: $$w(r,t) = \frac{h^3}{3\mu_s}\biggl\{ \frac{1}{3r}\frac{\partial}{\partial r}\Bigl(r\frac{\partial P}{\partial r}(r,t)\Bigr) +\frac{\mu_s \kappa}{h^2}\int\limits_0^t \frac{1}{r}\frac{\partial}{\partial r}\Bigl(r\frac{\partial P}{\partial r}(r,\tau)\Bigr)d\tau\biggr\}. \label{1vI(9.1)}$$ Here, $P(r,t)$ is the contact pressure. It is assumed that the cartilage layer is bonded to a rigid impermeable substrate, that is there is no solid displacement at the cartilage-bone interface and no fluid flow through the bone [@Ateshian1994]. In view of (\[1vI(9.1)\]), the contact condition that the boundary points of the cartilage layer acquire a constant vertical displacement $-\delta_0(t)$ (due to the action of the indenter) can be written as $$w(r,t) = -\delta_0(t), \quad r\leq a. \label{1vI(9.2)}$$ The substitution of (\[1vI(9.1)\]) into Eq. (\[1vI(9.2)\]) results in an integro-differential equation $$\frac{1}{r}\frac{\partial}{\partial r}\Bigl(r\frac{\partial P}{\partial r}(r,t)\Bigr) +\frac{3\mu_s \kappa}{h^2}\int\limits_0^t \frac{1}{r}\frac{\partial}{\partial r}\Bigl(r\frac{\partial P}{\partial r}(r,\tau)\Bigr)d\tau =-\frac{3\mu_s}{h^3}\delta_0(t), \label{1vI(9.2a)}$$ which requires imposing a suitable boundary condition at the edge of the contact zone, i.e., at $r=a$ . In order to impose the mentioned boundary condition, we note that at the initial moment of contact $t=0$, formula (\[1vI(9.1)\]) simplifies as follows: $$w(r,0) = \frac{h^3}{3\mu_s}\frac{1}{3r}\frac{\partial}{\partial r}\Bigl(r\frac{\partial P}{\partial r}(r,0)\Bigr). \label{1vI(9.3)}$$ Comparing formula (\[1vI(9.3)\]) with the known asymptotic solutions for thin elastic layers [@Barber1990; @Chadwick2002; @ArgatovMishuris2011ve], we conclude that the instantaneous deformational response of a thin biphasic layer coincides with the response of a thin bonded incompressible elastic layer. Thus, by this analogy, we will require that $P(r,t)\to 0$ as $r\to a$, that is the contact pressure is assumed to vanish at the edge of the contact area. As a result of integration of Eq. (\[1vI(9.2a)\]) with respect to the radial coordinate, we arrive at the following integral equation: $$P(r,t)+\frac{3\mu_s \kappa}{h^2}\int\limits_0^t P(r,\tau)\,d\tau =\frac{3\mu_s}{4h^3}\delta_0(t)(a^2-r^2). \label{1vI(9.4)}$$ Now, in order to derive the relationship between the indenter displacement and the contact force $$F(t)=2\pi\int\limits_0^a P(r,t)r\,dr,$$ we multiply both sides of Eq. (\[1vI(9.4)\]) by $2\pi r$ and after that we integrate the equation obtained with respect to $r$ from $0$ to $a$. As a results of this operation, we get $$F(t)+\frac{3\mu_s \kappa}{h^2}\int\limits_0^t F(\tau)\,d\tau =\frac{3\mu_s a^4}{16h^3}\delta_0(t). \label{1vI(9.5)}$$ Further, by inverting the Volterra integral operator on the right-hand side of Eq. (\[1vI(9.5)\]), we obtain $$F(t)=\frac{3\mu_s a^4}{16h^3}\biggl\{\delta_0(t)- \chi\int\limits_0^t e^{-\chi(t-\tau)}\delta_0(\tau)\,d\tau\biggr\}, \label{1vI(9.6)}$$ where we introduced the shorthand notation $$\chi=\frac{3\mu_s \kappa}{h^2}. \label{1vI(9.6b)}$$ Finally, after integrating by parts, Eq. (\[1vI(9.6)\]) yields $$F(t)=\frac{3\mu_s a^4}{16h^3}\biggl\{\delta_0(0)+ \int\limits_0^t e^{-\chi(t-\tau)}\frac{d\delta_0}{d\tau}(\tau)\,d\tau\biggr\}. \label{1vI(9.7)}$$ In impact problems, under the assumption that $$\delta_0(0)=0,$$ the force-displacement relationship (\[1vI(9.7)\]) takes the form $$F(t)=\frac{3\mu_s a^4}{16h^3}\int\limits_0^t \exp\Bigl\{-\frac{(t-\tau)}{\tau_R}\Bigr\}\frac{d\delta_0}{d\tau}(\tau)\,d\tau\biggr\}. \label{1vI(9.8)}$$ Here we introduced the notation $\tau_R=1/\chi$. In view of (\[1vI(9.6b)\]), we have $$\tau_R=\frac{h^2}{3\mu_s \kappa}, \label{1vI(9.9)}$$ while comparing (\[1vI(9.9)\]) with (\[1vI(2.2)\]), we get the stiffness coefficient $$k=\frac{3\mu_s a^4}{16h^3}. \label{1vI(9.90)}$$ It should be emphasized that Eq. (\[1vI(9.8)\]) represents a short-time asymptotic approximation, which is valid for moments of time $t$ such that $H_A\kappa t/h^2\ll 1$. For typical human cartilage material properties, $H_A=0{.}5$ MPa and $\kappa=2\times 10^{-15}$ ${\rm m}^4/{\rm Ns}$. Thus, assuming a typical cartilage thickness $h=1$ mm, we get $h^2/(H_A\kappa)=10^3$ s; thus, the asymptotic model (\[1vI(9.8)\]) certainly remains valid for up to 100 s, which is well in the range of usual values of impact durations. Comparing Eq. (\[1vI(9.8)\]) with Eq. (\[1vI(2.2)\]), we see that the short-time deformational response of a thin biphasic layer bonded to a rigid impermeable substrate under the action of a frictionless flat-ended indenter is mathematically equivalent to that of a thin incompressible layer following the Maxwell viscoelastic model. Note that the Maxwell’s model based perturbation model considered in Section \[1dsSection5\] could be useful for modeling the impact response of articular cartilage (or artificial tissues for its replacement) in the whole time range, i.e. in the short-, medium- and long-time range. We also emphasize that the biphasic model is not equivalent to a viscoelastic model, because the biomechanical response of a poroelastic material such as articular cartilage is crucially dependent on the boundary conditions for the sample. In particular, viscoelastic equivalents of the deformational response of an articular cartilage sample subjected to the same simple loading protocols in confined and unconfined conditions will be essentially different, especially in the short-time range. Thus, in comparing experimental results from different sources, a particular attention should be paid to the fixation conditions for tissue samples. Key features of non-linear impact {#1dsSection10} ================================= To illustrate the application of the developed linear theory of viscoelastic impact, let us analyze the experimental data obtained in [@BurginAspden2008] for drop-weight impact testing (with the impactor mass $m=100$ g) of isolated bovine articular cartilage samples of 5 mm diameter (correspondingly, the radius of the samples is $a=2{.}5$ mm). In [@BurginAspden2008], the force data, $F(t)$, were converted to engineering stress, $\sigma(t)$, by dividing them by the original cross-section area of the sample, $\pi a^2$, i.e., $$\sigma(t)=\frac{F(t)}{\pi a^2}.$$ The effective strain, $\epsilon(t)$, was evaluated by dividing the measured impactor displacement, $x(t)$, by the sample thickness, $h$, which is assumed to be $0{.}5\pm 0{.}11$ mm, as follows: $$\epsilon(t)=\frac{x(t)}{h}.$$ (Here, stress and strain are assumed to be positive in compression.) The stress-strain relationship was differentiated to obtain the incremental dynamic modulus $$E_{\rm dyn}=\frac{d\sigma}{d\epsilon}.$$ The maximum incremental dynamic modulus, $E_{\rm max}$, was found, and the modulus $E_{10}$ at stresses of 10 MPa was determined to enable comparison of dynamic moduli at constant value of stress. The initial impact velocity was calculated from the drop height, $h_0$, by the well-known formula $v_0=\sqrt{2gh_0}$. The incremental dynamic modulus can be evaluated as a function of time in the form $$E_{\rm dyn}(t)=\frac{\dot{\sigma}(t)}{\dot{\epsilon}(t)}=\frac{h}{\pi a^2}\frac{\dot{F}(t)}{\dot{x}(t)}. \label{1vI(E.1)}$$ In the case of the Maxwell model (see, Section \[1dsSection2\]), we will have $$\frac{\dot{F}(t)}{\dot{x}(t)}=k\frac{\cos\omega t-(\zeta\omega_0/\omega)\sin\omega t }{\cos\omega t+(\zeta\omega_0/\omega)\sin\omega t}, \label{1vI(E.2)}$$ where $k$ is the stiffness coefficient. First of all, observe that in view of (\[1vI(E.1)\]) and (\[1vI(E.2)\]), the variation of $E_{\rm dyn}(t)$ does not depend on the impact velocity $v_0$. In other words, the time variation of the incremental dynamic stiffness in the linear viscoelastic impact tests remains the same for different initial impact velocities. We emphasize that this conclusion is valid for a general linear viscoelastic law. Second, from (\[1vI(E.1)\]) and (\[1vI(E.2)\]), it follows that the value of $E_{\rm dyn}(t)$ gradually decreases to zero with increasing contact force $F(t)$ (when $\dot{F}(t)>0$). Thus, we arrive at the formula $$E_{\rm max}=E_{\rm dyn}(0). \label{1vI(E.0)}$$ Further, in order to evaluate $E_{10}$, we need first solve the equation $$F(t_{10})=\pi a^2\sigma_{10}, \label{1vI(E.3)}$$ where $\sigma_{10}=10$ MPa. In view of (\[1vI(2.7)\]), Eq. (\[1vI(E.3)\]) takes the form $$\exp(-\zeta\omega_0 t_{10})\sin\omega t_{10}=\frac{\pi a^2\omega\sigma_{10}}{kv_0}. \label{1vI(E.4)}$$ Here, $\zeta$, $\omega_0$, and $\omega$ are independent of $v_0$, and are determined by formulas (\[1vI(2.6)\]). Now, from (\[1vI(E.4)\]), it is seen that the value of the time moment $t_{10}$ depends on the initial velocity $v_0$. Thus, the Maxwell impact model (and generally speaking, any linear viscoelastic model of impact) predicts that the value of $E_{10}$ increases with increasing impact velocity $v_0$. $h_0$ (mm) $v_0$ (m/s) $E_{\rm max}$ (MPa) $E_{10}$ (MPa) $\sigma_{\rm max}$ (MPa) $\epsilon_{\max}$ $e_$ $\Delta m$ (%) ------------ ------------- --------------------- ---------------- -------------------------- -------------------- -------------------- ---------------- 25 $0{.}70$ $86\pm 22$ $75\pm 13$ $15{.}6\pm 2{.}9$ $0{.}48\pm 0{.}06$ $0{.}64\pm 0{.}08$ $2{.}2$ 50 $0{.}99$ $100\pm 32$ $71\pm 16$ $24{.}5\pm 3{.}5$ $0{.}60\pm 0{.}13$ $0{.}46\pm 0{.}14$ $2{.}5$ 80 $1{.}25$ $118\pm 33$ $73\pm 12$ $34{.}2\pm 5{.}0$ $0{.}62\pm 0{.}11$ $0{.}47\pm 0{.}05$ $5{.}7$ 100 $1{.}40$ $128\pm 28$ $72\pm 13$ $40{.}5\pm 4{.}6$ $0{.}68\pm 0{.}09$ $0{.}41\pm 0{.}08$ $9{.}9$ : Impact parameters for isolated bovine articular cartilage samples [@BurginAspden2008][]{data-label="1vITable1"} Table \[1vITable1\] shows that the impact testing [@BurginAspden2008] was performed in the non-linear regime with maximum compressive strains of 50–60%. That is why, the prediction of the linear impact model concerning $E_{\rm max}$ are not fulfilled. Furthermore, the linear theories of impact predict that the maximum contact force $F_M$ (correspondingly, the maximum contact stress $\sigma_{\rm max}=F_M/(\pi a^2)$) and the maximum displacement $x_m$ (correspondingly, the maximum strain $\epsilon_{\rm max}=x_m/h$) are proportional to $v_0$. On the other hand, the data from Table 1 show that the ratio $\sigma_{\rm max}/\epsilon_{\rm max}$ increases with increasing $v_0$. This fact also clearly indicates the non-linear deformational behavior of cartilage at high level of strain. Note here that the ratio $\sigma_{\rm max}/\epsilon_{\rm max}$ is ralted to the so-called pulsatile dynamic modulus (see, in particular, [@Argatov2012sine]). Concerning the coefficient of restitution $e_$ note that it is not constant, as it would be if the cartilage deformation were described by the Maxwell model (see formula (\[1vI(2.9)\])). Finally, the last column of Table \[1vITable1\] gives the values of percentage increase in mass of each sample after 24 h immersed in PBS following impact loading. This is indicative of increasing amounts of damage in the cartilage samples [@BurginAspden2008]. Discussion {#1dsSectionD} ========== Consider now the general case of linear viscoelastic force-displacement relationship $$F=\int\limits_0^t k(t-s)\frac{dx}{ds}(s)\,ds \label{1vI(7.1)}$$ with the relaxation stiffness $$k(t)=k_0\Psi\Bigl(\frac{t}{\tau_R}\Bigr).$$ Here, $k_0$ is the initial stiffness, $\tau_R$ is the characteristic relaxation time, $\Psi(\tau)$ is the dimensionless relaxation function with $\tau$ being a dimensionless independent time-like variable. Making use of the change of variables $$t=\tau_R\tau,\quad x=v_0\tau_R\xi, \label{1vI(7.2)}$$ we transform the impact equation $m\ddot{x}+F=0$ and the initial conditions $x(0)=0$, $\dot{x}=v_0$ into the following problem: $$\xi^{\prime\prime}+\alpha\int\limits_0^\tau \Psi(\tau-\sigma)\frac{d\xi}{d\sigma}(\sigma)\,d\sigma=0, \label{1vI(7.3)}$$ $$\xi(0)=0,\quad \xi^\prime(0)=1. \label{1vI(7.4)}$$ Here prime denotes differentiation with respect to $\tau$, and we introduced the notation $$\alpha=\frac{k_0\tau_R^2}{m}. \label{1vI(7.5)}$$ Note that for the Maxwell model (see Section \[1dsSection2\], Eq. (\[1vI(2.2)\])) we have $k_0=k$, $\tau_R=b/k$, and $\alpha=1/(4\zeta^2)$. Furthermore, according to Eqs. (\[1vI(7.2)\])), the variable impact velocity is $$\dot{x}(t)=v_0\xi^\prime(\tau).$$ Let $\tau_c$ be the dimensionless duration of the impact process. Then, the coefficient of restitution can be found as $$e_=-\xi^\prime(\tau_c). \label{1vI(7.6)}$$ From Eqs. (\[1vI(7.3)\])) and (\[1vI(7.4)\])), it is evident that $\tau_c$ is a function of $\alpha$ only and does not depend on $v_0$. Thus, in view of (\[1vI(7.6)\])), we conclude that the coefficient of restitution $e_$ is constant with respect to the initial impact velocity $v_0$. It can be shown that the same qualitative conclusions are drawn from the linear biphasic model [@MowKueiLaiArmstrong1980] for articular cartilage deformation. In this case, the parameter $\tau_R$, which enters Eqs. (\[1vI(7.2)\])), can be defined as a typical diffusion time $\tau_D=h^2/(\kappa H_A)$, where $h$ is the cartilage layer thickness, $\kappa$ is the cartilage permeability, and $H_A$ is the aggregate modulus. Observe that the biphasic theory incorporating Lamé parameters assumes that the material of solid phase is linearly elastic in order for these to have unique values. But if the material is viscoelastic these parameters are difficult to define and they become functions of deformation and/or time, if they are meaningful at all. Furthermore, there is an intrinsic circularity problem associated with using the aggregate modulus $H_A$, which is evaluated at equilibrium after the interstitial water is squeezed out, to define the mechanical properties that are then assumed to pertain during the impact deformation. Thus, the fact that the biphasic theory provides a good fit to measured curves in the creep and stress relaxation tests can be basically considered as a consequence of a curve-fitting procedure with a minimum of three free parameters rather than a derivation from first principles. In other words, it remains to be an open question on the efficiency of mixture models for articular cartilage at high strain rates. In the present study we addressed the question of whether the main features of articular impact observed in [@Varga2007; @Edelsten2010] could be qualitatively predicted using a linear viscoelastic theory or the linear biphasic theory. It is to note that the deformations encountered in impact tests should be small enough for the linear theories to apply. With respect to engineering polymers note that the linear theory of viscoelasticity may hold reasonably well even up to some $5-10\%$ extension, in particular for certain rubbers [@Tschoegl1997]. Conclusions {#1dsSectionC} =========== The results of this study based on the linear viscoelasticity imply the following properties of the linear impact models: 1\. The coefficient of restitution $e_$ is a function of the damping ratio $\zeta$ alone. This means that $e_$ does not depend on the impact velocity $v_0$, but it depends on the impactor mass $m$ and the sample thickness (through the stiffness $k$). 2\. The impact duration $t_c$ is inversely proportional to $\omega_0$, that is $t_c$ is proportional to $\sqrt{m}$, and depends on the damping ratio as well. The impact duration does not depend on the impact velocity $v_0$. 3\. The maximum displacement, $x_m$, and the maximum contact force, $F_M$, are proportional to $\sqrt{m}$ and $v_0$. 4\. The time variation of the incremental dynamic stiffness $dF/dx$ remains the same for different initial impact velocities. 5\. In the drop weight impact test, the gravitational effect increases the impact duration $t_c$ and decreases the coefficient of restitution $e_$. At that, the coefficient of restitution increases with the impact velocity $v_0$. Acknowledgment {#acknowledgment .unnumbered} ============== The financial support from the European Union Seventh Framework Programme under contract number PIIF-GA-2009-253055 is gratefully acknowledged. 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--- address: \| Physique Théorique et Mathématique,\ Université Libre de Bruxelles, C.P. 231, 1050 Brussels, Belgium.\ International Solvay Institutes, Brussels, Belgium.\ Department of Fundamental Physics,\ Chalmers University of Technology, 412 96 Göteborg, Sweden.\ $\mathrm{Email: [email protected],[email protected]}$ author: - CHRISTOFFER PETERSSON title: MULTILEPTON AND MULTIPHOTON SIGNATURES OF SUPERSYMMETRY AT THE LHC --- Introduction ============ The discovery of a scalar boson, with properties in agreement with the predictions of the Standard Model (SM), reinforces the desire to understand the origin of the electroweak scale. The quadratic sensitivity of the scalar mass to ultraviolet physics suggests the presence of new states beyond the SM (BSM) at or below the TeV scale. But, in spite of the impressive range of searches performed by the LHC collaborations during Run I, no such new states have been observed so far. While these null results may be taken as an indication of the absence of new states in this energy range, they may also be taken as a motivation to push forward in new directions in the exploration of TeV scale physics. Supersymmetric (SUSY) extensions of the SM have the potential to both stabilize the electroweak scale and explain why it is hierarchically smaller than the Planck scale. The current bounds on superpartner masses are discomforting, but one should perhaps keep in mind that many searches are designed to probe particular SUSY extensions which are [minimal]{} in terms of their particle content and the underlying assumptions. However, given the non-minimality of, for example, the particle content of the SM, with three generations of quarks and leptons with a hierarchical mass spectrum, it could be that minimality is not a good guiding principle. By going beyond minimality in terms of model building, non-standard phenomenology can easily arise, with new search channels opening up and/or with standard search channels closing down. In the fist part of this note, by allowing for spectra beyond those of minimal models, we discuss an example of a BSM model that both fits a slight excess in the data and that predicts non-standard signatures which are currently not being targeted at the LHC. In the second part, we discuss a scenario where a deviation from the minimal model building assumptions opens up new search channels, while evading constraints coming from the standard ones. We consider simplified models based on the framework of gauge mediated SUSY breaking (GMSB), with R-partity conservation. In Section \[multileptons\], based on the paper [@multileptons], we do the exercise of explaining a slight excess in terms of multilepton events observed by the CMS collaboration [@CMSmultileptons]. We provide a simple model that can explain the excess, without being excluded by any other data, and we discuss how to best probe this model. In Section \[multiphotons\], based on the paper [@multiphotons], we discuss how the standard phenomenology of GMSB is modified if SUSY is broken in more than one hidden sector.[^1] In such multiple hidden sector models, the final state spectrum is typically softer than in standard GMSB, which implies that existing LHC searches are not very sensitive to these kind of models. The upshot is that these models typically give rise to additional (soft) photons in the final state, and we propose new searches designed to probe them. Multilepton signatures {#multileptons} ====================== Let us start by discussing the small excess observed by the CMS collaboration in a search for events with three or more leptons with 19.5 fb${}^{-1}$ of data at $\sqrt{s}=8$ TeV [@CMSmultileptons]. This small excess was seen in the final state category of events with three electrons or muons,[^2] one hadronically decaying tau lepton ($\tau_h$), low hadronic activity[^3] and no tagged b-jets. In this category, CMS observed (expected) in the three bins of missing transverse energy $\MET{<}50$GeV, 50${<}\MET{<}100$GeV and $\MET{>}100$GeV, respectively. The probability to observe 22 events in the combined $\MET$-range, when $10.1\pm2.4$ events were expected, is about 1%. However, when taking into account the fact that they search in 64 independent categories, the probability for this fluctuation in the combined $\MET$-range is about 50%, while the joint probability to observe such an excess in all the three $\MET$-bins is about 5% [@CMSmultileptons]. The most likely explanation for this slight excess is that it is due to a statistical fluctuation and that it will go away with more data. Nevertheless, we take the opportunity to perform the exercise of trying to fit this excess with some BSM physics. We consider two simplified models of GMSB, denoted by [M.I]{} and [M.II]{}, with spectra given in Figure \[fig:models\]. These models were studied in [@CMSmultileptons]. Here we extend that study by taking into account the exclusion bounds arising from pair production of the next-to-lightest SUSY particle (NLSP), determining the best fit model, considering other relevant searches and discussing prospects and possible new searches designed to probe the best fit model. Concerning the particle content of the models in Figure \[fig:models\], as always in GMSB, the lightest SUSY particle (LSP) is the nearly massless gravitino $\widetilde{G}$. In model [M.I]{}/[M.II]{} we take the NLSP to be the right-handed stau/sleptons, $\tilde{\tau}_R/\tilde{\ell}_R$, where “slepton" refers to either a selectron or a smuon, $\tilde{\ell}_R=\tilde{e}_R,\tilde{\mu}_R$. The next-to-NLSP (NNLSP) is the $\tilde{\ell}_R/\tilde{\tau}_R$, while the Bino $\widetilde{B}$ is taken to be heavier. All remaining superpartners are assumed to be sufficiently heavy and effectively decoupled. While such a decoupling is typically not possible in minimal GMSB models, where the relations among the soft masses are completely determined in terms of the gauge quantum numbers, it is possible to realize such spectra within the framework of See [@selectronNLSP] for a complete characterization of models realizing the non-standard GMSB spectrum of the simplified model [M.II]{} in Figure \[fig:models\]. Concerning the decay channels, since we assume R-parity, the NLSP only has one decay mode, i.e. to its SM partner and the gravitino. In contrast, the NNLSP has two possible decay channels, either the two-body decay to its SM partner and the gravitino, or the three-body decay, via an off-shell Bino, to the NLSP. For the parameter space region we are interested in, where the gravitino mass is in the range 0.1eV${<} \,m_{3/2} {<} \,10$eV, the NNLSP coupling to the gravitino is strongly suppressed compared to the gauge couplings entering the three-body decay, and the dominant NNLSP decay mode is the three-body decay. At the LHC, the models [M.I]{} and [M.II]{} give rise to the processes shown in Figure \[fig:processes\], and the final states $4\tau+2\ell+\MET$ and $2\tau+4\ell+\MET$, respectively. Hence, NNLSP pair production gives rise to multilepton events which could be relevant for the CMS search [@CMSmultileptons]. In order to see if we can fit the excess in [@CMSmultileptons], we simulate the two processes in Figure \[fig:processes\] at the LHC and analyze of data at $\sqrt{s}=8$ TeV, with kinematic and geometric cuts applied in accordance with the CMS search. ![Spectra for the simplified models [M.I]{} and [M.II]{}.[]{data-label="fig:models"}](Petersson_models.pdf){width="0.55\linewidth"} ![\[fig:processes\] The NNLSP pair production processes for models [M.I]{} (left) and [M.II]{} (right).](Petersson_slep.pdf "fig:"){width=".37\textwidth"} ![\[fig:processes\] The NNLSP pair production processes for models [M.I]{} (left) and [M.II]{} (right).](Petersson_stau.pdf "fig:"){width=".37\textwidth"} In Figure \[fig:results\], we show the number of signal events the processes in Figure \[fig:processes\] give rise to in the stau/slepton mass plane, where model [M.I]{}/[M.II]{} corresponds to the lower/upper triangular half plane. Figure \[fig:results\] (left) corresponds to the final state category where CMS observed the excess, and we see that both model [M.I]{} and [M.II]{} contain regions in the mass plane where the number of signal events fill the gap between the observed and expected number of events. So far we have only discussed the multilepton final states arising from the pair production of the NNLSP. Of course, in these models, there will also be pair production of the NLSP. In model [M.II]{}, the pair produced NLSP sleptons decay to the final state $\ell^+\ell^- {+}\MET$. The current bound on such right-handed sleptons is $m_{\tilde{\ell}_R}>245$ GeV [@ATLASsleptons], which actually excludes the entire parameter space region of [M.II]{} relevant for explaining the CMS excess. In model [M.I]{}, the pair produced NLSP staus decay to the final state $\tau^+\tau^-{+}\MET$. The current strongest bound on such right-handed staus is still the one set by LEP at As can be seen from Figure \[fig:results\], even if this stau mass bound imposes a non-trivial constraint on [M.I]{}, there still remains a parameter space region of [M.I]{} that can explain the CMS excess. The best fit model, obtained by considering the three $\MET$-bins individually, has $m_{\tilde{\ell}_R}=145$GeV and $m_{\tilde{\tau}_R}=90$GeV. ![\[fig:results\] The number of signal events in the $m_{\tilde{\ell}_R}/m_{\tilde{\tau}_R}$ mass plane, where we have combined the three $\MET$-bins. The left plot corresponds to the category discussed in the text, where CMS observed an excess. The middle plot corresponds to a similar category, but where the invariant mass of the OSSF lepton pair is instead within a window of $\pm15$GeV around the $Z$ boson mass. The right plot corresponds to the category where the final state consists of four electrons or muons, but no hadronic tau. In all three plots, the number of expected and observed events are indicated, as well as the LHC and LEP exclusion bounds arising from NLSP pair production.](Petersson_1tau-OSSF1-offZ.pdf "fig:"){width=".32\textwidth"} ![\[fig:results\] The number of signal events in the $m_{\tilde{\ell}_R}/m_{\tilde{\tau}_R}$ mass plane, where we have combined the three $\MET$-bins. The left plot corresponds to the category discussed in the text, where CMS observed an excess. The middle plot corresponds to a similar category, but where the invariant mass of the OSSF lepton pair is instead within a window of $\pm15$GeV around the $Z$ boson mass. The right plot corresponds to the category where the final state consists of four electrons or muons, but no hadronic tau. In all three plots, the number of expected and observed events are indicated, as well as the LHC and LEP exclusion bounds arising from NLSP pair production.](Petersson_1tau-OSSF1-onZ.pdf "fig:"){width=".32\textwidth"} ![\[fig:results\] The number of signal events in the $m_{\tilde{\ell}_R}/m_{\tilde{\tau}_R}$ mass plane, where we have combined the three $\MET$-bins. The left plot corresponds to the category discussed in the text, where CMS observed an excess. The middle plot corresponds to a similar category, but where the invariant mass of the OSSF lepton pair is instead within a window of $\pm15$GeV around the $Z$ boson mass. The right plot corresponds to the category where the final state consists of four electrons or muons, but no hadronic tau. In all three plots, the number of expected and observed events are indicated, as well as the LHC and LEP exclusion bounds arising from NLSP pair production.](Petersson_0tau-OSSF1-offZ.pdf "fig:"){width=".32\textwidth"} In Figure \[fig:results\] (middle) and (right), we show the number of signal events we obtain for two of the other final state categories in [@CMSmultileptons]. We see that the best fit model gives rise to a number of signal events that is within the 1$\sigma$ variation from the SM prediction in the middle plot. Also in the right plot there is no conflict with the data. We refer the reader to the paper [@multileptons] for discussions about data from other categories and searches. Concerning the possibility to probe this best fit model, it is important to realize that, in the last decay step of the process in Figure \[fig:processes\] (left), each of the 90 GeV staus decays to two approximately massless particles, a gravitino and a tau, which therefore roughly share the energy. Since the taus dominantly decay hadronically, many of these signal processes give rise to at least two hadronic taus which are hard enough to allow for reconstruction. We find that the most promising search channel for the best fit model would be in the final state involving two hadronic taus, two or three electrons or muons and $\MET$. In paper [@multileptons] we estimate the number of signal events the best fit model would contribute with to these final states, and we find that, already with the existing data set, such a search could have very good sensitivity. Multiphoton signatures {#multiphotons} ====================== If SUSY is realized in Nature it must be in a broken phase at low energies. A model-independent consequence of (global) SUSY breaking is the presence of a spin 1/2 Goldstone mode, the goldstino. Upon coupling to gravity, the goldstino is eaten by the spin 3/2 gravitino, becoming its longitudinal components, and the gravitino becomes massive. This is in analogy with electroweak symmetry breaking in the SM, where the Goldstone bosons are eaten by the $Z$ and $W$ bosons, which become massive. When the mass of the gravitino is small compared to the energy scale under consideration, in analogy with the equivalence theorem in the SM, the gravitino can be replaced by its longitudinal goldstino components. We consider this case, where the gravitino mass is small and where the communication of SUSY breaking to the visible sector, which we take to be the MSSM, is done via gauge interactions (i.e. within the framework of GMSB). In this section we investigate how the usual phenomenology of GMSB models is modified if we make the non-minimal assumption that SUSY is broken in more than one hidden sector.[^4] If SUSY is broken in $n$ hidden sectors, there will be $n$ neutral spin 1/2 goldstino-like fermions in the spectrum. However, there is only one particular linear combination that corresponds to the true goldstino mode, i.e. the one that is eaten by the gravitino. The remaining $n{-}1$ linear combinations correspond to so called pseudo-goldstini which, in contrast to the true goldstino, are not protected by the Goldstone shift symmetry and therefore they acquire masses, both at the tree and the radiative level. In comparison to standard GMSB models, where it is assumed that SUSY is broken in only one hidden sector and that there is only the nearly massless gravitino below the SM superpartners, multiple hidden sector models will, in addition, contain a tower of these massive pseudo-goldstini. In the case where the lightest SM superpartner is a Bino-like neutralino, its dominant decay channel is generically to a photon and the heaviest of the pseudo-goldstini. The reason is that the strength of the neutralino couplings are related to the mass of the pseudo-goldstini and the larger the mass, the stronger the coupling. In paper [@multiphotons] we also show that, in models with more than two hidden sectors, i.e. with more than one pseudo-goldstino, the heaviest pseudo-goldstino can decay promptly to a photon and a lighter pseudo-goldstino.[^5] Hence, in comparison to standard GMSB, the final states will contain additional photons. In order to illustrate the characteristic features of GMSB models with multiple hidden sectors and with a Bino-like neutralino being the lightest SM superpartner, let us for concreteness take the number of hidden sectors to be three, and thereby the number of massive pseudo-goldstini to be two. Above the Bino-like neutralino, depending on the ultraviolet model one has in mind, one could consider different superpartners which could be produced at the LHC. Here, as an example, we include the right-handed sleptons in the simplified model we consider. For simplicity, we take all three families of sleptons to be mass-degenerate, and with a change of notation with respect to the Section \[multileptons\], “slepton" here refers to a selectron, smuon or stau, $\tilde{\ell}_R=\tilde{e}_R,\tilde{\mu}_R,\tilde{\tau}_R$. ![\[Spec+proc\] The spectrum of the simplified model we consider, and the process it gives rise to.](Petersson_modelsPGLD2.pdf "fig:"){width=".35\textwidth"} ![\[Spec+proc\] The spectrum of the simplified model we consider, and the process it gives rise to.](Petersson_sleptonPGLD2.pdf "fig:"){width=".45\textwidth"} In the simplified model depicted in Figure \[Spec+proc\] (left), the main production mode is slepton pair production and the relevant process is shown in Figure \[Spec+proc\] (right). In this model, the heaviest pseudo-goldstino, to which the neutralino decays via a photon, is denoted by $\tilde{G}''$. This pseudo-goldstino subsequently decays promptly to the lighter pseudo-goldstino $\tilde{G}'$ which, in contrast to $\tilde{G}''$, is stable on collider time scales. Eventually $\tilde{G}'$ will decay to the gravitino $\tilde{G}$ but since that coupling is very small, this decay takes place outside the detector. Thus, from the point of view of collider physics, the gravitino plays no role and therefore it is not included in Figure \[Spec+proc\] (left). The process in Figure \[Spec+proc\] (right) gives rise to the final state $\ell^+\ell^- + 4\gamma+\MET$. Note that, if we would consider a different production mechanism by replacing the sleptons by other SM superpartners, then the OSSF lepton pair in Figure \[Spec+proc\] (right) would be replaced by the corresponding SM partners. In order to be as model-independent as possible in terms of the production mode, let us focus on the last two decay steps in the process, from which the photons and $\MET$ originate. In comparison to standard GMSB, where each of the two neutralinos decays to a photon and a nearly massless gravitino, since the pseudo-goldstini are massive, the emitted photons will here be softer. Moreover, since some of the $\MET$ that would be carried away by $\tilde{G}''$ is transformed into soft photon energy in the last decay step, also the amount of $\MET$ will here be smaller. In paper [@multiphotons] we show that, due to the hard cuts on the transverse momentum of the photons ($p_T^\gamma$) and the $\MET$, the existing LHC searches for GMSB are not very sensitive to multiple hidden sector models. Instead, what one should do in order to probe these models is to relax the $p_T^\gamma$ and $\MET$ cuts, but require additional soft photons in the final state. In [@multiphotons] we find that a search for four photons, each with $p_T^\gamma>20$GeV, and $\MET>$50GeV could easily lead to a discovery (or very strong constraints) already with the existing amount data, i.e. Conclusions =========== In the first part of this note we provided a possible explanation of a slight excess, observed by the CMS collaboration, in terms of a simplified model of GMSB, with a gravitino LSP, a stau NLSP at around 90GeV and mass-degenerate selectron/smuon at around 145GeV. Since the stau mass is close to the LEP bound it will be interesting to see if the LHC at some point will be able to set a stronger bound on the stau mass. The search we propose that would best probe this model is in the final state $2\tau_h+(2/3)\ell+\MET$. In the second part, we discussed GMSB models in which SUSY is broken in more than one hidden sector. The two key features of such multiple hidden sector models were that the final state spectrum was generically softer than in standard GMSB, making the existing LHC searches poorly sensitive, and the presence of additional decay steps, where soft photons are emitted, opening up new search channels. The search we propose, that would be model-independent in terms of the production mode, is an inclusive search in the final state $(3/4)\gamma+\MET$, with minimal $p_T$-requirements on the photons. We focused on the example of slepton pair production which, due to the presence of a lepton pair in the final state, could be probed by a search in the final state $2\ell +2\gamma+\MET$, which would capture also the case where one or two photons are too soft to allow for reconstruction. Acknowledgments {#acknowledgments .unnumbered} =============== I would like to thank J.D’Hondt, K.De Causmaecker, G.Ferretti, B.Fuks, A.Mariotti, K.Mawatari and D.Redigolo for the collaborations on the projects I presented here. This work is supported by the Swedish Research Council (VR) under the contract 637-2013-475, by IISN-Belgium (conventions 4.4511.06, 4.4505.86 and 4.4514.08) and by the “Communauté Française de Belgique" through the ARC program and by a “Mandat d’Impulsion Scientifique" of the F.R.S.-FNRS. Finally I would like to thank the organizers of the conference “Rencontres de Moriond 2014, Electroweak Session" for their effort in organizing a very nice and fruitful meeting. References {#references .unnumbered} ========== [99]{} D.Redigolo, [Phys.Lett.]{}B 731, 7 (2014), \[arXiv:1310.0018 \[hep-ph\]\]. CMS Collaboration, arXiv:1404.5801 \[hep-ex\]. G.Ferretti, A.Mariotti, K.Mawatari and C.Petersson, [JHEP]{}, 1404:126 (2014), \[arXiv:1312.1698 \[hep-ph\]\]. K.Benakli, C.Moura, [Nucl.Phys.B]{} 791:125 (2008), \[arXiv:0706.3127 \[hep-th\]\]. C.Cheung [et al.]{}, [JHEP]{}, 1003:073 (2010), \[arXiv:1002.1967 \[hep-ph\]\]. C.Cheung [et al.]{}, [JHEP]{}, 1007:035 (2010), \[arXiv:1004.4637 \[hep-ph\]\]. N.Craig [et al.]{}, [JHEP]{}, 1010:095 (2010), \[arXiv:1007.1239 \[hep-ph\]\]. M.McCullough, [Phys.Rev.D]{} 82:115016 (2010), \[arXiv:1010.3203 \[hep-ph\]\]. H.C.Cheng [et al.]{}, [JHEP]{}, 1103:019 (2011), \[arXiv:1012.5300 \[hep-ph\]\]. K.I.Izawa [et al.]{}, [JHEP]{}, 1103:007 (2011), \[arXiv:1101.4633 \[hep-ph\]\]. J.Thaler and Z.Thomas, [JHEP]{}, 1107:060 (2011), \[arXiv:1103.1631 \[hep-ph\]\]. C.Cheung [et al.]{}, [JHEP]{}, 1108:115 (2011), \[arXiv:1104.2600 \[hep-ph\]\]. D.Bertolini [et al.]{}, [JHEP]{}, 1204:130 (2012), \[arXiv:1111.0628 \[hep-ph\]\]. K.Benakli, arXiv:1402.4286 \[hep-ph\]. R.Argurio [et al.]{}, [Phys.Rev.Lett.]{}107:061601 (2011), \[arXiv:1102.2386 \[hep-th\]\]. R.Argurio [et al.]{}, [JHEP]{}, 1206:096 (2012), \[arXiv:1112.5058 \[hep-ph\]\]. T.Liu [et al.]{}, [Phys.Lett.B]{} 726:228 (2013), \[arXiv:1301.5479 \[hep-ph\]\]. K.Hikasa [et al.]{}, arXiv:1403.5731 \[hep-ph\]. P.Meade, N.Seiberg and D.Shih, [Prog.Theor.Phys.Suppl.]{}177, 143 (2009), \[arXiv:0801.3278 \[hep-ph\]\]. L.Calibbi, A.Mariotti, C.Petersson and D.Redigolo, arXiv:1405.4859 \[hep-ph\]. ATLAS Collaboration, arXiv:1403.5294 \[hep-ex\]. OPAL Collaboration, [Eur.Phys.J.]{}C46, 307 (2006), \[hep-ex/0507048\]. I.Antoniadis [et al.]{}, [Nucl.Phys.B]{} 841:157 (2010), \[arXiv:1006.1662 \[hep-ph\]\]. C.Petersson [et al.]{}, [JHEP]{}, 1202:142 (2012), \[arXiv:1111.3368 \[hep-ph\]\]. C.Petersson [et al.]{}, [JHEP]{}, 1210:016 (2012), \[arXiv:1203.4563 \[hep-ph\]\]. B.Bellazzini [et al.]{}, [Phys.Rev.D]{} 86:033016 (2012), \[arXiv:1207.0803 \[hep-ph\]\]. C.Petersson [et al.]{}, [Phys.Rev.D]{} 87:013008 (2013), \[arXiv:1211.2114 \[hep-ph\]\]. E.Dudas [et al.]{}, [Nucl.Phys.B]{} 870:353 (2013), \[arXiv:1211.5609 \[hep-ph\]\]. E.Dudas [et al.]{}, arXiv:1309.1179 \[hep-ph\]. I.Antoniadis [et al.]{}, arXiv:1405.4314 \[hep-ph\]. [^1]: For different discussions and aspects of multiple hidden sector models in the context of gravity mediation and gauge mediation, see [@goldstini] and [@goldstiniGMSB], respectively. [^2]: Out of these three electrons or muons, there is one opposite sign same flavor (OSSF) lepton pair whose invariant mass is outside a window of $\pm$15 GeV around the $Z$ boson mass. [^3]: The scalar sum of the jet $p_T$ values, denoted by $H_T$, is required to be below 200 GeV. [^4]: Also for such a multiple breaking of a symmetry, there is a SM analogy since the electroweak symmetry is broken both by the vacuum expectation value of the SM scalar field and by the chiral condensate in QCD. [^5]: In order for this decay to be prompt, the SUSY breaking scales should be hierarchical with at least two of them having values around or below $5{-}10$ TeV.	ArXiv
[Comment on “Phase Coexistence in Multifragmentation”]{}\ In their letter Moretto et al. [@moretto96] propose the fragment charge distribution in nuclear multifragmentation to give a signal for the coexistence of nuclear liquid and vapor phase. To our opinion this signal is not usefull and misleading as fluctuations of different origin spoil it. Phase transitions in macro-physics are usually indicated by a peak in the specific heat e.g. $c_p(T)$ or by an anomaly of the caloric equation of state ($C\!E\!S$) $T(E)$ e.g. at constant pressure or volume. In closed finite systems, as e.g. highly excited nuclei, phase transitions are well indicated by the shape of the $C\!E\!S$ c.f.[@gross72; @gross95]. Inherent to phase transitions are large fluctuations at the transition which do not allow a clear phase separation in space or any other observable in small finite systems because of the nonvanishing coherence length of the phase fluctuations c.f. [@janke95; @gross150] and which differ at const.E and at const.T. E.g. even though the backbending of the $C\!E\!S$ is clearly seen for a 10-state Potts model at a lattice size of $100100$ and the area under the oscillation of $T^{-1}(E)$ is close to the asymptotic value of the surface entropy no phase separation can be seen in the configurations. Therefore, the interpretation by ref.[@moretto96] is too naive and suffers from several further difficulties: Equations like formulas (1-3) of ref.[@moretto96] notice charge conservation only via the mean value but leave its fluctuation unrestricted. These fluctuations are usually substantial especially near to phase transitions. Moreover, in nuclear fragmentation one has to take care of the indistinguishability of identical fragments and the partition problem is not the Euler problem as is suggested in [@phair95]. The correct formula for the quantum partition of an integer $Z_0$ is given in [@gross110]. The experimental method used in ref.[@phair95; @moretto96] is of course not ideally suited to look for a phase transition in equilibrized nuclear systems. First of all this system is generated in a collision of two sizeable nuclei. The transverse energy $E_t$ does not give the total excitation energy $\varepsilon^$ of the system nor is it neccessarily proportional to it. In fact the width in $\varepsilon^$ at low $E_t$ can easily be of the order of a few MeV/nucleon[@moretto94a]. I.e. a fixed value of $E_t$ allows for considerable fluctuations of $\varepsilon^$. It is therefore neccessary to investigate the signal of ref.[@moretto96] in a situation where we definitely [have an equilibrized]{} nuclear system with a [sharply defined excitation energy]{}. Lacking experimental data of this kind we investigated a model system from which we know by experience that it reproduces nuclear multifragmentation and which shows a nuclear phase-transition of first order towards fragmentation: The Berlin microcanonical Metropolis Monte Carlo $M\!M\!M\!C$-model [@gross95]. Here as also in other versions of statistical multifragmentation models like the Copenhagen model [@bondorf95] the phase transition towards fragmentation is clearly seen as anomaly in the $C\!E\!S$. In fig.1 we show the resultant parameter $cZ$ or better the quantity $cZ=ln\{P_n(Z)/P_{n+1}(Z)\}$ averaged over the IMF-multiplicities $n$ to get better statistics vs.$Z$. Here $P_n(Z)$ is the probability to find one fragment with charge $Z$ in events with n IMF’s. The two panels at $\varepsilon^=5$ and $=6$ MeV/nucleon resemble the findings of Moretto. At lower excitation in contrast to ref.[@moretto96] $cZ$ is forced to rise with $Z$ as the emission of a second fragment with larger $Z$ is prohibited due to limited energy resources. We guess that at low [transverse]{} energy the experimental data of ref.[@moretto96] are overshadowed by deep inelastic collisions where some of the small fragments are likely from projectile break-up which as such have small transverse energies. Consequently, low total transverse energies do not really characterize the limitation to low excitation energies as indicated by the large width in $\varepsilon^(E_t)$ [@moretto94a]. This is probably the reason for the vanishing quantity $c$ found in ref.[@moretto96] at low transverse energies. Conclusion: From all experience of microcanonical first order phase transitions in small systems one knows that it is normally rather difficult to see a clear phase separation even though the caloric equation of state gives an unambiguous signal, phase fluctuations are usually too large. Within the arguments of ref.[@phair95; @moretto96] there are at least [two]{} important conservation laws to be observed by the reaction: Conservation of charge [and energy]{}. The latter forces the “chemical potential” $c$ to rise again at low excitation energy. The observation of an anomaly in the caloric equation of state [@pochodzalla95] is still the best signal for a phase transition as was predicted in [@gross72; @bondorf95; @gross95]. Since long this is one of the classical signals for phase-transitions.\ \ D.H.E. Gross and A.S. Botvina Hahn-Meitner-Institut Berlin, Bereich Theoretische Physik,14109 Berlin, Germany\ L.G. Moretto et al. , 76:372–375, 1996. D.H.E. Gross et al. , 56:1544, 1986. D.H.E. Gross. , 53:605–658, 1990. D.H.E. Gross et al. , in print, 1996. W. Janke and S. Kappler. , 197:227, 1995. L. Phair et al. , 75:213, 1995. D.H.E. Gross et al. , 1:340–358, 1992. L.Moretto, private communication 1994 J.P. Bondorf et al. , 257:133–221, 1995. J. Pochodzalla et al. , 75:1040, 1995.	ArXiv
--- author: - 'Justin C. Smith, Francisca Sagredo, and Kieron Burke' bibliography: - 'main.bib' - 'renew.bib' - 'exp.bib' - 'Master.bib' title: Warming Up Density Functional Theory --- Introduction {#intro} ============ [Warm dense matter:]{} The study of warm dense matter (WDM) is a rapidly growing multidisciplinary field that spans many branches of physics, including for example astrophysics, geophysics, and attosecond physics[@MD06; @DOE09; @LHR09; @KDBL15; @KD09; @KDP15; @HRD08; @KRDM08; @RMCH10; @SEJD14; @GDRT14]. Classical (or semiclassical) plasma physics is accurate for sufficiently high temperatures and sufficiently diffuse matter[@I04]. The name WDM implies too cool and too dense for such methods to be accurate, and this regime has often been referred to as the malfunction junction, because of its difficulty[@DOE09]. Many excellent schemes have been developed over the decades within plasma physics for dealing with the variety of equilibrium and non-equilibrium phenomena accessed by both people and nature under the relevant conditions[@BL04]. These include DFT at the Thomas-Fermi level (for very high temperatures) and use of the local density approximation (LDA) within Kohn-Sham (KS) DFT at cold to moderate temperatures (at very high temperatures, sums over unoccupied orbitals fail to converge). The LDA can include thermal XC corrections based on those of the uniform gas, for which simple parametrizations have long existed[@SD13b; @KSDT14]. [Electronic structure theory:]{} On the other hand, condensed matter physicists, quantum chemists, and computational materials scientists have an enormously well-developed suite of methods for performing electronic structure calculations at temperatures at which the electrons are essentially in their ground-state (GS), say, 10,000K or less[@B12]. The starting point of many (but not all) such calculations is the KS method of DFT for treating the electrons[@KS65]. Almost all such calculations are within the Born-Oppenheimer approximation, and ab initio molecular dynamics (AIMD) is a standard technique, in which KS-DFT is used for the electronic structure, while Newton’s equations are solved for the ions[@CP85]. [DFT in WDM:]{} In the last decade or so, standard methods from the electronic structure of materials have had an enormous impact in warm dense matter, where AIMD is often called QMD, quantum molecular dynamics[@GDRT14]. Typically a standard code such as VASP is run to perform MD[@KRDM08]. In WDM, the temperatures are a noticeable fraction of the Fermi energy, and thus the generalization of DFT to thermal systems must be used. Such simulations are computationally demanding but they have the crucial feature of including realistic chemical structure, which is difficult to include with any other method while remaining computationally feasible. Moreover, they are in principle exact[@M65; @KS65], if the exact temperature-dependent exchange-correlation free energy could be used because of Mermin’s theorem establishing thermal DFT(thDFT). In practice, some standard ground-state approximation is usually used. (There are also quantum Monte Carlo calculations which are typically even more computationally expensive[@MD00; @FBEF01; @M09b; @SBFH11; @DM12; @SGVB15; @DGSM16]. The beauty of the QMD approach is that it can provide chemically realistic simulations at costs that make useful applications accessible[@MMPC12].) There have been many successes, such as simulation of Hugoniot curves measured by the $Z$ machine[@RMCH10] or a new phase diagram for high density water which resulted in improved predictions for the structure of Neptune[@MD06]. Because of these successes, QMD has rapidly become a standard technique in this field. [Missing temperature dependence:]{} However, the reliability and domain of applicability of QMD calculations are even less well understood than in GS simulations. At the equilibrium level of calculation, vital for equations of state under WDM conditions and the calculation of free-energy curves, a standard generalized gradient approximation (GGA) calculation using, e.g., PBE[@PBE96], is often (but not always) deemed sufficient, just as it is for many GS materials properties. Such a calculation ignores thermal exchange-correlation (XC) corrections, i.e., the changes in XC as the temperature increases, which are related to entropic effects. We believe we know these well for a uniform gas (although see the recent string of QMC papers[@SGVB15; @DGSM16] and parametrizations[@KSDT14]), but such corrections will be unbalanced if applied to a GGA such as PBE. So how big a problem is the neglect of such corrections? [(A little) beyond equilibrium:]{} On the other hand, many experimental probes of WDM extract response functions such as electrical or thermal conductivity[@MD06]. These are always calculated from the equilibrium KS orbitals, albeit at finite temperature. Work on molecular electronics shows that such evaluations suffer both from inaccuracies in the positions of KS orbitals due to deficiencies in XC approximations, and also require further XC corrections, even if the [exact]{} equilibrium XC functional were used[@TFSB05; @QVCL07; @KCBC08]. Acronym Meaning Acronym Meaning --------- ------------------------------ --------- ---------------------------- GGA Generalized Gradient Approx. RPA Random Phase Approximation GS ground-state TDDFT Time-dependent DFT HXC Hartree XC thDFT thermal DFT KS Kohn-Sham unif uniform gas LDA Local Density Approx. XC exchange-correlation PBE Perdew-Burke-Ernzerhof ZTA Zero-Temperature Approx. QMC quantum Monte Carlo : Acronyms frequently used in this chapter. \[acrodef\] ł[\^]{} Background ========== [Generalities:]{} Everything described within uses atomic units, is non-relativistic and does not include external magnetic fields. Unless otherwise noted, all results are for the electronic contributions within the Born-Oppenheimer approximation. While all results are stated for density functionals, in practice, they are always generalized to spin-density functionals in the usual way. Ground-state DFT ---------------- [Hohenberg-Kohn functional:]{} Just over 50 years ago, in 1964, Hohenberg and Kohn wrote down the foundations of modern DFT[@HK64]. They start with the many-body Hamiltonian = + + , where $\hat{T}$, $\hat{V}\ee$, and $\hat{V}$ are the kinetic, electron-electron, and potential energy operators, respectively. Assuming a non-degenerate ground-state, they proved by reductio ad absurdum that the external potential, $v(\br)$ is a unique functional of the density $\n(\br)$, and therefore all observables are also density functionals. More directly Levy defines the functional F\[\] = \_ \| + \| , \[Ffun\] where $\Psi$ is normalized and antisymmetric, and uses it to define the energy functional E\_v\[\] = F\[\] + v() (), whose minimization over normalized non-negative densities with finite kinetic energy yields the ground-state energy and density[@L81]. [Kohn-Sham scheme:]{} In 1965, Mermin generalized the Hohenberg-Kohn theorems for electrons in the grand canonical potential with fixed non-zero temperature $\tau$ and chemical potential $\mu$[@M65]. Later in 1965, Kohn and Sham created an exact method to construct the universal functional (see Eq. (\[FfunKS\])). The Kohn-Sham scheme imagines a system of $N$ non-interacting electrons that yield the electronic density of the original interacting $N$ electron system. These fictitious electrons sit in a new external potential called the KS potential. The KS scheme is written as a set of equations that must be solved self-consistently: {-\^2 + v() }\_i() = \_i \_i(), () = \_i\^N \|\_i()\|\^2, \[KSeq\] v() = v() + v() + v(), v() = , \[XCpot\] where $\phi_i(\br)$ and $\epsilon_i$ are the KS orbitals and energies, $v\H(\br)$ is the classical Hartree potential, and $v\xc(\br)$ is the exchange-correlation potential defined by the unknown XC energy, $E\xc$, in Eq. (\[XCpot\]). These must be solved self-consistently since the Hartree potential and $E\xc$ depend explicitly on the density. Lastly, the total energy can be found via F\[\] = T+ U+ E\[FfunKS\] where $T\s$ is the kinetic energy of the KS electrons and $U\H$ is the Hartree energy. In practice, an approximation to $E\xc$ must be supplied. There exists a wealth of approximations for $E\xc$[@MOB12]. The simplest, LDA, uses the XC per electron of the homogeneous electron gas[@PW92]: E= e(()) \[LDAdef\] where $e\xc\unif$ is the XC energy density of a uniform gas with density $\n(\br)$. We can imagine going up a ladder by adding in more ingredients (like gradients of the density[@PBE96]) and obeying different or additional conditions to make more complicated and more accurate functionals[@PRCS09]. For the exact $E\xc$, these equations have been proven to converge[@WSBW13]. Asymmetric Hubbard dimer and its relevance ------------------------------------------ Throughout this chapter we illustrate results with the simplest interesting model of an interacting system. This model is the asymmetric Hubbard dimer[@CFSB15]. The Hubbard dimer crosses the divide between the weakly and strongly correlated communities. Previous work has shown that the basic concepts of ground-state KS-DFT apply to this model and here we demonstrate similar applicability to thermal DFT. The Hamiltonian is given by $$\hat{H} = -t \sum_\sigma (\hat{c}_{1\sigma}^\dagger \hat{c}_{2\sigma} + H.c.) + \sum_{i=1}^2(U \hat{n}_{i\uparrow} \hat{n}_{i\downarrow} + v_i \hat{n}_i) \label{HubbHam}$$ where $\hat{c}_{i\sigma}^\dagger (\hat{c}_{i\sigma})$ are electron creation (annihilation) operators and $\hat{n}_{i\sigma} = \hat{c}_{i\sigma}^\dagger\hat{c}_{i\sigma}$ are number operators. $t$ is the strength of electron hopping between sites, $U$ is the ultra-short range Coulomb repulsion for when both electrons are on the same site, and $v_i$ is the on-site external potential. We choose, without loss of generality, $v_1 +v_2 =0$ then $\Delta v = v_2 - v_1$ and the occupation difference, the analog of density, is $\Delta n = n_2 - n_1$. ![ On-site occupations as a function of on-site potential difference for $U=1$ and 5 in the asymmetric Hubbard dimer. The HK theorem guarantees that each function is invertible. There is a simple analytic result for $U=0$, and for large $U$, the relation tends toward a (smoothed) step function, with the step at $\dv=U$. []{data-label="hubfigdnGS"}](chapfigdnGS.png){width=".8\textwidth"} In Fig. \[hubfigdnGS\] we plot the density $\dn$ versus asymmetry $\dv$ in the Hubbard dimer. The Hohenberg-Kohn theorem applies to this Hamiltonian, and guarantees $\dn(\dv)$ is an invertible function for any value of $U$. The main physics is a competition between asymmetry and interaction strength. The weakly correlated regime is $U < \Delta v$ and the opposite is strong correlation. Increasing $\dv$ pushes the electrons onto a single site, thus $\dn$ approaches 2. Likewise, for small $\dv$ or large $U$ the electrons are apart and $\dn$ tends to 0. This is made most clear by the extreme cases, i.e., \[dnU\] \|\_[U=0]{}()\| = 2/, \|\_[U]{}()\| 2(-U). The ability to vary $U$ and move continuously from weak to strong correlation in a model that is analytically solvable makes the Hubbard dimer an excellent illustrator of how KS-DFT works[@CFSB15]. Ensemble DFT as a route to excitation energies ---------------------------------------------- In this section we take a quick aside to overview ensemble DFT (eDFT), a close cousin of thermal DFT. [Excitations in DFT:]{} Although time-dependent DFT (TDDFT) is the standard method used to determine the excited states of a system [@M16], there are still many deficiencies, due to crude approximations to the XC functional as well as being unable to approximate multiple excitations, charge transfer excitations, canonical intersections, and polarizabilities of long-chain polymers; all things that can be important for photochemistry[@MZCB04; @DWH03; @T03; @LKQM06; @FBLB02]. Ensemble DFT is a time-independent alternative to the standard TDDFT that can be a useful method for extracting excited states. Naturally, since eDFT and TDDFT are based on two different fundamental theories, it is possible to use eDFT on different systems to those of the traditional method and expect different successes and likewise different failures. eDFT is based on a variational principle made up of ensembles of ground and excited states [@T79]. These ensembles are made of decreasing weights, with the ground state always having the highest weight. E\^ \_[k=0]{}\^[M-1]{} w\_[k]{} \_[k]{}\| \| \_[k]{}, w\_[0]{} w\_[1]{} ... w\_[k]{} 0 \[eDFT1eq\] where all $\Psi_k$ are normalized, antisymmetric, and mutually orthogonal, $\bw=(w_{0}, w_{1}, ... w_{k} )$, and the sum of all weights is 1. The ensemble-weighted density is \^() = \_[k=0]{}\^[M-1]{} w\_[k]{} \_[\_[k]{}]{}(). \[eDFT2\] Just as in the ground-state case, a one-to-one correspondence from the weighted density to the potential can be established [@GOK88], and applying this to a non-interacting system of the same weighted density can be used to construct a KS eDFT. From this KS system it is in principle possible to extract the exact excited states of the system. [Relation to thermal DFT:]{} The connection to thermal DFT is natural and straight forward. Thermal DFT is a special case of eDFT. In thDFT, one chooses the ensemble to be the grand canonical ensemble with the usual Boltzmann factors for the weights. However, unlike eDFT, the weights themselves depend on the eigenvalues of the Hamiltonian, including the strength of the interaction. Thus the weights in the KS system are different from those of the interacting system. In most applications of eDFT, the weights are chosen to be the same in both the physical and non-interacting systems. [History:]{} Ensemble DFT was originally proposed by Gross-Olivera-Kohn in 1988[@GOK88], but, like thermal DFT, there has been slow progress over the last 30 years due to a lack of useful approximations to the XC functional. Many of these difficulties arise from the so-called ghost interaction errors that occur in the Hartree energy[@GPG96]. More specifically these ghost-interaction errors appear when only using the ground state definition of the Hartree energy, which causes unphysical contributions and must therefore be accounted for by using a more accurate definition of the Hartree energy for ensembles[@PYTB14; @YTPB14]. [Recent progress:]{} More recently, work has been done to extract the weight dependence of the KS eigenvalues, which are required in order to extract accurate transition frequencies[@PYTB14; @YTPB14]. It was also found that a large cancellation of the weight-dependence occurs in the exact ensembles. Further, a new numerical method for inverting ensemble densities was derived for spherically symmetric systems, and this method was also tested for cylindrically symmetric systems. This inversion of densities to extract potentials provides a useful test of eDFT approximations. Recent work combines linear interpolation with an extrapolation method in eDFT to extract excited states that are independent of ensemble weights[@SHMK16]. Also, an exact analytical expression for the exchange energy was derived, and a generalized adiabatic connection for ensembles (GACE) was used to connect the ensemble XC functional with the ground state functional[@DMF16]. Finally, a ghost interaction correction has been developed for range-separated eDFT[@AKF16]. [A simple exercise:]{} Here we show an example of the importance of the weight-dependence of functionals in eDFT, in a seemingly simple system. We put only one electron in the Hubbard dimer, so there is no interaction, and its a simple tight-binding Hamiltonian. We will derive the exact kinetic energy functional, which is an example of the non-interacting KS kinetic energy functional. We will also approximate it, as if we were interested in orbital-free eDFT. There are only two levels, the ground state and a first excited state. Thus we can make only a bi-ensemble. The ensemble-weighted ground-state density is \[dnw\] \^[w]{}= (1-w) \_[0]{} +w \_[1]{} = (1-2 w) \_[0]{}(x), where $\dn_{0}$, and $\dn_{1}$ correspond to the occupational difference of the two sites for the ground and first-excited state, respectively, and $\dn_{0}(x)$ is given by Eq. (\[dnU\]) with $U=0$. This last result is true only because $\dn_{1}=-\dn_0$ in this simple model. The weight as previously stated is $w \leq 0.5$. Similarly, the kinetic energy for a single particle in the ground-state is known, and $T_{\sss S,1}=-T_{\sss S,0}$, so \[eq:Tswn\] T\^[w]{}= (1-w)T\_[S, 0]{} + w T\_[S,1]{}= (1-2w) T\_[S,0]{}= (1-2w) where $x=\dv/(2\,t)$. Using $V^w= \dv\dn^w/2$ and adding it to $T\s^w$ yields the ensemble energy, which is exactly linear in $w$, and passes through $E_0$ at $w=0$ and (would pass through) $E_1$ at $w=1$. This simple linearity with $w$ is true by construction of the ensemble, when energies are plotted against $w$ for a fixed potential. But now we show that things get complicated when we consider them as density functionals. Inverting the relation between potentials and densities we find \[xw\] x= and inserting this into the kinetic energy yields \[eq:Tswn2\] T\^[w]{}\[\]= / [2]{} Even in this trivial case, the the $w$-dependence of the kinetic energy density functional is non-linear. ![The exact energy curve for a single particle in a Hubbard dimer $E_{exact}= T\s^{w}[\dn^{w}] +V^{w}$ (red), when $x=1$, in units of $2\,t$. The blue line is the approximate energy when $T\s^w$ is replaced by its ground-state analog, $T\s^0$. Notice that $w=0$ corresponds to $E_0$, which is $1/{\sqrt{2}}$ in these units. []{data-label="ensemble2"}](ensemble2.png){width=".8\textwidth"} Next, we make the most naive approximation, namely to replace $T\s^w[\dn]$ with its ground-state counterpart. This yields an approximate eDFT theory from which, in principle, we can estimate the energy of the first excited state. To do this, we insert the exact $w$-dependent density of Eq. (\[dnw\]) into the ground-state functional, add the exact $V^w$, and plot the resulting energy. The exact and approximate results are shown in Fig. \[ensemble2\]. The approximation is very bad, yielding an excited state energy of about $-0.3$ instead of $0.71$, by using the value at $w=1/2$. But it at least illustrates the difficulties of capturing an accurate $w$-dependence in an ensemble density functional. Thermal DFT in a nutshell ------------------------- [Mermin-Kohn-Sham equations:]{} In a thermal system, Eq. (\[KSeq\]) and (\[XCpot\]) are generalized such that the density and XC potential become $\n^\tau(\br)$ and $v\xc^\tau(\br)$, i.e. temperature dependent, and $E\xc[\n]$ in Eq. (\[XCpot\]) becomes $A\xc^\tau[\n]$, the exchange-correlation free energy density functional. The density becomes \^() = \_i f\_i \|\^\_i()\|\^2 where the sum is now over all states and $f_i = (1 + e^{(\epsilon^\tau_i -\mu)/\tau})^{-1}$, the Fermi occupation factors. One of the core difficulties in thermal DFT calculations is this sum, since a huge number of states are required once the temperature is sufficiently high. This leads to large computational demands and convergence issues. We call these the Mermin-Kohn-Sham equations. To extract the total free energy from the MKS equations, we write A\^= A\^- U+ A\^- ()v\^(), where the MKS free energy is A\^= \_i \_i\^- S\^, S\^= - \_i, and $S\s$ is the MKS entropy. [Exchange-correlation free energy:]{} Compared to ground-state DFT, relatively few approximations have been developed for $A\xc\t$. In active use are two approximations: thermal LDA (thLDA) and the Zero-Temperature Approximation (ZTA). The former uses the temperature-dependent XC free energy of the uniform gas instead of its ground-state analog in Eq. (\[LDAdef\]). The ZTA means simply using any ground-state XC functional instead of a temperature-dependent one. We denote use of the exact ground-state XC functional as exact ZTA (EZTA). ![ Effects of temperature on the on-site occupations as a function of on-site potential difference for $U=1$ and 5 in the asymmetric Hubbard dimer (see Sec. \[simp\]) with and without temperature. Increasing temperature pushes the electrons towards opposite sites and lowers $\Delta n$ while increasing $\Delta v$ pushes electrons to the same site and raises $\Delta n$. []{data-label="hubfigdn"}](chapfigdn.png){width=".8\textwidth"} In Fig. \[hubfigdn\] we demonstrate the effects of turning on temperature for the Hubbard dimer (see Sec. \[simp\] for more information). Even a small increase in temperature can have a big impact on a system regardless of the strength of correlation. Some recent developments in thermal DFT ======================================= The formalism for thermal DFT was originally developed alongside that of ground-state DFT, but in the intervening decades ground-state DFT (and TDDFT, for that matter) has received significantly more attention and consequently more developments[@WGB05; @B12]. However, In the past couple decades thermal DFT has seen more use, and with that much more development in the past ten years. In this section we outline some recent developments to thermal DFT from our group. Exact conditions and their relevance ------------------------------------ [Zero temperature:]{} One of the most crucial steps in understanding and developing functionals beyond LDA is exact conditions. These conditions take many forms with some common examples being coordinate and interaction scaling conditions[@LP85], and bounds on the XC energy[@LO81]. There are well over a dozen conditions in ground-state DFT (a recent meta-GGA functional even uses seventeen[@SRP15]!), but the use of exact conditions is much more nascent in thermal DFT. ### Coordinate-temperature scaling and the thermal connection formula {#scale} [Uniform coordinate scaling:]{} The most straightforward application of exact conditions to thermal DFT is by uniform scaling of the density[@LP85]. The very basic conditions that this procedure generates in ground-state DFT are built in to almost all modern approximations. In a sense, this is simply dimensional analysis, but while keeping the density fixed (which is the tricky bit). Early work on exact conditions for thermal DFT[@PPFS11; @PPGB13] derived basic conditions such as the signs of correlation quantities, including the separation into kinetic and potential contributions, and the adiabatic connection formula at finite temperature. More conditions come from coordinate scaling of the density, showing that is intimately related to temperature dependence. Examples of a few of these conditions are F\^= \^2 F\^[/\^2]{}\[\_[1/]{}\], S\^= S\^[/\^2]{}\[\_[1/]{}\], A\^\[\] = A\^[/\^2]{}\[\_[1/]{}\], where $\n\g(\br)=\gamma^3\n(\gamma\br)$. For any of these functionals, this means that, if you know the functional at [any]{} one finite temperature, the functional at [all]{} possible temperatures is available via scaling of the density. [New formulas:]{} In recent work, many new formulas relating correlation components of the energy to one another were derived[@PB16], such as Kc\^[,]{}\[\] = Ac\^[,]{}\[\] - , where $K\c^{\tau,\lambda}[n] = T\c^{\tau,\lambda}[\n] - \tau S\c^{\tau,\lambda}[\n]$ is the correlation kentropy. There was also a rewriting of the adiabatic connection formula[@PPFS11; @LP75], using the relation to scaling mentioned above, yielding the XC free energy at temperature $\tau$: =\_ \_\^ U\^[’]{}\[n\], \_()=\^3 (), \[Axcth\] where $U\xc\t[\n]$ is the purely potential contribution to the XC free energy, and the scaling is the usual coordinate scaling of the density introduced by Levy and Perdew[@LP85] for the GS problem. Note that this thermal connection formula uses only information between the desired temperature and higher ones, allowing approximations that begin from the high-temperature end instead of the low-temperature end[@PB16]. A second set of formulas give the many relations among the different correlation energy components (total, potential, and kentropic). These are very important in ground-state DFT[@FTB00; @PEB96] for understanding the origins of different physical contributions to the correlation free energy and have guided the construction of many approximations. [Entropy:]{} Lastly for this section, we look at a new set of exact conditions for the electronic entropy as a functional of the density[@BSGP16]. The most important is that the universal functional can be written solely in terms of a temperature integral over entropy, such as F=F\^0\[\]-\_0\^d’ S, A=E-\_0\^d’ S, \[thent\] i.e., the universal contribution to the free energy functional is a simple integral over the electronic entropy, and the second shows that all thermal corrections to the XC free energy are given by an integral over the XC entropy. These formulas have no analog in ground-state DFT. They also lead to fundamental inequalities on the various thermal derivatives of both interacting and KS quantities. Such conditions have long been known for the uniform gas[@I82], but our results are their generalization to inhomogeneous systems. Analogs are also easily derived from statistical mechanics, but again, the tricky part is to deduce their behavior as functionals of the density rather than the external potential. This is why, for example, all derivatives are total with respect to temperature. The particle number is fixed by the density, so temperature is the sole remaining variable. Our work uses the formalism and methods of ground-state DFT, generalized to finite temperature, but the same results can also be extracted in the language of statistical mechanics[@DT11; @DT16]. [Tiny violations:]{} A minor illustration of the relevance of these conditions is that we found that a recent parameterization of the thermal XC free energy of the uniform gas[@KSDT14] violates one of our conditions for low densities[@BSGP16]. This violation is slight, and unlikely to ever influence the results of any thLDA calculation. Nonetheless, it is always better to build parameterizations that satisfy known conditions, so that the corresponding approximate calculations are guaranteed to satisfy such conditions[@SGVB15; @DGSM16]. [Zero-temperature approximations:]{} This work also showed that any ZTA calculation automatically satisfies most of our conditions, whereas the inclusion of thermal XC corrections risks violating them for specific systems. For example, all approximations in Fig. \[hubfig\] (discussed below) are guaranteed to satisfy these conditions. But practical calculations including approximate thermal XC corrections should be checked for possible violations of exact conditions in the future. Exact calculations on a simple model system {#simp} ------------------------------------------- [Importance for ground-state DFT:]{} A crucial step in all DFT development is the exact solution of simple systems and the test of approximate functionals against exact quantities. There are large databases of molecular properties, based either on highly reliable and accurate experimental measurements, or on far more accurate quantum chemical calculations[@SGB97; @ZKP98; @KPB99; @TS09]. But these databases usually contain at most a few numbers per system, such as the atomization energy and bond length(s). A substantially more sophisticated test occurs when a highly accurate calculation is performed such as QMC[@CKB79; @NU99] or DMRG[@W92], and an inversion of the KS equations[@SV09], so that essentially exact KS potentials, eigenvalues, orbitals, etc, can all be deduced. This is a much more powerful test of a DFT approximation, and usually provides detailed insight into its limitations. The QMC calculations of Umrigar and collaborators[@UG94; @FUT94; @HU97] and their high impact, testify to this fact. [Difficulty of exact thermal calculations:]{} Although almost all practical calculations of WDM are in a condensed phase (with hot ions), almost all high level inversions yielding exact KS quantities are for atoms or small molecules. But even for such systems, it is difficult to imagine accurate inversions at finite temperature, as only the average particle number is fixed, and all possible particle numbers must be considered. [Exact calculations for Hubbard dimer:]{} Exact calculations are only possible for this model because the Hilbert space is severely truncated which allows us to compute all energies analytically (see Fig. \[spectrum\] for complete diagram of the energy spectrum). However, this means the model is not even a qualitatively realistic representation of very high temperatures (though we choose parameters such that the ceiling of the Hilbert space does not effect results). But we are able to do the inversion exactly, and so extract all the different contributions to X and C as a function of both $\tau$ and $U$. These are the first exact inversions of an interacting system at finite temperature. They show us the structure of the underlying functionals, but cannot tell us which approximations will be accurate. For example, there is no real analog of LDA for this system (although BALDA[@LSOC03] somewhat plays this role). ![ Energy spectrum for the Fock space of the asymmetric Hubbard dimer at various parameters ranging from weakly to strongly correlated. The x-axis is labeled by the number of particles. The dashed lines correspond to the non-interacting (tight-binding) symmetric case. The labels in the figures denote degeneracy. The triplet for $N=2$ is always at $E=0$ due to symmetry. The top left is symmetric and weakly correlated, so the spectral lines are close to the dashed ones. The top right is symmetric but strongly correlated, and the energies for $N=2$ are substantially raised. We also see pairs of levels pushed together. The Hubbard bands of the infinite chain roughly run between these levels. In the lower left panel, we turn on asymmetry, and show that it lessens the effects of $U$ shown in the upper panels. []{data-label="spectrum"}](spectrum.png){width="100.00000%"} ![ Difference in on-site densities as a function of temperature for an asymmetric Hubbard dimer with $U=2$ and site-potential difference $2$ (in units where the hopping parameter $t=1/2$). The approximate calculations are all MKS-DFT-ZTA equilibrium calculations where HF denotes Hartree-Fock, $E\c{(2)}$ includes the leading correlation correction to HF in powers of the interaction, while EZTA denotes using the exact ground-state XC functional[@CFSB15]. []{data-label="hubfig"}](propfig3.png){width=".8\textwidth"} [Paradox:]{} To see why such simplistic calculations are important, consider the bottom panel of Fig. \[hubfig\]. The black line shows the exact density difference in the dimer versus temperature for moderate correlation and asymmetry. The blue curve is a Hartree-Fock calculation, while the red curve adds in the high-density limit of GS correlation. Finally, EZTA in green uses the exact GS functional (i.e. the best possible ZTA), which we had already found in Ref. [@CFSB15], in the MKS equations, which therefore is the best possible calculation that ignores thermal XC contributions. By construction, this becomes exact in the zero-temperature limit. But, to our surprise, we found that the relative error in the free energy and density vanishes in the [high]{} temperature limit. In fact, as temperature increases, the fractional errors at first increase, and then start to lessen. ![ Correlation energy components as a function of temperature. All inequalities proven in Ref. [@PPFS11] are satisfied. This figure shows that the naive assumption that $A\c\t$ is bounded by $E\c = A\c(\tau=0)$ is not true. Fig. 1 of Ref. [@SPB16] shows that the total free energy increases in magnitude as temperature increases, making the XC contribution relatively less important. Thus ZTA (or any reasonably bounded approximation) will yield relatively exact free energies, densities, and KS orbitals, in the limit of high temperatures. []{data-label="hubXC"}](chapfigXC.png){width=".8\textwidth"} [Resolution of paradox:]{} How can this counterintuitive result be right? The answer is very simple. For high temperatures, the XC components of the energy remain finite (and actually shrink, see Fig. \[hubXC\]), while the KS contributions grow, at least in this simple case. Thus [any]{} XC approximation will produce the same effect. This is why all the approximations merge onto the exact line in Fig. \[hubfig\] for sufficiently high temperatures. Of course, there can still be a significant absolute error in the free energy which might have important effects on quantities of interest. But the principle is clear: EZTA becomes relatively exact in both the low- and [high]{}-temperature limits. This is also trivially true for the uniform gas, once the (infinite) Hartree energy is included, and we suspect it to be true for all systems. [Relevance for response functions:]{} The increasing accuracy of the density with temperature has very important implications for calculations of conductivity. This means that the error in the KS orbitals, used in the construction of KS conductivities, starts to decrease beyond some temperature. This is true for [any]{} GS approximation for XC (within reason). All these conclusions may explain the tremendous success so far gotten by ignoring the thermal XC effects, especially for conductivities. Beyond equilibrium: Linear response thermal time-dependent DFT -------------------------------------------------------------- [Zero temperature:]{} There are many applications in WDM where the system is perturbed away from equilibrium. At zero temperature, the standard approach to such problems is to apply TDDFT or many-body non-equilibrium Green’s function methods[@SV13]. TDDFT in particular[@RG84] can handle both strong perturbations, such as atoms and molecules in intense laser fields, and weak perturbations, where the linear response formulation yields excitation energies and oscillator strengths[@BWG05]. [General case too difficult:]{} Unfortunately, the situation is very complicated if the perturbation is strong, as then a non-equilibrium treatment is needed. Theories in which the temperature is held fixed do not apply. This is the situation for example in calculations of stopping power[@GSK96]. There are many fine attempts to overcome these difficulties under a variety of practically useful conditions[@RTKC16], but we have not seen a way to construct a general DFT treatment of such problems. [Linear response:]{} For a finite system (which has to be very carefully defined in the thermal case), we proved a limited theorem for the linear density response to a time-dependent external field[@PGB16]. This proof allows for (finite numbers of) degeneracies in the excited states, but not in the equilibrium state. Armed with such a theorem, all the usual XC response properties, such as the XC kernel, can be defined at finite temperature. Combined with our thermal connection formula, we have the finite-temperature generalization of the Gross-Kohn response equation[@GK85]: (12)=(12) + d3d4 (13) f(34) (42), \[thGK\] where $1$ denotes the coordinates $\br,t$, and 2 another pair[@KBP94], $\chi\t(12)$ is the density-density response function at temperature $\tau$, $\chi\s$ its KS counterpart, and $f\Hxc\t(12)$ the thermal Hartree-XC kernel. This becomes the Random Phase Approximation (RPA) when $f\xc =0$. Insertion of this into the thermal connection formula yields an RPA-type equation for the XC free energy[@PGB16]: A= \_[”]{} \_\^[”]{} \_0\^ dd’ . \[AcFD\] If XC contributions to the kernel are neglected, this becomes the long-known random-phase approximation to the XC free energy, albeit using the KS orbitals. Since random-phase approximation calculations have become standard within the GS materials world[@SHSG10], there is little additional computational demand in performing them at moderate finite temperatures. Inclusion of any approximate treatment of the XC kernel yields an entirely novel approach to XC approximations for equilibrium thermal DFT. In particular, one can consider making a uniform approximation in both space and time, and also decide whether or not to include thermal corrections in an approximate kernel. All such treatments can be first tested on a uniform gas, for which the XC free energy is accurately known from QMC calculations[@DGSM16]. Recent applications of DFT in WDM {#apps} ================================= [Planetary science:]{} The materials of interest in WDM research span the periodic table. Accurate thermal conductivities for inertial confinement fusion fuel materials such as deuterium and tritium are needed to find the calculated total neutron yield in fusion science (the simulated mixing between the fuel and coating on ICF capsules is very sensitive to thermal conductivities)[@HCBK14]. Accurate iron thermal conductivities are used to determine whether the conventional model for how the Earth’s core developed is valid because heat flux contributes to the Earth’s geomagnetic field. Differential heating experiments at ALS, LCLS, Omega, and Titan facilities are all done slightly differently (heating via optical lasers, XFEL, x-rays, or proton heating), to fit hydrodynamics models because of their high accuracy and because other approaches (SESAME, Purgatorio, and Lee-More) all differ[@BJKR78; @LM84; @WSSI06]. Our work suggesting that ignoring XC thermal corrections nevertheless yields accurate KS eigenstates and eigenvalues[@SPB16] helps explain why conductivities can be accurate in these calculations. Much WDM research is motivated by the desire to understand planetary interiors. The Juno mission is measuring Jupiter’s gravitational field extremely accurately, constraining theories of its interior[@MGF16], while Kepler has shown that many notions of planetary formation must be rethought with our new data on extra-solar planets[@C16]. But there is limited understanding of whether initial planetary protocores remain stable during accretion or if they dissolve into outer metallic hydrogen layers. Recent DFT-MD calculations show that MgO is surprisingly soluble in hydrogen under these conditions[@WM12]. Similarly, the moon is thought to have formed in an enormous impact, but such a scenario depends crucially on the equation of state of MgO under extreme conditions. Recent DFT calculations and Z-machine experiments have nailed this EOS more accurately than before, and far better than unreliable extrapolations from more mundane conditions[@RSLD15]. [Alternate methods:]{} Path integral Monte Carlo is an excellent tool for studying WDM, and has been recently extended beyond small atoms to include water and carbon, and has recently been shown to match reasonably well with DFT calculations at lower temperatures[@DM12], validating both. Meanwhile, DFT calculations have predicted new superionic phases of H$_2$O, under conditions relevant to Uranus and Neptune interiors[@WWM13]. [DFT failure:]{} A less successful application of DFT in WDM is to the liquid insulator to liquid metal transition in dense D$_2$, at about 1000K and 300 GPa. DFT calculations with several different functionals yield very different results, none of which are in satisfactory agreement with experiment. The interpretation also depends on the accuracy of the conductivity from the DFT calculations. This system remains a challenge to WDM simulations. [X-ray Thompson scattering:]{} Some of the most exciting recent experiments have been from the LINAC at SLAC, allowing X-ray Thompson scattering (XRTS) measurements of shocked materials. These include the first highly resolved measurements of the plasmon spectrum in an ultrafast heated solid[@SGLC15]. Ref. [@DDRF16] gives x-ray scattering results from plasmons in dynamically compressed deuterium, from which one can deduce the ionization state as a function of compression. Ionization begins at about the pressure that DFT-MD calculations show molecular dissociation. In a completely different material, X-ray diffraction showed diamond formation on nanosecond timescales, caused by shock compression to about 200 GPa[@KRGG16]. This helps explain why the lonsdaleite crystal structure occurs naturally close to meteor impacts. XRTS has been performed on a variety of materials including Be, Li, C, CH shells, and Al. Most experiments probe the electron dynamic structure factor, which is decomposed via the somewhat ad-hoc Chihara decomposition into bound, loosely bound, and free electrons[@BSDH16]. But by running TDDFT at finite temperatures, one directly calculates the densities, and can then test the accuracy of Chihara for determining the ionization state. The results of Ref. [@PGB16] are already being used to justify thTDDFT calculations such as Ref. [@BSDH16]. Relation of thermal DFT to quantum chemistry ============================================ At first glance, it would appear that warm dense matter has little or nothing to do with chemistry. In fact, this is not true, it is simply chemistry in an exciting new regime with which we are relatively unfamiliar. To see this, we first note that the plasma physicists who usually study WDM think in terms of average properties of their systems, such as mean densities and numbers of electrons ionized. They are familiar with density functional methods, but traditionally only at the level of the LDA. Successes with such an approximation are often attributed to systems being somehow ‘locally uniform’. But the success of DFT methods in chemistry can be directly correlated with the arrival of the GGA and hybrids of it with Hartree-Fock. These approximations were tested on the G2 data set, and shown to yield much better energetics than LDA, because the G2 data set had already been carefully constructed and benchmarked, using both quantum chemical methods and experimental information[@CRRP97; @CRRP98; @PHMK05]. This vote of confidence led to their widespread adoption in many branches of chemistry, and also led to the confidence that GGAs were better than LDA for many materials problems. It is the same GGAs, used in MD simulations, that have led to the revolution in WDM simulations over the past decade or so[@LBKC00]. The improved accuracy due to GGAs implies that the details of the electronic structure matter, and that these systems are in no way locally uniform. In fact, in many cases, there are large evanescent regions of the HOMO, just as in gas-phase molecules. The KS system is ideal for computing this, and GGAs and hybrid account for the energetic consequences. So the very success of DFT-MD for WDM implies that the detailed chemistry is vital, even if it is happening within simulations of extended systems under high temperature and pressure. The recent work in our group is almost entirely focused on bringing GGA-level XC technology to the WDM field. Being able to distinguish among different components of the correlation energy, and switch from one to another, is a crucial part of the exact conditions that were used to construct GGAs[@PB16]. The adiabatic connection formula is often invoked in modern DFT research to understand both exact DFT and approximations, and its recasting as a temperature integral should prove useful in the search for accurate thermal XC approximations. On the other hand, ground-state DFT has benefited enormously from testing on benchmark data[@SGB97; @ES99]. But for thermal effects, even a simple H atom is difficult, as one must include sums over all possible particle numbers in the partition function. The asymmetric Hubbard dimer is the simplest imaginable exactly-solvable model, and can be considered a model for H$_2$ in a minimal basis. While the truncated Hilbert space makes it unrealistic at higher temperatures, it also makes it practical to solve exactly. Thus our calculations on this model demonstrate the behavior of correlation at finite temperatures in one simple case. Unfortunately, due to the lack of a continuum, this cannot be used to check the performance of LDA or GGA. Thermodynamics tells us simple relations between entropy and free energy and other quantities. But it requires very careful reasoning to deduce the corresponding relations among density functionals, as the density must be held fixed, not the external potential. Our relations between entropy and the universal part of the Mermin functional show this, and subtraction of the corresponding KS contributions yields crucial relations among correlation contributions. They also yield simple inequalities that are not automatically satisfied once thermal XC contributions are approximated. Lastly, the recent proof of TDDFT for finite temperatures within linear response justifies the extraction of conductivities within the Kubo response formalism from KS orbitals and energies. It also shows that the random-phase approximation, which is now routinely calculated for inhomogeneous systems in many codes in both quantum chemistry and materials science[@KF96; @F01; @F08; @EYF10; @EBF12; @F16], might be an excellent starting point for more accurate approximations to the XC thermal corrections, using approximations to the temperature-dependent XC kernel. Finally, our recent work explains how ignoring thermal XC effects, which is usually done in practical DFT WDM calculations, might not be as poor an approximation as it first appears. Calculations on the Hubbard dimer show that XC effects become relatively less important as the temperature increases. Thus the errors in the self-consistent density and orbitals caused by any approximation to the XC lessen with increasing temperature, so that calculations of the KS conductance should be more accurate as temperature increases (somewhat counterintuitively). To summarize, the success of modern density functional approximations in WDM simulations strongly implies the importance of chemical phenomena in such simulations, and the need to accurately approximate the energetics. Conclusion ========== Thermal density functional theory is an increasingly utilized tool for calculations of hot systems such as warm dense matter. These WDM systems include inertial confinement fusion, planetary interiors, and shock experiments. There have been many recent developments ranging from exact conditions, improved understanding, and extensions beyond equilibrium with more foreseeable, and exciting, improvements on the horizon. These steps forward set up the foundation for further future success of thermal DFT in the years to come. The authors acknowledge support from the US Department of Energy (DOE), Office of Science, Basic Energy Sciences under Award No. DE-FG02-08ER46496. J.C.S. acknowledges support through the NSF Graduate Research fellowship program under Award No. DGE-1321846.	ArXiv
--- abstract: 'We prove geometric superrigidity for actions of cocompact lattices in semisimple Lie groups of higher rank on infinite dimensional Riemannian manifolds of nonpositive curvature and finite telescopic dimension.' address: \| Einstein Institute of Mathematics\ Edmond J. Safra Campus, Givat Ram\ The Hebrew University of Jerusalem\ Jerusalem, 91904, Israel author: - Bruno Duchesne bibliography: - 'biblio.bib' title: \| Superrigidity in infinite dimension and finite rank\ via harmonic maps --- [^1] Introduction ============ Geometric Superrigity --------------------- In the nineteen seventies, Margulis proved his famous superrigidity theorem to show that irreducible lattices in higher rank semisimple Lie groups and algebraic groups are arithmetic. \[Margulis\]Let $G,H$ be semisimple algebraic groups over local fields without compact factors. Assume that the real rank of $G$ is at least 2 and let $\Gamma$ be an irreducible lattice of $G$. Any homomorphism $\Gamma\to H$ with unbounded and Zariski dense image extends to a homomorphism $G\to H$. Using the dictionary between semisimple algebraic groups over local fields and symmetric spaces of noncompact type (in the Archimedean case) and Euclidean buildings (in the non-Archimedean case), Theorem \[Margulis\] can be interpreted in a geometric way. This is the subject of the so-called geometric superrigidity, see [@MR2655318] for a survey in french or the older [@MR1168043]. Using this geometric interpretation, Corlette [@MR1147961] (in the Archimedean case) and later Gromov and Schoen [@MR1215595] extended Margulis superridigity theorem in the case where $G$ is a simple Lie group of rank 1 that is the isometry group of a quaternionic hyperbolic space or the isometry group of the Cayley hyperbolic plane. The main tool in these two former results are harmonic maps. Some time later, Mok, Siu and Yeung gave a very general statement [@MR1223224] of geometric superrigidity in the Archimedean case.\ The framework of geometric superrigidity was extended [@MR2219304; @MR2377496] to nonpositively curved metric spaces, which may be not locally compact, in the particular higher rank case where $\Gamma$ is a lattice in a product.\ In [@MR1253544 6.A], Gromov invited geometers to study some “cute and sexy" infinite dimensional symmetric spaces of nonpositive curvature and finite rank. The geometry of these spaces $X_p({\mathbb{R}})=$O$(p,\infty)/($O$(p)\times$O$(\infty))$ and their analogs, $X_p({\mathbb{K}})$ over the field ${\mathbb{K}}$ of complex or quaternionic numbers, were studied in [@Duchesne:2011fk]. Gromov also conjectured that actions of lattices in semisimple Lie groups on some $X_p({\mathbb{R}})$ should be subject to geometric superrigidity.\ In this article, a Riemannian manifold will be a connected smooth manifold modeled on a separable Hilbert space and endowed with a smooth Riemannian metric. In particular, such a manifold may have infinite dimension. See [@MR1330918] or [@MR1666820] for an accurate definition.\ The main result of this paper is the following theorem. \[theorem\]Let $\Gamma$ be an irreducible torsion free uniform lattice in a connected higher rank semisimple Lie group with finite center and no compact factor $G$. Let $N$ be a simply connected complete Riemannian manifold of nonpositive sectional curvature and finite telescopic dimension. If $\Gamma$ acts by isometries on $N$ without fixed point in $N\cup{\partial}N$ then there exists a $\Gamma$-equivariant isometric totally geodesic embedding of a product of irreducible factors of the symmetric space of $G$ in $N$. In the unpublished paper [@ks99], Korevaar and Schoen introduced the notion of FR-spaces (Finite Rank spaces). Later Caprace and Lytchak introduced the notion of spaces of finite telescopic dimension in [@MR2558883], without knowing [@ks99]. The two notions are the same for complete CAT(0) spaces and can be defined by an inequality at large scale inspired by Jung Inequality. For any bounded subset $Y\subset{\mathbb{R}}^n$, Jung proved that $$\mathrm{rad}(Y)\leq\sqrt{\frac{n}{2(n+1)}}\mathrm{diam}(Y),$$ see [@MR1456512] and references therein. A complete CAT(0) space $X$ has telescopic dimension less than $n$ if for any $\delta>0$ there exists $D>0$ such that for any bounded subset $Y\subseteq X$ of diameter larger than $D$, we have $$\mathrm{rad}(Y)\leq\left(\delta+\sqrt{\frac{n}{2(n+1)}}\right)\mathrm{diam}(Y).$$\ Theorem \[theorem\] applies to the particular case where $N$ is a symmetric space of noncompact type and finite rank (see [@sym] for the meaning of noncompact type in infinite dimension). The fact that symmetric spaces of noncompact type and finite rank have finite telescopic dimension was expected in [@ks99] and proved in Corollary 1.8 of [@sym]. Actually, it is proved that a symmetric space of noncompact type is a finite product of irreducible symmetric spaces of noncompact type and that irreducible factors of infinite dimension are some $X_p(\mathbb{K})$.\ This theorem implies that there is no geometrically Zariski-dense (see [@Monod:2012fk 5.B]) action of a uniform lattice as in Theorem \[theorem\], on a symmetric space of noncompact type, infinite dimension and finite rank. In rank 1, it was shown that the isometry group of the real hyperbolic space $\mathbf{H}^n$ has geometrically Zariski-dense actions on the infinite dimensional hyperbolic $\mathbf{H}^\infty$, see [@Monod:2012fk].\ The strategy to prove Theorem \[theorem\] goes as follows. We first use a result of existence of harmonic maps due to Korevaar and Schoen (see Theorem \[existence\]) for CAT(0) spaces targets. When the target is moreover a Riemannian manifold, the unique harmonic map is smooth, as it was proved for $\mathbf{H}^\infty$ in Proposition 7 of [@Delzant:2010fk]. We conclude that the harmonic map is totally geodesic, thanks to an argument from [@MR1223224]. \[complex\]Theorem \[theorem\] extends to the case where $G$ is the connected component of the isometry group of the quaternionic hyperbolic space or of the Cayley hyperbolic plane and $N$ has nonpositive complexified sectional curvature. This last condition is satisfied when $N$ is a symmetric space of noncompact type. A flat torus theorem for parabolic isometries --------------------------------------------- In the last section, we include an extension for parabolic isometries of the well-known flat torus theorem [@MR1744486 Chapter II.7]. This extension allows us to obtain a rigidity statement easily.\ Let $\gamma$ be an isometry of a CAT(0) space $X$. The translation length of $\gamma$ is the number $\|\gamma\|=\inf_{x\in X}d(\gamma x,x)$; $\gamma$ is said to be ballistic if $\|\gamma\|>0$ and neutral otherwise. Since the infimum in the definition of $\|\gamma\|$ may or may not be achieved, it is usual to distinguish between semisimple isometries, for which the infimum is a minimum, and parabolic isometries, for which the infimum is not a minimum. Let $\varphi\colon G\to$Isom$(X)$ be a homomorphism. We say that $G$ acts by ballistic isometries on $X$ if $\varphi(g)$ is a ballistic isometry for any $g\neq e$.\ A CAT(0) space $X$ is said to be $\pi$-visible if any points $\xi,\eta\in{\partial}X$ that satisfy $\angle(\xi,\eta)=\pi$ are extremities of a geodesic line. For example, Hilbert spaces, Euclidean buildings and symmetric spaces of noncompact type are $\pi$-visible. Let $H$ be a subset of Isom$(X)$, we denote by $\mathcal{Z}_{\mathrm{Isom}(X)}(H)$ the centralizer of $H$, that is the set of elements in Isom$(X)$ that commute with all elements in $H$. \[ftt\]Let $X$ be a complete $\pi$-visible CAT(0) space and let $A$ an abelian free group of rank $n$ acting by ballistic isometries on $X$. Then there exists a $A$-invariant closed convex subspace $Y\subseteq X$. The space $Y$ decomposes as $Z\times{\mathbb{R}}^n$, $\mathcal{Z}_{\textrm{Isom}(X)}(A)$ preserves this decomposition and the action $\mathcal{Z}_{\textrm{Isom}(X)}(A){\curvearrowright}Y$ is diagonal. Moreover, the action $A{\curvearrowright}{\mathbb{R}}^n$ is given by a lattice of ${\mathbb{R}}^n$ acting by translations and for any $a\in A$, the action of $a$ on $Z$ is neutral. \[rr\]Let $\Gamma$ be a lattice in a semi-simple Lie group of real rank $r$. If $\Gamma$ acts by ballistic isometries on a symmetric space of nonpositive curvature $X$ then $r\leq$rank$(X)$. The author thanks Pierre Pansu for suggesting this approach to superrigidity in infinite dimension and thanks Pierre Py for pleasant and useful discussions about regularity of harmonic maps in infinite dimension. Harmonic maps ============= In this section, we recall the standard notions of totally geodesic maps and harmonic maps between Riemannian manifolds (maybe of infinite dimension). We refer to [@MR1896863], among others, for an introduction to these notions in finite dimension.\ Let $(M,g)$ be a smooth Riemannian manifold with Levi-Civita connection $\nabla$. Let $u$ be a chart form an open subset $U\subset M$ to an open subset $V$ of a Hilbert space $\mathcal{H}$. The restriction to $U$ of any vector field $X\in\Gamma(TM)$ can be thought as a smooth map $V\to\mathcal{H}$ and thus we can consider the differential $DX$ of $X$ as a linear map from $\mathcal{H}$ to itself. The Christoffel symbol of $\nabla$, $\Gamma(u)$ with respect to $u$, is defined by the relation $$\nabla_YX=D_YX+\Gamma(u)(X,Y),$$ see [@MR1330918 1.5]. Let $f$ be a smooth map between Riemannian manifolds $(M,g)$ and $(N,h)$ with Levi-Civita connections $^M\nabla$ and $^N\nabla$. The vector bundle $f^{-1}TN$, which is the vector bundle over $M$ with fibers $T_{f(x)}N$ for $x\in N$, is endowed with the connection induced from $^N\nabla$, which we denote also by $^N\nabla$. In charts $(u,U),(v,V)$ of $M$ such that $f(U)\subset V$, this connection is given by the formula $$^N\nabla_XY=D_{dfX}Y+\Gamma(v)(dfX,Y)$$ for $X\in\Gamma(TM)$ and $Y\in\Gamma(f^{-1}TN)$. The vector bundle $TM^$ is also endowed with a connection $^M\nabla^$ induced from $^M\nabla$. For $\omega\in\Gamma(TM^)$ and $X,Y\in\Gamma(TM)$, $$^M\nabla^_X\omega(Y)=X\cdot\omega(Y)-\omega(\nabla_XY).$$ The vector bundle $f^{-1}TN\otimes TM$ is endowed with the connection $\nabla\colon\Gamma(TM^\otimes f^{-1}TN)\to\Gamma(TM^\otimes TM^\otimes f^{-1}TN)$ induced by $^M\nabla^$ and $^N\nabla$. This connection is defined by the formula $$\nabla_X(\omega\otimes V)= ^M\negthinspace\nabla^_X\omega\otimes V+\omega\otimes ^N\negthinspace\nabla_XV$$ for $X\in\Gamma(TM)$, $\omega\in\Gamma(TM^)$ and $V\in\Gamma(f^{-1}TN)$.\ The differential $df$ of a smooth map $f$ is a section of $TM^\otimes f^{-1}TN$ and $f$ is called totally geodesic* if $\nabla df=0$. One can think of this property in two equivalent ways. A map $f$ is totally geodesic if and only if it preserves the connections, that is $^N\nabla_XdfY=df(^M\nabla_XY)$ for $X,Y\in\Gamma(TM)$. And $f$ is totally geodesic if and only if it maps geodesics to geodesics.\ When $M$ is finite dimensional, there is a more general notion. Let $\tau(f)$ be the trace of $\nabla df$. It is a section of the vector bundle $TM^\otimes TM^\otimes f^{-1}TN$ defined by $$\tau(f)=\sum_{i}\nabla df(e_i,e_i)$$ for any orthonormal base $(e_i)$ of $T_xM$. The map $f$ is harmonic if $\tau(f)=0$. Harmonic maps are important because they are solutions of a variational problem. Let $\|\|\ \|\|$ be the norm associated to the Riemannian metric $g\otimes h$ on $TM^\otimes f^{-1}TN$. Actually for $x\in M$, $\|\|d_xf\|\|$ is the Hilbert-Schmidt norm of the linear map $d_xf\colon T_xN\to T_{f(x)}N$. This norm is well defined because $T_xM$ is finite dimensional. If $M$ is complete and has finite Riemannian volume, the energy* of $f$ is $$E(f)=\int_M\|\|df\|\|^2.$$ Harmonic maps are exactly critical points of the energy. There exists an equivariant version of this variational problem. Let $\Gamma$ the fundamental group of a compact Riemann manifold $M$ acting by isometries on $N$ and let $f$ be a $\Gamma$-equivariant map $f\colon \tilde{M}\to N$. Since $\|\|df\|\|^2$ is $\Gamma$-invariant, one can define the energy of $f$ by $E(f)=\int_M\|\|df\|\|^2.$ In the case where $N$ is finite dimensional and non positively curved, the existence of equivariant harmonic maps was considered in [@MR965220; @MR1049845]. Harmonic maps for metric spaces targets ======================================= In [@MR1266480; @MR1483983], Korevaar and Schoen developed a theory of harmonic maps with metric spaces targets (Jost developed also a similar theory, see [@MR1451625] or [@MR2829653 8.2]). We recall the definitions (not in full generality but in a framework convenient to our purpose) and refer to the original papers for details.\ Let $(\Omega,\mu)$ be a standard measure space with finite measure and let $(X,d)$ be a complete separable metric space with base point $x_0$. The space $L^p(\Omega,X)$ for $1\leq p\leq \infty$ is the space of measurable maps $u\colon \Omega\to X$ such that $\int_\Omega d\left(u(\omega),x_0\right)^pd\mu(\omega)$. This space is a complete metric space with distance satisfying $d(u,v)^p=\int_\Omega d\left(u(\omega),v(\omega)\right)^pd\mu(\omega)$ and if $(X,d)$ is CAT(0) then so is $L^2\left(\Omega,X\right)$.\ Let $\Gamma$ be the fundamental group of a compact Riemannian manifold $(M,g)$ and let $\rho\colon \Gamma\to$Isom$(X)$ be a representation of $\Gamma$. The group $\Gamma$ acts by deck transformations on the universal covering $\tilde{M}$ of $M$. We denote by $L^p_\rho(\tilde{M},X)$ the space of measurable equivariant map $u\colon\tilde{M}\to X$ such that the restriction of $u$ to a compact fundamental domain $K\subset \tilde{M}$ is in $L^p(K,X)$. For two maps $u,v\in L^p_\rho(\tilde{M},X)$, the function $x\mapsto d(u(x),v(x))$ is $\Gamma$-invariant and thus can be seen as a function on $M$. The distance on $ L^p_\rho(\tilde{M},X)$ is given by the relation $d(u,v)^p=\int_Md(u(x),v(x))^pd\mu(x)$ where $\mu$ is the measure associated to the Riemannian metric $g$.\ For $u\in L^p_\rho(\tilde{M},X)$ and $\varepsilon>0$ the $\varepsilon$-approximate energy at $x\in \tilde{M}$ is defined by $$e_\varepsilon(x)=\int_{S(x,\varepsilon)}\frac{d(u(x),u(y))^p}{\varepsilon^p}d\sigma(y)$$ where $S(x,{\varepsilon})$ is the $\varepsilon$-sphere around $x$ and $d\sigma$ is the measure induced by $g$ on $S(x,\varepsilon)$ divided by $\varepsilon^{(\mathrm{dim}(M)-1)}$. Now $u$ is said to have finite energy if $e_\varepsilon$ converges weakly to a density energy $e$, which is absolutely continuous with respect to $d\mu$ and has finite $L^1$-norm, when $\varepsilon$ goes to $0$. In this case, the energy of $u$ is $E(u)=\int_Me(x)d\mu(x)$. A minimizer of the energy functional is called a harmonic map. In [@MR1483983 Theorem 2.3.1], Korevaar and Schoen proved the existence of an equivariant harmonic map when the target is a CAT(-1) space under the assumption there is no fixed point at infinity. Actually, a Gromov-hyperbolic metric space, for example a CAT(-1) space, is nothing else than a metric space of telescopic dimension, or rank, at most 1 [@MR2558883 Introduction]. In the unpublished paper [@ks99], the analog in the higher rank (but finite !) case is proved. We include the original argument. \[existence\] Let $\Gamma$ be the fundamental group of a compact Riemannian manifold $M$ with universal covering $\tilde{M}$ and let $X$ be a complete CAT(0) space of finite telescopic dimension. If $\Gamma$ acts by isometries on $X$ without fixed point at infinity then there exists a unique equivariant harmonic map $f\colon\tilde{M}\to X$. Moreover, this harmonic map is Lipschitz. For $L>0$, let $\mathcal{C}_L$ be the set of $\Gamma$-equivariant maps from $\tilde{M}$ to $X$ that are $L$-Lipschitz and have finite energy. Thanks to Theorem 2.6.4 in [@MR1266480], we fix $L>0$ such that $\mathcal{C}_L$ is not empty. We claim that $\mathcal{C}_L$ is a closed convex subset of $L^2_\rho(\tilde{M},X)$. Let $u,v\in L^2_\rho(\tilde{M},X)$ and let $t\mapsto u_t$ be the geodesic segment with endpoints $u$ and $v$. If $u,v$ are $L$-Lipschitz, then the convexity of distance function on $X$ [@MR1744486 Proposition II.2.2] shows that $u_t$ is also convex for any $t$. Now, the L$^2$-convergence of a sequence with a common Lipschitz bound implies the uniform convergence of this sequence and since a uniform limit of a sequence of $L$-Lipschitz maps is also $L$-Lipschitz, we obtain that $\mathcal{C}_L$ is a closed convex subset of $L^2_\rho(\tilde{M},X)$.\ Let $x_0\in \tilde{M}$ and let $X'=\{x\in X\|\ u(x_0)=x,\ u\in\mathcal{C}_L\}$. The convexity of $\mathcal{C}_L$ implies that $X'$ is a convex subset of $X$. We want to show that for any $x\in X'$, there exists a unique map $u\in\mathcal{C}_L$ that minimizes the energy among maps in $\mathcal{C}_L$ such that $u(x_0)=x$. Let $u,v\in\mathcal{C}_L$ such that $u(x_0)=v(x_0)=x\in X_0$ then we have $$\label{ineq} \int_Md(u,v)^2\underset{(\mathrm{PI})}{\leq}C\int_M\|\|\nabla d(u,v)\|\|^2\underset{(\mathrm{CI})}{\leq}C\left[\frac{1}{2}\left(E(u)+E(v)\right)-E(m)\right]$$ where $C$ is some positive number and $m$ is the midpoint of the segment $[u,v]$. Actually, Inequality (PI) is a Poincaré inequality (Lemma \[poincaré\]) for the function $d(u,v)$, which is $2L$-Lipschitz and vanishes at $x_0$, and Inequality (CI) is [@MR1266480 Inequality (2.6ii)]. Inequality (\[ineq\]) shows that an energy minimizing sequence $(u_n)$ with $u_n(x_0)=x$ for any $n$ is Cauchy and thus an energy minimizing map in $\{u\in\mathcal{C}_l\ \|\ u(x_0)=x\}$ exists and is unique. Let us denote by $f_x$ this map.\ We define $I(x)$ to be $E(f_x)$. Now, we aim to show that $I\colon X'\to{\mathbb{R}}^+$ is a convex lower semicontinuous function. Assume this is the case, since $I$ is $\Gamma$-invariant and lower semicontinuous, its lower levelsets $X_r:=\{x\in X'\|\ I(x)\leq r\}$ are $\Gamma$-invariant closed convex subsets of $X$. Proposition 4.4 in [@Duchesne:2011fk] implies that the intersection $\cap_{r>\inf I}I_r$ is non empty otherwise the center of directions associated to $\{I_r\}_{r>\inf I}$ would be a $\Gamma$-fixed point at infinity. Since $\cap_{r>\inf I}I_r\neq\emptyset$, there is an energy minimizing $\Gamma$-equivariant map, which is unique thanks to Inequality (\[ineq\]).\ From the convexity of $E$, it is clear that $I$ is also convex. Let $r>\inf I$ let $x\in X$ be a limit point of a sequence $(x_n)$ in $X_r$. It suffices to show that $f_n:=f_{x_n}$ is a Cauchy sequence in $\mathcal{C}_L$ to obtain that $I_r$ is closed. Let $I_n=\inf_{X_0\cap B(x,1/n)} I$. We may assume that $x_n\in X_0\cap B(x,1/n)$ and $E(f_n)\leq I_n+1/n$. Now, Inequality (CI) in (\[ineq\]) implies that $$\int_M\|\nabla d(f_n,f_m)\|^2\underset{n,m\to \infty}{\longrightarrow}0$$ and Lemma \[poincaré\] applied to the function $d(f_n,f_m)-d(f_n(x_0),f_m(x_0))$ allows us conclude that $(f_n)$ is a Cauchy sequence. \[poincaré\]If $f\colon M\to{\mathbb{R}}$ if a $L$-Lipschitz function that vanishes at some point $x_0\in {M}$ then there exists $C>0$ which depends only on ${M}$ and $L$, such that $$\int_Mf^2\leq C\int_M \|\|\nabla f\|\|^2.$$ Let $R$ be the diameter of $M$. By an abuse of notation, we also denote by $f$ the function $f\circ\exp\colon T_{x_0}M\to{\mathbb{R}}$ and we denote by $\mu$ the pull-back, by the exponential map, of the measure associated to the Riemannian metric on $M$ and we denote by $dx$ the Lebesgue measure on $T_{x_0}M$. The measure $\mu$ is absolutely continuous with respect to $dx$ and there are positive numbers $c$ and $C_1$ such that the density $\upsilon$ of $\mu$ satisfies $c<\upsilon(x)<C_1$ for any $x$ in the $R$-ball around the origin in $T_{x_0}M$. We have $$\int_Mf^2\leq\int_{B(0,R)}f^2(x)d\mu(x)\leq C_1\int_{B(0,R)}f(x)^2dx.$$ Moreover, $$\begin{aligned} \int_{B(0,R)}f(x)^2dx&=\int_{B(0,R)}\int_0^{\|\|x\|\|}\left.\frac{d}{du}\right\|_{u=t}f(ux/\|\|x\|\|)^2\ dt\ dx\\ &\leq \int_{B(0,R)}\int_0^R2f(tx/\|\|x\|\|)\nabla_{\frac{tx}{\|\|x\|\|}}f\cdot\frac{x}{\|\|x\|\|}\ dt\ dx.\end{aligned}$$ Now, let $n$ be the dimension of $M$ and let $\sigma$ be the Lebesgue measure on $S^{n-1}$. Using polar coordinates, the fact that $\|\|\nabla f\|\|\leq L$ and Hölder inequality, we have for some $C_2>0$ $$\begin{aligned} \int_Mf^2&\leq 2C_1 L\int_0^R\int_{S^{n-1}}\int_0^Rt\ \|\|\nabla_{tv}f\|\|\ dt\ d\sigma(v)r^{n-1}dr\\ &\leq C_2\int_{S^{n-1}}\int_0^R \|\|\nabla_{tv}f\|\|^{n-1} t^{n-1}dt\ d\sigma(v)=C_2\int_{B(0,R)}\|\|\nabla_{x}f\|\|^{n-1}dx.\end{aligned}$$ Once again, using Hölder inequality and the fact that the exponential map is finite to one on $B(x_0,R)$, we have for some $C_3,C>0$, $$\int_Mf^2\leq C_3\int_{B(x_0,R)}\|\|\nabla_xf\|\|^2dx\leq C\int_M\|\|\nabla f\|\|^2.$$ Smoothness ========== It is a standard fact that the most difficult part to obtain smoothness of weak harmonic maps is the first regularity step, which is the continuity of the harmonic map (see for example [@MR2829653 8.4]). In our situation, we already know that the harmonic map is Lipschitz and we can easily adapt the argument given in [@Delzant:2010fk], where the target is the infinite dimensional hyperbolic space. \[smooth\]Let $\Gamma$ be the fundamental group of a compact Riemannian manifold $M$ with universal covering $\tilde{M}$ and let $N$ be a simply connected complete Riemannian manifold of nonpositive sectional curvature. If $\Gamma$ acts by isometries on $N$ and $f\colon M\to N$ is a $\Gamma$-equivariant harmonic map in the sense of Korevaar and Schoen then $f$ is a smooth harmonic map. We only sketch the proof with the slight modifications to adapt [@Delzant:2010fk Proposition 7]. We already know that $f$ is Lipschitz. Choose a point $x\in N$. Since $N$ is a simply connected Riemannian manifold of nonpositive curvature, the Cartan-Hadamard Theorem [@MR1666820 IX.3.8] implies that the exponential map at $x$ is a diffeomorphism from the tangent space $T_xN$ to $N$. This gives us a global chart and we can think of $N$ as a Hilbert space $(\mathcal{H},<\ ,\ >):=(T_xN,h_x)$ with a non constant Riemannian metric $h$. Moreover, since $N$ has nonpositive sectional curvature, for any $v\in\mathcal{H}$ and any point $y\in N$, $$\label{Rauch} h_y(v,v)\geq<v,v>,$$ see [@MR1666820 Theorem IX.3.6]. In this chart, the covariant derivative can be expressed by $$\nabla_YX=D_YX+\Gamma(\exp)(X,Y)$$ where $\Gamma(\exp)$ is the Christoffel symbol of this chart. Let $B$ be a ball in $\tilde{M}$ of radius less than the injectivity radius of $M$. This way, the projection $\tilde{M}\to M$ identifies $B$ with a ball in $M$. Consider $f$ as a map from $\tilde{M}$ to $\mathcal{H}$. Inequality (\[Rauch\]) shows that $f\|_B$, which has finite energy for the distance induced by $h$, has finite energy for the one induced by $<\ ,\ >$, too. Thus, $f\|_B$ is in the usual Sobolev space (for vector valued maps) $W^{1,2}(B,\mathcal{H})$. Since $f$ is harmonic, it satisfies the equation $$\Delta_hf+\sum_{i,j=1}^{\mathrm{dim}(M)}h^{ij}\Gamma(exp)\left(\frac{{\partial}f}{{\partial}x_i},\frac{{\partial}f}{{\partial}x_j}\right)=0$$ weakly. An induction on $k$ shows that $f\|_B$ is in $W^{k,p}(B,\mathcal{H})$ for any $k\in{\mathbb{N}}$ and $p>1$. This shows that $f$ is actually smooth. A vanishing theorem =================== Let $\tilde{M}$ be an irreducible symmetric space of noncompact type that is not the real or complex hyperbolic space and let $\Gamma$ be a uniform torsion free lattice of Isom$(\tilde{M})$. In order to prove a geometric statement of superrigidity in the cocompact Archimedean case, Mok, Siu and Yeung proved the existence [@MR1223224] of a 4-tensor $Q$ on $\tilde{M}$ that satisfies strong conditions. They also proved the following formula [@MR1223224 Theorem 3] for an equivariant map $f\colon \tilde{M}\to N$ where $N$ is a smooth Riemannian manifold of finite dimension $$\int_M\left<\left(Q\circ\sigma_{2\, 4}\right),\nabla df\otimes\nabla df\right>=1/2\int_M\left<Q,f^R^N\right>.$$ In this formula, $Q\circ\sigma_{2\, 4}(X,Y,Z,T)=Q(X,T,Z,Y)$ and the scalar products are those induced by the Riemannian metrics of $M$ and $N$ on $(T^M)^{\otimes4}\otimes(f^{-1}TN)^{\otimes2}$ and $(T^M)^{\otimes4}$. Actually, the proof of this formula goes through in the case where $N$ has infinite dimension, without modification. This formula, conditions satisfied by $Q$ and the harmonicity of $f$ imply that $\nabla df$ vanishes, that is $f$ is totally geodesic. Let $\tilde{M}$ be the symmetric space associated to $G$. Since $\Gamma$ is a torsion free uniform lattice, the quotient space $\Gamma\backslash \tilde{M}$ is a compact manifold. Since $\tilde{M}$ has no fixed point at infinity of $N$, there exists a equivariant harmonic map $f\colon \tilde{M}\to N$ by Theorem \[existence\]. Thanks to Proposition \[smooth\], we know that $f$ is a smooth equivariant harmonic map.\ Assume first that $G$ is simple, that is to say $\tilde{M}$ is irreducible. Now, Mok-Siu-Yeung above argument implies that $f$ is totally geodesic. Since $\Gamma$ does not fix a point in $N$, $f(N)$ is not reduced to a point. Now, since $M$ is irreducible, $f$ is an isometry up to rescaling the metric on $M$ (see for example [@MR0262984]).\ Now, if $\tilde{M}\simeq\tilde{M}_1\times\dots\times\tilde{M}_n$ with $n\geq2$ then thanks to a Bochner formula [@MR1223224 11], it is proved that the restriction of $f$ to any fiber $x_1\times\dots\times x_{i-1}\times \tilde{M_i}\times x_{i+1}\times \dots\times x_n$ is harmonic. The irreducibility of $\Gamma$ allows the authors of [@MR1223224] to prove that $f$ is actually totally geodesic and thus $f$ factorizes through $$\tilde{M}\overset{\pi}{\longrightarrow}\prod_{i\in I}\tilde{M_i}\overset{f'}{\longrightarrow}N$$ where $I$ is a non empty subset of $\{1,\dots,n\}$, $\pi$ is the canonic projection and $f'$ is an isometry (after renormalization of the metric on each factor $\tilde{M_i}$ for $i\in I$). We now explain Remark \[complex\]. Let $(N,h)$ be a Riemannian manifold with Riemann tensor $R$. Let $X,Y$ be vectors in the complexified tangent space $T_xN\otimes{\mathbb{C}}$ at $x\in N$. We also denote by $R$ and $h$ the linear extensions of the Riemann tensor and the metric to the complexification of $T_xN$. The complexified sectional curvature* between $X$ and $Y$ is $$\mathrm{Sec}_{\mathbb{C}}(X,Y)=\frac{R(X,Y,\overline{X},\overline{Y})}{\|\|X\wedge Y\|\|^2_{\mathbb{C}}}$$ where $\|\|\ \|\|_{\mathbb{C}}$ is the Hermitian norm on $\wedge^2(T_xN\otimes{\mathbb{C}})$ induced by $h$, which is the norm associated to the scalar product $$<X\wedge Y,Z\wedge T>_h=\det\left[\begin{array}{cc} h(X,Z)&h(X,T)\\ h(Y,Z)&h(Y,T) \end{array}\right]$$ on $\wedge^2T_xN$. The Riemannian manifold $N$ is said to have nonpositive complexified sectional curvature if Sec$_{\mathbb{C}}(X,Y)\leq0$ for any $X,Y\in T_xN\otimes{\mathbb{C}}$.\ The result of [@MR1223224], which is the existence of a tensor $Q$ that implies the vanishing of $\nabla df$ for a harmonic map $f$, is true when $N$ has nonpositive complexified sectional curvature and $G$ is the connected component of the isometry group of the quaternionic hyperbolic space or the Cayley hyperbolic plane. Thus Theorem \[theorem\] is also true in this case.\ Let $C$ be the curvature operator as introduced in [@sym 3.2]. We also denote by $C$ its ${\mathbb{C}}$-linear extension to $T_xN\otimes{\mathbb{C}}$. For $X,Y\in T_xN\otimes{\mathbb{C}}$, $$\mathrm{Sec}_{\mathbb{C}}(X,Y)=\frac{<C(X\wedge Y),\overline{X\wedge Y}>_h}{\|\|X\wedge Y\|\|^2_{\mathbb{C}}}.$$ In the case where $N$ is a symmetric space of noncompact type, $C$ is nonpositive and thus, the complexified sectional curvature is nonpositive. A flat torus theorem for parabolic isometries ============================================= We start with some preliminary results. \[lem\]Let $X$ be a $\pi$-visible complete CAT(0) space. If $Y\subseteq X$ is closed and convex then it is also $\pi$-visible. Let $\xi,\eta\in{\partial}Y$ such that $\angle(\xi,\eta)=\pi$. There exists a geodesic $c\colon{\mathbb{R}}\to X$ such that $c(\infty)=\xi$ and $c(-\infty)=\eta$. Let $x$ be the projection of $c(0)$ on $Y$. We define $c_+$ (respectively $c_-$) to be the geodesic ray from $x$ toward $\xi$ (respectively $\eta$). By definition of the boundary, $d(c(t),c_+(t))$ and $d(c(-t),c_-(t))$ are bounded for $t\geq0$. The real function $t\mapsto d(c(t),Y)$ is convex bounded and thus constant equal to some $d_0\geq0$. Now let $c'(t)$ be the projection of $c(t)$ on $Y$ for $t\in{\mathbb{R}}$. For $s,t\in{\mathbb{R}}$ and $x\in[c(t),c(s)]$, $d(x,[c'(t),c'(s)])=d_0$. By the same argument as above, for all $x\in[c'(t),c'(s)]$, $d(x,[c(t),c(s)])=d_0$ and we are in position to apply the Sandwich Lemma [@MR1744486 II.2.12.(2)], which shows that the convex hull of $c(t),c(s),c'(s),c'(t)$ is a Euclidean rectangle. In particular, $c'\colon{\mathbb{R}}\to Y$ is a geodesic with $c'(\infty)=\xi$ and $c'(-\infty)=\eta$. We recall that a ballistic isometry $\gamma$ of a complete CAT(0) space $X$ has two canonical fixed points at infinity, which we denote by $\omega_\gamma$ and $\omega_{\gamma^{-1}}$. They are limit points at infinity of $\gamma^nx$ and $\gamma^{-n}x$ for any $x\in X$ (see [@MR2574740 3.C], for example). \[prop\] Let $X$ be a $\pi$-visible complete CAT(0) space and let $\gamma$ be a ballistic isometry of $X$. Then there exists a closed convex subspace $Y\subseteq X$ that splits as $Z\times{\mathbb{R}}$. Moreover, $\mathcal{Z}_{\mathrm{Isom}(X)}(\gamma)$ preserves $Y$ and acts diagonally. In particular, $\gamma\|_Y$ acts as a translation of length $\|\gamma\|$ along the factor ${\mathbb{R}}$. Let $Y$ be the union of geodesics with endpoints $\omega_\gamma$ and $\omega_{\gamma^{-1}}$. Since $X$ is $\pi$-visible, $Y$ is nonempty and $\gamma$-invariant. Moreover, $Y$ is a closed subspace of $X$. Let $x\in X$ be a limit point of a sequence $(x_n)$ of points in $Y$. Let $c_n$ be the geodesic such that $c_n(0)=x_n$, $c_n(-\infty)=\omega_{\gamma^{-1}}$ and $c_n(\infty)=\omega_{\gamma}$. Thanks to [@MR1744486 Proposition II.9.22], $c_n$ converges to a geodesic $c$ such that $c(0)=x$, $c(-\infty)=\omega_{\gamma^{-1}}$ and $c(\infty)=\omega_{\gamma}$.\ Since $Y$ is closed, convex and $\gamma$-invariant, $\left\|\gamma\|_Y\right\|=\|\gamma\|$. The subspace $Y$ decomposes as a product $Y\simeq Z\times{\mathbb{R}}$ (see [@MR1744486 Theorem II.2.14]) and $\gamma$ preserves this decomposition. Thus $\gamma\|_Y$ can be written $\gamma_0\times\gamma_1$. A simple computation shows that $\|\gamma\|^2=\|\gamma_0\|^2+\|\gamma_1\|^2$. Assume for contradiction that $\|\gamma_0\|>0$ then there exists $\omega_{\gamma_0}\in{\partial}Z$ such that $\gamma_0^nx_0\to \omega_{\gamma_0}$ for any $x_0\in Z$. Thus, for any $x\in Y$, $\gamma^nx\to(\arccos(\|\gamma_0\|/\|\gamma_1\|),\omega_{\gamma_0},\omega_\gamma)$ in the spherical join ${\partial}Z\ast{\partial}{\mathbb{R}}={\partial}Y$. \[add\]Let $A$ be an abelian group acting by isometries on a CAT(0) space $X$. The set $N$ of neutral elements in $A$ is a subgroup of $A$. Moreover, if $A\simeq{\mathbb{R}}^n$ and the action is continuous then it is a linear subspace of $A$. We recall that $\|a\|=\lim_{n\to\infty}\frac{d(nax,x)}{n}$ for any $x\in X$. Thus $$\|a+b\|=\lim_n\frac{d(n(a+b)x,x)}{n} \leq\lim_n\frac{d(na(nbx),nax)+d(nax,x)}{n} \leq\|a\|+\|b\|.$$ Moreover, $\|a\|=\|-a\|$ and $\|na\|=n\|a\|$ for any $n\in{\mathbb{N}}$. Thus, if $A\simeq{\mathbb{R}}^n$, by continuity $\|\lambda a\|=\|\lambda\|\|a\|$ for any $\lambda\in{\mathbb{R}}$ and $a\in A$. We are ready to prove Theorem \[ftt\]. We prove this theorem by induction on $n$. For $n=1$, this is Proposition \[prop\]. Now suppose $n\geq2$ and choose a primitive element $a\in A$ and apply Proposition \[prop\]. We obtain an $\mathcal{Z}_{\mathrm{Isom}(X)}(a)$-invariant closed convex subspace $Y_1\simeq Z_1\times{\mathbb{R}}$ and $a$ acts as a translation of length $\|a\|$ on the ${\mathbb{R}}$-factor. The group $\mathcal{Z}_{\mathrm{Isom}(X)}(a)$ acts also diagonally on $Y_1$. Let $N$ be the subset of $A$ formed by elements $b=(b_1,b_2)$ of $A$ such that $\|b_1\|=0$. Lemma \[add\] shows $N$ is a subgroup of $A$. The subgroup $N$ acts properly on ${\mathbb{R}}$ and thus is cyclic. It contains $a$ and since $a$ is primitive in $A$, $N=a{\mathbb{Z}}$. Now let $B$ be a free abelian group of rank $n-1$ such that $A=N\oplus B$. Observe that $B$ acts by ballistic isometries on $Z_1$. We can now apply an induction for $B{\curvearrowright}Z_1$ and we obtain $Y_2\simeq Z\times{\mathbb{R}}^{n-1}\subseteq Z_1$. By induction $Y_2$ is $\mathcal{Z}_{\mathrm{Isom}(Z_1)}(B)$-invariant and $\mathcal{Z}_{\mathrm{Isom}(Z_1)}(B)$ preserves this decomposition. Moreover, for any $\gamma\in\mathcal{Z}_{\mathrm{Isom}(X)}(A)$, $\gamma_{Z_1}\in\mathcal{Z}_{\mathrm{Isom}(Z_1)}(B)$ and thus $\gamma_{Z_1}$ preserves $Y_2$ and acts diagonally on it. In particular, $a_{Z_1}$ (which is neutral) has a trivial part on ${\mathbb{R}}^{n-1}$. Now if we set $Y=Y_2\times{\mathbb{R}}\simeq Z\times{\mathbb{R}}^{n}\subseteq X$, $Y$ has the desired properties. Thanks to [@MR0302822 Corollary 2.9], $\Gamma$ contains an Abelian free group of rank $r$. Since this Abelian free group acts also by ballistic isometries, it suffices to apply Theorem \[ftt\] to find a Euclidean subspace of $X_p(\mathbb{K})$ of dimension $r$ and thus $p\geq r$. [^1]: The author is supported by a postdoctoral fellowship of the Swiss National Science Foundation.	ArXiv
--- abstract: 'We derive the braid relations of the charged anyons interacting with a magnetic field on Riemann surfaces. The braid relations are used to calculate the quasiparticle’s spin in the fractional quantum Hall states on Riemann surfaces. The quasiparticle’s spin is found to be topological independent and satisfies physical restrictions.' address: 'International School for Advanced Studies, SISSA, I-34014 Trieste, Italy ' author: - Dingping Li title: 'Intrinsic quasiparticle’s spin and fractional quantum Hall effect on Riemann surfaces' --- The possibility of fractional statistics on two dimensional surfaces was discovered in Ref.[@lei]. When two fractional-statistics particles (anyons) are interchanged, the wave function changes by a phase $\exp(i\theta)$, where $\theta$ is neither $\theta =0$ (Bose statistics) nor $\theta =\pi$ (Fermion statistics). The quasiparticles in fractional quantum Hall systems (for a review on the fractional quantum Hall effect (FQHE), see Ref.[@girvin]) are anyons [@arovas; @halp] and this picture had been used to construct the hierarchical wave function in the FQHE [@halp]. Anyons may also find applications in some condensed matter systems [@sup]. On higher dimension spaces ($D>2$), only Fermion and Boson exist and the Fermion’s spin is an half-integer, the Boson’s spin is an integer. It will be very interesting to know what is the spin of anyons and the spin-statistics relation of anyons. The spin-statistics relation of anyons is generalized one if the spin $s$ equals to $s={\case\theta/2\pi}$. Anyons in various models, for example, in non-linear sigma models, Chern-Simons field theories and relativistic quantum field theories on $2D$ dimension spaces indeed have the generalized spin-statistics relation [@rdt; @fro]. Naturally, we will ask the question what is the spin-statistics relation of the quasiparticle in the FQHE. The recent discussion about the quasiparticle’s spin (QPS) can be found in Refs.[@lidp; @wen; @kiv]. Reference.[@lidp] calculated the QPS by analyzing the hierarchical wave function or by calculating the Berry phase of the quasiparticles moving in a closed path on the sphere. Reference.[@wen] obtained the QPS by analyzing the Ginzburg-Laudau-Chern-Simons (GLCS) theory of the FQHE on the sphere. On the other hand, Reference.[@kiv] calculated the QPS based on the GLCS theory on the disc geometry. However, the results in Refs.[@lidp; @wen; @kiv] are different from each other. The ambiguity of the QPS in the literature is due to lack of a good definition of the QPS. In this paper, we will calculate the QPS by using braid relations of anyons on Riemann surfaces [@wu; @thou; @ein; @wend; @eina; @imbo; @bla]. We will show that the QPS calculated in the following is consistent with physical restrictions and is intrinsic in the sense of its topological independence. The results of the present paper is agreed with the results of Ref.[@lidp]. Let us consider $N$ spinning anyons on an oriented compact Riemann surface with $g$ handles[@eina; @imbo; @bla]. To define the anyon’s spin, we attach an oriented local frame to every particle. When a particle moves on a curved surface (the torus is a flat surface), the attached frame is parallel transported and a path-dependent frame-rotation is associated with the particle transport. Let us denote the clockwise $2\pi$ rotation of the frame attached to the particle by $R_{2\pi}$. The action of the operator $R_{2\pi}$ on the wave function will give a phase $\exp (i2\pi S)$. We define $S$ as the spin of the particle. $S$ is equal to $1/2$ for the electron by this definition. The braiding operators are $\sigma_i ,\, \rho_{n,i} ,\, \tau_{n,i}$, where $\sigma_i$ interchanges (clockwise) particle $i$ and particle $i+1$ and $\rho_{n,i} ,\, \tau_{n,i}$ take particle $i$ around noncontractable loops on the $m^{,}th$ handle. Here we use the same definition of operators $\rho_{n,i} ,\, \tau_{n,i}$ as Ref. [@eina]. The braid relations for spinning anyons on the Riemann surface with $g$ handles are \[brarel\] $$\sigma_j\sigma_{j+1}\sigma_j =\sigma_{j+1}\sigma_j\sigma_{j+1} , \label{brarel:1}$$ $$\tau_{m,j+1}= \sigma^{-1}_j \tau_{m,j}\sigma_j , \rho_{m,j+1}=\sigma^{-1}_j\rho_{m,j}\sigma_j , \label{brarel:2}$$ $$\rho^{-1}_{m,j}\tau_{m,j+1}\sigma^{-2}_j\rho_{m,j} \sigma^2_j\tau^{-1}_{m,j+1}=\sigma^2_j , \label{brarel:3}$$ $$\sigma_1 \sigma_2 \cdots \sigma^2_{N-1}\cdots \sigma_2 \sigma_1=R^{2(g-1)}_{2\pi} \prod^g_n\rho^{-1}_{n,1}\tau^{-1}_{n,1}\rho_{n,1}\tau_{n,1}. \label{relation}$$ For the spinless anyons on the sphere, Equation (\[relation\]) becomes $\sigma_1 \sigma_2 \cdots \sigma^2_{N-1}\cdots \sigma_2 \sigma_1=1 $, which had been derived in Ref. [@thou]. It expresses the fact that a closed (clockwise) loop of particle $1$ around all the other particles can be continuously shrunk to a point on the rear side of the sphere. Eq. (\[relation\]) is the generalization of the case of the spinless anyons on the sphere to the spinning anyons on general Riemann surfaces[@eina; @imbo]. When we deform the loop of the left side of Eq. (\[relation\]) to the loop of the right side of Eq. (\[relation\]), described by $\prod^g_n\rho^{-1}_{n,1}\tau^{-1}_{n,1}\rho_{n,1}\tau_{n,1}$, the attached spin frame of is rotated ($4\pi (g-1)$ rotation), we obtain a phase $R^{2(g-1)}_{2\pi}$. We need to include this phase in the right side of the equation. However for the charged anyons in magnetic field, which is the case of quasiparticles in FQHE, the braid relation (\[relation\]) should be changed. We also need to include the Aharonov-Bohm phase $\exp (2\pi iq\Phi )$ in the right of Eq. (\[relation\]) because the charged anyon interacts with the magnetic field, where $q$ is the anyon’s charge and $\Phi$ is the magnetic flux out of the surface. Thus in stead of Eq. (\[relation\]), for the charged anyons on the magnetic field, we have $$\begin{aligned} \sigma_1 \sigma_2 \cdots \sigma^2_{N-1}\cdots \sigma_2 \sigma_1 & = & \exp (2\pi iq\Phi ) R^{2(g-1)}_{2\pi} \nonumber \\ & \times & \prod^g_n\rho^{-1}_{n,1} \tau^{-1}_{n,1}\rho_{n,1}\tau_{n,1} . \label{relation1}\end{aligned}$$ We will only consider the Abelian fractional statistics, which means that the representation of operator $\sigma_i$ is given by $\sigma_i =\sigma =\exp{i\theta}{\bf 1}_M$, where ${\bf 1}_M$ is the $M \times M$ identity matrix. Inserting $\sigma_i =\sigma =\exp{i\theta}{\bf 1}_M$ in Eq. (\[brarel:2\]) and Eq. (\[brarel:3\]), one obtains that $\tau_{m,j}=\tau_m , \, \rho_{m,j}=\rho_m ,\, \tau_m \rho_m =\sigma^2\rho_m \tau_m $. These relations and Eq. (\[relation1\]) yield $$\begin{aligned} \exp [2i(N-1)\theta] & = & \exp [2\pi iq\Phi +4\pi(g-1)S] \nonumber \\ & \times & \exp (-2ig\theta) . \label{brarel3}\end{aligned}$$ If there are several kinds of anyons, we need to introduce mutual statistics[@mutual]. The mutual statistics $\theta_{i,j}$ means that when a particle of the $i^{,}{th}$ kind moves clockwise around a particle of the $j^{,}{th}$ kind, we get a phase $\exp (2\theta_{i,j})$. $\theta_{i,i}=\theta_i$ actually is the fractional statistics parameter of the particle of the $i^{,}{th}$ kind. The left hand of Eq. (\[brarel3\]) is a phase which is obtained by moving one particle (clockwise) around all other particles. If there exist other kinds of particles, instead of Eq. (\[brarel3\]), we have $$\begin{aligned} \exp [2i(N_i-1)\theta_i & + & 2i\sum_{j\not= i}^l N_j\theta_{i,j}] =\exp (-2ig\theta_i) \nonumber \\ & \times & \exp [2\pi iq_i\Phi +4\pi(g-1)S_i], \label{mainre}\end{aligned}$$ where there are $l$ different kinds of particles and the spin of the particle of the $i^{,}{th}$ kind is $S_i$ and the charge is $q_i$. Eq. (\[mainre\]) gives a constraint on the parameters (numbers, statistics and spin ) of anyons that $$[((N_i-1+g)\theta_i + \sum_{j\not= i}^l N_j\theta_{i,j}) / {\pi} ] -q_i\Phi - 2(g-1)S_i \label{const}$$ is an integer. Now we consider the FQHE on a surface with $g$ handles. We use the metric $ds^2=g_{z\bar z}dzd{\bar z}$ in complex coordinates. The volume form is $dv=[ig_{z\bar z} / 2]dz\wedge d{\bar z} = g_{z\bar z}dx\wedge dy$. As in the case of the FQHE on the disc, sphere and torus, we apply a constant magnetic field on the surface. The natural generalization of the constant magnetic field to high genus Riemann surfaces[@bolte] is $F=Bdv=(\partial_zA_{\bar z}- \partial_{\bar z}A_z)dz\wedge d{\bar z}$. Thus $ \partial_zA_{\bar z}- \partial_{\bar z}A_z =ig_{z\bar z}B / 2$. The flux $\Phi$ is given by $2\pi \Phi = \int F =BV$, where $V$ is the area of the surface and we assume here $B>0$ ($\Phi >0$). The Hamiltonian of an electron on the surface under the magnetic field is given by the Laplace-Beltrami operator, $$\begin{aligned} H & = & [1/ 2m \sqrt{g}] (P_{\mu}-A_{\mu})g^{\mu \nu}\sqrt{g}(P_{\nu}-A_{\nu}) \nonumber \\ & = & [ g^{z\bar z} / m] [(P_z-A_z)(P_{\bar z}-A_{\bar z}) \nonumber \\ & + & (P_{\bar z}-A_{\bar z})(P_z-A_z)] \\ \label{hamil} & = & [2g^{z\bar z}/ m] (P_z-A_z)(P_{\bar z}-A_{\bar z})+[B/ 2m] \nonumber\end{aligned}$$ where $g^{z\bar z}=[1 / g_{z\bar z}]$ and $P_{z}=-i\partial_z ,\, P_{\bar z}=-i\partial_{\bar z}$. The inner product of two wave functions is defined as $<\psi_1 \| \psi_2 >=\int dv {\bar \psi_1 }\times \psi_2$. $H^{\prime}=[2g^{z\bar z} / m] (P_z-A_z)(P_{\bar z}-A_{\bar z})$ is a positive definite hermitian operator Because $<\psi \| H^{\prime} \|\psi > \, \geq 0$ for any $\psi$. Thus if $H^{\prime} \psi =0$, $\psi$ satisfies $(P_{\bar z}-A_{\bar z})\psi =0$. The solutions of this equation are the ground states of the Hamiltonian $H$ or $H^{\prime}$, or the lowest Landau levels (LLL). The existence of the solutions of this equation is guaranteed by Riemann-Roch theorem[@griff]. The solutions belong to the holomorphic line bundle under the gauge field. Riemann-Roch theorem tells us that $h^0(L)-h^1(L)=deg(L)-g+1$, where $h^0(L)$ is the dimension of the holomorphic line bundle or the degeneracy of the ground states of the Hamiltonian $H$, $h^1(L)$ is the dimension of the holomorphic differential and $deg(L)$ is the degree of the line bundle which is equal to the first Chern number of the gauge field, or the magnetic flux out of the surface, $\Phi$. Because $deg(L) >2g-2$ (the magnetic field is very strong in the FQHE), $h^1(L)$ is equal to zero [@griff] and $h^0(L)=\Phi-g+1$. As a consistent check, $h^0(L)$ indeed gives the right degeneracy of the ground states in the case of a particle on the sphere and torus interacting with a magnetic-monopole field. In the case of high genus surfaces, Ref. [@avron] had discussed the degeneracy of the LLL for the leaky tori (see also Ref. [@asory] for the related discussions). If the filling factor is $1 / m$ in the state of the FQHE [@laughlin], we suppose to have relation $m(N+\Delta_g)=\Phi$. We assume that $\Delta_g$ is independent of $m$, and this assumption will be justified by the explicit constructions of some examples of Laughlin wave functions on high genus surfaces [@iengoli]. If $m=1$, we have an integer quantum Hall state and expect that the LLL are completely filled. In this case, $N=\Phi-\Delta_g$, and $N$ is also the degeneracy of the LLL, which should be equal to $\Phi-g+1$ (according to the discussions above). Thus $\Delta_g$ is equal to $g-1$. The relation between the electron numbers and the flux for the FQHE at the filling factor as $1 / m$ is then $m(N+g-1)=\Phi$, which indeed gives correct results in the case of the Laughlin state on the sphere and torus [@haldane]. If $N_q$ quasiparticles are created, one has $m(N+\Delta_g)+N_q=\Phi$. The mutual statistics between the electron and quasiparticle is $2\theta_{mut}=2\pi$, the charge of the quasiparticle is $1 / m$ and the statistics parameters is $\pi / m$. We remark that, if $2\theta_{i,j}=2\pi, i\not= j$ in Eq. (\[const\]) or Eq. (\[mainre\]), we can simply omit the term $N_j\theta_{i,j}$. By applying Eq. (\[const\]) to those $N_q$ quasiparticles, one can show that $$[(N_q-1+g)/ m]+N- [\Phi / m] -2(g-1)S_q \label{spin1}$$ is an integer. From Eq. (\[spin1\]) and the equation $m(N+\Delta_g)+N_q=\Phi$, the QPS turns out to be $S_q=[1 / 2m]+[n / 2(g-1)]$ ($n$ is an integer and $g \not= 1$ is assumed). Let us discuss how to fix $n$. We consider a cluster of particles which contains $n_i$ particles of the $i^{,}{th}$ kind with the mutual statistics $\theta_{i,j}$. By using the method developed in Ref. [@thou], we get the statistics, spin and charge of the cluster, $$S_c = \sum_i [n_i(n_i-1)\theta_i / 2\pi ] +n_iS_i + \sum_{i \not= j} n_in_j\theta_{i,j} / 2\pi \label{culster:2}$$ and $q_c=\sum n_iq_i$ and $\theta_c=\sum_i n_i^2\theta_i+ \sum_{i \not= j}n_in_j\theta_{i,j}$ are the charge and statistics of the cluster respectively. If the cluster’s charge is an odd (even) integer and the cluster satisfies the fermionic (bosonic) statistics, we suppose that the cluster contains only an odd (even) number of electrons (for example, see Ref. [@frozee]), and thus the cluster’s spin is an half-integer (integer). If the cluster contains $m$ quasiparticles in the above example, the cluster’s charge is $1$ and its statistics of is fermionic. This cluster shall be the hole of the electron and the cluster’s spin is an half-integer (see also Ref. [@fro]). By using Eq. (\[culster:2\]) for this cluster, we get a restriction for the QPS, $[mn / 2(g-1)]=integer$. If $m$ and $g-1$ are coprime to each other (for example, $g$ is equal to $0,2,3$), $n$ must be equal to $2n^{\prime}(g-1)$ ($n^{\prime}$ is an integer). Thus $S_q$ is equal to $1 / 2m$ (up to an integer) and the spin-statistics relation is the standard one. However, when $m$ and $g-1$ are not coprime to each other for some high genus surfaces, there exist other solutions for the spin except $1 / 2m$. We write $m=kp$ and $g-1=kq$, where $p$ and $q$ are coprime to each other, we have solutions $n=2n^{\prime}q$. The other solutions of the QSP is $S_q=[1 / 2m]+[n^{\prime}/ k]$ where $n^{\prime}=1,\cdots , k-1$. Therefor $S_q$ can not be completely fixed by using the braid group analysis. However, we shall point out that it is highly unlikely that those [other]{} solutions are the [true]{} QPS, as we expect that the value of the spin shall be intrinsic and does not depend on the surface where quasiparticles live. To completely fix the QPS on Riemann surfaces, we can obtain the QPS by analyzing the wave function of the quasiparticles on Riemann surfaces (we plan to do so elsewhere), as it was done for the case on the sphere[@lidp]. Let us calculate the QPS in the standard hierarchical state[@halp; @lidp; @haldane; @blok; @read; @gread; @lidpt]. We remark that the above method can be used to calculate the QPS in other kinds of quantum Hall fluids, for example, the multi-layered FQHE or Jain state[@jain] etc.. The hierarchical state is described by a symmetric matrix $\Lambda_{i,j}, i,j=1,2,\cdots , l$, where $\Lambda_{i,i+1}=\Lambda_{i+1,i}=\pm 1$, $\Lambda_{1,1}$ is an odd integer and $\Lambda_{i,i}, i\not= 1$ are even integers, where $l$ is the level of the hierarchical state and $N_i$ is the number of the particles in level $i$ ($N_1$ is the number of the electrons, $N_2$ is the number of the condensed quasiparticles (or holes) of the first level (Laughlin) state, etc.). On the torus[@haldane; @lidpt], we have a relation, $\sum_j \Lambda_{i,j}N_j=\delta_{i,1}\Phi$, and on the sphere[@lidp; @haldane], the relation is $\sum_j \Lambda_{i,j}N_j -\Lambda_{i,i}=\delta_{i,1}\Phi$. Following the discussion about the Laughlin state on Riemann surfaces, we expect that the relation is $\sum_j \Lambda_{i,j}N_j + (g-1) \Lambda_{i,i}= \delta_{i,1}\Phi$ for the hierarchical state on Riemann surfaces. We define a $l$ dimension integer lattice with bases $E_i$ and the inner products $E_i\cdot E_j =\Lambda_{i,j}$ (see Ref. [@lidpt]) The above equation can be rewritten as $$\sum_{i=1}^{l} N_iE_i + (g-1) (E_i\cdot E_i)E_i^{\star}=E_1^{\star}\Phi , \label{hierar:1}$$ where $E_i^{\star}$ are the bases of the inverse lattice $E_i$ and defined by $E_i^{\star}\cdot E_j =\delta_{i,j}$. It can be verified that $E_i^{\star}\cdot E_j^{\star} =\Lambda_{i,j}^{-1}$. The quasiparticle is described by a vector ${\cal Q}_k =k_iE_i^{\star}$ ($k_i$ is an integer) on the lattice $E_i^{\star}$ (see Refs. [@blok; @read]). The statistics parameter of this quasiparticle is $\theta_k = {\cal Q}_k\cdot {\cal Q}_k \pi=\sum_{i,j}k_i\Lambda_{i,j}^{-1}k_j$ and the charge is $Q_k={\cal Q}_k \cdot E_1^{\star}=\sum_ik_i \Lambda_{i,1}^{-1}$. The mutual statistics between the quasiparticles ${\cal Q}_{k}$ and ${\cal Q}_{k^{\prime}}$ is $\theta_{k,k^{\prime}}= {\cal Q}_{k}\cdot {\cal Q}_{k^{\prime}}\pi =\sum_{i,j}k_i\Lambda_{i,j}^{-1}k^{\prime}_j$. If $N$ quasiparticles denoted by the vector ${\cal Q}_k$, is created, one has $\sum_{i=1}^{l} N_iE_i + (g-1) (E_i\cdot E_i)E_i^{\star}+ N {\cal Q}_k=E_1^{\star}\Phi$. Making inner product on two sides of this equation with ${\cal Q}_k$, one yields $${\frac {N\theta_k}{\pi}}+(g-1)\Delta \cdot {\cal Q}_k- Q_k \Phi =integer, \label{delt}$$ where $\Delta = \sum_{i=1}^{l}(E_i \cdot E_i)E_i^{\star}$. Applying Eq. (\[const\]) to those quasiparticles and comparing it with Eq. (\[delt\]), one gets $$S_k={\theta_k \over 2\pi}-{\frac {\Delta \cdot {\cal Q}_k} {2}} +{\frac {n}{2(g-1)}}, \, n=integer.\, g\not= 1. \label{result}$$ By using the argument in Ref. [@thou] (which we did for the quasiparticle of the Laughlin state), we can fix $n$ in some cases. The charge of the quasiparticle $E_i$ (denoted by the vector $E_i$) is $\delta_{i,1}$ and the statistics is $\theta = \delta_{i,1}\pi$. Thus the spin of this quasiparticle is ${\frac {\delta_{i,1}} {2}}+integer$. Because $E_i=\sum_j \Lambda_{i,j}E_j^{\star}$, the quasiparticle $E_i$ is a cluster which contains $\Lambda_{i,j}$ quasiparticles with the vector as $E_j^{\star}$. If $\det{\Lambda}$ and $g-1$ are coprime to each other, by using Eq. (\[culster:2\]) for the cluster, we find that $S_i={\frac {E_i \cdot E_i} {2}}- {\frac {\Delta \cdot E_i} {2}}-{1\over 2}$ for the quasiparticle $E_j^{\star}$ when $l-i$ is an even integer and $S_i={\frac {E_i \cdot E_i} {2}}- {\frac {\Delta \cdot E_i} {2}}$ when $l-i$ is an odd integer. We will use notation $i \in i_{e}$ if $l-i$ is an even integer. Generally, for the quasiparticle ${\cal Q}$, we find that $$S_{\cal Q}={\frac { {\cal Q}\cdot {\cal Q}} {2}}- {\frac {\Delta \cdot {\cal Q}} {2}}-{1\over 2}\Delta^{\prime} \cdot {\cal Q}, \label{result:1}$$ where $\Delta^{\prime}=\sum_{i} E_i, i\in i_e$. If the quasiparticles have the standard spin-statistics relation, it is required that ${\frac {\Delta \cdot {\cal Q}} {2}} +{1\over 2}\Delta^{\prime} \cdot {\cal Q}$ is an integer. Indeed, this number is always an integer for the Laughlin state. However, in the hierarchical state, this number may not be an integer. Thus the quasiparticles in the hierarchical state usually do not have the standard spin-statistics relation. If $\det{\Lambda}$ and $g-1$ are not coprime to each other, there exist other solutions, not only Eq. (\[result:1\]). As we argued in the case of the Laughlin state, these [other]{} solutions is [unlikely]{} the [true]{} QPS. Thus we suppose that Eq. (\[result:1\]) is the spin for quasiparticles and it is topological independent. The above method does not give any information about the QPS in the FQHE on the torus ($g=1$). However, the above discussion strongly suggests that the QPS in the FQHE on the torus is also given by Eq. (\[result:1\]) which suppose to be the QPS on any Riemann surfaces. The Lagrangian for the long-distance physics of the Hall fluid on Riemann surfaces is [@wen], $$\begin{aligned} {\cal L}& = & {1\over 4\pi}(\alpha_{\mu , i}\Lambda_{i,j}\epsilon^{\mu \nu \lambda}\partial_{\nu}\alpha_{\lambda , j} + 2A_{\mu}t_i\epsilon^{\mu \nu \lambda} \partial_{\nu}\alpha_{\lambda , i} \nonumber \\ & + & 2\omega s_i\epsilon^{0 \nu \lambda} \partial_{\nu}\alpha_{\lambda , i}),\end{aligned}$$ where $\omega$ is the connection one form (the curvature is given by $R=d\omega$). In the case of the hierarchical state, $t_i$ is equal to $\delta_{i,1}$ and the matrix $\Lambda$ is one we gave in the previous discussion. By using Eq. (\[hierar:1\]), we can show that $s_i=\Lambda_{i,i}$. So $s_i$ is a topological independent constant. It is reasonable to believe that $s_i$ is a topological independent constant for any kinds of quantum Hall fluids. Due to the presence of the third term in the Lagrangian, the spin-statistics relation usually is not generalized one [@wen]. 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--- abstract: \| We show that, with indivisible goods, the existence of competitive equilibrium fundamentally depends on agents’ substitution effects, not their income effects. Our Equilibrium Existence Duality allows us to transport results on the existence of competitive equilibrium from settings with transferable utility to settings with income effects. One consequence is that net substitutability—which is a strictly weaker condition than gross substitutability—is sufficient for the existence of competitive equilibrium. We also extend the “demand types” classification of valuations to settings with income effects and give necessary and sufficient conditions for a pattern of substitution effects to guarantee the existence of competitive equilibrium. JEL Codes: C62, D11, D44 author: - Elizabeth Baldwin - Omer Edhan - Ravi Jagadeesan - Paul Klemperer - Alexander Teytelboym bibliography: - 'bib.bib' date: 17th June 2020 title: \| The Equilibrium Existence Duality:\ Equilibrium with Indivisibilities & Income Effects --- Introduction ============ This paper shows that, when goods are indivisible and there are income effects, the existence of competitive equilibrium fundamentally depends on agents’ substitution effects—i.e., the effects of compensated price changes on agents’ demands. We provide general existence results that do not depend on income effects. In contrast to the case of divisible goods, competitive equilibrium does not generally exist in settings with indivisible goods [@henry1970indivisibilites]. Moreover, most previous results about when equilibrium does exist with indivisible goods assume that utility is transferable—ruling out income effects but allowing tractable characterizations of (Pareto-)efficient allocations and aggregate demand that can be exploited to analyze competitive equilibrium.[^1] But understanding the role of income effects is important for economies with indivisible goods, as these goods may comprise large fractions of agents’ budgets. Furthermore, in the presence of income effects, the distribution of wealth among agents affects both Pareto efficiency and aggregate demand, making it necessary to develop new methods to analyze competitive equilibrium with indivisible goods. The cornerstone of our analysis is an application of the relationship between Marshallian and Hicksian demand. As in classical demand theory, Hicksian demand is defined by fixing a utility level and minimizing the expenditure of obtaining it. We combine Hicksian demands to construct a family of “Hicksian economies” in which prices vary, but agents’ utilities—rather than their endowments—are held constant. Our key result, which we call the Equilibrium Existence Duality, states that competitive equilibria exist for all endowment allocations if and only if competitive equilibria exist in the Hicksian economies for all utility levels. Preferences in each Hicksian economy reflect agents’ substitution effects. Therefore, by the Equilibrium Existence Duality, the existence of competitive equilibrium fundamentally depends on substitution effects. Moreover, as fixing a utility level precludes income effects, agents’ preferences are quasilinear in each Hicksian economy. Hence, the Equilibrium Existence Duality allows us to transport (and so generalize) any necessary or sufficient condition for equilibrium existence from settings with transferable utility to settings with income effects.[^2] In particular, our most general existence result gives a necessary and sufficient condition for a pattern of agents’ substitution effects to guarantee the existence of competitive equilibrium in the presence of income effects. Consider, for example, the case of substitutable goods in which each agent demands at most one unit of each good. With transferable utility, substitutability is sufficient for the existence of competitive equilibrium [@KeCr:82] and defines a maximal domain for existence [@GuSt:99]. With income effects, @FlJaJaTe:19 showed that competitive equilibrium exists under gross substitutability. The Equilibrium Existence Duality tells us that, with income effects, competitive equilibrium in fact exists under net substitutability and that net substitutability defines a maximal domain for existence. Moreover, we show that gross substitutability implies net substitutability; the reverse direction is not true in the presence of income effects. An implication of our results is that it is unfortunate that [@KeCr:82], and much of the subsequent literature, used the term “gross substitutes” to refer to a condition on quasilinear preferences. Indeed, gross and net substitutability are equivalent without income effects, and our work shows that it is net substitutability, not gross substitutability, that is critical to the existence of competitive equilibrium with substitutes.[^3] To appreciate the distinction between gross and net substitutability, suppose that Martine owns a house and is thinking about selling her house and buying one of two different other houses: a spartan one and a luxurious one [@quinzii1984core]. If the price of her own house increases, she may wish to buy the luxurious house instead of the spartan one—exposing a gross complementarity between her existing house and the spartan one. However, Martine regards the houses as net substitutes: the complementarity emerges entirely due an income effect. Competitive equilibrium is therefore guaranteed to exist in economies with Martine if all other agents see the goods as net substitutes, despite the presence of gross complementarities. Our most general equilibrium existence theorem characterizes the combinations of substitution effects that guarantee the existence of competitive equilibrium. It is based on classification of valuations into “demand types.” A demand type is defined by the set of vectors that summarize the possible ways in which demand can change in response to a small generic price change. For example, the set of all substitutes valuations forms a demand type, as does the set of all complements valuations, etc. Applying @BaKl:19’s taxonomy to changes in Hicksian demands, we see that their definition easily extends to general utility functions, capturing agents’ substitution effects. Examples of demand types in our setting with income effects, therefore, include the set of all net substitutes preferences, the set of all net complements preferences, etc. The Equilibrium Existence Duality then makes it straightforward that the Unimodularity Theorem[^4]—which encompasses many standard results on the existence of competitive equilibrium as special cases[^5]—is unaffected by income effects. Therefore, as with the case of substitutes, conditions on complementarities and substitutabilities that guarantee the existence of competitive equilibrium in settings with transferable utility translate to conditions on net complementarities and substitutabilities that guarantee the existence of competitive equilibrium in settings with income effects. In particular, there are patterns of net complementarities that are compatible with the existence of competitive equilibrium. Our results may have significant implications for the design of auctions that seek competitive equilibrium outcomes, and in which bidders face financing constraints. For example, they suggest that versions of the Product-Mix Auction [@klemperer2008new], used by the Bank of England since the Global Financial Crisis, may work well in this context. Several other papers have considered the existence of competitive equilibrium in the presence of indivisibilities and income effects. [@quinzii1984core], [@gale1984equilibrium], and [@svensson1984competitive] showed the existence of competitive equilibrium in a housing market economy in which agents have unit demand and endowments. Building on those results, [@kaneko1986existence], [@van1997existence; @van2002existence], and [@yang2000equilibrium] analyzed settings with multiple goods, but restricted attention to separable preferences. By contrast, our results—even for the case of substitutes—allow for interactions between the demand for different goods. We also clarify the role of net substitutability for the existence of competitive equilibrium. In a different direction, [@DaKoMu:01] proved a version of the sufficiency direction of the Unimodularity Theorem for settings with income effects. [@DaKoMu:01] also defined domains of preferences using an optimization problem that turns out to be equivalent to the expenditure minimization problem. However, they did not note the connection to the expenditure minimization problem or Hicksian demand, and, as a result, did not interpret their sufficient conditions in terms of substitution effects or establish the role of substitution effects in determining the existence of equilibrium. We proceed as follows. Section \[sec:setting\] describes our setting—an exchange economy with indivisible goods and money. Section \[sec:EED\] develops the Equilibrium Existence Duality. Since the existing literature has focused mostly on the case in which indivisible goods are substitutes, we consider that case in Section \[sec:subst\]. Section \[sec:demTypes\] develops demand types for settings with income effects and states our Unimodularity Theorem with Income Effects. Section \[sec:auctions\] remarks on implications for auction design, and Section \[sec:conclusion\] is a conclusion. Appendix \[app:EEDproof\] proves the Equilibrium Existence Duality. Appendix \[app:grossToNet\] proves the connection between gross and net . Appendices \[app:dualDemPrefs\] and \[app:maxDomain\] adapt the proofs of results from the literature to our setting. The Setting {#sec:setting} =========== We work with a model of exchange economies with indivisibilities—adapted to allow for income effects. There is a finite set $J$ of agents, a finite set $I$ of indivisible goods, and a divisible numéraire that we call “money.” We allow goods to be undesirable, i.e., to be “bads." We fix a total endowment $\tot \in \mathbb{Z}^I$ of goods in the economy.[^6] Preferences and Marshallian Demand ---------------------------------- Each agent $j \in J$ has a finite set $\Feas{j} \subseteq \Z^I$ of feasible bundles of indivisible goods and a lower bound $\feas{j} \ge -\infty$ on her consumption of money. As bundles that specify negative consumption of some goods can be feasible, our setting implicitly allows for production.[^7] The principal cases of $\feas{j}$ are $\feas{j} = -\infty$, in which case all levels of consumption of money are feasible, and $\feas{j} = 0$, in which case the consumption of money must be positive. Hence, the set of feasible consumption bundles for agent $j$ is $\Feans{j} = (\feas{j},\infty) \times \Feas{j}$. Given a bundle $\bunn \in \Feans{j},$ we let $\numer$ denote the amount of money in $\bunn$ and $\bun$ denote the bundle of goods specified by $\bunn,$ so $\bunn = (\defbunn)$. The utility levels of agent $j$ lie in the range $\feasUtil{j}$, where $-\infty \le \minu{j} < \maxu{j} \le \infty.$ Furthermore, each agent $j$ has a utility function $\utilFn{j}: \Feans{j} \to \feasUtil{j}$ that we assume to be continuous and strictly increasing in $\numer$, and to satisfy $$\label{eq:ulimits} \lim_{\numer \to (\feas{j})^+} \util{j}{\defbunn} = \minu{j} \quad \text{and} \quad \lim_{\numer \to \infty} \util{j}{\defbunn} = \maxu{j}$$ for all $\bun \in \Feas{j}.$ Condition (\[eq:ulimits\]) requires that some consumption of money above the minimum level $\feas{j}$ be essential to agent $j$.[^8] We let $\pnumer = 1$. Given an endowment $\bunndow = (\defbunndow) \in \Feans{j}$ of a feasible consumption bundle and a price vector $\p \in \mathbb{R}^I,$ agent $j$’s Marshallian demand for goods is $$\dM{j}{\p}{\bunndow} = \left\{\bun^* \lgiv \bunn^* \in \argmax_{\bunn \in X^j \mid \pall \cdot \bunn \le \pall \cdot \bunndow} \util{j}{\bunn}\lgivend\right\}.$$ As usual, Marshallian demand is given by the set of bundles of goods that maximize an agent’s utility, subject to a budget constraint, given a price vector and an endowment. An income effect is a change in an agent’s Marshallian demand induced by a change in her money endowment, holding prices fixed.[^9] Our setup is flexible enough to capture a wide range of preferences with and without income effects, as the following two examples illustrate. \[eg:quasilin\] Given a valuation $\valFn{j}: \Feas{j} \to \mathbb{R},$ letting $\feas{j} = \minu{j}= -\infty$ and $\maxu{j} =\infty,$ one obtains a quasilinear utility function given by $$\util{j}{\defbunn} = \numer + \val{j}{\bun}.$$ When agents utility functions are quasilinear, they do not experience income effects. When all agents have quasilinear utility functions, we say that utility is transferable. \[eg:quasilog\] Given a function $\quasivalFn{j}: \Feas{j} \to (-\infty,0)$, which we call a quasivaluation,[^10] and letting $\minu{j}= -\infty$, $\maxu{j} =\infty,$ and $\feas{j} = 0,$ there is a quasilogarithmic utility function given by $$\util{j}{\bunn} = \log \numer - \log(- \quasival{j}{\bun}).$$ Unlike with quasilinear utility functions, agents with quasilogarithmic utility functions exhibit income effects. Hicksian Demand, , and the --------------------------- The concept of Hicksian demand from consumer theory plays a key role in our analysis. Given a utility level $\ub \in \feasUtil{j}$ and a price vector $\p,$ agent $j$’s Hicksian demand for goods is $$\label{eq:costMin} \dH{j}{\p}{\ub} = \left\{\bun^* \lgiv \bunn^* \in \argmin_{\bunn \in \Feans{j} \mid \util{j}{\bunn} \ge \ub} \pall \cdot \bunn\lgivend\right\}.$$ As in the standard case with divisible goods, Hicksian demand is given by the set of bundles of goods that minimize the expenditure of obtaining a utility level given a price vector. A substitution effect is a change in an agent’s Hicksian demand induced by a change in prices, holding her utility level fixed. As in classical demand theory, Marshallian and Hicksian demand are related by the duality between the utility maximization and expenditure minimization problems. Specifically, a bundle of goods is expenditure-minimizing if and only if it is utility-maximizing.[^11] \[fac:dualDem\] Let $\p$ be a price vector. 1. For all endowments $\bunndow$, we have that $\dM{j}{\p}{\bunndow} = \dH{j}{\p}{\ub},$ where $$\ub = \max_{\bunn \in \Feans{j} \mid \pall \cdot \bunn \le \pall \cdot \bunndow} \util{j}{\bunn}.$$ 2. For all utility levels $\ub$ and endowments $\bunndow$ with $$\pall \cdot \bunndow = \min_{\bunn \in \Feans{j} \mid \util{j}{\bunn} \ge \ub} \pall \cdot \bunn,$$ we have that $\dH{j}{\p}{\ub} = \dM{j}{\p}{\bunndow}.$ If an agent has a quasilinear utility function, then, as she experiences no income effects, her Marshallian and Hicksian demands coincide and do not depend on endowments or utility levels. Under quasilinearity, we therefore refer to both Marshallian and Hicksian demand simply as demand, which we denote by $\dQL{j}{\p}$. Formally, if $j$ has quasilinear utility with valuation $\valFn{j},$ defining $\dQL{j}{\p}$ as the solution to the quasilinear maximization problem $$\label{eqn:quasilin} \dQL{j}{\p} = \argmax_{\bun \in \Feas{j}} \{\val{j}{\bun} - \p \cdot \bun\},$$ we have that $\dM{j}{\p}{\bunndow} = \dQL{j}{\p}$ for all endowments $\bunndow$ and that $\dH{j}{\p}{\ub} = \dQL{j}{\p}$ for all utility levels $\ub$. We next show that the interpretation of the expenditure minimization problem as a quasilinear maximization problem persists in the presence of income effects. Specifically, we can rewrite the expenditure minimization problem of Equation (\[eq:costMin\]) as a quasilinear optimization problem by using the constraint to solve for $\numer$ as a function of $\bun$. Formally, for a bundle $\bun \in \Feas{j}$ of goods and a utility level $\ub \in \feasUtil{j},$ we let $\cf{j}{\bun}{\ub} = \util{j}{\cdot,\bun}^{-1}(\ub)$ denote the level of consumption of money (or savings) needed to obtain utility level $\ub$ given $\bun.$[^12] By construction, we have that $$\dH{j}{\p}{\ub} = \argmin_{\bun \in \Feas{j}} \left\{\cf{j}{\bun}{\ub} + \p \cdot \bun\right\}.$$ It follows that agent $j$’s expenditure minimization problem at utility level $\ub$ can be written as a quasilinear maximization problem for the valuation $-\cf{j}{\cdot}{\ub}$, which we therefore call the Hicksian valuation. The ** of agent $j$ at utility level $u$ is $\valHDef{j} = -\cf{j}{\cdot}{\ub}$. Note that $\cf{j}{\cdot}{\ub}$ is continuous and strictly increasing in $\ub,$ and hence $\valHDef{j}$ is continuous and strictly decreasing in $\ub$. The following lemma formally states that agent $j$’s Hicksian demand at utility level $\ub$ is the demand correspondence of an agent with valuation $\valHDef{j}$. \[lem:dHvalH\] For all price vectors $\p$ and utility levels $\ub$, we have that $$\dH{j}{\p}{\ub} = \argmin_{\bun \in \Feas{j}} \left\{\cf{j}{\bun}{\ub} + \p \cdot \bun\right\} = \argmax_{\bun \in \Feas{j}} \left\{\valH{j}{\bun}{\ub} - \p \cdot \bun\right\}.$$ As $\utilFn{j}(\bunn)$ is strictly increasing in $\numer,$ we have that $$\dH{j}{\p}{\ub} = \left\{\bun^* \lgiv \bunn^* \in \argmin_{\bunn \in \Feans{j} \mid \util{j}{\bunn} = \ub} \pall \cdot \bunn\lgivend\right\}.$$ Applying the substitution $\numer = \cf{j}{\bun}{\ub}=-\valH{j}{\bun}{\ub}$ to remove the constraint from the minimization problem yields the lemma. It follows from Lemma \[lem:dHvalH\] that an agent’s at a utility level gives rise to a quasilinear utility function that reflects the agent’s substitution effects at that utility level. Lemma \[lem:dHvalH\] also yields a relationship between the family of and income effects. Indeed, by Fact \[fac:dualDem\], an agent’s income effects correspond to changes in her Hicksian demand induced by changes in her utility level, holding prices fixed. By Lemma \[lem:dHvalH\], these changes in Hicksian demand reflect the changes in the that are induced by the changes in utility levels. Hence, the at each utility level determine an agent’s substitution effects, while the variation of the with the utility level captures her income effects. To illustrate how an agent’s family of reflects her income effects, we consider the cases of quasilinear and quasilogarithmic utility. With quasilinear utility, the at utility level $\ub$ is $\valH{j}{\bun}{\ub} = \val{j}{\bun} - \ub.$ Changes in $\ub$ do not affect the relative values of bundles under $\valH{j}{\cdot}{\ub}$, so changes in the utility level do not affect Hicksian demand. Indeed, there are no income effects. By construction, a utility function $\util{j}{\bunn}$ is quasilinear in $\numer$ if and only if $\cf{j}{\bun}{\ub}$ is quasilinear in $\ub$—or, equivalently, $\valH{j}{\bun}{\ub}$ is quasilinear in $-\ub$. In general, it follows from Fact \[fac:dualDem\] and Lemma \[lem:dHvalH\] that agent $j$’s preferences exhibit income effects if and only if $\cf{j}{\bun}{\ub}$—or, equivalently, $\valH{j}{\bun}{\ub}$—is not additively separable between $\bun$ and $\ub$. \[eg:quasilogDualVal\] With quasilogarithmic utility, the at utility level $\ub$ is $\valH{j}{\bun}{\ub} = e^{\ub} \quasival{j}{\bun}.$ In this case, each is a positive linear transformation of $\quasivalFn{j}$. Income effects are reflected by the fact that $\valH{j}{\bun}{\ub}$ is not additively separable between $\bun$ and $\ub$. We use Lemma \[lem:dHvalH\] to convert preferences with income effects into families of valuations. It turns out that each continuously decreasing family of valuations is the family of of a utility function, so a utility function can be represented equivalently by a family of . \[fac:dualPrefs\] Let $F: \Feas{j} \times \feasUtil{j} \to (-\infty,-\feas{j})$ be a function. There exists a utility function $\utilFn{j}: \Feans{j} \to \feasUtil{j}$ whose at each utility level $\ub$ is $F(\cdot,\ub)$ if and only if for each $\bun \in \Feas{j},$ the function $F(\bun,\cdot)$ is continuous, strictly decreasing, and satisfies[^13][^14] $$\label{eq:vlimits} \lim_{\ub \to (\minu{j})^+} F(\bun,\ub) = -\feas{j} \quad \text{and} \quad \lim_{\ub \to (\maxu{j})^-} F(\bun,\ub) = -\infty.$$ Finally, we combine the families of to form a family of , in each of which utility is transferable and agents choose consumption bundles to minimize the expenditure of obtaining given utility levels. The * for a profile of utility levels $(\ubj)_{j \in J}$* is the transferable utility economy in which agent $j$’s valuation is $\valH{j}{\cdot}{\ubj}$. The family of consists of the “duals" of the original economy in which income effects have been removed and are given by substitution effects. Like the construction of , the construction of the allows us to convert economies with income effects to families of economies with transferable utility and is a key step of our analysis. The Equilibrium Existence Duality {#sec:EED} ================================= We now turn to the analysis of in exchange economies. consists of an endowment $\bunndowj \in \Feans{j}$ for each agent $j$ such that $\sum_{j \in J} \bundowj = \tot,$ where $\tot$ is the total endowment. Given , a specifies a price vector such that markets for goods clear when agents maximize utility. By Walras’s Law, it follows that the market for money clears as well. Given $(\bunndowj)_{j \in J}$, a ** consists of a price vector $\p$ and a bundle $\bunj \in \dM{j}{\p}{\bunndowj}$ for each agent such that $\sum_{j \in J} \bunj = \tot$. In transferable utility economies, a consists of a price vector $\p$ and a bundle $\bunj \in \dQL{j}{\p}$ for each agent such that $\sum_{j \in J} \bunj = \tot$. In this case, the does not affect because endowments do not affect (Marshallian) demand. We therefore omit the when considering in transferable utility economies in which exists—i.e., $\tot \in \sum_{j \in J} \Feas{j}$. On the other hand, the total endowment $\tot$ affects even when utility is transferable. Recall that utility is transferable in the . Furthermore, by Lemma \[lem:dHvalH\], a in the for a profile $(\ubj)_{j \in J}$ of utility levels consists of a price vector $\p$ and a bundle $\bunj \in \dH{j}{\p}{\ubj}$ for each agent such that $\sum_{j \in J} \bunj = \tot$. Thus, agents act as if they minimize expenditure in in the .[^15] Building on Fact \[fac:dualDem\] and Lemma \[lem:dHvalH\], our Equilibrium Existence Duality connects the equilibrium existence problems in the original economy (which can feature income effects) and the (in which utility is transferable). Specifically, we show that always exists in the original economy if and only if it always exists in the . Here, we hold agents’ preferences and the total endowment (of goods) fixed but allow the to vary. \[thm:existDualExchange\] Suppose that the total endowment and the sets of feasible bundles are such that exists. exist for all if and only if exist in the for all profiles of utility levels. By Lemma \[lem:dHvalH\], agents’ substitution effects determine their preferences in each . Therefore, Theorem \[thm:existDualExchange\] tells us that any condition that ensures the existence of can be written as a condition on substitution effects alone. That is, substitution effects fundamentally determine whether exists. Both directions of Theorem \[thm:existDualExchange\] also have novel implications for the analysis of in economies with indivisibilities. As demands in the are given by Hicksian demand in the original economy (Lemma \[lem:dHvalH\]), the “if" direction of Theorem \[thm:existDualExchange\] implies that every condition on demand $\dQLFn{j}$ that guarantees the existence of in settings with transferable utility translates into a condition on Hicksian demand $\dHFn{j}$ that guarantees the existence of in settings with income effects. In Sections \[sec:subst\] and \[sec:demTypes\], we use the “if" direction of Theorem \[thm:existDualExchange\] to obtain new domains for the existence of with income effects from previous results on the existence of in settings with transferable utility [@KeCr:82; @BaKl:19]. Conversely, the “only if" direction of Theorem \[thm:existDualExchange\] shows that if a condition on demand defines a maximal domain for the existence of in settings with transferable utility, then the translated condition on Hicksian demand defines a maximal domain for the existence of in settings with income effects. In Sections \[sec:subst\] and \[sec:demTypes\], we also use this implication to derive new maximal domain results for settings with income effects. To prove the “only if" direction of Theorem \[thm:existDualExchange\], we exploit a version of the Second Fundamental Theorem of Welfare Economics for settings with indivisibilities. To understand connection to the existence problem for the , note that the existence of in the is equivalent to the conclusion of the Second Welfare Theorem—i.e., that each Pareto-efficient allocation can be supported in an equilibrium with endowment transfers—as the following lemma shows.[^16] \[lem:dualEconSWT\] Suppose that the total endowment and the sets of feasible bundles are such that exists. exist in the for all profiles of utility levels if and only if, for each Pareto-efficient allocation $(\bunnj)_{j \in J}$ with $\sum_{j \in J} \bunj = \tot$, there exists a price vector $\p$ such that $\bunnj \in \dM{j}{\p}{\bunnj}$ for all agents $j$. We prove Lemma \[lem:dualEconSWT\] in Appendix \[app:EEDproof\]. Intuitively, as utility is transferable in the , variation in utility levels between plays that same role as endowment transfers in the Second Welfare Theorem. It is well-known that the conclusion of the Second Welfare Theorem holds whenever exist for all [@maskin2008fundamental].[^17] It follows that always exists in the whenever it always exists in the original economy, which is the “only if" direction of Theorem \[thm:existDualExchange\]. We use a different argument to prove the “if" direction. Our strategy is to show that there exists a profile of utility levels and a in the corresponding in which all agents’ expenditures equal their budgets in the original economy. To do so, we apply a topological fixed-point argument that is similar in spirit to standard proofs of the existence of . Specifically, we consider an auctioneer who, for a given profile of candidate equilibrium utility levels, evaluates agents’ expenditures over all in the and adjusts candidate equilibrium utility levels upwards (resp. downwards) for agents who under- (resp. over-) spend their budgets.[^18] The existence of in the ensures that the process is nonempty-valued, and the transferability of utility in the ensures that the process is convex-valued. Kakutani’s Fixed Point Theorem implies the existence of a fixed-point utility profile. By construction, there exists a in the corresponding at which agents’ expenditures equal the values of their endowments. By Lemma \[lem:dHvalH\], agents must be maximizing utility given their endowments at this equilibrium, and hence once obtains a in the original economy. The details of the argument are in Appendix \[app:EEDproof\]. Examples -------- We next illustrate the power of Theorem \[thm:existDualExchange\] using the two examples. Our first example is a “housing market” in which agents have unit-demand preferences, may be endowed with a house, and can experience arbitrary income effects. We can use Theorem \[thm:existDualExchange\] to reduce the existence problem to the assignment game of [@koopmans1957assignment]—reproving a result originally due to [@quinzii1984core]. \[eg:house\] For each agent $j,$ let $\Feas{j} \subseteq \{\zero\} \cup \{\e{i} \mid i \in I\}$ be nonempty. In this case, in , utility is transferable and agents have unit demand for the goods. As the does not affect when utility is transferable, the results of [@koopmans1957assignment] imply that exist in the for all profiles of utility levels (provided that exists). Hence, Theorem \[thm:existDualExchange\] implies that exist for all endowment allocations—even in the presence of income effects. In the second example, we revisit the quasilogarithmic utility functions from Example \[eg:quasilog\]. We provide sufficient conditions on agents’ quasivaluations for to exist. These conditions are related to, but not in general implied by, the conditions developed in Sections \[sec:subst\] and \[sec:demTypes\]. \[eg:quasilogExist\] For each agent $j,$ let $\quasivalFn{j}: \Feas{j} \to (-\infty,0)$ be a quasivaluation. Let agent $j$’s utility function be quasilogarithmic for the quasivaluation $\quasivalFn{j}$, as in Example \[eg:quasilog\]. In this case, agent $j$’s at each utility level is a positive linear transformation of $\quasivalFn{j}$ (Example \[eg:quasilogDualVal\]). Hence, by Theorem \[thm:existDualExchange\], exist for all as long as exists when utility is transferable and each agent $j$’s valuation is an (agent-dependent) positive linear transformation of $\quasivalFn{j}$—e.g., if the quasivaluations $\quasivalFn{j}$ are all valuations [@MiSt:09], or all valuations of a unimodular demand type [@BaKl:19]. Additionally, in the case in which one unit of each good is available in total (i.e., $\totComp{i} = 1$ for all goods $i$), [@CaOzPa:15] showed that exists when utility is transferable and all agents have sign-consistent tree valuations. Hence, if one unit of each good is available in total, then Theorem \[thm:existDualExchange\] implies that exist with quasilogarithmic utility for all if all agents’ quasivaluations are sign-consistent tree valuations. In the remainder of the paper, we use Theorem \[thm:existDualExchange\] to develop novel conditions on preferences that ensure the existence of . The Case of Substitutes {#sec:subst} ======================= In this section, we apply the Equilibrium Existence Duality (Theorem \[thm:existDualExchange\]) to prove a new result regarding the existence of with substitutable indivisible goods and income effects: we show that a form of net is sufficient for, and in fact defines a maximal domain for, the existence of . We begin by reviewing previous results on the existence of under (gross) . We then derive our existence theorem for and relate it to the previous results. In this section, we focus on the case in which each agent demands at most one unit of each good. Formally, we say that an agent $j$ demands at most one unit of each good if $\Feas{j} \subseteq \{0,1\}^I$. We extend to the case in which agents can demand multiple units of some goods in Section \[sec:ssub\]. and the Existence of {#sec:subOld} ---------------------- We recall a notion of for preferences over indivisible goods from [@FlJaJaTe:19], which extends the condition from classical demand theory. It requires that uncompensated increases in the price of a good weakly raise demand for all other goods. With quasilinear utility, the modifier “gross” can be dropped—as in classical demand theory (see also Footnote 1 in [@KeCr:82]). \[def:gsub\] Suppose that agent $j$ demands at most one unit of each good. 1. \[part:gsub\] A utility function $\utilFn{j}$ is a * utility function at endowment $\bundow \in \Feas{j}$ of goods* if for all money endowments $\numerdow > \feas{j}$, price vectors $\p$, and $\lambda > 0,$ whenever $\dM{j}{\p}{\bunndow} = \{\bun\}$ and $\dM{j}{\p + \lambda \e{i}}{\bunndow} = \{\bunpr\},$ we have that $\bunprComp{k} \ge \bunComp{k}$ for all goods $k \not= i$.[^19] 2. A * valuation* is a valuation for which the corresponding quasilinear utility function is a utility function.[^20] Technically, Definition \[def:gsub\] imposes a condition on the locus of prices at which Marshallian demand is single-valued—following [@AuMi:02], [@HaKoNiOsWe:11], [@BaKl:19], and [@FlJaJaTe:19].[^21] It is well-known that when utility is transferable, exists under . \[fac:subExist\] Suppose that utility is transferable and that exists. If each agent demands at most one unit of each good and has a valuation, then exists.[^22] Moreover, the class of valuations forms a maximal domain for the existence of in transferable utility economies. Specifically, if an agent has a non- valuation, then may not exist when the other agents have valuations. Technically, we require that one unit of each good be present among agents’ endowments (i.e., that $\totComp{i} = 1$ for all goods $i$) as complementarities between goods that are not present are irrelevant for the existence of . \[fac:subMaxDomain\] Suppose that $\totComp{i} = 1$ for all goods $i$. If $\|J\| \ge 2$, agent $j$ demands at most one unit of each good, and $\valFn{j}$ is not a valuation, then there exist sets $\Feas{k} \subseteq \{0,1\}^I$ of feasible bundles and valuations $\valFn{k}: \Feas{k} \to \mathbb{R}$ for agents $k \not= j$, for which there exists but no .[^23] While Fact \[fac:subMaxDomain\] shows that there is no domain strictly containing the domain of valuations for which the existence of can be guaranteed in transferable utility economies, it does not rule out the existence of other domains for which the existence of can be guaranteed. For example, [@SuYa:06], [@CaOzPa:15], and [@BaKl:19] gave examples of domains other than for which the existence of is guaranteed. Generalizing Fact \[fac:subExist\] to settings with income effects, [@FlJaJaTe:19] showed that exists for $(\bunndowj)_{j \in J}$ if each agent $j$’s utility function is a utility function at her endowment $\bundowj$ of goods.[^24] However, [@FlJaJaTe:19] did not offer a maximal domain result for . In the next section, we show that does not actually drive existence of with substitutable indivisible goods. and the Existence of {#sec:nsub} ---------------------- In light of Theorem \[thm:existDualExchange\] and Fact \[fac:subExist\], exists if agents’ Hicksian demands satisfy an appropriate condition—i.e., if preferences satisfy a net analogue of . We build on Definition \[def:gsub\] to define a concept of for settings with indivisibilities. is a version of the condition from classical consumer theory. It requires that compensated increases in the price of a good (i.e., price increases that are offset by compensating transfers) weakly raise demand for all other goods. \[def:nsub\] Suppose that agent $j$ demands at most one unit of each good. A utility function $\utilFn{j}$ is a * utility function* if for all utility levels $\ub$, price vectors $\p$, and $\lambda > 0,$ whenever $\dH{j}{\p}{\ub} = \{\bun\}$ and $\dH{j}{\p + \lambda \e{i}}{\ub} = \{\bunpr\}$, we have that $\bunprComp{k} \ge \bunComp{k}$ for all goods $k \not= i$. For quasilinear utility functions, coincides with (gross) . More generally, can be expressed as a condition on . \[rem:netSubDual\] By Lemma \[lem:dHvalH\], if an agent demands at most one unit of each good, then she has a utility function if and only if her at all utility levels are valuations. We can apply Fact \[fac:dualPrefs\] and Remark \[rem:netSubDual\] to construct large classes of preferences with income effects from families of valuations. There are several rich families of valuations, including endowed assignment valuations [@HaMi:05] and matroid-based valuations [@ostrovsky2015gross]. This leads to a large class of quasilogarithmic utility functions. \[eg:quasilogSubs\] A quasilogarithmic utility function $\utilFn{j}$ is a utility function if and only if the quasivaluation $\quasivalFn{j}$ is a valuation.[^25] More generally, in light of Fact \[fac:dualPrefs\] and Remark \[rem:netSubDual\], each family of valuations leads to a class of utility functions with income effects consisting of the utility functions whose all belong to the family. These classes are defined by conditions on substitution effects and do not restrict income effects. By contrast, places substantial restrictions on the form of income effects.[^26] To understand the difference between gross and , we compare the conditions in a setting in which agents have unit demand for goods. \[eg:houseGrossNet\] Consider an agent, Martine, who owns a house $i_1$ and is considering selling it to purchase (at most) one of houses $i_2$ and $i_3$. If Martine experiences income effects, then her choice between $i_2$ and $i_3$ generally depends on the price she is able to procure for her house $i_1$. For example, if $i_3$ is a more luxurious house than $i_2,$ then Martine may only demand $i_3$ if the value of her endowment is sufficiently large—i.e., if the price of her house $i_1$ is sufficiently high. As a result, when Martine is endowed with $i_1$, she does not generally have preferences: increases in the price of $i_1$ can lower Martine’s demand for $i_2$. That is, Martine can regard $i_2$ as a gross complement for $i_1$. In contrast, Martine has preferences—no compensated increase in the price of $i_1$ could make Martine stop demanding $i_2$—a condition that holds generally in the housing market economy.[^27] Note also that, unlike , generally depends on endowments: if Martine were not endowed a house, she would have preferences [@kaneko1982central; @kaneko1983housing; @DeGa:85]. While Example \[eg:houseGrossNet\] shows that does not imply , it turns out that implies . \[prop:ssubst\] If agent $j$ demands at most one unit of each good and there exists an endowment $\bundow$ of goods at which $\utilFn{j}$ is a utility function, then $\utilFn{j}$ is utility function. Proposition \[prop:ssubst\] and Example \[eg:houseGrossNet\] show that (at any one endowment of goods) implies but places additional restrictions on income effects. Nevertheless, the restrictions on substitution effects alone, entailed by , are sufficient for the existence of . \[thm:netSubExist\] If all agents demand at most one unit of each good and have utility functions, then exist for all . Theorem \[thm:netSubExist\] is an immediate consequence of the Equilibrium Existence Duality and the existence of in transferable utility economies under . Remark \[rem:netSubDual\] implies that the agents’ at all utility levels are valuations. Hence, Fact \[fac:subExist\] implies that exist in the for all profiles of utility levels if exists. The theorem follows by the “if” direction of Theorem \[thm:existDualExchange\]. As implies (Proposition \[prop:ssubst\]), the existence of under is a special case of Theorem \[thm:netSubExist\]. But Theorem \[thm:netSubExist\] is more general: as Example \[eg:houseGrossNet\] shows, allows for forms of gross complementarities between goods, in addition to . The following example illustrates how the distinction between and relates to the existence of when agents caan demand multiple goods. \[eg:grossVersusNetNumerical\] There are two goods and the total endowment is $\tot = (1,1)$. There are two agents, which we call $j$ and $k,$ and $j$’s feasible set of consumption bundles of goods is $\Feas{j} = \{0,1\}^2$. We consider the price vectors $\p = (2,2)$ and $\ppr = (4,2)$ and consider two examples in which agent $j$’s Marshallian demand changes from $(1,1)$ to $(0,0)$ as prices change from $\p$ to $\ppr$—a . But the consequences for the existence of are different across the two cases. In Case \[eg:numericalComp\], the reflects a net complementarity for $j$, and may not exist if $k$ sees goods as net substitutes. In Case \[eg:numericalLog\], the reflects only an income effect for $j$, as in Example \[eg:houseGrossNet\], so is guaranteed to exist if $k$ sees goods as net substitutes. 1. \[eg:numericalComp\] Suppose that $j$ has a quasilinear utility function with valuation given by $$\val{j}{\bun} = \begin{cases} 0 & \text{if } \bun = (0,0),(0,1),(1,0)\\ 5 & \text{if } \bun = (1,1). \end{cases}$$ Here, $\valFn{j}$ is not a valuation because $\dQL{j}{\p} = \{(1,1)\}$ while $\dQL{j}{\ppr} = \{(0,0)\}$: i.e., increasing the price of the first good can lower $j$’s demand for the second good. If $\Feas{k} = \{(0,0),(0,1),(1,0)\}$ and agent $k$ has a quasilinear utility function with a valuation given by $$\label{eq:kUnitDem} \val{k}{\bun} = \begin{cases} 0 & \text{if } \bun = (0,0)\\ 4 & \text{if } \bun = (1,0)\\ 3 & \text{if } \bun = (0,1), \end{cases}$$ then no exists.[^28] 2. \[eg:numericalLog\] Suppose instead that $\utilFn{j}$ is quasilogarithmic (as defined in Example \[eg:quasilog\]) with quasivaluation given by $$\quasival{j}{\bun} = \begin{cases} -11 & \text{if } \bun = (0,0)\\ -7 & \text{if } \bun = (0,1)\\ -4 & \text{if } \bun = (1,0)\\ -1 & \text{if } \bun = (1,1). \end{cases}$$ At the endowment $\bundowj = (0,1)$ of goods, $\utilFn{j}$ is not a utility function as, letting $\numerdowj = 3,$ we have that $\dM{j}{\p}{\bunndowj} = \{(1,1)\}$ while $\dM{j}{\ppr}{\bunndowj} = \{(0,0)\}$.[^29] That is, increasing the price of the first good can lower $j$’s Marshallian demand for the second good. By contrast, as $\quasivalFn{j}$ is a valuation, Example \[eg:quasilogSubs\] implies that $\utilFn{j}$ is a utility function: the is entirely due to an income effect. For example, at the utility level $$\ub = \max_{\bunn \in \Feans{j} \mid \pprall \cdot \bunn \le \pprall \cdot \bunndowj} \util{j}{\bunn} = \log \frac{5}{11},$$ we have that $\dH{j}{\p}{u} = \{(1,0)\}$ and that $\dH{j}{\ppr}{\ub} = \{(0,0)\}$,[^30] so the decrease in the Marshallian demand for the second good as prices change from $\p$ to $\ppr$ at the endowment $\bunndowj$ reflects an income effect. By Theorem \[thm:netSubExist\], exists whenever $k$ has a utility function. For example, if $k$ has a quasilinear utility function with a valuation given by Equation (\[eq:kUnitDem\]), then for the defined by $\bundowj = (0,1)$, $\bundowag{k} = (1,0)$, and $\numerdowj = \numerdowag{k} = 3,$ the price vector $(3,2)$ and the allocation of goods defined by $\bunj = (1,0)$ and $\bunag{k} = (0,1)$ comprise a .[^31] In Case \[eg:numericalLog\], agent $j$ has preferences—leading to the guaranteed existence of when agent $k$ has preferences. By contrast, in Case \[eg:numericalComp\], agent $j$ does not have preferences—and may not exist when $k$ has preferences. In general, net substitutability forms a maximal domain for the existence of . Specifically, if an agent does not have preferences, then may not exist when the other agents have quasilinear preferences. \[prop:netSubstMaxDomain\] Suppose that $\totComp{i} = 1$ for all goods $i$. If $\|J\| \ge 2$, agent $j$ demands at most one unit of each good, and $\utilFn{j}$ is not a utility function, then there exist sets $\Feas{k} \subseteq \{0,1\}^I$ of feasible bundles and valuations $\valFn{k}: \Feas{k} \to \mathbb{R}$ for agents $k \not= j$, and for which no exists. Proposition \[prop:netSubstMaxDomain\] is an immediate consequence of the Equilibrium Existence Duality and the fact that defines a maximal domain for the existence of with transferable utility. By Remark \[rem:netSubDual\], there exists a utility level $\ub$ at which agent $j$’s $\valHDef{j}$ is not a valuation. Fact \[fac:subMaxDomain\] implies that there exist feasible sets $\Feas{k} \subseteq \{0,1\}^I$ and valuations $\valFn{k}$ for agents $k \not= j$, for which exists but no would exist with transferable utility if agent $j$’s valuation were $\valHDef{j}$. With those sets $\Feas{k}$ of feasible bundles and valuations $\valFn{k}$ for agents $k \not= j,$ the “only if” direction of Theorem \[thm:existDualExchange\] implies that there exists for which no exists. Proposition \[prop:netSubstMaxDomain\] entails that any domain of preferences that contains all quasilinear preferences and guarantees the existence of must lie within the domain of preferences. Therefore, Proposition \[prop:netSubstMaxDomain\] and Theorem \[thm:netSubExist\] suggest that is the most general way to incorporate income effects into a condition to ensure the existence of . By contrast, the relationship between the nonexistence of and failures of depends on why fails. can fail due to substitution effects that reflect net complementarities, as in Example \[eg:grossVersusNetNumerical\]\[eg:numericalComp\], or due to income effects, as in Example \[eg:grossVersusNetNumerical\]\[eg:numericalLog\]. If the failure of reflects a net complementarity, then Proposition \[prop:netSubstMaxDomain\] tells us that may not exist if the other agents have quasilinear preferences, as in Example \[eg:grossVersusNetNumerical\]\[eg:numericalComp\]. On the other hand, the failure of is only due to income effects, then Theorem \[thm:netSubExist\] tells us that exists if the other agents have preferences (e.g., quasilinear preferences), as in Example \[eg:grossVersusNetNumerical\]\[eg:numericalLog\]. Demand Types and the Unimodularity Theorem {#sec:demTypes} ========================================== In this section, we characterize exactly what conditions on patterns of substitution effects guarantee the existence of . Specifically, we consider classification of valuations into “demand types” based on sets of vectors that summarize the possible ways in which demand can change in response to a small generic price change. We first review the definition of demand types from [@BaKl:19]. We then extend the concept of demand types to settings with income effects, and develop a version of the Unimodularity Theorem that allows for income effects and characterizes which demand types guarantee the existence of (see also [@DaKoMu:01]). A special case of the Unimodularity Theorem with Income Effects extends Theorem \[thm:netSubExist\] to settings in which agents can demand multiple units of some goods. Demand Types and the Unimodularity Theorem with Transferable Utility {#sec:demTypesTU} -------------------------------------------------------------------- We first review the concept of demand types for quasilinear settings, as developed by @BaKl:19. An integer vector is primitive if the greatest common divisor of its components is 1. By focusing on the directions of demand changes, we can restrict to primitive demand change vectors. A ** is a set $\mathcal{D} \subseteq \mathbb{Z}^I$ of primitive such that if $\dvec \in\mathcal{D}$ then $- \dvec \in\mathcal{D}$. \[def:demTypeTU\] Let $\valFn{j}$ be a valuation. 1. A bundle $\bun$ is uniquely demanded by agent $j$ if there exists a price vector $\p$ such that $\dQL{j}{\p} = \{\bun\}.$ 2. A pair $\{\bun,\bunpr\}$ of uniquely demanded bundles are adjacently demanded by agent $j$ if there exists a price vector $\p$ such that $\dQL{j}{\p}$ contains $\bun$ and $\bunpr$ but no other bundle that is uniquely demanded by agent $j$. 3. If $\mathcal{D}$ is a , then $\valFn{j}$ is of demand type $\mathcal{D}$ if for all pairs $\{\bun,\bunpr\}$ that are adjacently demanded by agent $j$, the difference $\bunpr - \bun$ is a multiple of an element of $\mathcal{D}$.[^32] For intuition, suppose that a small price change causes a change in demand. Then, generically, demand changes between adjacently demanded bundles. Thus, the demand type vectors represent the possible directions of changes in demand in response to small generic price changes (see Proposition 3.3 in [@BaKl:19] for a formal statement). To illustrate Definition \[def:demTypeTU\], we consider an example. \[eg:demTypeDef\] Suppose that there are two goods and let $$\Feas{j} = \{0,1,2,3\}^2 \ssm \{(2,3),(3,2),(3,3)\}.$$ Consider the valuation defined by $\val{j}{\bun} = \bunComp{1} + \bunComp{2}$. As Figure \[fig:demTypeDef\] illustrates, the uniquely demanded bundles are $(0,0)$, $(0,3),$ $(1,3),$ $(3,0)$, and $(3,1)$. When $1 = \pComp{1} < \pComp{2},$ agent $j$’s demand is $\dQL{j}{\p} = \{(0,0),(1,0),(2,0),(3,0)\}$. Hence, as the bundles $(1,0)$ and $(2,0)$ are not uniquely demanded, the bundles $(0,0)$ and $(3,0)$ are adjacently demanded. As a result, for $\valFn{j}$ to be of demand type $\mathcal{D}$, the set $\mathcal{D}$ must contain the vector $(1,0)$, which is the primitive proportional to the demand change $(3,0) - (0,0) = (3,0)$. Similarly, the bundles $(0,0)$ and $(0,3)$ are adjacently demanded, and any $\mathcal{D}$ such that $\valFn{j}$ is of demand type $\mathcal{D}$ must contain the vector $(0,1)$. When $\pComp{1} < \pComp{2} = 1,$ demand is $\dQL{j}{\p} = \{(3,0),(3,1)\}$. Hence, the bundles $(3,0)$ and $(3,1)$ are adjacently demanded. Similarly, the bundles $(0,3)$ and $(1,3)$ are adjacently demanded. These facts respectively imply, again, that $(0,1)$ and $(1,0)$ are in any $\mathcal{D}$ such that $V^j$ is of demand type $\mathcal{D}$. Last, when $p_1 = p_2 < 1,$ agent $j$’s demand is $\dQL{j}{\p} = \{(1,3),(2,2),(3,1)\}$. Hence, as the bundle $(2,2)$ is not uniquely demanded, the bundles $(1,3)$ and $(3,1)$ are adjacently demanded. As a result, for $\valFn{j}$ to be of demand type $\mathcal{D}$, the set $\mathcal{D}$ must contain the vector $(1,-1)$, which is the primitive proportional to the demand change $(3,1) - (1,3) = (2,-2)$. By contrast, the bundles $(0,0)$ and $(3,1)$ are not adjacently demanded: the only price vector at which agent $j$ demands them both is $\p = (1,1),$ but $\dQL{j}{1,1}$ also contains the uniquely demanded bundles $(0,3)$, $(1,3),$ and $(3,0)$. Similarly, the bundles $(0,0)$ and $(1,3)$ are not adjacently demanded. Hence, $$\mathcal{D} = \pm \left\{\begin{bmatrix}1 \\ 0\end{bmatrix}, \begin{bmatrix}0 \\ 1\end{bmatrix}, \begin{bmatrix} 1 \\ -1\end{bmatrix}\right\}$$ is the minimal $\mathcal{D}$ such that $\valFn{j}$ is of demand type $\mathcal{D}$. Consider any valuation of the same demand type $\mathcal{D}$ as in Example \[eg:demTypeDef\], and a change in price from $\p$ to $\ppr=\p+\lambda \e{1}$ for some $\lambda>0$. For generic choices of $\p$ and $\lambda$, the demand at any price on the straight line from $\mathbf{p}_I$ to $\mathbf{p}_I'$ either is unique, or demonstrates the adjacency of two bundles uniquely demanded at prices on this line. The change in demand between such bundles must therefore be a multiple of an element of $\mathcal{D}$ (by Definition \[def:demTypeTU\]). Moreover, since only the price of good 1 is changing and that price is increasing, the law of demand entails that demand for good 1 must strictly decrease upon any change in demand.[^33] Thus, the change in demand between the two consecutive uniquely demanded bundles must be a positive multiple of either $(-1,0)$ or $(-1,1)$. Therefore, demand for good 2 must (weakly) increase, reflecting between the goods. This two-good example is a special case of an important class of demand types. \[eg:ssubDemType\] The strong substitutes consists of all vectors in $\Z^I$ with at most one $+1$ component, at most one $-1$ component, and no other nonzero components. As illustrated in Example \[eg:demTypeDef\], this captures one-to-one substitution between goods through demand type vectors with one component of $1$ and one component of $-1$. Furthermore, if an agent $k$ demands at most one unit of each good, then $\valFn{k}$ is a valuation if and only if it is of the strong substitutes demand type (see Theorems 2.1 and 2.4 in [@fujishige2003note]). In settings in which agents can demand multiple units of each good, a form of concavity is needed to ensure the existence of . A valuation is concave if, under that valuation, each bundle of goods that is a convex combination of feasible bundles of goods is demanded at some price vector. For the formal definition, we let $\conv(T)$ denote the convex hull of a set $T \subseteq \mathbb{R}^I.$ \[def:concave\] A valuation $\valFn{j}$ is concave if for each bundle $\bun \in \conv(\Feas{j}) \cap \mathbb{Z}^n,$ there exists a price vector $\p$ such that $\bun \in \dQL{j}{\p}$. In Section \[sec:subOld\], we discussed that guarantees the existence of in transferable utility economies when agents demand at most one unit of each good. Generalizing that result, @BaKl:19 identified a necessary and sufficient condition for the concave valuations of a demand type to form a domain for the guaranteed existence of . A set of vectors in $\Z^I$ is unimodular if every linearly independent subset can be extended to be a basis for $\R^I$, of , such that any square matrix whose columns are these vectors has determinant $\pm 1$. For example, the in Example \[eg:demTypeDef\] is unimodular, while the $$\label{eq:subsComp} \pm \left\{\begin{bmatrix} 1 \\ -1\end{bmatrix}, \begin{bmatrix}1 \\ 1\end{bmatrix}\right\}$$ is not unimodular, because $$\left\|\begin{matrix} 1 & 1\\ -1 & 1 \end{matrix}\right\| = 2.$$ The in (\[eq:subsComp\]) represents that the two goods can be substitutable or complementary for agents—a possibility that can cause to fail to exist, as in Example \[eg:grossVersusNetNumerical\]\[eg:numericalComp\]. [@BaKl:19] showed that the unimodularity of a is precisely the condition for the corresponding demand type to guarantee the existence of . \[fac:unimod\] Let $\mathcal{D}$ be a . exist for all finite sets $J$ of agents with concave valuations of demand type $\mathcal{D}$ and for all total endowments for which exist if and only if $\mathcal{D}$ is unimodular.[^34] [@DaKoMu:01] used conditions on the ranges of agents’ demand correspondences to describe classes of concave valuations, which correspond to the concave valuations of unimodular demand types;[^35] they formulated a version of the “if" direction of Fact \[fac:unimod\] with those conditions.[^36] As [@poincare1900second] showed, the strong substitutes is unimodular. Therefore, in light of Example \[eg:ssubDemType\], the existence of in transferable utility economies in which agents demand at most one unit of each good and have substitutes valuations (Fact \[fac:subExist\]) is a special case of Fact \[fac:unimod\]. Moreover, Fact \[fac:unimod\] is strictly more general: as [@BaKl:19] showed, there are unimodular for which the existence of cannot be deduced from the corresponding result for strong substitutes by applying a change of basis to the space of bundles of goods.[^37] To illustrate the additional generality, we discuss an example of such a demand type.[^38] \[eg:5D\] There are five goods. Consider the $$\mathcal{D} = \pm\left\{\begin{bmatrix}1 \\ 0 \\ 0 \\ 0 \\ 0\end{bmatrix}, \begin{bmatrix}0 \\ 1 \\ 0 \\ 0 \\ 0\end{bmatrix}, \begin{bmatrix}0 \\ 0 \\ 1 \\ 0 \\ 0\end{bmatrix}, \begin{bmatrix}0 \\ 0 \\ 0 \\ 1 \\ 0\end{bmatrix}, \begin{bmatrix}0 \\ 0 \\ 0 \\ 0 \\ 1\end{bmatrix}, \begin{bmatrix}1 \\ -1 \\ 1 \\ 0 \\ 0\end{bmatrix}, \begin{bmatrix}0 \\ 1 \\ -1 \\ 1 \\ 0\end{bmatrix}, \begin{bmatrix}0 \\ 0 \\ 1 \\ -1 \\ 1\end{bmatrix}, \begin{bmatrix}1 \\ 0 \\ 0 \\ 1 \\ -1\end{bmatrix}, \begin{bmatrix}-1 \\ 1 \\ 0 \\ 0 \\ 1\end{bmatrix}\right\}.$$ Intuitively, this allows for independent changes in the demand for each good (through the first five vectors), as well as for substitution from a good to the bundle consisting of its two neighbors if the goods are arranged in a circle (through the last five vectors). This is unimodular, and cannot be obtained from the strong substitutes by a change of basis of the space of integer bundles of goods (see, e.g., Section 19.4 of [@schrijver1998theory]). Moreover, the demand types defined by maximal, unimodular turn out to define maximal domains for the existence of in settings with transferable utility. Here, we say that a unimodular is maximal if it is not strictly contained in another unimodular . \[fac:unimodMaxDomain\] Let $\mathcal{D}$ be a maximal unimodular . If $\|J\| \ge 2$ and $\valFn{j}$ is non-concave or not of demand type $\mathcal{D}$, then there exist sets $\Feas{k}$ of feasible bundles and concave valuations $\valFn{k}: \Feas{k} \to \mathbb{R}$ of demand type $\mathcal{D}$ for agents $k \not= j$, as well as a total endowment, for which there exists but no .[^39] While Fact \[fac:unimod\] shows that there exist valuations in each non-unimodular demand type for which does not exist, Fact \[fac:unimodMaxDomain\] shows that for every valuation outside a maximal unimodular demand type, there exist concave valuations within the demand type that lead to non-existence. Hence, the necessity direction of Fact \[fac:unimod\], together with Fact \[fac:unimodMaxDomain\], provide complementary perspectives on the way in which can fail to exist outside the context of unimodular demand types. Demand Types and the Unimodularity Theorem with Income Effects -------------------------------------------------------------- We now use Fact \[fac:dualPrefs\] to extend the demand types framework to settings with income effects. \[def:demTypeIncEff\] An agent’s preferences are of demand type $\mathcal{D}$ if her at all utility levels are of demand type $\mathcal{D}$. Lemma \[lem:dHvalH\] leads to an economic interpretation of Definition \[def:demTypeIncEff\]: a utility function is of demand type $\mathcal{D}$ if $\mathcal{D}$ summarizes the possible ways in which Hicksian demand can change in response to a small generic price change. In particular, Definition \[def:demTypeIncEff\] extends the concept of demand types to settings with income effects by placing conditions on substitution effects. Indeed, Definition \[def:demTypeIncEff\] considers only the properties of at each utility level (which, by Lemma \[lem:dHvalH\], reflect substitution effects), and not how an agent’s vary with her utility level (which, by Fact \[fac:dualDem\] and Lemma \[lem:dHvalH\], reflects income effects). [@DaKoMu:01] translated their conditions on the ranges of agents’ demand correspondences from quasilinear settings to settings with income effects by using Fact \[fac:dualPrefs\] in an analogous manner (see Assumption 3$'$ in [@DaKoMu:01]). However, the economic interpretation in terms of substitution effects that Lemma \[lem:dHvalH\] leads to was not clear from formulation. As with the case of transferable utility, a concavity condition is needed to ensure the existence of . With income effects, the relevant condition is a version of the quasiconcavity condition from classical demand theory for settings with indivisible goods. We define quasiconcavity based on concavity and duality.[^40] \[def:quasiConc\] An agent’s utility function is quasiconcave if her at all utility levels are concave. As with the case of transferable utility, unimodularity is a necessary and sufficient condition for the existence of to be guaranteed for all quasiconcave preferences of a demand type when income effects are present. \[thm:unimod\] Let $\mathcal{D}$ be a . exist for all finite sets $J$ of agents with quasiconcave utility functions of demand type $\mathcal{D}$, for all total endowments, and for all if and only if $\mathcal{D}$ is unimodular. The “only if” direction of Theorem \[thm:unimod\] is a special case of the Unimodularity Theorem with Transferable Utility (Fact \[fac:unimod\]). The “if” direction of Theorem \[thm:unimod\] is an immediate consequence of the Equilibrium Existence Duality and Fact \[fac:unimod\]. Consider a finite set $J$ of agents with quasiconcave preferences of demand type $\mathcal{D}$ and a total endowment for which exists. By definition, the agents’ at all utility levels are concave and of demand type $\mathcal{D}$. Hence, exist in the for all profiles of utility levels by the “if” direction of Fact \[fac:unimod\]. By the “if” direction of Theorem \[thm:existDualExchange\], must therefore exist in the original economy for all . [@DaKoMu:01] proved a version of the “if” direction of Theorem \[thm:unimod\] under the assumptions that utility functions are monotone in goods, that consumption of goods is nonnegative, and that the total endowment is strictly positive (see Theorems 2 and 4 in [@DaKoMu:01]).[^41] Note that they formulated their result in terms of Fact \[fac:dualPrefs\] and a condition on the ranges of demand correspondences (see their Assumption 3$'$) instead of in terms of unimodular demand types. approach was to show the existence of in a convexified economy and that, under unimodularity, in the convexified economy give rise to in the original economy. In contrast, our approach of using the Equilibrium Existence Duality illuminates the role of substitution effects in ensuring the existence of . Moreover, it yields a maximal domain result for unimodular demand types with income effects. \[prop:netUnimodMaxDomain\] Let $\mathcal{D}$ be a maximal unimodular . If $\|J\| \ge 2$ and $\utilFn{j}$ is not quasiconcave or not of demand type $\mathcal{D}$, then there exist sets $\Feas{k}$ of feasible bundles and concave valuations $\valFn{k}: \Feas{k} \to \mathbb{R}$ of demand type $\mathcal{D}$ for agents $k \not= j$, as well as a total endowment and , for which no exists. Proposition \[prop:netUnimodMaxDomain\] is an immediate consequence of the Equilibrium Existence Duality and the maximal domain result for unimodular demand types under the transferability of utility. By definition, there exists a utility level $\ub$ at which agent $j$’s $\valHDef{j}$ is non-concave or not of demand type $\mathcal{D}$. In either case, Fact \[fac:unimodMaxDomain\] implies that there exist sets $\Feas{k}$ of feasible bundles and concave valuations $\valFn{k}: \Feas{k} \to \mathbb{R}$ of demand type $\mathcal{D}$ for agents $k \not= j$, and a total endowment for which exists but no would exist with transferable utility if agent $j$’s valuation were $\valHDef{j}$. With those sets $\Feas{k}$ of feasible bundles and valuations $\valFn{k}$ for agents $k \not= j$ and that total endowment, the “only if" direction of Theorem \[thm:existDualExchange\] implies that there exists for which no exists. Intuitively, Proposition \[prop:netUnimodMaxDomain\] and Theorem \[thm:unimod\] suggest that Definition \[def:demTypeIncEff\] is the most general way to incorporate income effects into unimodular demand types from the quasilinear setting and ensure the existence of . Indeed, Proposition \[prop:netUnimodMaxDomain\] entails that any domain of preferences that contains all concave quasilinear preferences of a maximal, unimodular demand type and guarantees the existence of must lie within the corresponding demand type constructed in Definition \[def:demTypeIncEff\]. The Strong Substitutes Demand Type and Net with Multiple Units {#sec:ssub} --------------------------------------------------------------- We now use the case of Theorem \[thm:unimod\] for the strong substitutes demand type to extend Theorem \[thm:netSubExist\] to settings in which agents can demand multiple units of some goods. In such settings, if utility is transferable, the condition needed to ensure the existence of is *—the condition requiring that agents see units of goods as [@MiSt:09]. As [@shioura2015gross] and [@BaKl:19] showed, there is a close relationship between strong (net) and the strong substitutes demand type.[^42] 1. A valuation is a valuation* if it corresponds to a valuation when each unit of each good is regarded as a separate good. 2. A utility function is a * utility function* if it corresponds to a utility function when each unit of each good is regarded as a separate good. \[fac:ssubDemTypeConc\] A valuation (resp. utility function) is a strong (net) valuation (resp. utility function) if and only if it is concave (resp. quasiconcave) and of the strong substitutes demand type.[^43][^44] As the strong substitutes is unimodular [@poincare1900second], the existence of under is therefore a special case of the Unimodularity Theorem with Income Effects. \[cor:snsubExist\] If all agents have utility functions, then exist for all . Corollary \[cor:snsubExist\] can also be proven directly using the Equilibrium Existence Duality and the existence of under in transferable utility economies [@MiSt:09; @ikebe2015stability]. Theorem \[thm:netSubExist\] is the special case of Corollary \[cor:snsubExist\] for settings in which agents demand at most one unit of each good. As there are unimodular unrelated to the strong substitutes (such as the one in Example \[eg:5D\]), Theorem \[thm:unimod\] is strictly more general than Corollary \[cor:snsubExist\] (and hence Theorem \[thm:netSubExist\]). In particular, Theorem \[thm:unimod\] also illustrates that certain patterns of net complementarities can also be compatible with the existence of . As the strong substitutes is maximal as a unimodular (see, e.g., Example 9 in [@danilov2004discrete]), Proposition \[prop:netUnimodMaxDomain\] yields a maximal domain result for . \[cor:snsubMax\] If $\|J\| \ge 2$ and $\utilFn{j}$ is not a utility function, then there exist valuations $\valFn{k}$ for agents $k \not= j$, as well as a total endowment and , for which no exists. Auction Design {#sec:auctions} ============== Our work has several implications for auction design. First, our perspective of analyzing preferences by using the expenditure/minimization problem may yield new approaches for extending auction bidding languages to allow for income effects. Second, our equilibrium existence results suggest that some auctions with competitive equilibrium pricing may work well for indivisible goods even in the presence of financing constraints. One set of examples are Product-Mix Auctions, such as the one implemented by the Bank of England[^45]—these implement competitive equilibrium allocations assuming that the submitted sealed bids represent bidders’ actual preferences, since truth-telling is a reasonable approximation in these auctions when there are sufficiently many bidders. However, while we have shown that gross complementarities do not lead to the nonexistence of competitive equilibrium, they do create problems for dynamic auctions. When agents see goods as gross substitutes, iteratively increasing the prices of over-demanded goods leads to a competitive equilibrium [@KeCr:82; @FlJaJaTe:19]. In contrast, when there are gross complementarities between goods, increases in the price of an over-demanded good can lead to other goods being under-demanded due to an income effect. So, even though competitive equilibrium always exists when agents see goods as (strong) net substitutes, it may not be possible to find a competitive equilibrium using a monotone, dynamic auction. In particular, simple “activity rules” that require bidders to bid on a smaller total number of units of goods as prices increase may result in inefficient outcomes. So, the Product-Mix Auction approach of finding competitive equilibrium based on a single round of sealed bids seems especially useful in the presence of income effects. Conclusion {#sec:conclusion} ========== The Equilibrium Existence Duality is a useful tool for analyzing economies with indivisible goods. It is based on the relationship between Marshallian and Hicksian demands, and on an interpretation of Hicksian demand in terms of a quasilinear maximization problem. The Equilibrium Existence Duality shows that competitive equilibrium exists (for all endowment allocations) if and only if competitive equilibrium exists in each of a family of Hicksian economies. An application is that it is net substitutability, not gross substitutability, that is relevant to the existence of equilibrium. And extending the demand types classification of valuations [@BaKl:19] allows us to state a Unimodularity Theorem with Income Effects that gives conditions on the patterns of substitution effects that guarantee the existence of competitive equilibrium. In short, with income effects, just as without them, existence does not depend on agents seeing goods as substitutes; rather, substitution effects are fundamental to the existence of competitive equilibrium. Our results point to a number of potential directions for future work. First, it would be interesting to investigate applications of the Equilibrium Existence Duality to other results on the existence of equilibrium with transferable utility—such as those of [@BiMa:97], [@Ma:98], and [@CaOzPa:15]. Second, our results could be used to further develop auction designs that find competitive equilibrium outcomes given the submitted bids, such as Product-Mix Auction. More broadly, our approach may lead to new results about the properties of economies with indivisibilities and income effects. @fact=@theorem @lemma=@theorem @proposition=@theorem @corollary=@theorem @definition=@theorem @claim=@theorem @remark=@theorem @example=@theorem Proof of Theorem \[thm:existDualExchange\] and Lemma \[lem:dualEconSWT\] {#app:EEDproof} ======================================================================== We prove the following result, which combines Theorem \[thm:existDualExchange\] and Lemma \[lem:dualEconSWT\]. \[thm:existDualExchangeSWT\] Suppose that the total endowment and the sets of feasible bundles are such that exists. The following are equivalent. 1. \[cond:marshall\] exist for all . 2. \[cond:SWT\] For each Pareto-efficient allocation $(\bunnj)_{j \in J}$ with $\sum_{j \in J} \bunj = \tot$, there exists a price vector $\p$ such that $\bunnj \in \dM{j}{\p}{\bunnj}$ for all agents $j$. 3. \[cond:hicks\] exist in the for all profiles of utility levels. The remainder of this appendix is devoted to the proof of Theorem \[thm:existDualExchangeSWT\]. Proof of the \[cond:marshall\] implies \[cond:SWT\] Implication in Theorem \[thm:existDualExchangeSWT\] ------------------------------------------------------------------------------------------------------- The proof of this implication is essentially identical to the proof of Theorem 3 in [@maskin2008fundamental]. Consider a Pareto-efficient allocation $(\bunnj)_{j \in J}$ with $\sum_{j \in J} \bunnj = \tot.$ Let agent $j$’s endowment be $\bunndowj = \bunnj$. By Statement \[cond:marshall\] in the theorem, there exists a , say consisting of the price vector $\p$ and the allocation $(\hbunj)_{j \in J}$ of goods. By the definition of , we have that $\hbunj \in \dM{j}{\p}{\bunj}$ for all agents $j$. In particular, letting $\hnumerj = \numerj - \p \cdot (\hbunj - \bunj)$ for each agent $j,$ we have that $\sum_{j \in J} \hbunnj = \sum_{j \in J} \bunnj$ and that $\util{j}{\hbunnj} \ge \util{j}{\bunnj}$ for all agents $j$. As the allocation $(\bunnj)_{j \in J}$ is Pareto-efficient, we must have that $\util{j}{\hbunnj} = \util{j}{\bunnj}$ for all agents $j$. It follows that $\bunj \in \dM{j}{\p}{\bunnj}$ for all agents $j$—as desired. Proof of the \[cond:SWT\] implies \[cond:hicks\] Implication in Theorem \[thm:existDualExchangeSWT\] ---------------------------------------------------------------------------------------------------- Let $(\ubj)_{j \in J}$ be a profile of utility levels. Consider any allocation $(\bunj)_{j \in J} \in \bigtimes_{j \in J} \Feas{j}$ of goods with $\sum_{j \in J} \bunj = \tot$ that minimizes $$\sum_{j \in J} \cf{j}{\bunj}{\ubj}$$ over all allocations $(\hbunj)_{j \in J} \in \bigtimes_{j \in J} \Feas{j}$ of goods with $\sum_{j \in J} \hbunj = \tot.$ Such an allocation exists because each set $\Feas{j}$ is finite and exists. For each agent $j,$ let $\numerj = \cf{j}{\bunj}{\ubj}$—so $\util{j}{\bunnj} = \ubj$. \[cl:dualPE\] The allocation $(\bunnj)_{j \in J}$ is Pareto-efficient. Consider any allocation $(\hbunnj)_{j \in J} \in \bigtimes_{j \in J} \Feans{j}$ with $\sum_{j \in J} \hbunj = \tot$, and $\util{j}{\hbunnj} \ge \util{j}{\bunnj} = \ubj$ for all agents $j$ with strict inequality for some $j = j_1$. As $\cf{j}{\hbunj}{\cdot}$ is strictly increasing for each agent $j$, we must have that $$\hnumerj = \cf{j}{\hbunj}{\util{j}{\hbunnj}} \ge \cf{j}{\hbunj}{\ubj}$$ for all agents $j$ with strict inequality for $j = j_1$. Hence, we must have that $$\sum_{j \in J} \hnumerj > \sum_{j \in J} \cf{j}{\hbunj}{\ubj} \ge \sum_{j \in J} \cf{j}{\bunj}{\ubj} = \sum_{j \in J} \numerj,$$ where the second inequality follows from the definition of $(\bunj)_{j \in J}$, so the allocation $(\bunnj)_{j \in J}$ cannot be Pareto-dominated. By Claim \[cl:dualPE\] and Statement \[cond:SWT\] in the theorem, there exists a price vector $\p$ such that $\bunj \in \dM{j}{\p}{\bunnj}$ for all agents $j$. Fact \[fac:dualDem\] implies that $\bunj \in \dH{j}{\p}{\ubj}$ for all agents $j$. By Lemma \[lem:dHvalH\], it follows that the price vector $\p$ and the allocation $\left(\bunj\right)_{j \in J}$ of goods comprise a in the for the profile $(\ubj)_{j \in J}$ of utility levels. Proof of the \[cond:hicks\] implies \[cond:marshall\] Implication in Theorem \[thm:existDualExchangeSWT\] --------------------------------------------------------------------------------------------------------- Let $(\bunndowj)_{j \in J}$ be an endowment allocation. For each agent $j,$ we define a utility level $\umin{j} = \util{j}{\bunndowj}$ and let $$\begin{aligned} K^j &= \numerdowj - \min_{\bun \in \Feas{j}} \cf{j}{\bun}{\umin{j}},\end{aligned}$$ which is non-negative by construction. Furthermore, let $K = 1 + \sum_{j \in J} K^j$ and let $$\umax{j} = \max_{\bun \in \Feas{j}} \util{j}{\numerdowj + K, \bun}.$$ Given a profile $\ubvec = (\ubj)_{j \in J}$ of utility levels, let $$\payoffs{\ubvec} = \left\{\left(\begin{array}{l} \cf{j}{\bunj}{\ubj} - \numerdowj\\ + \p \cdot (\bunj - \bundowj) \end{array}\right)_{j \in J} \lgiv \begin{array}{l} \left(\p,(\bunj)_{j \in J}\right) \text{ is a competitive}\\ \text{equilibrium in the \dualecon}\\ \text{for the profile } (\ubj)_{j \in J} \text{ of utility levels} \end{array}\lgivend \right\}$$ denote the set of profiles of net expenditures over all in the for the profile $(\ubj)_{j \in J}$ of utility levels. As discussed in Section \[sec:EED\], the strategy of the proof is to solve for a profile $\ubvec = (\ubj)_{j \in J}$ of utility levels such that $\zero \in \payoffs{\ubvec}$. We first show that the correspondence $\payoffFn: \bigtimes_{j \in J} [\umin{j},\umax{j}] \toto \mathbb{R}^J$ is upper hemicontinuous and has compact, convex values. We then apply a topological fixed point argument to show that there exists a profile $\ubvec = (\ubj)_{j \in J} \in \bigtimes_{j \in J} [\umin{j},\umax{j}]$ of utility levels such that $\zero \in \payoffs{\ubvec}$. We conclude the proof by constructing a for the endowment allocation $(\bunndowj)_{j \in J}$ in the original economy from a in the for the profile $(\ubj)_{j \in J}$ of utility levels. ### Proof of the Regularity Conditions for $\payoffFn$. {#proof-of-the-regularity-conditions-for-payofffn. .unnumbered} We begin by proving that the correspondence $\payoffFn: \bigtimes_{j \in J} [\umin{j},\umax{j}] \toto \mathbb{R}^J$ is upper hemicontinuous and has compact, convex values. We actually give explicit bounds for the range of $\payoffFn$. Let $$\overline{M} = \max_{j \in J} \left\{\cf{j}{\bundowj}{\umax{j}} - \numerdowj\right\}$$ and let $$\underline{M} = \sum_{j \in J} \left(\min_{\bun \in \Feas{j}} \left\{\cf{j}{\bun}{\umin{j}}\right\} - \numerdowj\right) - (\|J\|-1) \overline{M}.$$ \[cl:regT\] The correspondence $\payoffFn: \bigtimes_{j \in J} [\umin{j},\umax{j}] \toto \mathbb{R}^J$ is upper hemicontinuous and has compact, convex values and range contained in $[\underline{M},\overline{M}]^J$. The proof of Claim \[cl:regT\] uses the following technical description of $\payoffFn$. \[cl:hicksEqHelp\] Let $\ubvec = (\ubj)_{j \in J} \in \bigtimes_{j \in J} [\umin{j},\umax{j}]$ be a profile of utility levels and let $(\bunj)_{j \in J} \in \bigtimes_{j \in J} \Feas{j}$ be an allocation of goods with $\sum_{j \in J} \bunj = \tot$. If $(\bunj)_{j \in J}$ minimizes $$\sum_{j \in J} \cf{j}{\hbunj}{\ubj}$$ over all allocations $(\hbunj)_{j \in J} \in \bigtimes_{j \in J} \Feas{j}$ of goods with $\sum_{j \in J} \hbunj = \tot$, then we have that $$\payoffs{\ubvec} = \left\{\rgivend \left(\cf{j}{\bunj}{\ubj} - \numerdowj + \p \cdot (\bunj - \bundowj)\right)_{j \in J} \rgiv \p \in \mathcal{P}\right\},$$ where $$\mathcal{P} = \left\{\p \lgiv \cf{j}{\bunj}{\ubj} + \p \cdot \bunj \le \cf{j}{\bunpr}{\ubj} + \p \cdot \bunpr \text{ for all } j \in J \text{ and } \bunpr \in \Feas{j} \lgivend\right\}.$$ By construction, we have that $$\mathcal{P} = \left\{\p \lgiv \begin{array}{l} \left(\p,(\bunnj)_{j \in J}\right) \text{ is a \ce\ in the}\\ \text{ \dualecon\ for the profile } (\ubj)_{j \in J} \text{ of utility levels} \end{array}\lgivend \right\}.$$ A standard lemma regarding in transferable utility economies shows that in the for the profile $(\ubj)_{j \in J}$ of utility levels, if $\left(\p,(\hbunnj)_{j \in J}\right)$ is a , then so is $\left(\p,(\bunj)_{j \in J}\right)$.[^46] In this case, we have that $$\cf{j}{\bunj}{\ubj} + \p \cdot \bunj = \cf{j}{\hbunj}{\ubj} + \p \cdot \hbunj,$$ and hence that $$\cf{j}{\bunj}{\ubj} - \numerdowj + \p \cdot (\bunj - \bundowj) = \cf{j}{\hbunj}{\ubj} - \numerdowj + \p \cdot (\hbunj - \bundowj),$$ for all agents $j$. The claim follows. It suffices to show that $\payoffFn$ has convex values, range contained in $[\underline{M},\overline{M}]^J$, and a closed graph. We first show that $\payoffs{\ubvec}$ is convex for all $\ubvec \in \bigtimes_{j \in J} [\umin{j},\umax{j}]$. We use the notation of Claim \[cl:hicksEqHelp\] to prove this assertion. Note that $\mathcal{P}$ is the set of solutions to a set of linear inequalities, and is hence convex. Claim \[cl:hicksEqHelp\] implies that $\payoffs{\ubvec}$ is the set of values of a linear function on $\mathcal{P}$—so it follows that $\payoffs{\ubvec}$ is convex as well. We next show that $\payoffs{\ubvec} \subseteq [\underline{M},\overline{M}]^J$ holds for all $\ubvec \in \bigtimes_{j \in J} [\umin{j},\umax{j}].$ We again use the notation of Claim \[cl:hicksEqHelp\]. Let $\ubvec \in \bigtimes_{j \in J} [\umin{j},\umax{j}]$ and $\payoffvec \in \payoffs{\ubvec}$ be arbitrary. By Claim \[cl:hicksEqHelp\], there exists $\p \in \mathcal{P}$ such that $$\payoff{j} = \cf{j}{\bunj}{\ubj} - \numerdowj + \p \cdot (\bunj - \bundowj)$$ for all agents $j$. Note that for all agents $j,$ we must have that $$\payoff{j} \le \cf{j}{\bundowj}{\ubj} - \numerdowj \le \cf{j}{\bundowj}{\umax{j}} - \numerdowj \le \overline{M},$$ where the first inequality holds due to the definition of $\mathcal{P}$, the second inequality holds because $\cf{j}{\bundowj}{\cdot}$ is strictly increasing, and the third inequality holds due to the definition of $\overline{M}$. Furthermore, as $\sum_{j \in J} \bunj = \tot = \sum_{j \in J} \bundowj$, we have that $$\begin{aligned} \sum_{j \in J} \payoff{j} &= \sum_{j \in J} (\cf{j}{\bunj}{\ubj} - \numerdowj).\end{aligned}$$ It follows that $$\begin{aligned} \payoff{j} &= \sum_{k \in J} (\cf{k}{\bunag{k}}{\ubag{k}} - \numerdowag{k}) - \sum_{k \in J \ssm \{j\}} \payoff{k}\\ &\ge \sum_{k \in J} (\cf{k}{\bunag{k}}{\umin{k}} - \numerdowag{k}) - \sum_{k \in J \ssm \{j\}} \payoff{k}\\ &\ge \sum_{k \in J} (\cf{k}{\bunag{k}}{\umin{k}} - \numerdowag{k}) - (\|J\|-1) \overline{M}\\ &\ge \underline{M}\end{aligned}$$ for all agents $j$, where the first inequality holds because $\cf{k}{\bunag{k}}{\cdot}$ is increasing for each agent $k$, the second inequality holds because $\payoff{k} \le \overline{M}$ for all agents $k$, and the third inequality holds due to the definition of $\underline{M}$. Last, we show that $\payoffFn$ has a closed graph. Our argument uses the following version of Farkas’s Lemma. \[fac:farkas\] Let $L_1,L_2$ be disjoint, finite sets and, for each $\ell \in L_1 \cup L_2,$ let $\v^\ell \in \R^I$ be a vector and let $\alpha_\ell$ be a scalar. There exist scalars $\lambda_\ell$ for $\ell \in L_1 \cup L_2$ with $\lambda_\ell \ge 0$ for $\ell \in L_2$ such that $$\sum_{\ell \in L_1 \cup L_2} \lambda_\ell \v^\ell = \zero \quad \text{and} \sum_{\ell \in L_1 \cup L_2} \lambda_\ell \alpha_\ell < 0$$ if and only if there does not exist a vector $\p \in \R^I$ such $\v^\ell \cdot \p \le \alpha_\ell$ for all $\ell \in L_1 \cup L_2$ with equality for all $\ell \in L_1.$ Consider a sequence ${\ubvec_{(1)}},{\ubvec_{(2)}},\ldots \in \bigtimes_{j \in J} [\umin{j},\umax{j}]$ of profiles of utility levels. For each $m$, let ${\payoffvec_{(m)}} \in \payoffs{{\ubvec_{(m)}}}$. Suppose that ${\ubvec_{(m)}} \to \ubvec$ and ${\payoffvec_{(m)}} \to \payoffvec$ as $m \to \infty$. We need to show that $\payoffvec \in \payoffs{\ubvec}$. As each set $\Feas{j}$ is finite and exists, by passing to a subsequence we can assume that there exists an allocation $(\bunj)_{j \in J} \in \bigtimes_{j \in J} \Feas{j}$ of goods with $\sum_{j \in J} \bunj = \tot$ that, for each $m,$ minimizes $$\sum_{j \in J} \cf{j}{\hbunj}{{\ubj_{(m)}}}$$ over all allocations $(\hbunj)_{j \in J} \in \bigtimes_{j \in J} \Feas{j}$ of goods with $\sum_{j \in J} \hbunj = \tot.$ By the continuity of $\cf{j}{\hbunj}{\ub}$ in $\ub$ for each agent $j$, the allocation $(\bunj)_{j \in J}$ minimizes $$\sum_{j \in J} \cf{j}{\hbunj}{\ubj}$$ over all allocations $(\hbunj)_{j \in J} \in \bigtimes_{j \in J} \Feas{j}$ of goods with $\sum_{j \in J} \hbunj = \tot.$ Suppose for sake of deriving a contradiction that $\payoffvec \notin \payoffs{\ubvec}$. Let $L_1 = J$ and let $L_2 = \bigcup_{j \in J} \{j\} \times \Feas{j}.$ Define vectors $\v^\ell \in \mathbb{R}^I$ for $\ell \in L_1 \cup L_2$ by $$\v^\ell = \begin{cases} \bunj - \bundowj & \text{ for } \ell = j \in L_1\\ \bunj - \bunpr & \text{ for } \ell = (j,\bunpr) \in L_2\\ \end{cases}$$ and scalars $\alpha_\ell$ for $\ell \in L_1 \cup L_2$ by $$\alpha_\ell = \begin{cases} \cf{j}{\bunj}{\ubj} - \numerdowj - \payoff{j} & \text{ for } \ell = j \in L_1\\ \cf{j}{\bunpr}{\ubj} - \cf{j}{\bunj}{\ubj} & \text{ for } \ell = (j,\bunpr) \in L_2. \end{cases}$$ By Claim \[cl:hicksEqHelp\], there does not exist a price vector $\p$ such that $\v^\ell \cdot \p \le \alpha_\ell$ for all $\ell \in L_1 \cup L_2$ with equality for all $\ell \in L_1.$ The “if” direction of Fact \[fac:farkas\] therefore guarantees that there exist scalars $\lambda_\ell$ for $\ell \in L_1 \cup L_2$ with $\lambda_\ell \ge 0$ for all $\ell \in L_2$ such that $$\sum_{\ell \in L_1 \cup L_2} \lambda_\ell \v^\ell = \zero \quad \text{and} \quad \sum_{\ell \in L_1 \cup L_2} \lambda_\ell \alpha_\ell < 0.$$ By the definition of the scalars $\alpha_\ell,$ we have that $$\sum_{j \in J} \lambda_j \left(\cf{j}{\bunj}{\ubj} - \numerdowj - \payoff{j}\right) + \sum_{j \in J} \sum_{\bunpr \in \Feas{j}} \lambda_{j,\bunpr} \left(\cf{j}{\bunpr}{\ubj} - \cf{j}{\bunj}{\ubj}\right) < 0.$$ Due the continuity of $\cf{j}{\hbunj}{\ub}$ in $\ub$ for each agent $j$ and because ${\ubvec_{(m)}} \to \ubvec$ and ${\payoffvec_{(m)}} \to \payoffvec$ as $m \to \infty$, there must exist $m$ such that $$\sum_{j \in J} \lambda_j \left(\cf{j}{\bunj}{{\ubj_{(m)}}} - \numerdowj - {\payoff{j}_{(m)}}\right) + \sum_{j \in J} \sum_{\bunpr \in \Feas{j}} \lambda_{j,\bunpr} \left(\cf{j}{\bunpr}{{\ubj_{(m)}}} - \cf{j}{\bunj}{{\ubj_{(m)}}}\right) < 0.$$ Defining scalars $\alpha'_\ell$ for $\ell \in L_1 \cup L_2$ by $$\begin{aligned} \alpha'_\ell &= \begin{cases} \cf{j}{\bunj}{{\ubj_{(m)}}} - \numerdowj - {\payoff{j}_{(m)}} & \text{ for } \ell = j \in L_1\\ \cf{j}{\bunpr}{{\ubj_{(m)}}} - \cf{j}{\bunj}{{\ubj_{(m)}}} & \text{ for } \ell = (j,\bunpr) \in L_2, \end{cases}\end{aligned}$$ we have that $$\sum_{\ell \in L_1 \cup L_2} \lambda_\ell \v^\ell = \zero \quad \text{and that} \quad \sum_{\ell \in L_1 \cup L_2} \lambda_\ell \alpha'_\ell < 0.$$ The “only if” implication of Fact \[fac:farkas\] therefore guarantees that there does not exist a price vector $\p$ such that $\v^\ell \cdot \p \le \alpha'_\ell$ for all $\ell \in L_1 \cup L_2$ with equality for all $\ell \in L_1.$ By Claim \[cl:hicksEqHelp\], it follows that ${\payoffvec_{(m)}} \notin \payoffs{{\ubvec_{(m)}}}$—a contradiction. Hence, we can conclude that $\payoffvec \in \payoffs{\ubvec}$—as desired. ### Completion of the Proof of the \[cond:marshall\] implies \[cond:SWT\] Implication in Theorem \[thm:existDualExchangeSWT\]. {#completion-of-the-proof-of-the-condmarshall-implies-condswt-implication-in-theoremthmexistdualexchangeswt. .unnumbered} We first solve for a profile $\ubvec = (\ubj)_{j \in J}$ of utility levels such that $\zero \in \payoffs{\ubvec}.$ \[cl:fpEq\] Under Statement \[cond:hicks\] in Theorem \[thm:existDualExchangeSWT\], there exists a profile $\ubvec = (\ubj)_{j \in J}$ of utility levels such that $\zero \in \payoffs{\ubvec}$. To prove Claim \[cl:fpEq\], we apply a topological fixed point argument. Consider the compact, convex set $$Z = [\underline{M},\overline{M}]^J \times \bigtimes_{j \in J} [\umin{j},\umax{j}].$$ As $\payoffs{\ubvec} \subseteq [\underline{M},\overline{M}]^J$ for all $\ubvec \in \bigtimes_{j \in J} [\umin{j},\umax{j}],$ we can define a correspondence $\Phi: Z \toto Z$ by $$\Phi(\payoffvec,\ubvec) = \payoffs{\ubvec} \times \argmin_{\hubvec \in \bigtimes_{j \in J} [\umin{j},\umax{j}]} \left\{\sum_{j \in J} \payoff{j} \hubj\right\}.$$ Claim \[cl:regT\] guarantees that $T: \bigtimes_{j \in J} [\umin{j},\umax{j}] \toto \mathbb{R}^J$ is upper hemicontinuous and has compact, convex values. Statement \[cond:hicks\] in Theorem \[thm:existDualExchangeSWT\] ensures that the correspondence $T$ has non-empty values. Because $\bigtimes_{j \in J} [\umin{j},\umax{j}]$ is compact and convex, it follows that the correspondence $\Phi$ is upper hemicontinuous and has non-empty, compact, convex values as well. Hence, Kakutani’s Fixed Point Theorem guarantees that $\Phi$ has a fixed point $(\payoffvec,\ubvec)$. By construction, we have that $\payoffvec \in \payoffs{\ubvec}$ and that $$\label{eq:fpUtil} \ubj \in \argmin_{\hubj \in [\umin{j},\umax{j}]} \payoff{j} \hubj$$ for all agents $j$. It suffices to prove that $\payoffvec = \zero$. Let $\left(\p,(\bunnj)_{j \in J}\right)$ be a in the for the profile $(\ubj)_{j \in J}$ of utility levels with $$\label{eq:fpPayoff} \cf{j}{\bunj}{\ubj} - \numerdowj + \p \cdot (\bunj - \bundowj) = \payoff{j}$$ for all agents $j$. As $\ubj \ge \umin{j}$ and $\cf{j}{\bunj}{\cdot}$ is increasing for each agent $j$, it follows from Equation (\[eq:fpPayoff\]) and the definition of $K^j$ that $$\begin{aligned} \payoff{j} &= \cf{j}{\bunj}{\ubj} - \numerdowj + \p \cdot (\bunj - \bundowj) \nonumber\\ &\ge \cf{j}{\bunj}{\umin{j}} - \numerdowj + \p \cdot (\bunj - \bundowj) \nonumber\\ &\ge \p \cdot (\bunj-\bundowj) - K^j \label{eq:tlb}\end{aligned}$$ for all agents $j$. Next, we claim that $\payoff{j} \le 0$ for all agents $j$. If $\payoff{j} > 0,$ then Equation (\[eq:fpUtil\]) would imply that $\ubj = \umin{j}$. But as $\payoffvec \in \payoffs{\ubvec},$ it would follow that $$\payoff{j} \le \cf{j}{\bundowj}{\umin{j}} - \numerdowj + \p \cdot (\bundowj - \bundowj) = \cf{j}{\bundowj}{\umin{j}} - \numerdowj = 0,$$ where the last equality holds due to the definitions of $\cfFn{j}$ and $\umin{j},$ so we must have that $\payoff{j} \le 0$ for all agents $j$. As $(\bunj)_{j \in J}$ is the allocation of goods in a , we have that $\sum_{j \in J} \bunj = \tot = \sum_{j \in J} \bundowj$ and hence that $$\sum_{j \in J} \p \cdot (\bunj - \bundowj) = 0 \ge \sum_{j \in J} \payoff{j},$$ where the inequality holds because $\payoff{j} \le 0$ for all agents $j$. It follows that for all agents $j,$ we have that $$\payoff{j} - \p \cdot (\bunj - \bundowj) \le \sum_{k \in J \ssm \{j\}} (\p \cdot (\bunag{k} - \bundowag{k}) - \payoff{k}) \le \sum_{k \in J \ssm \{j\}} K^k \le \sum_{k \in J} K^k < K,$$ where the second inequality follows from Equation (\[eq:tlb\]), the third inequality holds because $K^j \ge 0,$ and the fourth inequality holds due to the definition of $K$. Hence, by Equation (\[eq:fpPayoff\]), we have that $$\cf{j}{\bunj}{\ubj} = \numerdowj + \payoff{j} - \p \cdot (\bunj - \bundowj) < \numerdowj + K$$ for all agents $j$. Since utility is strictly increasing in the consumption of money, it follows that $$\ubj = \util{j}{\cf{j}{\bunj}{\ubj},\bunj} < \util{j}{\numerdowj + K,\bunj} \le \umax{j},$$ where the equality holds due to the definition of $\cfFn{j}$ and the second inequality holds due to the definition of $\umax{j}$. Equation (\[eq:fpUtil\]) then implies that $\payoff{j} \ge 0$ for all agents $j$, so we must have that $\payoff{j} = 0$ for all agents $j$. By Claim \[cl:fpEq\], there exists a profile $\ubvec = (\ubj)_{j \in J}$ of utility levels and a $(\p,(\bunj)_{j \in j})$ in the corresponding with $$\label{eq:tFpFinal} \numerdowj = \cf{j}{\bunj}{\ubj} + \p \cdot (\bunj - \bundowj)$$ for all agents $j$. Lemma \[lem:dHvalH\] implies that $\bunj \in \dH{j}{\p}{\ubj}$ for all agents $j$, and we have that $\util{j}{\numerdowj - \p \cdot (\bunj - \bundowj),\bunj} = \ubj$ for all agents $j$ by Equation (\[eq:tFpFinal\]) and the definition of $\cfFn{j}$. It follows from Fact \[fac:dualDem\] that $\bunj \in \dM{j}{\p}{\bunndowj}$ for all agents $j,$ so the price vector $\p$ and the allocation $(\bunj)_{j \in J}$ of goods comprise a in the original economy for the endowment allocation $(\bunndowj)_{j \in J}$. Proof of Proposition \[prop:ssubst\] {#app:grossToNet} ==================================== We actually prove a stronger statement. \[cl:marshallNetWeak\] Suppose that agent $j$ demands at most one unit of each good and let $\bundow \in \Feas{j}$. A utility function $\utilFn{j}$ is a utility function if for all money endowments $\numerdow > \feas{j},$ price vectors $\p,$ and $0 < \mu < \lambda$, whenever 1. \[cond:start\] $\dM{j}{\p}{\bunndow} = \{\bun\}$, 2. \[cond:end\] $\dM{j}{\p + \lambda \e{i}}{\bunndow} = \{\bunpr\}$, 3. \[cond:middle\] $\{\bun,\bunpr\} \subseteq \dM{j}{\p + \mu \e{i}}{\bunndow}$, and 4. \[cond:ineq\] $\bunprComp{i} < \bunComp{i}$, we have that $\bunprComp{k} \ge \bunComp{k}$ for all goods $k \not= i.$ To complete the proof of the proposition from Claim \[cl:marshallNetWeak\], we work in the setting of Claim \[cl:marshallNetWeak\]. Note that, for the endowment $\bundow$ of goods, $\utilFn{j}$ is a utility function when $\bunprComp{k} \ge \bunComp{k}$ holds for all goods $k \not= i$ under Conditions \[cond:start\] and \[cond:end\]. This property clearly implies that $\bunprComp{k} \ge \bunComp{k}$ holds for all goods $k \not= i$ under Conditions \[cond:start\], \[cond:end\], \[cond:middle\], and \[cond:ineq\], and hence that $\utilFn{j}$ is utility function by Claim \[cl:marshallNetWeak\]. The proposition therefore follows from Claim \[cl:marshallNetWeak\]. It remains to prove Claim \[cl:marshallNetWeak\]. In the argument, we use the following characterization of valuations. \[fac:subsDemCplx\] Suppose that agent $j$ demands at most one unit of each good. A valuation $\valFn{j}$ is a valuation if and only if for all price vectors $\p$ with $\|\dQL{j}{\p}\| = 2,$ writing $\dQL{j}{\p} = \{\bun,\bunpr\},$ the difference $\bunpr - \bun$ is a vector with at most one positive component and at most one negative component. We prove the contrapositive. Suppose that $\utilFn{j}$ is not a utility function. We show that there exists a money endowment $\numerdow,$ a price vector $\p,$ price increments $0 < \mu < \lambda$, and goods $i \not= k$ such that Conditions \[cond:start\], \[cond:end\], \[cond:middle\], and \[cond:ineq\] from the statement hold but $\bunprComp{k} < \bunComp{k}.$ By Remark \[rem:netSubDual\], there exists a utility level $\ub$ such that $\valHDef{j}$ is not a valuation. Hence, by Lemma \[lem:dHvalH\] and the “if" direction of Fact \[fac:subsDemCplx\] for $\valFn{j} = \valHDef{j}$, there exists a price vector $\hp$ such that $\|\dH{j}{\hp}{\ub}\| = 2$, and writing $\dH{j}{\hp}{\ub} = \{\bun,\bunpr\},$ the difference $\bunpr - \bun$ has at least two positive components or at least two negative components. Without loss of generality, we can assume that the difference $\bunpr - \bun$ has at least two negative components. Suppose that $\bunprComp{i} < \bunComp{i}$ (so Condition \[cond:ineq\] holds) and that $\bunprComp{k} < \bunComp{k},$ where $i,k \in I$ are distinct goods. Define a money endowment $\numerdow$ by $$\numerdow = \cf{j}{\bun}{\ub} + \hp \cdot (\bun - \bundow) = \cf{j}{\bunpr}{\ub} + \hp \cdot (\bunpr - \bundow);$$ Fact \[fac:dualDem\] implies that $\dM{j}{\hp}{\bunndow} = \{\bun,\bunpr\}$. Let $\mu$ be such that $$\dM{j}{\hp - \mu \e{i}}{\bunndow}, \dM{j}{\hp + \mu \e{i}}{\bunndow} \subseteq \{\bun,\bunpr\};$$ such a $\mu$ exists due to the upper hemicontinuity of $\dMFn{j}$. Let $\p = \hp - \mu \e{i},$ let $\lambda = 2 \mu,$ and let $\ppr = \p + \lambda \e{i} = \hp + \mu \e{i}$. By construction, we have that $\{\bun,\bunpr\} \subseteq \dM{j}{\p + \mu \e{i}}{\bunndow} = \dM{j}{\hp}{\bunndow}$, so Condition \[cond:middle\] holds. It remains to show that $\dM{j}{\p}{\bunndow} = \{\bun\}$ and that $\dM{j}{\ppr}{\bunndow} = \{\bunpr\}$. As $j$ demands at most one unit of each good, we must have that $\bunComp{i} = 1$ and that $\bunprComp{i} = 0.$ We divide into cases based on the value of $\bundowComp{i}$ to show that $$\label{eq:subsIneqs} \begin{aligned} \util{j}{\numerdow - \p \cdot (\bun - \bundow),\bun} &> \util{j}{\numerdow - \p \cdot (\bunpr - \bundow),\bunpr}\\ \util{j}{\numerdow - \ppr \cdot (\bunpr - \bundow),\bunpr} &> \util{j}{\numerdow - \ppr \cdot (\bun - \bundow),\bun}. \end{aligned}$$ $\bundowComp{i} = 0$. In this case, we have that $$\begin{aligned} \util{j}{\numerdow - \p \cdot (\bun - \bundow),\bun} &> \util{j}{\numerdow - \hp \cdot (\bun - \bundow),\bun}\\ &= \util{j}{\numerdow - \hp \cdot (\bunpr - \bundow),\bunpr}\\ &=\util{j}{\numerdow - \p \cdot (\bunpr - \bundow),\bunpr},\end{aligned}$$ where the inequality holds because $\pComp{i} < \hpComp{i}$ and $\bunComp{i} > \bundowComp{i}$, the first equality holds because $\{\bun,\bunpr\} \subseteq \dM{j}{\hp}{\bundow},$ and the second equality holds because $\bunprComp{i} = \bundowComp{i}.$ Similarly, we have that $$\begin{aligned} \util{j}{\numerdow - \ppr \cdot (\bun - \bundow),\bun} &< \util{j}{\numerdow - \hp \cdot (\bun - \bundow),\bun}\\ &= \util{j}{\numerdow - \hp \cdot (\bunpr - \bundow),\bunpr}\\ &= \util{j}{\numerdow - \ppr \cdot (\bunpr - \bundow),\bunpr},\end{aligned}$$ where the inequality holds because $\pprComp{i} > \hpComp{i}$ and $\bunComp{i} > \bundowComp{i}$, the first equality holds because $\{\bun,\bunpr\} \subseteq \dM{j}{\hp}{\bundow},$ and the second equality holds because $\bunprComp{i} = \bundowComp{i}.$ $\bundowComp{i} = 1$. In this case, we have that $$\begin{aligned} \util{j}{\numerdow - \p \cdot (\bunpr - \bundow),\bunpr} &< \util{j}{\numerdow - \hp \cdot (\bunpr - \bundow),\bunpr}\\ &= \util{j}{\numerdow - \hp \cdot (\bun - \bundow),\bun}\\ &= \util{j}{\numerdow - \p \cdot (\bun - \bundow),\bun}\end{aligned}$$ where the inequality holds because $\pComp{i} < \hpComp{i}$ and $\bunprComp{i} < \bundowComp{i}$, the first equality holds because $\{\bun,\bunpr\} \subseteq \dM{j}{\hp}{\bundow},$ and the second equality holds because $\bunComp{i} = \bundowComp{i}.$ Similarly, we have that $$\begin{aligned} \util{j}{\numerdow - \ppr \cdot (\bunpr - \bundow),\bunpr} &> \util{j}{\numerdow - \hp \cdot (\bunpr - \bundow),\bunpr}\\ &= \util{j}{\numerdow - \hp \cdot (\bun - \bundow),\bun}\\ &= \util{j}{\numerdow - \ppr \cdot (\bun - \bundow),\bun},\end{aligned}$$ where the inequality holds because $\pprComp{i} > \hpComp{i}$ and $\bunprComp{i} < \bundowComp{i}$, the first equality holds because $\{\bun,\bunpr\} \subseteq \dM{j}{\hp}{\bundow},$ and the second equality holds because $\bunComp{i} = \bundowComp{i}.$ As $\bundow \in \Feas{j} \subseteq \{0,1\}^I,$ the cases exhaust all possibilities. Hence, we have proven that Equation (\[eq:subsIneqs\]) must hold. As $\dM{j}{\p}{\bunndow},\dM{j}{\ppr}{\bunndow} \subseteq \{\bun,\bunpr\},$ we must have that $\dM{j}{\p}{\bunndow} = \{\bun\}$ and that $\dM{j}{\ppr}{\bunndow} = \{\bunpr\}$—so Conditions \[cond:start\] and \[cond:end\] hold, as desired. [FOR ONLINE PUBLICATION]{} Proofs of Facts \[fac:dualDem\] and \[fac:dualPrefs\] {#app:dualDemPrefs} ===================================================== Proof of Fact \[fac:dualDem\] ----------------------------- We begin by proving two technical claims. \[cl:dualDemHelpMtoH\] Let $\bunndow \in \Feans{j}$ be an endowment and let $\ub$ be a utility level. If $$\label{eq:marshallForProofOfDual1} \ub = \max_{\bunn \in \Feans{j} \mid \pall \cdot \bunn \le \pall \cdot \bunndow} \util{j}{\bunn},$$ then we have that $$\pall \cdot \bunndow = \min_{\bunn \in \Feans{j} \mid \util{j}{\bunn} \ge \ub} \pall \cdot \bunn$$ and that $\dM{j}{\p}{\bunndow} \subseteq \dH{j}{\p}{\ub}$. Letting $\bunpr \in \dM{j}{\p}{\bunndow}$ be arbitrary and $\numerpr = \numerdow - \p \cdot (\bunpr - \bundow),$ we have that $\util{j}{\bunnpr} = u$ and that $\pall \cdot \bunnpr \le \pall \cdot \bunndow$ by construction. It follows that $$\pall \cdot \bunndow \ge \min_{\bunn \in \Feans{j} \mid \util{j}{\bunn} \ge \ub} \pall \cdot \bunn.$$ Suppose for the sake of deriving a contradiction that there exists $\bunndpr \in \Feans{j}$ with $\pall \cdot \bunndpr < \pall \cdot \bunndow$ and $\util{j}{\bunndpr} \ge \ub.$ Then, we have that $\numerdpr < \numerdow + \p \cdot (\bundpr - \bundow)$; write $\numertpr = \numerdow + \p \cdot (\bundpr - \bundow),$ so $\numertpr > \numerdpr.$ Since $\utilFn{j}$ is strictly increasing in consumption of money, it follows that $\util{j}{\numertpr,\bundpr} > u$—contradicting Equation (\[eq:marshallForProofOfDual1\]) as $\numertpr + \p \cdot \bundpr = \pall \cdot \bunndow.$ Hence, we can conclude that $$\pall \cdot \bunndow = \min_{\bunn \in \Feans{j} \mid \util{j}{\bunn} \ge \ub} \pall \cdot \bunn.$$ Since $\util{j}{\bunnpr} = \ub$ and $\pall \cdot \bunnpr = \pall \cdot \bunndow,$ it follows that $\bunpr \in \dH{j}{\p}{\ub}$. Since $\bunpr \in \dM{j}{\p}{\bunndow}$ was arbitrary, we can conclude that $\dM{j}{\p}{\bunndow} \subseteq \dH{j}{\p}{\ub}$. \[cl:dualDemHelpHtoM\] Let $\bunndow \in \Feans{j}$ be an endowment and let $\ub$ be a utility level. If $$\label{eq:hicksForProofOfDual1} \pall \cdot \bunndow = \min_{\bunn \in \Feans{j} \mid \util{j}{\bunn} \ge \ub} \pall \cdot \bunn,$$ then we have that $$\ub = \max_{\bunn \in \Feans{j} \mid \pall \cdot \bunn \le \pall \cdot \bunndow} \util{j}{\bunn}$$ and that $\dH{j}{\p}{\ub} \subseteq \dM{j}{\p}{\bunndow}$. Let $\bunpr \in \dH{j}{\p}{\ub}$ be arbitrary and $\numerpr = \cf{j}{\bunpr}{\ub}$. We have that $\util{j}{\bunnpr} \ge \ub$ and that $\pall \cdot \bunnpr = \pall \cdot \bunndow$ by construction. It follows that $$\ub \le \max_{\bunn \in \Feans{j} \mid \pall \cdot \bunn \le \pall \cdot \bunndow} \util{j}{\bunn}.$$ We next show that $$\ub = \max_{\bunn \in \Feans{j} \mid \pall \cdot \bunn \le \pall \cdot \bunndow} \util{j}{\bunn}.$$ Suppose for sake of deriving a contradiction that there exists $\bunndpr \in \Feans{j}$ with $\pall \cdot \bunndpr \le \pall \cdot \bunndow$ and $\util{j}{\bunndpr} > \ub.$ By definition of $\cfFn{j}$, we know that $\numerdpr > \cf{j}{\bundpr}{\ub}$. Letting $\numertpr = \cf{j}{\bundpr}{\ub}$, we have that $\numertpr + \p \cdot \bundpr < \pall \cdot \bunndow$, which contradicts Equation (\[eq:hicksForProofOfDual1\]) as $\util{j}{\numertpr,\bundpr} = \ub.$ Hence, we can conclude that $$\ub = \max_{\bunn \in \Feans{j} \mid \pall \cdot \bunn \le \pall \cdot \bunndow} \util{j}{\bunn}.$$ Since $\util{j}{\bunpr} = u$ and $\pall \cdot \bunpr = \pall \cdot \bunndow,$ it follows that $\bunpr \in \dM{j}{\p}{\bunndow}$. Since $\bunpr \in \dH{j}{\p}{\ub}$ was arbitrary, we can conclude that $\dH{j}{\p}{\ub} \subseteq \dM{j}{\p}{\bunndow}$. Let $\bunndow \in \Feans{j}$ be an endowment and let $\ub$ be a utility level. By Claims \[cl:dualDemHelpMtoH\] and \[cl:dualDemHelpHtoM\], Conditions (\[eq:marshallForProofOfDual1\]) and (\[eq:hicksForProofOfDual1\]), are equivalent, and under these equivalent conditions, we have that $\dM{j}{\p}{\bunndow} \subseteq \dH{j}{\p}{\ub}$ and that $\dH{j}{\p}{\ub} \subseteq \dM{j}{\p}{\bunndow}$. Hence, we must have that $\dM{j}{\p}{\bunndow} = \dH{j}{\p}{\ub}$ under the equivalent Conditions (\[eq:marshallForProofOfDual1\]) and (\[eq:hicksForProofOfDual1\])—as desired. Proof of Fact \[fac:dualPrefs\] ------------------------------- We prove the “if” and “only if” directions separately. ### Proof of the “If" Direction. {#proof-of-the-if-direction. .unnumbered} We define a utility function $\utilFn{j}$ implicitly by $$\util{j}{\bunn} = F(\bun,\cdot)^{-1}(-\numer),$$ which is well-defined, continuous, and strictly increasing in $\numer$ by the Inverse Function Theorem because $F(\bun,\cdot)$ is continuous, strictly decreasing, and satisfies Condition (\[eq:vlimits\]). Condition (\[eq:ulimits\]) holds because $F$ is defined over the entirety of $\Feas{j} \times \feasUtil{j}.$ ### Proof of the “Only If" Direction. {#proof-of-the-only-if-direction. .unnumbered} We define $F:\Feas{j} \times (\minu{j},\maxu{j}) \to (-\infty,-\feas{j})$ implicitly by $$F(\bun,\ub) = -\util{j}{\cdot,\bun}^{-1}(\numer),$$ which is well-defined, continuous, and strictly decreasing in $\ub$ by the Inverse Function Theorem because $\util{j}{\cdot,\bun}$ is continuous, strictly increasing, and satisfies Condition (\[eq:ulimits\]). Condition (\[eq:vlimits\]) holds because $\utilFn{j}$ is defined over the entirety of $\Feans{j}$. Proofs of the Maximal Domain and Necessity Results for Settings with Transferable Utility {#app:maxDomain} ========================================================================================= In this appendix, we supply proofs of Facts \[fac:subMaxDomain\] and \[fac:unimodMaxDomain\], as well as the “only if" direction of Fact \[fac:unimod\]. Utility is transferable throughout this appendix. We use the concept of a pseudo-equilibrium price vector. Suppose that utility is transferable. A pseudo-equilibrium price vector is a price vector $\p$ such that $$\tot\in \conv\left(\sum_{j \in J} D^j(\p)\right).$$ There is a connection between pseudo-equilibrium price vectors, , and the existence problem. \[fac:pseudoEquil\] If utility is transferable and the total endowment is such that a competitive equilibrium exists, then, for each pseudo-equilibrium price vector $\p$, there exists an allocation $(\bunj)_{j \in J}$ such that $\p$ and $(\bunj)_{j \in J}$ comprise a . The nonexistence of may therefore be demonstrated by using the contrapositive of Fact \[fac:pseudoEquil\]. Our arguments use valuations that are linear on their domain. That is, let $\mathbf{t}_I\in\R^I$, let $X^j_I\subseteq\Z^n$ be finite, and let $V^j=V^{j,\mathbf{t}_I}\mathbf:X^j_I\rightarrow\R$ be given by $V^{j,\mathbf{t}_I}(\mathbf{x}_I)\coloneq\mathbf{t}_I\cdot\mathbf{x}_I$ for all $\mathbf{x}_I\in X^j_I$. Recalling Equation (\[eqn:quasilin\]) for demand sets in the quasilinear case, we observe that, for each $\mathbf{s}_I\in\R^I$, we have that $$\label{eqn:faces} D^j(\mathbf{t}_I-\mathbf{s}_I)=\argmax_{\mathbf{x}_I\in X^j_I}\left(\mathbf{t}_I\cdot\mathbf{x}_I-(\mathbf{t}_I-\mathbf{s}_I)\cdot\mathbf{x}_I\right)=\argmax_{\mathbf{x}_I\in X^j_I}\mathbf{s}_I\cdot\mathbf{x}_I.$$ \[lem:linConv\] If $\conv(X^j_I)\cap\Z^I=X^j_I$, then $V^{j,\mathbf{t}_I}$ is concave for all $\mathbf{t}_I\in\R^I$. Observe by Equation (\[eqn:faces\]) that $D^j(\mathbf{t}_I)=\argmax_{\mathbf{x}_I\in X^j_I}\mathbf{0}\cdot\mathbf{x}_I=X^j_I$. So, if $\mathbf{x}_I\in\conv(X^j_I)\cap\Z^I=X^j_I$ then $\mathbf{x}_I\in D^j(\mathbf{t}_I)$. By Definition \[def:concave\], we know $V^{j,\mathbf{t}_I}$ is concave. We will also make use of an alternative characterization of concavity. \[fac:conc\] A valuation $\valFn{j}$ is concave if and only if $\conv\left(\dQL{j}{\p}\right) \cap \Z^I = \dQL{j}{\p}$ for all price vectors $\p$. Additional Facts regarding Unimodularity and Demand Types --------------------------------------------------------- The following results are especially useful in the proof of the “only if” direction of Fact \[fac:unimod\], and the proof of Fact \[fac:unimodMaxDomain\]. We seek to construct pseudo-equilibrium price vectors (the total endowment is in the convex hull of aggregate demand) that are not price vectors (the total endowment is not demanded on aggregate). Failure of unimodularity allows such constructions because of the following property. \[fac:unimodSet\] A demand type vector set $\mathcal{D}$ is unimodular if and only if there is no linearly independent subset $\{\mathbf{d}^1,\ldots,\mathbf{d}^r\}$ of $\mathcal{D}$ such that there exists $\mathbf{z}=\sum_{\ell=1}^r \alpha_\ell \dvec^\ell\in\Z^I$ with $\alpha_\ell\in(0,1)$ for $\ell=1,\ldots,r$. To see the connection to Fact \[fac:pseudoEquil\] and to existence of competitive equilibrium, suppose that $\{\mathbf{d}^1,\ldots,\mathbf{d}^r\}$ and $\mathbf{z}$ are as in Fact \[fac:unimodSet\]. If $\tot=\mathbf{z}$ and if $D^j_M(\p,\bunndowj)=\{\mathbf{0},\mathbf{d}^j\}$ for $j=1,\dots,r$, then $\p$ is a pseudo-equilibrium price vector but there is no competitive equilibrium at $\p$. [@BaKl:19] generalized Fact \[fac:subsDemCplx\] to the general case of transferable utility. \[fac:demTypeCplx\] Let $\valFn{j}$ be a valuation of demand type $\mathcal{D}$. For any price $\ppr$, if $\conv(\dQL{j}{\ppr})$ has an edge $E$, then the difference between the extreme points of $E$ is proportional to a demand type vector, and there exists a price $\p$ such that $\conv(\dQL{j}{\p})=E$. Moreover if $\dvec$ is in the minimal $\mathcal{D}$, such that $\valFn{j}$ is of demand type $\mathcal{D}$, then there exists a price vector $\p$ such that $\conv(\dQL{j}{\p})$ is a line segment, the difference between whose endpoints is proportional to $\dvec.$ We also demonstrate now the following useful corollary of Fact \[fac:demTypeCplx\]. \[cor:edges\] Let $V^j=V^{j,\mathbf{t}_I}$ for some $\mathbf{t}_I\in\R^I$, and let $\mathcal{D}$ be the minimal demand type vector set such that $V^j$ is of demand type $\mathcal{D}$. Then $\mathcal{D}$ consists of the primitive integer vectors in the directions of the edges of the polytope $\conv(X^j_I)$. Observe that $D^j(\mathbf{t}_I)=X^j_I$ and so, by Fact \[fac:demTypeCplx\], each edge of $\conv(X^j_I)$ is proportional to a vector in $\mathcal{D}$. Conversely, if $\mathbf{d}\in\mathcal{D}$ then, by Fact \[fac:demTypeCplx\], there exists a price $\p$ such that $\conv(D^j(\p))$ is a line segment, the difference between whose endpoints is proportional to $\mathbf{d}$. But writing $\mathbf{s}_I=\mathbf{t}_I-\p$, we see from Equation (\[eqn:faces\]) that $D^j(\p)=\argmax_{x_I\in X^j_I} \mathbf{s}_I\cdot\mathbf{x}_I$ which tells us (cf. e.g. @Gruenbaum1967, Section 2.4) that $E$ is an edge of $\conv(X^j_I)$. Our proofs of Facts \[fac:subMaxDomain\] and \[fac:unimodMaxDomain\], and the “only if” direction of Fact \[fac:unimod\], now follow the same structure. Within each argument, we address a demand type which is not unimodular. Observe by Fact \[fac:unimodSet\] that when unimodularity fails for a set of vectors $\mathcal{D}$, then there exist polytopes, with integer vertices and whose edge directions are in $\mathcal{D}$, that contain a non-vertex integer vector, $\mathbf{z}$. We use Corollary \[cor:edges\] construct valuations of the appropriate demand type such that, at some price $\p$, the convex hull of the aggregate demand set is a polytope with these properties; and such that there exists a feasible endowment allocation is the total endowment is the non-vertex integer vector $\mathbf{z}$. Thus $\p$ is a pseudo-equilibrium price. Moreover, we design our individual valuations so that $\p$ is not a competitive equilibrium. The contrapositive of Fact \[fac:pseudoEquil\] can then be applied to show the non-existence of competitive equilibrium. Proof of Fact \[fac:subMaxDomain\] ---------------------------------- By Fact \[fac:subsDemCplx\], there exists a price vector $\p$ such that $\dQL{j}{\p} = \{\bunpr,\bunpr+\norm\}$, where $\norm$ has at least two positive components or at least two negative components. Identify $I$ with $\{1,\ldots,\|I\|\}$ and without loss of generality assume that $g_1,g_2<0$. Because agent $j$ demands at most one unit of each good, we know that $\bunpr,\bunpr+\norm\in\{0,1\}^{\|I\|}$ and so $\norm\in \{-1,0,1\}^{\|I\|}$. We conclude both that $g_1=g_2=-1$ and that $x'_1=x'_2=1$. Let $k \in J \ssm \{j\}$ be arbitrary. For agents $j' \in J \ssm \{j,k\},$ let $\Feas{j'} = \{\zero\}$, let $\valFn{j'}$ be arbitrary, and let $\bundowag{j'} = 0$. Let $X^k_I\coloneq\{\rgivend\mathbf{x}_I\in\{0,1\}^{\|I\|}\rgiv x_1+x_2\leq 1\}$ and let $\mathbf{t}_I\coloneq\p-\mathbf{e}^1-\mathbf{e}^2$. Let $V^k = V^{k,{\bf t}_I},$ which is a valuation by Example 11 and Corollary D.2, because each edge of $\conv(X^k_I)$ is proportional to either ${\bf e}^1 - {\bf e}^2$ or to ${\bf e}^\ell$ for some $\ell \in I$; or, alternatively, by Theorem 4 in Hatfield et al. (2019) because, like assignment valuations, it is the supremal convolution of $\|I\|-1$ unit-demand valuations. Set $\mathbf{w}^j_I\coloneq\bunpr\in X^j_I$ and set $\mathbf{w}^k_I\coloneq\tot-\bunpr$. Since $\bunpr\in\{0,1\}^{\|I\|}$ it follows that $\mathbf{w}^k_I\in\{0,1\}^{\|I\|}$, and moreover since $x'_1=x'_2=1$ we know $w^k_1=w^k_2=0$; thus $\mathbf{w}^k_I\in X^k_I$. Now $(\mathbf{w}^{j'}_I)_{j'\in J}$ is clearly an endowment allocation. By Equation (\[eqn:faces\]) we know that $$D^k(\mathbf{p}_I)=\argmax_{\mathbf{x}_I\in X^k_I}(\mathbf{e}^1+\mathbf{e}^2)\cdot\mathbf{x}_I=\{\mathbf{x}_I\in\{0,1\}^{\|I\|}\|x_1+x_2=1\}.$$ Observing that $\mathbf{e}^2\in D^k(\mathbf{p}_I)$ and considering the vectors from $\mathbf{e}^2$ to other elements of the demand set, we can write $D^k(\mathbf{p}_I)$ as $$D^k(\p) = \e{2} + \left\{\rgivend \alpha_2 (\e{1} - \e{2}) + \sum_{\ell = 3}^{\|I\|} \alpha_\ell \e{\ell} \rgiv \alpha_\ell \in \{0,1\} \text{ for } 2 \le \ell \le \|I\|\right\}.$$ Combining this with agent $j$, and recalling other agents’ demand sets are identically zero, we conclude that $$\sum_{j'\in J} D^{j'}(\mathbf{p}_I)=\bunpr+\mathbf{e}^2+\left\{\rgivend\alpha_1\norm+\alpha_2(\mathbf{e}^1-\mathbf{e}^2)+\sum_{\ell=3}^{\|I\|}\alpha_\ell\mathbf{e}^\ell\rgiv\alpha_\ell\in\{0,1\}\text{ for }1\leq \ell\leq \|I\|\right\}.$$ The convex hull of this set can be expressed very similarly, but the weights $\alpha_\ell$ are allowed to lie in $[0,1]$. Since $\bunpr, \bunpr+\mathbf{g}\in\{0,1\}^{\|I\|},$ we have that if $g_i=1$ (resp. $g_i = -1$), then $x'_i=0$ (resp. $x'_i = 1$). Taking $$\alpha_\ell = \begin{cases} \dfrac{\|\normComp{\ell}\|}{2} & \text{ if } \normComp{\ell} \not= 0\\ 1 - \bunprComp{\ell} & \text{if } \normComp{\ell} = 0 \end{cases}$$ for $1 \le \ell \le \|I\|$, we have that $$\begin{aligned} \bunprComp{i} + \alpha_1 \normComp{i} + \alpha_i &= 1 - \frac{1}{2} + \frac{1}{2} = 1 & \text{ for all } i \in I \text{ with } \normComp{i} = -1\\ \bunprComp{i} + \alpha_1 \normComp{i} + \alpha_i &= \bunprComp{i} + 0 + (1 - \bunprComp{i}) = 1 & \text{ for all } i \in I \text{ with } \normComp{i} = 0\\ \bunprComp{i} + \alpha_1 \normComp{i} + \alpha_i &= 0 + \frac{1}{2} + \frac{1}{2} = 1 & \text{ for all } i \in I \text{ with } \normComp{i} = 1.\end{aligned}$$ As $\bunprComp{1} = \bunprComp{2} = 1$ and $\normComp{1} = \normComp{2} = -1,$ it follows that $$\bunpr + \e{2} + \alpha_1\norm+\alpha_2(\mathbf{e}^1-\mathbf{e}^2)+\sum_{\ell=3}^{\|I\|}\alpha_\ell\mathbf{e}^\ell = \tot.$$ As $\alpha_\ell \in [0,1]$ for all $1 \le \ell \le \|I\|$, we therefore have that $\tot \in \conv\left(\sum_{j'\in J} D^{j'}(\mathbf{p}_I)\right)$, so $\p$ is a pseudo-equilibrium price vector. But as $\alpha_1 \in (0,1)$ and the vectors $\mathbf{g},\mathbf{e}^1-\mathbf{e}^2,\mathbf{e}^3,\ldots,\mathbf{e}^{\|I\|}$ are linearly independent, we have that $\tot \notin \sum_{j'\in J} D^{j'}(\mathbf{p}_I),$ so there is no at $\p$.[^47] Therefore, by the contrapositive of Fact \[fac:pseudoEquil\], no can exist. Proof of the “Only If” Direction of Fact \[fac:unimod\] ------------------------------------------------------- Let $\mathcal{D}$ be a that is not unimodular. We need to show that there exists a finite set $J$ of agents with concave valuations of demand type $\mathcal{D}$, as well as a total endowment, for which there exists but no . We will use $J=\{j,k\}$. Let $\linset = \{\dvec^1,\ldots,\dvec^n\} \subseteq \mathcal{D}$ be a minimal non-unimodular subset. By construction, $\linset$ is linearly independent, and $\{\dvec^1,\ldots,\dvec^{n-1}\}$ is unimodular. Let $$\mathcal{P} = \left\{\rgivend\sum_{\ell=1}^n \alpha_\ell \dvec^\ell \rgiv 0 \le \alpha_\ell \le 1\text{ for }\ell=1,\ldots,n\right\}$$ denote the parallelepiped spanned by $\linset$. By Fact \[fac:unimodSet\], there exists $\mathbf{z} = \sum_{\ell=1}^n \beta_\ell \dvec^\ell \in \mathcal{P} \cap \mathbb{Z}^I$ with $\beta_\ell \in (0,1)$ for all $\ell=1,\ldots,n$. Let $X^j_I\coloneq \mathcal{P}\cap\Z^n$ and let $V^j\coloneq V^{j,\mathbf{0}}$ be the linear valuation which is identically zero on its domain. Recall Equation (\[eqn:faces\]): we know $D^j(\mathbf{0})=X^j_I$. Observe that $\mathbf{z}\in X^j_I$. Clearly $\conv(X^j_I)\cap\Z^I=X^j_I$ and so $V^j$ is concave by Lemma \[lem:linConv\]. Let $\mathbf{s}_I$ satisfy $\mathbf{s}_I\cdot \dvec^\ell= 0$ for $\ell=1,\ldots,n-1$ and $\mathbf{s}_I\cdot\dvec^n>0$. (Such an $\mathbf{s}_I$ exists as $\linset$ is linearly independent.) Then, for $\mathbf{x}_I=\sum_{\ell=1}^n \alpha_\ell \dvec^\ell\in X^j_I$, we have $$\mathbf{s}_I\cdot\mathbf{x}_I=\sum_{\ell=1}^n \alpha_\ell \mathbf{s}_I\cdot\dvec^\ell=\alpha_n \mathbf{s}_I\cdot\dvec^n.$$ We assumed that $\mathbf{s}_I\cdot\dvec^n>0$, so $\mathbf{s}_I\cdot\mathbf{x}_I$ is minimized when $\alpha_n=0$; equivalently $-\mathbf{s}_I\cdot\mathbf{x}_I$ is maximized when $\alpha_n=0$. So, by Equation (\[eqn:faces\]), we know that $$D^j(\mathbf{s}_I)=\argmax_{\mathbf{x}_I\in X^j_I}-\mathbf{s}_I\cdot\mathbf{x}_I=\left\{\rgivend\sum_{\ell=1}^{n-1} \alpha_\ell \dvec^\ell \rgiv 0 \le \alpha_\ell \le 1 \text{ for }\ell=1,\ldots,n-1\right\}\cap\Z^I.$$ Now set $X^k_I\coloneq\{\mathbf{0},\dvec^n\}$ and let $V^k\coloneq V^{k,\mathbf{s}_I}$. By Equation (\[eqn:faces\]) again, we know that $D^k(\mathbf{s}_I)=X^k_I$. As $\mathbf{d}^n\in\mathcal{D}$, which is a , we know that $\mathbf{d}^n$ is a primitive integer vector, from which it follows that $\conv(X^k_I)\cap\Z^I=X^k_I$. Thus, by Lemma \[lem:linConv\], we know that $V^k$ is concave. Observe that $$D^j(\mathbf{s}_I)+D^k(\mathbf{s}_I)=\left\{\rgivend\sum_{\ell=1}^n \alpha_\ell \dvec^\ell \rgiv 0 \le \alpha_\ell \le 1 \text{ for }\ell=1,\ldots,n-1 \text{ and }\alpha_n\in\{0,1\}\right\} \cap \Z^I.$$ So $\conv(D^j(\mathbf{s}_I)+D^k(\mathbf{s}_I))=\mathcal{P}$. Let the total endowment $\tot$ be $\mathbf{z}$. Set $\mathbf{w}^j_I\coloneq\mathbf{z}\in X^j_I$, and set $\mathbf{w}^k_I\coloneq\mathbf{0}\in X^k_I$. This is clearly an endowment allocation. Since $\tot\in\mathcal{P}=\conv(D^j(\mathbf{s}_I)+D^k(\mathbf{s}_I))$, we see $\mathbf{s}_I$ is a pseudo-equilibrium price vector. But, since $\linset$ is linearly independent and since $0<\beta_n<1$, we know $\tot\notin D^j(\mathbf{s}_I)+D^k(\mathbf{s}_I)$, so there is no competitive equilibrium at $\mathbf{s}_I$. It follows, by the contrapositive of Fact \[fac:pseudoEquil\], that no can exist. Proof of Fact \[fac:unimodMaxDomain\] ------------------------------------- We will use the following claim. \[cla:maximal\] If $\mathcal{D}$ is a maximal unimodular then $\mathcal{D}$ spans $\mathbb{R}^I$. Let $\linset \subseteq \mathcal{D}$ be a maximal, linearly independent set. As $\mathcal{D}$ is unimodular, there exists a set $T$ of such that $\linset \cap T = \emptyset$ and $\linset \cup T$ is a basis of $\mathbb{R}^I$ with determinant $\pm 1.$ We claim that $\mathcal{D}_0 = \mathcal{D} \cup T \cup -T$ is unimodular. To see why, let $L' \subseteq \mathcal{D} \cup T\cup -T$ be a maximal linearly independent set. As $\mathcal{D}_0$ spans $\mathbb{R}^I$ by construction, $L'$ must span $\mathbb{R}^I$. Due to the maximality of $\linset$, we must have that $\|L' \cap (T \cup -T)\| = \|T\|.$ It follows that $L' \cap \mathcal{D}$ is a basis for the span of $\mathcal{D}$. As $\mathcal{D}$ is unimodular, $L' \cap \mathcal{D}$ must be the image of $\linset$ under a unimodular change of basis of the span of $\mathcal{D}$. It follows that $L'$ is a basis for $\mathbb{R}^I$ with determinant $\pm 1$—so $\mathcal{D}_0$ is unimodular. Due to the maximality of $\mathcal{D},$ we must have that $T = \emptyset$, and hence $\mathcal{D}$ must span $\mathbb{R}^I$. As $\mathcal{D}$ is unimodular, it follows that $\mathcal{D}$ must integrally span $\mathbb{Z}^I$. We next divide into cases based on whether $\valFn{j}$ is non-concave and of demand type $\mathcal{D}$, or not of demand type $\mathcal{D}$, to construct concave valuations $\valFn{k}$ of demand type $\mathcal{D}$ for agents $k \not= j$ and a total endowment for which no exists. $\valFn{j}$ is not concave but is of demand type $\mathcal{D}$. By Fact \[fac:conc\], there exists a price vector $\p$ such that $\dQL{j}{\p} \not= \conv\left(\dQL{j}{\p}\right) \cap \Z^I.$ Let $\bunpr \in \dQL{j}{\p}$ be an extreme point of $\conv\left(\dQL{j}{\p}\right)$, so there exists $\mathbf{s}_I\in\R^n$ satisfying $$\label{eqn:xprime} \{\bunpr\}=\argmax_{\mathbf{y}_I\in \dQL{j}{\p}}\mathbf{s}_I\cdot\mathbf{y}_I.$$ Let $\bundpr \in \left(\conv\left(\dQL{j}{\p}\right) \cap \Z^I\right) \ssm \dQL{j}{\p}$ be arbitrary. Let $k \in J \ssm \{j\}$ be arbitrary. Let $X^k_I=(\conv(D^j(\p))\cap\Z^I)+\{-\bunpr\}$. Since $V^j$ is of demand type $\mathcal{D}$, it follows by Fact \[fac:demTypeCplx\] that every edge of $\conv(D^j(\p))$ is a multiple of a vector in $\mathcal{D}$, and so the same is true of $\conv(X^k_I)$. Moreover, by definition of $X^k_I$ it is clear that $\conv(X^k_I)\cap\Z^I=X^k_I$. Fix $\mathbf{t}_I\coloneq\p+\mathbf{s}_I$ and let $V^k\coloneq V^{\mathbf{t}_I,k}$, which is concave by Lemma \[lem:linConv\] and of demand type $\mathcal{D}$ by Corollary \[cor:edges\]. By Equation (\[eqn:faces\]) we know $D^k(\p)=\argmax_{\mathbf{x}_I\in X^k_I}\mathbf{s}_I\cdot\mathbf{x}_I$, and so by Equation (\[eqn:xprime\]) and the definition of $X^k_I$, it follows that $D^k(\p)=\{\bunpr-\bunpr\}=\{\mathbf{0}\}$. Let the total endowment $\tot$ be $\mathbf{x}''_I$, let $\bundowj \coloneq \bunpr\in X^j_I$, and let $\mathbf{w}^k_I\coloneq\bundpr-\bunpr\in X^k_I$. For agents $j' \in J \ssm \{j,k\}$, let $\Feas{j'} = \{\zero\}$, let $\valFn{j'}$ be arbitrary, and let $\bundowag{j'} = \zero.$ Thus $(\mathbf{w}^{j'}_I)_{j'\in J}$ is an endowment allocation. Moreover, $$\sum_{j'\in J}D^{j'}(\p)=D^j(\p).$$ Thus $\tot=\bundpr\in \conv\left(\sum_{j'\in J}D^{j'}(\p)\right)$ and so $\p$ is a pseudo-equilibrium price vector. But $\tot=\bundpr\notin \sum_{j'\in J}D^{j'}(\p)=D^j(\p)$ by definition of $\bundpr$, and so there is no at $\p$. Therefore, by the contrapositive of Fact \[fac:pseudoEquil\], no can exist. $V^j$ is not of demand type $\mathcal{D}$. By Fact \[fac:demTypeCplx\] there exists a primitive integer vector $\norm\notin\mathcal{D}$ and a price vector $\p\in\R^n$ such that $D^j(\p)\subseteq\{\bunpr+\alpha\norm\ \| \ \alpha=0,\ldots,r\}$ where $r\geq 1$ and $\bunpr,\bunpr+r\norm\in D^j(\p)$. As $\mathcal{D}$ is not strictly contained in any unimodular , and as $\norm\notin\mathcal{D}$, the set $\mathcal{D}\cup\{\norm\}$ is not unimodular. Let $\{\mathbf{d}^1,\ldots,\mathbf{d}^m,\norm\}$ be a minimal non-unimodular subset of $\mathcal{D}\cup\{\norm\}$. Thus the set $\{\mathbf{d}^1,\ldots,\mathbf{d}^m,\norm\}$ is linearly independent and, by Fact \[fac:unimodSet\], there exists $$\label{eqn:zinP} \mathbf{z}=\beta_0\norm+\sum_{\ell=1}^m\beta_\ell \mathbf{d}^\ell\in \cap\Z^I\text{ with }0<\beta_\ell<1\text{ for }\ell=0,\ldots,m.$$ By Claim \[cla:maximal\], we know that $\mathcal{D}$ spans $\R^I$. Since $\mathcal{D}$ is also unimodular, by Fact \[fac:unimodSet\] there exist $\mathbf{d}^{m+1},\ldots,\mathbf{d}^n\in\mathcal{D}$ for some $n\geq m$ such that $\mathbf{d}^1,\ldots,\mathbf{d}^n$ are linearly independent and $$\mathbf{z}=\sum_{\ell=1}^n\gamma_\ell \mathbf{d}^\ell \text{ with }\gamma_\ell \in\Z \text{ for all }\ell=1,\ldots,n.$$ Moreover, by replacing $\mathbf{d}^{m+1},\ldots,\mathbf{d}^n$ with their negations if necessary, we can assume that $\gamma_{m+1},\ldots,\gamma_n \geq 0$. Let $k \in J \ssm \{j\}$ be arbitrary. Let $X^k_I=Y^k_I+Z^k_I$, where $$\begin{aligned} Y^k_I &=\left\{\rgivend\sum_{\ell=1}^m\alpha_\ell\mathbf{d}^\ell\rgiv -\|\gamma_\ell\|\leq\alpha_\ell\leq \|\gamma_\ell\|+1 \text{ for }\ell=1,\ldots,m\right\}\cap\Z^I\\ Z^k_I &=\left\{\rgivend\sum_{\ell=m+1}^n\alpha_\ell\mathbf{d}^\ell\rgiv 0\leq\alpha_\ell\leq \gamma_\ell \text{ for }\ell=m+1,\ldots,n\right\}\cap\Z^I.\end{aligned}$$ Observe that $\mathbf{z}\in X^k_I$. Moreover, $\conv(X^k_I)\cap\Z^I=X^k_I$, as we may see by writing $X^k_I\coloneq\{\sum_{\ell=1}^n\alpha_\ell \mathbf{d}^\ell\|c_\ell\leq\alpha_\ell\leq d_l \text{ for }\ell=1,\ldots,n\}\cap\Z^I$ for suitably chosen $c_\ell$ and $d_\ell$. Choose $\mathbf{s}_I$ such that $\mathbf{s}_I\cdot\mathbf{d}^\ell=0$ for $\ell=1,\ldots,m$ and $\mathbf{s}_I\cdot\mathbf{d}^\ell<0$ for $\ell=m+1,\ldots,n$. (Such an $\mathbf{s}_I$ exists because $\mathbf{d}^1,\ldots,\mathbf{d}^n$ are linearly independent.) Set $\mathbf{t}_I\coloneq\p+\mathbf{s}_I$ and set $V^k\coloneq V^{k,\mathbf{t}_I}$. Then $V^k$ is concave by Lemma \[lem:linConv\]. By Equation (\[eqn:faces\]) and the definition of $X^k_I$, we deduce that $$D^k(\p)=\argmax_{\mathbf{x}_I\in X^k_I}\mathbf{s}_I\cdot\mathbf{x}_I =Y^k_I.$$ Moreover, the edges of $X^k_I$ are parallel to $\mathbf{d}^1,\ldots,\mathbf{d}^n$ and so by Corollary \[cor:edges\], the valuation $V^k$ is of demand type $\mathcal{D}$. For agents $j' \in J \ssm \{j,k\},$ let $\Feas{j'} = \{\zero\}$, let $\valFn{j'}$ be arbitrary, and let $\bundowag{j'} = \mathbf{0}$. Let the total endowment $\mathbf{y}_I$ be $\mathbf{x}'_I + \mathbf{z}$. Set $\mathbf{w}^j_I \coloneq \mathbf{x}'_I \in X^j_I$ and $\mathbf{w}^k_I = \mathbf{z} \in X^k_I,$ so $(\mathbf{w}^{j'}_I)_{j' \in J}$ is . Now see that $$\label{eqn:demSet} \sum_{j'\in J}D^j(\p)\subseteq\{\bunpr+\alpha\norm\ \| \ \alpha=0,\ldots,r\}+Y^k_I$$ while, since $\bunpr+r\norm\in D^j(\p)$, we have the equality $$\conv\left(\sum_{j'\in J}D^j(\p)\right)=\{\bunpr+\alpha\norm\ \| \ 0\leq\alpha\leq r\}+\conv(Y^k_I)$$ Recalling Equation (\[eqn:zinP\]), we conclude that $\tot = \bunpr+\mathbf{z}\in \conv\left(\sum_{j'\in J}D^j(\p)\right)$, so $\p$ is a pseudo-equilibrium price vector. But, since $0<\beta_0<1$ in Equation (\[eqn:zinP\]) and since the set $\{\mathbf{d}^1,\ldots,\mathbf{d}^m,\mathbf{g}\}$ is linearly independent, we conclude from Equation (\[eqn:demSet\]) that $\tot = \bunpr+\mathbf{z}\notin \sum_{j'\in J}D^j(\p)$, so there is no at $\p$. Therefore, by the contrapositive of Fact \[fac:pseudoEquil\], no can exist. As the cases exhaust all possibilities, we have proven the fact. [^1]: For example, methods based on integer programming (see, e.g., [@koopmans1957assignment], [@BiMa:97], [@Ma:98], [@CaOzPa:15], and [@tran2019product]) rely on characterizations of the set of Pareto-efficient allocations as the solutions to a welfare maximization problem, while methods based on convex programming (see, e.g., [@Muro:2003], [@ikebe2015stability], and [-@CaEpVo:17]) and tropical geometry [@BaKl:14; @BaKl:19] rely on representing aggregate demand as the demand of a representative agent. [^2]: Outside the case of substitutes (which we describe in detail), [@BiMa:97] and [@Ma:98] gave necessary and sufficient conditions on profiles of valuations, and [@CaOzPa:15] gave sufficient conditions on agents’ individual valuations, for the existence of competitive equilibrium in transferable utility economies. [^3]: [@KeCr:82] were aware of the equivalence between gross and net substitutability in their setting (see their Footnote 1) but used the term “gross substitutes” due to an analogy of their arguments for existence with tâtonnement from general equilibrium theory. [^4]: See Theorem 4.3 of [@BaKl:19]; an earlier version was given by [@DaKoMu:01]. [^5]: It generalizes the quasilinear case of [@KeCr:82], and results of [@SuYa:06], [@MiSt:09], [@HaKoNiOsWe:11], and [@Teyt:14]. [^6]: In particular, we allow for multiple units of some goods to be present in the aggregate, unlike [@GuSt:99] and [@CaOzPa:15]. [^7]: Technological constraints on production (in the sense of [@HaKoNiOsWe:11] and [@FlJaJaTe:19]) can be represented by the possibility that some bundles of goods are infeasible for an agent to consume (see Example 2.15 in [@BaKl:14]). [^8]: @henry1970indivisibilites [pages 543–544], @mas1977indivisible [Theorem 1(i)], and @DeGa:85 [Equation (3.1)] made similar assumptions. If consuming money is inessential but consumption of money must be nonnegative, then it is known that may not exist [@mas1977indivisible]—even in settings in which agents have unit demand for goods (see, e.g., [@herings2019competitive]). However, the existence of can be guaranteed when the agents trade lotteries over goods [@Gul.etal2020]. [^9]: Note that income effects also correspond to changes in an agent’s Marshallian demand induced by changes in the value of her endowment, holding prices fixed. [^10]: Here, we call $\quasivalFn{j}$ a quasivaluation, and denote it by $\quasivalFn{j}$ instead of $\valFn{j}$, to distinguish it from the valuation of an agent with quasilinear preferences. [^11]: Although Fact \[fac:dualDem\] is usually stated with divisible goods (see, e.g., Proposition 3.E.1 and Equation (3.E.4) in [@MaWhGr:95]), the standard proof applies with multiple indivisible goods and money under Condition (\[eq:ulimits\]). For sake of completeness, we give a proof of Fact \[fac:dualDem\] in Appendix \[app:dualDemPrefs\]. [^12]: The function $\cfFn{j}$ is the compensation function of [@DeGa:85] (see also [@DaKoMu:01]). [^13]: A version of Fact \[fac:dualPrefs\] for the function $\cfFn{j}$ in a setting in which utility is increasing in goods is proved in Lemma 1 in [@DaKoMu:01]. For sake of completeness, we give a proof of Fact \[fac:dualPrefs\] in Appendix \[app:dualDemPrefs\]. Fact \[fac:dualPrefs\] is also similar in spirit to the duality between utility functions and expenditure functions (see, e.g., Propositions 3.E.2 and 3.H.1 in [@MaWhGr:95]). However, the arguments of the expenditure function (at each utility level) are prices, while the arguments of the (at each utility level) are quantities. [^14]: Condition (\[eq:vlimits\]) is analogous to Condition (\[eq:ulimits\]) and ensures that the corresponding utility function is defined everywhere on $\Feans{j}$. Note that Condition (\[eq:vlimits\]) is essentially automatic in the context of [@DaKoMu:01] and therefore does not appear explicitly in their result (Lemma 1 in [@DaKoMu:01]). [^15]: \[fn:quasiEquil\]As a result, in the coincide with quasiequilibria with transfers from the modern treatment of the Second Fundamental Theorem of Welfare Economics (see, e.g., Definition 16.D.1 in [@MaWhGr:95]). As the set of feasible levels of money consumption is open, agents always can always reduce their money consumption slightly from a feasible bundle to obtain a strictly cheaper feasible bundle. Hence, quasiequilibria with transfers coincide with equilibria with transfers in the original economy (see, e.g., Proposition 16.D.2 in [@MaWhGr:95] for the case of divisible goods). If the endowments of money were fixed in the , this concept would coincide with the concept of compensated equilibrium of [@arrow1971general] and the concept of quasiequilibrium introduced by [@debreu1962new]. [^16]: Recall that an allocation $(\bunnj)_{j \in J} \in \bigtimes_{j \in J} \Feans{j}$ is Pareto-efficient if there does not exist an allocation $(\hbunnj)_{j \in J} \in \bigtimes_{j \in J} \Feans{j}$ such that $$\sum_{j \in J} \hbunnj = \sum_{j \in J} \bunnj,$$ and $\util{j}{\hbunnj} \ge \util{j}{\bunnj}$ for all agents $j$ with strict inequality for some agent. [^17]: While [@maskin2008fundamental] assumed that goods are divisible, their arguments apply even in the presence of indivisibilities—as we show in Appendix \[app:EEDproof\]. [^18]: This approach is similar in spirit to proof of the existence of with divisible goods. [@negishi1960welfare] instead applied an adjustment process to the inverses of agents’ marginal utilities of money. However, approach does not generally yield a convex-valued adjustment process in the presence of indivisibilities. [^19]: Our definition of holds the endowment of goods fixed, but, unlike [@FlJaJaTe:19], imposes a condition at every feasible endowment of money. Imposing the “full substitutability in demand language” condition from Assumption D.1 in Supplemental Appendix D of [@FlJaJaTe:19] at every money endowment is equivalent to our condition. [^20]: Note that is independent of the endowment of goods as endowments do not affect the demands of agents with quasilinear utility functions. Our definition of coincides with definition [@DaKoLa:2003]. [^21]: By contrast, [@KeCr:82] imposed a condition at all price vectors. Imposing condition at every money endowment leads to a strictly stronger condition than Definition \[def:gsub\]\[part:gsub\] in the presence of income effects [@schlegel2018trading]. [^22]: Fact \[fac:subExist\] is a version of Theorem 1 in [@HaKoNiOsWe:11] for exchange economies and follows from Proposition 4.6 in [@BaKl:19]. See [@KeCr:82] and [@GuSt:99] for earlier versions that assume that valuations are monotone. [^23]: Fact \[fac:subMaxDomain\] is a version of Theorem 2 in [@GuSt:99] and Theorem 4 in [@yang2017maximal] that applies when $\Feas{k}$ can be strictly contained in $\{0,1\}^I$, as well as a version of Theorem 7 in [@HaKoNiOsWe:11] for exchange economies. For sake of completeness, we give a proof of Fact \[fac:subMaxDomain\] in Appendix \[app:maxDomain\]. The proof shows that the statement would hold if $\|J\| \geq \|I\|$ and agents $k \not= i$ were restricted to unit-demand valuations—as in Theorem 2 in [@GuSt:99]. [^24]: [@FlJaJaTe:19] worked with a matching model and considered equilibrium with personalized pricing, but their arguments also apply in exchange economies without personalized pricing. However, [@FlJaJaTe:19] only required that each agent sees goods as for a fixed endowment of goods and money. Our notion of considers a fixed endowment of goods but a variable endowment of money, and therefore the existence result of [@FlJaJaTe:19] is not strictly a special case of Theorem \[thm:netSubExist\]. Moreover, [@FlJaJaTe:19] also allowed for frictions such as transaction taxes and commissions in their existence result. [^25]: Indeed, recall that Example \[eg:quasilogDualVal\] tells us that agent $j$’s at each utility level is a positive linear transformation of $\quasivalFn{j}$. The conclusion follows by Remark \[rem:netSubDual\]. [^26]: See Remark E.1 in the Supplemental Material of [@FlJaJaTe:19]). [^27]: @DaKoMu:01 [Example 2] also showed the connection between housing market economy and a condition, but formulated their discussion in terms of the shape of the convex hull at domains at which demand is multi-valued instead of . Their discussion is equivalent to ours by Corollary 5 in [-@DaKoLa:2003] and Remark \[rem:netSubDual\]. [^28]: The existence of a feasible set of bundles of goods and a valuation for $k$ for which no exists follows from Fact \[fac:subMaxDomain\]. To check that $\valFn{k}$ is an example of such a valuation, suppose, for sake of deriving a contradiction, that $(\bunj,\bunag{k})$ is the allocation of goods in a . The First Welfare Theorem implies that $\bunj = (1,1)$ and that $\bunag{k} = (0,0)$. But for agent $j$ to demand $(1,1),$ the equilibrium prices would have to sum to at most 5, while for agent $k$ to demand $(0,0)$, the equilibrium prices would both have to be at least 3—a contradiction. Hence, we can conclude that no exists. [^29]: \[fn:evalU\]To show this, note that $\numerdowj - \ppr \cdot ((1,1) - \bundowj) = -1,$ so it would violate $j$’s budget constraint to demand $(1,1)$ at the price vector $\ppr$. For the other bundles, note that $$\begin{array}{c\|c\|c\|c\|c} \bun & (0,0) & (0,1) & (1,0) & (1,1)\\ \hline \util{j}{\numerdowj - \p \cdot (\bun - \bundowj),\bun} & \log \frac{5}{11} & \log \frac{3}{7} & \log \frac{3}{4} & \log 1\\ \hline \util{j}{\numerdowj - \ppr \cdot (\bun - \bundowj),\bun} & \log \frac{5}{11} & \log \frac{3}{7} & \log \frac{1}{4} & \text{undef.,} \end{array}$$ so $\dM{j}{\p}{\bunndowj} = \{(1,1)\}$ and $\dM{j}{\ppr}{\bunndowj} = \{(0,0)\}$. [^30]: The expressions for $\dH{j}{\p}{\ub}$ and $\dH{j}{\ppr}{\ub}$ hold because agent $j$’s Hicksian valuation at utility level $\ub$ is $\frac{5}{11}$ times the quasivaluation $\quasivalFn{j}$ (by Example \[eg:quasilogDualVal\]). [^31]: To show this, let $\hp = (3,2)$. It is clear that $(0,1) \in \dQL{k}{\hp}$. It remains to show that $(1,0) \in \dM{j}{\hp}{\bunndowj}$. Note that $\numerdowj - \hp \cdot ((1,1) - \bundowj) = 0,$ so it would violate $j$’s budget constraint to demand $(1,1)$ at the price vector $\hp$. For the other bundles, note that $$\util{j}{\numerdowj - \hp \cdot (\bun - \bundowj),\bun} = \begin{cases} \log \frac{5}{11} & \text{if } \bun = (0,0)\\ \log \frac{3}{7} & \text{if } \bun = (0,1)\\ \log \frac{1}{2} & \text{if } \bun = (1,0), \end{cases}$$ so $\dM{j}{\hp}{\bunndowj} = \{(1,0)\}$. [^32]: Definition \[def:demTypeTU\](c) coincides with Definition 3.1 in [@BaKl:19] by Proposition 2.20 in [@BaKl:19]. [^33]: As there are no income effects here, the compensated law of demand (see, e.g., Proposition 3.E.4 in [@MaWhGr:95]) reduces to the law of demand. [^34]: \[fn:classDiscConv\]The “if” direction of Fact \[fac:unimod\] is a case of the “if" direction of Theorem 4.3 in [@BaKl:19]. The “only if” direction of Fact \[fac:unimod\], which we prove in Appendix \[app:maxDomain\], is a mild strengthening of the “only if" direction of Theorem 4.3 in [@BaKl:19] that applies in exchange economies. [^35]: To understand the correspondence, let $\mathcal{D}$ be a unimodular . In the terminology of [@DaKoMu:01], a valuation $\valFn{j}$ is $\mathscr{D}(\mathscr{P}t(\mathcal{D},\mathbb{Z}))$-concave if, for each price vector $\p$, we have that $\dQL{j}{\p} = \conv(\dQL{j}{\p}) \cap \Z^I$ and each edge of $\conv(\dQL{j}{\p})$ is parallel to an element of $\mathcal{D}$ (see Definition 4 and pages 264–265 in [@DaKoMu:01]). It follows from Lemma 2.11 and Proposition 2.16 in [@BaKl:19] that a valuation is $\mathscr{D}(\mathscr{P}t(\mathcal{D},\mathbb{Z}))$-concave if and only if it is concave and of demand type $\mathcal{D}$. [^36]: See Definition 4, Theorem 3, and pages 264–265 in [@DaKoMu:01]. [^37]: By contrast, the existence results of [@SuYa:06] and [@Teyt:14] can be deduced from Fact \[fac:subExist\] applying an appropriate change of basis. Those results are also special cases of Fact \[fac:unimod\]. [^38]: Section 6.1 in [@BaKl:19] provides another example that includes only complements valuations. [^39]: Fact \[fac:unimodMaxDomain\] is related to Proposition 6.10 in [@BaKl:14], which connects failures of unimodularity to the non-existence of in specific economies. We supply a proof of Fact \[fac:unimodMaxDomain\] in Appendix \[app:maxDomain\]. [^40]: It is equivalent to define quasiconcavity in terms of the convexity of the upper contour sets, but Definition \[def:quasiConc\] is more immediately applicable for us. [^41]: existence result is not formally a special case of ours because they allowed for unbounded sets $\Feas{j}$ of feasible bundles of goods. [^42]: Requiring that different goods, rather than different units of goods, be leads to a condition called **. However, does not ensure the existence of when agents can demand multiple units of some goods [@DaKoLa:2003; @MiSt:09; @BaKl:19]. in turn corresponds to an “ordinary substitutes” demand type (see Definitions 3.4 and 3.5 and Proposition 3.6 in [@BaKl:19]). [^43]: The quasilinear case of this fact is part of Theorem 4.1(i) in [@shioura2015gross] (see also Proposition 3.10 in [@BaKl:19]). The general case follows from the quasilinear case by Lemma \[lem:dHvalH\] and Remark \[rem:netSubDual\]. [^44]: In particular, if agent $j$ demands at most one unit of each good, then $\utilFn{j}$ is a utility function if and only if it is of the strong substitutes demand type. [^45]: See [@klemperer2008new; @klemperer2010product; @klemperer2018product] and . Iceland planned a Product-Mix Auction for bidders with budget constraints [@klemperer2018product], but that auction was for a setting with divisible goods. [^46]: The lemma is due to @shapley1964values [page 3]; see also [@BiMa:97] and [@HaKoNiOsWe:11]. @jagadeesan2020lone [Lemma 1] proved the lemma in a setting with multiple units that allows for non-monotone valuations. [^47]: The existence of an integer vector that is in $\conv\left(\sum_{j'\in J} D^{j'}(\mathbf{p}_I)\right)$ but not $\sum_{j' \in J} D^{j'}(\mathbf{p}_I)$ follows from Fact \[fac:unimodSet\] as the vectors $\mathbf{g},\mathbf{e}^1-\mathbf{e}^2,\mathbf{e}^3,\ldots,\mathbf{e}^{\|I\|}$ do not comprise a unimodular set.	ArXiv
--- abstract: 'This paper leverages heterogeneous auxiliary information to address the data sparsity problem of recommender systems. We propose a model that learns a shared feature space from heterogeneous data, such as item descriptions, product tags and online purchase history, to obtain better predictions. Our model consists of autoencoders, not only for numerical and categorical data, but also for sequential data, which enables capturing user tastes, item characteristics and the recent dynamics of user preference. We learn the autoencoder architecture for each data source independently in order to better model their statistical properties. Our evaluation on two [MovieLens]{} datasets and an e-commerce dataset shows that mean average precision and recall improve over state-of-the-art methods.' author: - title: \| Deep Heterogeneous Autoencoders\ for Collaborative Filtering\ --- Deep Autoencoder, Heterogeneous Data, Shared Representation, Sequential Data Modeling, Collaborative Filtering Introduction ============ Although Collaborative Filtering (CF) techniques achieve good performance in many recommender systems [@Hu2008], their performance degrades significantly when historical data is sparse. In order to alleviate this problem, features from auxiliary data sources that reflect user preference have been extracted [@Oord2013; @Porteous2010], as shown in Fig. \[fig:auxiliary\_usage\]. How to represent data from different sources is still a research problem, and it has been shown that the representation itself substantially impacts performance [@Loyola2017; @Goodfellow2016]. Recently, representation learning that automatically discovers hidden factors from raw data has become a popular approach to remedy the data sparsity issue of recommender systems [@wangweiran2015; @Zheng2017]. Many online shopping platforms gather not only user profiles and item descriptions, but various other types of data, such as product reviews, tags and images. Recent research has added textual and visual information to recommender systems [@Fuzheng2016; @Oramas2017]. However, in many cases sequential data, such as user purchase and browsing history, which carries information about trends in user tastes, have largely been neglected in CF-based recommender systems. ![[Auxiliary information usage in recommender systems.]{} Item descriptions and user profiles are typically being used for feature extraction to alleviate the data sparsity problem. Our proposal also leverages sequential data, such as purchase histories, to reflect user preferences.[]{data-label="fig:auxiliary_usage"}](heterogeneous2recommender.png){width="\columnwidth"} In this paper we propose Deep Heterogeneous Autoencoders (DHA) for Collaborative Filtering to combine information from multiple domains. We use Stacked Denoising Autoencoders (SDAE) to extract latent features from non-sequential data, and Recurrent Neural Network Encoder-Decoders (RNNED) to extract features from sequential data. The model is able to capture both user preferences and potential shifts of interest over time. Each data source is modeled using an independent encoder-decoder mechanism. Different encoders can have different number of hidden layers and an arbitrary number of hidden units in order to deal with the intrinsic difference of data sources. For instance, user demographic data and item content are typically categorical, while user comments or item tags are textual. After pre-processing, such as one hot encoding, bag-of-words and word2vec computation, representation vectors are on a different level of abstraction. Owing to its flexible structure, our model is able to learn suitable latent feature vectors for each component. These local representations from each data source are joined to form a shared feature space, which couples the joint learning of the representation from heterogeneous data and the collaborative filtering of user-item relationships. The contributions of this paper are summarized as follows: 1. A method for modeling both static and sequential data in a consistent way for recommender systems in order to capture the trend in user tastes, and 2. Adaptation of the autoencoder architecture to accurately model each data source by considering their distinct abstraction levels. We show improvements in terms of mean average precision and recall on three different datasets. Related work ============ Incorporating side information into recommender systems ------------------------------------------------------- In order to improve recommendation performance, research has been focusing on using side information, such as user profiles and reviews [@Fuzheng2016; @Porteous2010]. In particular, deep learning models have been widely studied [@He2017; @Wu2017]. AutoRec first proposed the use of autoencoders for recommender systems [@Sedhain2015]. In more recent work, representations are learned via stacked autoencoders (SAE), and fed into conventional CF models, either loosely or tightly coupled [@szhang2017; @hwang2015]. Deep models that integrate autoencoders into collaborative filtering have shown state-of-the-art performance. Recurrent Neural Network Encoder-Decoder ---------------------------------------- Recurrent neural networks (RNNs) process sequential data one element at each step to capture temporal dynamics. The encoder-decoder mechanism was initially applied to RNN for machine translation [@Cho2014]. Recently, RNN encoder-decoders (RNNED) have been used to learn features from a series of actions and have successfully been applied in other areas. It was shown that Long Short-Term Memory (LSTM) networks have the ability to learn on data with long range temporal dependencies, and we adopt LSTMs for modeling sequential data. ![[Deep Heterogeneous Autoencoders and the integration with collaborative filtering.]{} The proposed model extracts a shared feature space from multiple sources of auxiliary information. It models non-sequential and sequential data to capture user preferences, item properties as well as temporal dynamics. It adopts independent encoder-decoder architectures for different data sources in order to better model their statistical properties. The product of $U \in \mathbb{R}^{m \times d}$ and $V \in \mathbb{R}^{n \times d}$ approximates the user-item interaction matrix.[]{data-label="fig:DHA"}](DHA_0606.png){width="45.00000%"} Deep Heterogeneous Autoencoders for Collaborative Filtering =========================================================== Overview -------- We propose a model that learns a joint representation from heterogeneous auxiliary information to mitigate the data sparsity problem of recommender systems. SDAEs are applied to numerical and categorical data for modeling the static tastes of users for items. We use RNNEDs to extract features from sequential data to reveal interest shifts over time. The model adopts an independent autoencoder architecture for each data source since the inputs are generally on a different level of abstraction, see Fig. \[fig:DHA\] for an overview. In order to discover the distinct statistical properties of every data source, our model takes the existing disparity of input abstraction levels into consideration, and applies autoencoders to each source independently by allowing distinct hidden layer numbers and arbitrary hidden units at every layer. Deep Heterogeneous Autoencoders ------------------------------- We define each source of auxiliary data as a component indexed by $c \in \{1,...,C\}$. $S_c$ denotes the input of component $c$. We pre-process non-sequential data like textual item descriptions by generating fixed-length embedding vectors. For sequential data, an embedding vector is learned for every time step after tokenization. We seperately describe the encoding-decoding outputs of the above two types of embedding vectors. As shown in Fig. \[fig:DHA\], SDAE is applied to fixed-length embedding vectors. Each component encoder takes the input $S_c$, generates a corrupted version of it, $\hat{S_c}$, and the first layer maps it to a hidden representation $h_c$, which captures the main factors of variation in the input data distribution[@vincent2008; @vincent2010]. More importantly, the number of component hidden layers in our model can differ from each other. The architecture is unique for each data source, where the number of layers of component $c$ is denoted as $L_c$. The representation at every layer is $S_{c,l}$. For the encoder of each component, given $l_c \in \{1, ..., L_c/2 \}$ and $c \in C$, the hidden representation $h_{c,l}$ is derived as: $$h_{c, l} = f \left(W_{c,l}h_{c,l-1} + b_{c,l} \right).$$ The decoder reconstructs the data at layer $L$ as follows: $$\bar{S}_c = g\left( W'_c h_{c,L} + b'_c \right).$$ The proposed model leverages sequential data by using two LSTMs for encoding and decoding one sequential data source. Specifically, the encoder reads a sequence with $T$ time steps. At the last time step, the hidden state $h_T$ is mapped to a context vector $c$, as a summary of the whole input sequence[@Cho2014]. The decoder generates the output sequence by predicting the next action $y_t$ given $h_t$. Both $y_t$ and $h_t$ are also conditioned on $y_{t-1}$ and the context vector $c$. To combine them, as shown in Fig. \[fig:DHA\], the first part of our model encodes all components to generate hidden representations $S_{c, L_c/2}$ of non-sequential data and $h_T$ of sequential data across all sources. These are merged to generate a joint latent representation, denoted as $h_{+,0}$. Analogous to the hidden layers of each component, the fusion model can have multiple hidden layers, the total number denoted as $L_+$. The representation of the first fusion hidden layer is $$h_{+,0} = f \left(\sum_{c\in C} W_{c, +}h_{c,L_c/2} + b_{+,0} \right).$$ The first hidden layer $h_{+,0}$ of the fused model is fed into the collaborative filtering model. After joint training, $h_{+,0}$ is the latent vector to generate recommendation results. DHA-based Collaborative Filtering --------------------------------- All data is fed into two DHAs for users and items, respectively. Fig. \[fig:DHA\] shows the process for items, and it is analogous for user data. Let $R \in \mathbb{R}^{m \times n}$ denote the rating matrix of users to items, $S_c^{\left(u \right)}$ being the component $c$ input for users and $S_c^{\left(v \right)}$ that for items. Then, $h_{+,0}^{\left(u \right)}$ and $h_{+,0}^{\left(v \right)}$ are the latent factors. The loss function of the proposed DHA based collaborative filtering is defined as: @size[8]{}@mathfonts \[dha\_cost\_func\] L = &\_[i,j]{} c\_[i,j]{} (r\_[i,j]{} - u\_i v\_j ) \^ 2 + \_f (\_i\|\|u\_i\|\|\^2 + \_j \|\|v\_j\|\|\^2)\ &+ \_[m]{} \_[cC\_u]{} loss(S\_c\^[(u)]{}, \|[S]{}\_c\^[(u)]{}) + \_[n]{} \_[cC\_i]{} loss(S\_c\^[(v)]{}, \|[S]{}\_c\^[(v)]{})\ &+ \_u \_i \|\| u\_i - h\_[+, 0]{}\^[u\_i]{} \|\|\^2 + \_v \_j \|\| v\_j - h\_[+, 0]{}\^[v\_j]{} \|\|\^2\ &+ \_w (\_c\_l (\|\|W\_[c,l]{}\^[(u)]{}\|\|\^2 + \|\|b\_[c,l]{}\^[(u)]{}\|\|\^2 )\ &+ \_c\_l (\|\|W\_[c,l]{}\^[(v)]{}\|\|\^2 + \|\|b\_[c,l]{}\^[(v)]{}\|\|\^2 ) ). The loss function includes reconstruction costs of user and item information sets, the error to predict $r_{i,j}$, and the approximation error between latent factor vectors of feature learning and collaborative filtering. The loss function is minimized to obtain parameters for the DHAs and the CF model. The mean squared error and the negative log-likelihood are used as cost functions for non-sequential and sequential data, separately. We use $\lambda_{m}$, $\lambda_{n}$, $\lambda_u$ and $\lambda_v$ to balance losses between users and items, $\lambda_f$, and $\lambda_w$ to regularize the weight matrix and bias vectors. Parameter learning ------------------ We apply coordinate descent to alternate the optimization between representation learning of heterogeneous data and user-item interaction, similar to [@hwang2015; @cwang2011]. Given $W$s and $b$s, the gradients of the loss function $L$ with respect to $u_i$ and $v_j$ are computed and set to 0, leading to the following updates: u\_i &(V\^TC\_i V +\_f I + \_u I)\^[-1]{}(V\^TC\_i R\_i + \_u h\_[+,0]{}\^[u\_i]{}),\ v\_j &(U\^TC\_j U +\_f I + \_v I)\^[-1]{}(U\^TC\_j R\_j + \_v h\_[+,0]{}\^[v\_j]{}), where $U \in \mathbb{R}^{m \times d}$ and $V \in \mathbb{R}^{n \times d}$ contain the user and item latent factor vectors, and $d$ is the vector dimensionality. Given $U$ and $V$, the weight matrix and bias vectors of every layer are learned by backpropagation with stochastic gradient descent (SGD). Gradients of $W$ and $b$ are calculated as follows: @size[8]{}@mathfonts =& \_w W\^u + \_m \_c loss(S\_c\^u, \|[S]{}\_c\^u) + \_u (U - h\_[+,0]{}\^u),\ =& \_w b\^u + \_m \_c loss(S\_c\^u, \|[S]{}\_c\^u) + \_u (U - h\_[+,0]{}\^u). A learning rate $\alpha$ is adopted to update all parameters using calculated gradients. Experiments =========== Experiments are conducted on three real world datasets, MovieLens-100k ([ml-100k]{}), MovieLens-10M ([ml-10m]{}), and one dataset from an e-commerce company (OfflinePay). We first investigate whether the flexible autoencoder architecture of our model can generate more accurate latent representations on non-sequential data. Experiments on OfflinePay evaluate the effectiveness of sequential data modeling. Datasets and preprocessing -------------------------- The first dataset, [ml-100k]{}, contains ratings from 943 users on 1,682 movies. It has demographic data for users and descriptions for movies. The second dataset, [ml-10m]{}, contains 10,000,054 ratings and 95,580 tags from 71,567 users for 10,681 movies. It contains item content information, but no demographic data. We employ user-added tags as an information source for users as well as for movies. OfflinePay is a dataset of user purchases in (offline) shops, paying with a plastic e-money card. The dataset contains a total of 67M transaction records from a four-month period. The goal of using the OfflinePay dataset is to recommend new shop genres to users, not individual products. After aggregating all transaction data into the format of (user $i$, shop genre $j$, number of transactions $r_{ij}$), and removing shoppers who used only one shop genre, the number of $r_{ij}$ values is 7,150,833 with 961,992 unique users and 105 shop genres. The auxiliary data sources include user registered information and shop genre textual descriptions. Additionally, we collect user purchase history on an e-commerce platform during the same time period. The sequence data contains the genres of purchased items online. The datasets are preprocessed to fixed-length embeddings for non-sequential data, and sequences of embedding vectors for sequential data, respectively. For [ml-100k]{}, we discretize continuous features like age to discrete values, compute a bag-of-words vector for each user and item. The vector dimensions are 821 for users, 2,482 for movies, respectively. For [ml-10m]{}, movie content description and tags that users give to items are textual information. We first tokenize texts, then train Doc2vec vectors for every data source with the embedding vector length set to 500. To generate shop genre embedding vectors for the OfflinePay dataset, all shop names that belong to same genre are grouped together and Doc2vec is applied to generate a 300-dimensional vector for each shop genre. User registered information is preprocessed the same way as [ml-100k]{}, and the vector length is 189. For the sequence of genre purchase history, Word2vec is adopted to build 100-d embedding vectors after tokenization. Genres in each sequence are mapped to the corresponding embedding vectors. In experiments, we rank predicted ratings of candidate items and recommend the top $M$ to each user. Mean average precision (MAP) and recall are used as evaluation metrics. [0.33]{}![image](ml100k_f50_recall_0824.png){width="\linewidth"} [0.33]{}![image](ml100k_f100_recall_0824.png){width="\linewidth"} [0.33]{}![image](ml100k_f150_recall_0824.png){width="\linewidth"} [0.33]{}![image](ml10m_f50_recall_0824.png){width="\linewidth"} [0.33]{}![image](ml10m_f100_recall_0824.png){width="\linewidth"} [0.33]{}![image](ml10m_f150_recall_0824.png){width="\linewidth"} Experimental setting -------------------- The number of hidden layers of each model is optimized on a validation dataset. The first fusion hidden layer of DHA is used to bridge the joint training between feature space learning and collaborative filtering. For other models, if the total number of hidden layers is $L$, we connect layer $L/2$ for joint training. The number of units in each hidden layer is incremented by $K$ from the middle of the autoencoder to both sides. For sequential data modeling, recent $T$ purchases is used in the experiments, and values $T \in \{5, 10\}$ are evaluated in our experiments. The mini-batch size is set to 50 and 1,000 for [ml-100k]{} and [ml-10m]{}, respectively. For the OfflinePay dataset, since the numbers of unique users and items differ significantly, it is set to 20 for items and 10,000 for users, separately. The model is implemented using the Theano library. Experiments on MovieLens datasets --------------------------------- We compare our model with the following algorithms. Note that experiments on [MovieLens]{} do not include sequential data. - AutoRec[@Sedhain2015]: I-AutoRec takes a partial item feedback vector as input and reconstructs at the output layer. - CDL[@hwang2015]: a hierarchical Bayesian model that jointly performs deep representation learning for content information and collaborative filtering for the ratings matrix. - DCF[@Lis2015]: a model that combines matrix factorization with marginalized denoising stacked autoencoders. We concatenate side information as input to DCF. - aSDAE[@Dong2017]: a hybrid model that integrates side information by an additional denoising autoencoder into the matrix factorization model. - DHA: the proposed model that applies independent autoencoder architecture to heterogeneous data sources. To compare different models, we repeat 80-20 splits of the data 5 times, run 5-fold cross validation and report average performance. Grid search is applied to find optimal hyperparameters for all models. We search the learning rate of SGD, $\alpha \in \{ 0.1, 0.05, 0.01, 0.001 \} $, the regularization of learned parameters, $\lambda_f$ and $\lambda_w$ of our model $ \in \{ 2.0, 0.1, 0.01, 0.001 \} $, the corruption level of masking noise $\in \{ 0.1, 0.3 \} $, the activation function $\in \{sigmoid, relu\} $, and the number of fusion hidden layers $\in \{ 1, 2 \} $. The parameters used to balance loss between user and item, $\lambda_{m}, \lambda_{n}, \lambda_{u}, \lambda_{v}$ are set to 1. For CDL, DCF and aSDAE, we search hidden layer number from 4 and 6. The joint training is alternated 5 times, and we run 5 epochs for learning features in each alteration. Before the joint training, layer-wise pretraining is conducted to initialize network weights. For the experiment on [ml-100k]{}, we input rating vectors, item content information and user demographic data to DHA and aSDAE. Rating vectors are not used in DCF and only item content information is used in CDL. I-AutoRec leverages no side information. After grid search, the adopted hidden layer number of CDL, DCF and aSDAE is 4. The number of fusion hidden layer is set to 1 for DHA. The parameter for regularizing learned parameters is set to 0.01 in DHA, 0.001 in CDL and aSDAE, and 0.1 in DCF, respectively. The optimal performance is found when the learning rate is set to 0.001 for CDL, DCF, 0.01 for aSDAE and DHA, and 0.1 for I-Autorec. \[tab: map@100 for ml-100k, ml-10m\] ----------- -- -------- -------- -------- -- -------- -------- -------- Model d=50 d=100 d=150 d=50 d=100 d=150 I-AutoRec 0.0573 0.0568 0.0572 0.0325 0.0323 0.0326 CDL 0.1896 0.1825 0.1685 0.1458 0.1532 0.1612 DCF 0.2012 0.2028 0.2069 0.1591 0.1620 0.1566 aSDAE 0.2161 0.2228 0.2142 0.1602 0.1560 0.1642 DHA 0.2236 0.2304 0.2258 0.1793 0.1774 0.1824 ----------- -- -------- -------- -------- -- -------- -------- -------- : [*MAP@100 comparison on [ml-100k]{} and [ml-10m]{} datasets.]{} Results are shown for three different settings of user and item latent factor vectors, $d$=50, 100, and 150.* As shown in Fig. \[fig:recall\_comparison\_ml-100k\_and\_ml-10m\], all models achieve better recall than I-AutoRec, showing the advantage of using side information. DHA and aSDAE perform better than CDL which only incorporates item content description. DHA outperforms aSDAE which integrates raw side information at every hidden layer. The MAP comparison in Table \[tab: map@100 for ml-100k, ml-10m\] shows our model obtains more precise results for all dimension settings. There are five sets of available inputs for the experiment on the [ml-10m]{} dataset. Users and movies have rating and tag vectors, movies also have content vectors. For CDL, DCF and aSDAE, different information vectors are concatenated as input. Our model uses all components, [i.e.]{} two components for users and three components for movies. In the experiment, the best performance is obtained when the number of hidden layers is set to 4 for CDL, aSDAE and to 6 for DCF. In our model, we use 2 fusion hidden layers and different layer numbers for components. The number of hidden layers, $L_c$, is set to 4 for users and movie rating vectors and to 2 for tag and content vectors. As shown in Fig. \[fig:recall\_comparison\_ml-100k\_and\_ml-10m\], DHA obtains better recall performance compared to other algorithms. aSDAE is competitive and outperforms both DCF and CDL in three dimension settings. The MAP comparison in Table \[tab: map@100 for ml-100k, ml-10m\] indicates that in addition to producing recommendation with better recall, our model also achieves better precision results. [0.34]{} ![image](edy_f50_recall_0910.png){width="\linewidth"} [0.34]{} ![image](edy_f100_recall_0910.png){width="\linewidth"} [0.34]{} ![image](edy_f150_recall_0910.png){width="\linewidth"} Experiments on OfflinePay dataset --------------------------------- Since the OfflinePay dataset involves user online purchase histories, we use the first 3-month data as training data, the following half month’s data as validation dataset to find optimal parameters, and data from the remaining half-month as test set. We compare the following algorithms: - implicit-cf[@Hu2008]: a matrix factorization model for implicit datasets. - CDL[@hwang2015]: a Bayesian model that learns a feature space from item information and jointly trains with CF. - DCF[@Lis2015]: a model that incorporates side information by marginalized denoising stacked autoencoders with a matrix factorization model. - DHA-RNNED-s10: our model that learns a latent representation only from the sequence of online purchases. The number of time steps in each sequence is 10. - DHA-RNNED-s5: our model with the same modeling process as DHA-RNNED-s10, but using 5 time steps in each sequence. - DHA-RNNED-item: our model extracts features from sequential online purchases at user side, and from shop genre descriptions at item side. - DHA-all: the proposed model that leverage non-sequential side information sets and sequential online purchase activities simultaneously. The used time step number of the purchase sequence is 10. In the experiment, the joint learning is alternated 3 times, and we run 3 epochs for feature extraction every time. The number of hidden layers for CDL, DCF and our model is set to 4, and 1 fusion hidden layer is used in DHA models. For the sequential modeling, we set the hidden units of LSTMs to be the same as the dimension of the user and item latent factor vector. The SGD learning rate and regularization parameters for each model are found by grid search on the validation set. We set the learning rate to 0.1 for implicit-cf and CDL, to 0.001 for DCF and to 0.01 for the other models. The parameter to regularize learned parameters is set to 2.0 for CDL, and 0.1 for DCF and DHA-all. There is no training alteration for implicit-cf, but we run 25 iterations to learn user and item latent factor vectors. DCF integrates both user registration information and shop genre descriptions, while CDL uses only the latter one. DHA-RNNED-s10 and DHA-RNNED-s5 do not include any side information except user online purchases. DHA-RNNED-item adopts sequential data and shop genre descriptions, and DHA-all utilizes all of the data. Note that since ratings are not used in any models, aSDAE is not applied on OfflinePay dataset. From Fig. \[fig:recall\_comparison\_offlinepay\], we observe that models taking advantage of side information have better recall than the baseline implicit-cf. CDL outperforms DCF which, in fact, uses more information sets. This may be due to the fact that many user registration records have outdated or missing values, making the feature extraction less accurate. Compared to CDL and DCF, the proposed models with sequential data modeling achieve better recall. This is due to the fact that offline shop genres in the dataset are included in the online purchased genres. This also indicates that the latent features is able to be extracted from recent online purchases accurately, and reflect the trends of user interests, then lead to better recommendations for offline products, as well. The MAP comparison in Table \[tab: map@100 for offlinepay\] shows that the models involving sequential modeling achieve higher precision. This consistently shows that the modeling of online purchases helps with offline product recommendation. The recall comparison in Fig. \[fig:recall\_comparison\_offlinepay\] shows that DHA-RNNED-s10 and DHA-RNNED-s5 have a similar trend as recommended item $M$ increases. These two models use only the sequence of purchased genres from an online e-commerce platform, but with different time steps in the sequence. it is also shown that DHA-all and DHA-RNNED-item have similar recalls. The difference between these two models is that the latter model does not include user registered data. Linking to the previous observation that CDL outperforms DCF, user data does not significantly contribute to the recommendation results. In order to compare the effect of purchase recency of the input sequence, we apply DHA-RNNED-s10 and DHA-RNNED-s5 to encode the recent ten and five purchases, respectively. Our hypothesis is that more recent online purchases are more representative of current user interests. Although the difference is not big, the recall and MAP comparisons support our hypothesis. The experiments demonstrate that with the independent autoencoder structure for user and item side information and the modeling of user online activities, our model is able to achieve competitive recall and MAP results. Conclusions =========== We proposed a model that incorporates multiple sources of heterogeneous auxiliary information in a consistent way to alleviate the data sparsity problem of recommender systems. 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--- abstract: 'We present near-infrared integral field spectroscopy data obtained with VLT/SINFONI of “the Teacup galaxy”. The nuclear K-band (1.95–2.45 ) spectrum of this radio-quiet type-2 quasar reveals a blueshifted broad component of FWHM$\sim$1600-1800 km s$^{-1}$ in the hydrogen recombination lines (Pa$\alpha$, Br$\delta$, and Br$\gamma$) and also in the coronal line \[Si VI\]$\lambda$1.963 . Thus the data confirm the presence of the nuclear ionized outflow previously detected in the optical and reveal its coronal counterpart. Both the ionized and coronal nuclear outflows are resolved, with seeing-deconvolved full widths at half maximum of 1.1$\pm$0.1 and 0.9$\pm$0.1 kpc along PA$\sim$72–74. This orientation is almost coincident with the radio axis (PA=77), suggesting that the radio jet could have triggered the nuclear outflow. In the case of the H$_2$ lines we do not require a broad component to reproduce the profiles, but the narrow lines are blueshifted by $\sim$50 km s$^{-1}$ on average from the galaxy systemic velocity. This could be an indication of the presence of a nuclear molecular outflow, although the bulk of the H$_2$ emission in the inner $\sim$2 ($\sim$3 kpc) of the galaxy follows a rotation pattern. We find evidence for kinematically disrupted gas (FWHM$>$250 km s$^{-1}$) at up to 5.6 kpc from the AGN, which can be naturally explained by the action of the outflow. The narrow component of \[Si VI\] is redshifted with respect to the systemic velocity, unlike any other emission line in the K-band spectrum. This indicates that the region where the coronal lines are produced is not co-spatial with the narrow line region.' author: - '\' title: 'An infrared view of AGN feedback in a type-2 quasar: the case of the Teacup galaxy' --- \[firstpage\] galaxies: active – galaxies: nuclei – galaxies: evolution – galaxies: individual – galaxies: jets. Introduction {#intro} ============ Cosmological simulations require active galactic nuclei (AGN) feedback to regulate black hole and galaxy growth [@diMatteo05; @Croton06]. This process occurs when the intense radiation produced by the active nucleus sweeps out and/or heats the interstellar gas, quenching star formation and therefore producing a more realistic number of massive galaxies in the simulations (see @Fabian12 for a review). Two major modes of AGN feedback are identified. Radio- or kinetic-mode feedback dominates in galaxy clusters and groups, where jet-driven radio bubbles heat the intra-cluster medium. This type of feedback is generally associated with powerful radio galaxies. Quasar- or radiative-mode feedback consists of AGN-driven winds of ionized, neutral, and molecular gas [@Fabian12; @Fiore17]. However, such a clear distinction between the two modes of feedback can be somewhat misleading. This is because it has been shown, through the detection of nuclear outflows, that radiative-mode feedback also acts in radio-galaxies (see e.g. @Emonts16 and references therein), whilst the presence of jets in quasars that are deemed to be radio-quiet can lead to faster and more turbulent AGN-driven winds [@Mullaney13; @Zakamska14]. Therefore, it is over-simplistic to consider the impact each mode of AGN-feedback has on its host galaxy in isolation. One of the most efficient ways to identify the imprint of outflows in large AGN samples at any redshift is to search for them in the warm ionized phase in the optical range (e.g. \[O III\]$\lambda$5007 Å). Indeed, during recent years it has become clear that ionized outflows are a ubiquitous phenomenon in type-2 quasars (QSO2s) at z$\la$0.7 [@Villar11; @Villar14; @Liu13; @Harrison14; @Karouzos16]. QSO2s are excellent laboratories to search for outflows and study their impact on their host galaxies, as the AGN continuum and the broad components of the permitted lines produced in the broad-line region (BLR) are obscured. Fast motions are often measured in QSO2s, with full-widths at half maximum (FWHM) $>$1000 km s$^{-1}$ and typical velocity shifts (V$_s$) of hundreds km s$^{-1}$. These outflows are likely triggered by AGN-related processes and they originate in the high-density regions (n$_e\ge 10^3~cm^{-3}$) within the central kiloparsecs of the galaxies. Optical integral field spectroscopy (IFS) studies have shown that these outflows can extend up to $\sim$15 kpc from the AGN [@Humphrey10; @Liu13; @Harrison14]. However, these results have been recently questioned, as the reported outflow extents could be overestimated due to seeing smearing effects [@Karouzos16; @Villar16; @Husemann16]. Now that ionized outflows have been identified as a common process in QSO2, the next goal is to investigate their impact on other gaseous phases, such as the molecular and coronal phases. Since H$_2$ is the fuel required to form stars and feed the SMBH, the impact of the outflows in this gaseous phase is what might truly affect how systems evolve. Detecting coronal outflows is also interesting because, due to the high ionization potentials (IP$\ga$100 eV; @Mullaney09 [@Rodriguez11; @Landt15]) of these lines, they are unequivocally associated with nuclear activity. Coronal lines have intermediate widths between those of the broad and the narrow emission lines (FWHM$\sim$500–1500 km s$^{-1}$) and are generally blueshifted and/or more asymmetric than lower-ionization lines [@Penston84; @Appenzeller91; @Rodriguez11]. This indicates that either coronal lines are produced in an intermediate region between the narrow-line region (NLR) and the BLR [@Brotherton94; @Mullaney08; @Denney12], and/or are related to outflows [@Muller06; @Muller11]. The near-infrared (NIR) range, and particularly the K-band, allows us to trace outflow signatures in the molecular, ionized and coronal phases simultaneously. In addition, because ionized outflows in QSO2 are heavily reddened [@Villar14], observing them in the NIR permits us to penetrate through the dust screen and trace the regions closer to the base of the outflow. The rest-frame NIR spectrum of QSO2s at z$<$0.7 has not been fully characterised yet. To the best of our knowledge, this has been done for one QSO2 so far: Mrk477 at z=0.037 [@Villar15]. Additionally, a NIR spectrum of the QSO2 SDSS J1131+1627 at z=0.173 was presented in @Rose11, but only Pa$\alpha$ was detected. Here we explore the NIR spectrum of the QSO2 SDSS J143029.88+133912.0 (J1430+1339; at $z$=0.0852). The Teacup galaxy ----------------- According to its \[O III\] luminosity (5$\times$10$^{42}~erg~s^{-1}$ = 10$^{9.1}L_{\sun}$; @Reyes08), J1430+1339 is a luminous QSO2, and considering its position in the 1.4GHz–\[O III\] luminosity plane [@Lal10] it is classified as radio-quiet (L$_{1.4GHz}=5\times 10^{23}~W Hz^{-1}$; Harrison et al. 2015, hereafter @Harrison15). Nonetheless, it is a factor of 10 above the radio-FIR correlation found for star-forming galaxies [@Villar14; @Harrison14], which makes it a “radio excess source”. The host galaxy shows clear signatures of a past interaction with another galaxy, in the form of shells, tails and chaotic dust lanes [@Keel15]. J1430+1339 was nicknamed “the Teacup galaxy” because of the peculiar appearance of its extended emission-line region (EELR) in SDSS and HST images [@Keel12; @Keel15]. This EELR is dominated by a filamentary bubble to the northeast (NE) with a radial extent of $\sim$12 kpc measured from the nucleus (see Figure \[fig1\]). In the opposite direction there is another knotty emission-line structure resembling a fan extending up to $\sim$7 kpc. The Teacup has been proposed as a fading AGN candidate [@Gagne14]. The NE emission-line bubble coincides with the radio-continuum structure detected in VLA maps @Harrison15. These radio maps also show another radio bubble extending $\sim$10 kpc to the west, as well as two compact radio structures: a brighter one coincident with the AGN position, and a fainter one located $\sim$0.8 kpc northeast from the AGN (PA$\sim$60), identified by @Harrison15 as high-resolution B (HR-B) region. According to the latter authors, this HR-B structure would be co-spatial with the base of the ionized nuclear outflow first reported by @Villar14 and @Harrison14 using the SDSS spectrum and IFU spectroscopy respectively. At the position of HR-B the outflow has an observed velocity of -740 km s$^{-1}$ relative to the narrow component of \[O III\]$\lambda$5007 Å, which @Harrison15 interpreted as gas accelerated by jets or quasar winds at that location. The latter authors also speculated that these jets/winds would be driving the 10–12 kpc radio bubbles. ![image](f1.eps){width="12cm"} Here we study the nuclear and extended NIR emission of the Teacup using seeing-limited SINFONI K-band data. In Section 2 we describe the observations and data reduction, in Section 3 we present the results on the nuclear and extended emission of the galaxy, and in Section 4 we discuss the observations and their implications. Throughout this paper we assume a cosmology with H$_0$=71 km s$^{-1}$ Mpc$^{-1}$, $\Omega_m$ = 0.27, and $\Omega_{\Lambda}$ =0.73. At the redshift of the galaxy (z=0.0852), the spatial scale is 1.591 kpc arcsec$^{-1}$. Observations and Data Reduction {#observations} =============================== We obtained K-band (1.95–2.45 ) observations of the Teacup with SINFONI on the 8 m Very Large Telescope (VLT). The data were taken during the night of 2015 March 7th in service mode (Program ID: 094.B-0189(A)) with a total on-source time of 1800 s and at an airmass of 1.3–1.4. Due to the strong and rapid variation of the IR sky emission, the observations were split into short exposures of 300 s each, following a jittering O-S-S-O pattern for sky and on-source frames. The observing conditions were clear and the seeing variation over the on-source observing period was small according to the DIMM seeing monitor[^1] (median optical seeing FWHM=1.05, standard deviation=0.21, and standard error=0.02). To calculate the seeing FWHM in the K-band we used the photometric standard star observed immediately after the target, which appears slightly elongated along PA=88.3$\pm$0.1, with a maximum FWHM=0.58. Along the minor axis, the FWHM=0.46. Therefore, the seeing error is dominated by the shape of the PSF rather than by the seeing variation during the observations. By averaging the maximum and minimum values of the FWHM measured for the star and by adding the seeing variation and PSF shape errors in quadrature, we get a seeing FWHM=0.52$\pm$0.06 ($\sim$830 pc resolution). We used the 0.125$\times$0.250 pixel$^{-1}$ configuration, which yields a field-of-view (FOV) of 8$\times$8 per single exposure. Due to the jittering process, the effective FOV in the case of our target is $\sim$9$\times$9 ($\sim$14$\times$14 kpc$^2$). The spectral resolution in the K-band is R$\sim$3300 ($\sim$75 km s$^{-1}$) and the instrumental broadening, as measured from the OH sky lines, is 6.0$\pm$0.5 Å with a dispersion of 2.45 Å pixel$^{-1}$. For the reduction of the data, we used the ESO pipeline ESOREX (version 3.8.3) and our own IDL routines for the telluric correction and flux calibration (see @Piqueras12). We applied the usual calibration corrections of dark subtraction, flat fielding, detector linearity, geometrical distortion, wavelength calibration, and subtraction of the sky emission to the individual frames. The individual cubes from each exposures were then combined into a single data cube. To estimate the uncertainty in the wavelength calibration we used the atmospheric OH lines, from which we measured an error of 7.8 km s$^{-1}$. The flux calibration was performed in two steps. First, to obtain the atmospheric transmission curves, we extracted the spectra of the standard stars with an aperture of 5$\sigma$ of the best 2D Gaussian fit of a collapsed image. The spectra were then normalised by a blackbody profile of the appropiate T$_{eff}$, taking the most relevant absorption spectral features of the stars into account. The result is a sensitivity function that accounts for the atmospheric transmission. Second, we flux-calibrated the spectra of the stars using their 2MASS K-band magnitudes. Every individual cube was then divided by the sensitivity function and multiplied by the conversion factor to obtain a fully-calibrated data cube. The uncertainty in the flux calibration is $\sim$15%. We refer the reader to @Piqueras12 [@Piqueras16] for further details on the data reduction. Results ======= Nuclear spectrum ---------------- We extracted two K-band spectra of the nuclear region of the Teacup in two circular apertures of 0.5 and 1.25 diameter ($\sim$0.8 and 2 kpc respectively), centred at the maximum of the Pa$\alpha$ emission (see Section \[nuclear\] for details). The minimum aperture was chosen to match the spatial resolution set by the seeing (FWHM=0.52$\pm$0.06). In the following we will refer to the spectrum extracted in this aperture as the nuclear spectrum. ### Continuum shape {#nuclear} Figure \[fig2\] shows the observed nuclear spectrum of the Teacup with the emission lines labelled. We extracted the spectra centred at the peak of the ionized gas emission (as traced by Pa$\alpha$) because it does not coincide with the maximum of the K-band continuum emission. The peaks of the ionized gas and continuum emission are spatially offset by 0.125 ($\sim$200 pc; i.e. the size of one spaxel) with PA=0. It is the case that 0.125 is 1/4 the seeing size and thus we cannot resolve two spectra spatially offset by 0.125. The continuum slope rises towards the red, showing a maximum at $\sim$2.35 . The red dashed line in the left panel of Figure \[fig2\] corresponds to a blackbody of T=1200 K, which better reproduces the K-band nuclear continuum of the Teacup. This K-band spectral shape is not common in type-2 AGN, which generally show the opposite slope, but it has been reported for a few Seyfert 2 galaxies (e.g. NGC7674 – @Riffel06; Mrk348 – @Ramos09; NGC4472 and NGC7743 – @Burtscher15) and it has been interpreted as emission from AGN-heated nuclear dust near the sublimation temperature. In the case of the Teacup, considering the relatively large area probed by the nuclear spectrum ($\sim$830 pc diameter), we are likely detecting hot polar dust within the ionization cones. The change of the continuum slope with increasing aperture, as shown in the right panel of Figure \[fig2\] is also noteworthy. The red excess is only observed in the nuclear spectrum, it then flattens if we consider intermediate apertures and the slope becomes negative in the case of the large aperture spectrum (1.25 diameter). The latter spectral shape resembles the typical K-band spectrum of Seyfert 2 galaxies [@Riffel06; @Ramos09; @Burtscher15]. This change of slope with increasing aperture is due to the extra contribution from stellar light included in the larger apertures. ### Emission line spectrum {#lines} By far the most prominent emission-line feature in the nuclear spectrum of the galaxy is Pa$\alpha$ (see Figure \[fig2\]), followed by Br$\delta$, He I$\lambda$2.060, Br$\gamma$, and the coronal line \[Si VI\]$\lambda$1.963 (all $\lambda$ given in unless otherwise specified). We also detect several H$_2$ emission lines, indicative of the presence of a nuclear molecular gas reservoir. We fitted the nuclear emission-line spectrum with Gaussian profiles using the Starlink program [dipso]{}. Since the \[Si VI\] and H$_2$ 1-0S(3) lines are blended (see central panel of Figure \[fig3\]) we fixed the FWHM of H$_2$ 1-0S(3) to match those of the other H$_2$ lines to enable us to obtain a reliable fit. In Table \[tab1\] we report the FWHMs corrected for instrumental broadening, velocity shifts (V$_s$) and fluxes resulting from our fits with [dipso]{} with their correspoding errors. The uncertainties in V$_s$ include the wavelength calibration error (7.8 km s$^{-1}$ as measured from the sky spectrum) and the individual fit uncertainties provided by [dipso]{}. In the case of the fluxes, the errors have been determined by adding quadratically the flux calibration error (15%) and the fit uncertainties. We require two Gaussians to reproduce the hydrogen recombination lines and the \[Si VI\]$\lambda$1.963 line profiles. This includes a narrow component of FWHM$\sim$400-460 km s$^{-1}$ and a broad blueshifted component of FWHM$\sim$1600–1800 km s$^{-1}$. We identify this broad component with the nuclear outflow reported from optical spectroscopy by @Villar14 and @Harrison15. We discard the possibility of a BLR origin because the broad components are significantly blueshifted from the narrow component. Additionally not only are they detected in the permitted lines, but also in the \[Si VI\] coronal line. In Figure \[fig3\] we show the profiles and corresponding fits of the emission lines showing blueshifted broad components in the nuclear spectrum, namely Pa$\alpha$, Br$\delta$, \[Si VI\], and Br$\gamma$. In the case of Pa$\alpha$ we fitted a broad component of FWHM=1800$\pm$90 km s$^{-1}$ with V$_s$=-234$\pm$35 km s$^{-1}$ relative to the central wavelength of the narrow component ($\lambda_c$=20353.39$\pm$0.54 Å, giving $z$=0.08516$\pm$0.00003). For Br$\delta$ and Br$\gamma$ we fitted blueshifted broad components consistent with that of Pa$\alpha$ within the uncertainties (see Table \[tab1\]). The coronal line \[Si VI\] also shows a broad component of FWHM=1600$\pm$120 km s$^{-1}$ blueshifted with respect to the narrow \[Si VI\] emission component. It is noteworthy that the narrow component is redshifted by V$_s$=54$\pm$11 km s$^{-1}$ from the narrow Pa$\alpha$ line, which is not observed in any other emission line in the K-band spectrum of the Teacup. As explained in Section \[intro\], coronal lines are generally blueshifted and slightly broader than lower ionization emission lines (e.g. @Rodriguez11). In the case of the Teacup, we detect the blueshifted broad component associated with the outflow, and a redshifted narrow component whose FWHM is the same as those of the recombination lines. This redshifted narrow component suggests that the coronal region and the region where the narrow core of the hydrogen lines is produced are different. Moreover, the lack of detection of coronal lines in the optical spectrum of the Teacup could be indicating that this coronal region is more reddened than the NLR. Using the SDSS spectrum presented in @Villar14 we measured \[Fe VII\]$\lambda$6087Å/\[O I\]$\lambda$6300Å=0.037$\pm$0.010, which is well below the range of $\sim$0.6-5.7 reported by @Rodriguez06b for Seyfert galaxies with strong coronal lines detected in their NIR spectra. Finally, in the case of the He I$\lambda$2.060 and the molecular lines, single Gaussians with FWHM$\sim$400–480 km s$^{-1}$ were sufficient to reproduce the profiles (see Table \[tab1\]). However, whilst neither the He I line nor the narrow core of the hydrogen recombination lines are shifted from the systemic velocity (as measured from the narrow component of Pa$\alpha$), all the H$_2$ lines are systematically blueshifted, with V$_s$=-51$\pm$32 km s$^{-1}$ on average. This could be a first indication of a molecular outflow in the Teacup (see e.g. @Muller16 and references therein), although deeper observations are required to confirm it. It should also be noted that the use of the narrow component of Pa$\alpha$ as a tracer of the systemic velocity is affected by uncertainties [@Villar14; @Muller16]. Accurate measurements of the systemic velocity are needed to understand the kinematic behaviour of the H$_2$ lines. ### Emission line diagnostics {#emission} To determine the degree of obscuration of the nuclear region of the Teacup we calculated the narrow and broad Pa$\alpha$/Br$\gamma$ ratios in the two apertures considered here. By comparing them (see Table \[tab2\]) with the theoretical value of 12.2 [@Hummer87] we can determine the optical and infrared extinction (A$_V$ and A$_K$ respectively) by using the parametrization A$_{\lambda}\propto\lambda^{-1.75}$ [@Draine89]. For the narrow component, we measure a maximum value of A$_V$=2.8$\pm$1.4 mag in the nucleus of the Teacup ($\sim$830 pc diameter). In the large aperture ($\sim$2 kpc diameter) the level of obscuration decreases to 0.55$\pm$1.14 mag. Following the same procedure, from the ratio of the broad components we measure A$_V$=3.5$\pm$1.9 mag in the nucleus of the Teacup and A$_V$=1.5$\pm$1.7 mag in the large aperture. The latter value of the extinction is consistent with A$_V$=1.9$\pm$0.3 mag reported by @Villar14 for the nuclear outflow as measured from the SDSS spectrum. The ratio H$_2$1-0S(1)/Br$\gamma$ can be used to disentangle the dominant excitation mechanism of the gas (see Table \[tab2\]). In the two apertures considered we measure values in the range 0.6–0.9, consistent with AGN photoinization (this ratio is lower than 0.6 for starburst galaxies and higher for LINERs; @Mazzalay13). Although we observe a tendency in 1-0S(1)/Br$\gamma$ to increase with the aperture, the values are consistent within the errors. ![Molecular line ratios measured in the two apertures considered here (0.5and 1.25; green triangles). Pink circles are the ratios measured from the nuclear spectra of the Seyfert 2 galaxies studied in @Ramos09. Yellow squares correspond to the ratios derived for the broad and narrow components of regions A and B of the luminous IR galaxy (LIRG) and gas-rich merger NGC3256 [@Emonts14]. The dashed line indicates the locus of T$_{vib}$=T$_{rot}$. Vertical dotted lines delimitate the regions of “thermal” and “non-thermal” excitation from @Mouri94.[]{data-label="fig4"}](f4.eps){width="6.5cm"} As can be seen from Figure \[fig2\], we detect several H$_2$ lines in the nuclear spectrum of the Teacup. In AGN, the lowest vibrational levels (v=1) of H$_2$ tend to be thermalized (i.e. excited by shocks and/or X-ray illumination; @Hollenbach89 [@Maloney96]), while higher level transitions are populated due to non-thermal processes such as UV fluorescence [@Black87]. The 1-0S(1)/2-1S(1) line ratio is an excellent discriminator between thermal and non-thermal processes. It is $\le$2 in gas excited by UV fluorescence and $\ge$5 in thermally-dominated gas [@Mouri94]. At the same time, the 1-0S(2)/1-0S(0) line ratio is sensitive to the strength of the incident radiation. In Figure \[fig4\] we show the line ratios that we have measured in the two apertures considered here. For comparison, we also plot the nuclear ratios of the five Seyfert 2 galaxies studied in @Ramos09 and those measured for the broad and narrow component of the lines in regions A and B of the gas-rich merger and luminous IR galaxy NGC3256 [@Emonts14]. We find 1-0S(1)/2-1S(1) ratios $\ge$5, consistent with thermal excitation. Indeed, it can be seen from Figure \[fig4\] that the position of the Teacup ratios in the diagram is very different from those measured in Seyfert 2 galaxies. On the other hand, the lower limits on the 1-0S(1)/2-1S(1) line ratio are consistent with the values reported by @Emonts14 for regions A and B of the LIRG NGC 3256, but the 1-0S(2)/1-0S(0) values are significantly higher. This implies that although the molecular gas in the Teacup and in NGC3256 is thermally excited, the strength of the incident radiation is not the same. This is expected considering that the presence of nuclear activity has not been confirmed yet in NGC3256 [@Emonts14], while the Teacup hosts a very luminous AGN. Using the two line ratios mentioned above we can derive the rotational and vibrational temperatures of the gas following @Reunanen02. Although we only have upper limits, in the case of the nuclear spectrum T$_{vib}\simeq T_{rot}\ga 3000$ K, which is characteristic of thermally excited gas. On the other hand, for the large aperture T$_{vib}<<T_{rot}$, indicating that gas excitation is more complex than local thermal equilibrium (LTE) conditions (T$_{vib}=T_{rot}$). Finally, we can use the H$_2$1-0S(1) luminosity to estimate the amount of molecular gas present in the nucleus of the Teacup. Following @Mazzalay13, the relation between the line flux and the warm molecular gas mass is $$M_{H_2}\simeq 5.0875 \times 10^{13} (\frac{D}{Mpc})^2 (\frac{F_{1-0S(1)}}{erg~s^{-1} cm^{-2}})10^{0.4A_K}, \label{eq1}$$ where D=387 Mpc and A$_K$ is the extinction reported in Table \[tab2\]. In the same Table we show M$_{H_2}$ measured in the two apertures considered, as well as the masses of cold molecular gas (M$_{cold}$) calculated by assuming a cold–to–warm mass ratio M$_{cold}$/M$_{H_2}\simeq0.7\times10^{6}$. This ratio was derived observationally by @Mazzalay13 by comparing values of M$_{cold}$ obtained from CO observations and H$_2$ luminosities for a large number of galaxies covering a wide range of luminosities, morphological types and nuclear activity. A similar ratio was reported by @Dale05 for a large sample of active and star-forming galaxies (M$_{cold}$/M$_{H_2}\simeq10^{5-7}$). In the nucleus of the Teacup we measure M$_{H_2}=(3.0\pm0.8)\times10^3 M_{\sun}$ and M$_{cold}=(2.2\pm0.6)\times10^9 M_{\sun}$. If instead of looking at the inner $\sim$830 pc of the Teacup we measure the molecular gas content in the inner 2 kpc, M$_{H_2}=(1.0\pm0.2)\times10^4 M_{\sun}$ and M$_{cold}=(7.4\pm1.8)\times10^9 M_{\sun}$. These cold gas masses are similar to those reported by @Villar13 for a sample of 10 QSO2s at z$\sim$0.2–0.3 with CO measurements. Ionized, coronal and molecular emission-line maps {#extended} ------------------------------------------------- In the previous sections we studied the spectra of the nuclear region of the Teacup. Here we take advantage of the spatial information provided by SINFONI and study the flux distribution and kinematics of Pa$\alpha$, \[Si VI\] and H$_2$1-0S(1). These three emission lines are the highest S/N representatives of the ionized, coronal and molecular phases of the gas respectively. Pa$\alpha$ is the only line that we detect in the extended emission-line structures (i.e. the NE bubble and the SW fan; see Section \[bubbles\]). We note, however, that the SINFONI FOV (9.2$\times$8.7) does not cover the full extent of the NE bubble, as shown in Figure \[fig1\]. In the case of the nuclear Pa$\alpha$ and \[Si VI\] emission we needed two Gaussians to reproduce the observed line profiles and obtain corresponding flux, velocity and velocity dispersion ($\sigma$) maps. For the H$_2$1-0S(1) emission line (hereafter H$_2$) a single Gaussian was sufficient. Since the \[Si VI\] and H$_2$1-0S(3) lines are blended, we fixed the FWHM of the latter emission line to match that of H$_2$1-0S(1) and get reliable fits for the \[Si VI\]. In Figure \[fig5\] we show the flux, velocity and $\sigma$ maps of the broad Pa$\alpha$, broad and narrow \[Si VI\] components, and H$_2$. These maps correspond to a 4$\times$4 (6.4$\times$6.4 kpc$^2$) FOV. Using the flux maps we can estimate the projected sizes of each emission-line region. In Table \[tab3\] we show the observed and seeing-deconvolved FWHMs along the major axis in each case. We note that the intrinsic, deconvolved sizes for the central distribution of the emission lines considered here are approximate sizes. They have been obtained by applying the standard Gaussian deconvolution method based on the well-known quadrature relation FWHM$_{int}^2$ = FWHM$_{obs}^2$ - FWHM$_{seeing}^2$. All the emission-line maps shown in Figure \[fig5\] are resolved except the narrow \[Si VI\] maps. The broad Pa$\alpha$ and broad \[Si VI\] flux maps show similar morphologies, and are both elongated roughly in the same direction (PA$\sim$72–75). We note that this elongation does not coincide with the orientation of the seeing major axis (PA=88.3$\pm$0.1), as measured from the standard star. In the case of the broad \[Si VI\] flux map, two peaks are observed, separated by $\sim$0.3 with a PA$\sim$70 (although it is difficult to distinguish them in the \[Si VI\] flux map shown in Figure \[fig5\], they are evident when displayed in contours). One of these flux peaks coincides with the AGN position, as traced by the maximum of the Pa$\alpha$ emission, and the other could be the coronal counterpart of the compact HR-B region detected by @Harrison15 in the high-angular resolution VLA radio maps. The HR-B compact region is located 0.5 northeast of the AGN position, with PA$\sim$60. As expected for outflow-related components, the maps of the broad Pa$\alpha$ and broad \[Si VI\] emission are dominated by blueshifted velocity components, reaching maximum values of -250 km s$^{-1}$. The narrow \[Si VI\] emission-line maps (third row of Figure \[fig5\]) are similar to those of the broad component, with the exception of the velocity map. The NE part of the emission is redshifted and the SW blueshifted, with maximum velocities of $\pm$150 km s$^{-1}$. From the narrow \[Si VI\] flux map we can set constraints on the size of the coronal line region (CLR) of the Teacup. The observed FWHM that we measure is consistent with the seeing FWHM within the errors (see Table \[tab3\]). Therefore the size of the CLR is formally unresolved and we can estimate an upper limit as \[(FWHM$_{seeing}$+3$\sigma_{seeing}$)$^2$-FWHM$_{seeing}^2]^{0.5}$ = 0.47$\sim$746 pc. The H$_2$ maps (bottom panels of Figure \[fig5\]) are completely different to those of the ionized and coronal lines. They are elongated almost in the N–S direction (PA=-7.7$\pm0.5\degr$), which is roughly perpendicular to the orientation of the Pa$\alpha$ and \[Si VI\] maps. The seeing-deconvolved FWHM along the major axis is 1.4$\pm$0.2 kpc (see Table \[tab3\]). The global H$_2$ velocity field seems to be dominated by rotation, with maximum velocities of $\pm$250 km s$^{-1}$. Deviations from this pattern are identified at some spatial locations, but they correspond to spaxels where the signal-to-noise is lower. The $\sigma$ values are at a maximum in the central region (FWHM$\sim$450 km s$^{-1}$) and decrease toward the edges (FWHM$\sim$150 km s$^{-1}$). ![image](f5.eps){width="18.0cm"} ### Extent of the nuclear outflow In Figure \[fig6\] we show Pa$\alpha$ flux maps extracted in consecutive velocity intervals of 500 km s$^{-1}$, centred at the maximum of the line profile in the central spaxel (see Figure \[fig7\]). These velocity cuts allow us to characterize the extent and orientation of the Pa$\alpha$ emission in the core and the wings of the line. In the central panel we see the Pa$\alpha$ emission corresponding to the core of the line, which we have assumed as systemic redshift. From this map we can distinguish the nuclear emission (inner $\sim$2) and the extent of the emission-line features covered by the SINFONI FOV (the NE bubble and the SW fan). The right and left middle panels correspond to the velocity bins centred at $\pm$500 km s$^{-1}$, so the Pa$\alpha$ wings dominate the emission. The nuclear emission appears elongated in the two panels and on larger scales, depending on whether we are sampling the blue or the red wing of Pa$\alpha$, we see the SW fan or the NE bubble, respectively. ![image](f6.eps){width="13cm"} ![Pa$\alpha$ line profile as extracted from the central spaxel. The continuum has been normalized (dashed line). The solid-green vertical lines indicate the velocity bins (from -2000 to 2000 km s$^{-1}$ in steps of 500 km s$^{-1}$) used for extracting the nine flux maps shown in Figure \[fig6\].[]{data-label="fig7"}](f7.eps){width="8cm"} The blueshifted broad component that we identified in the nuclear spectrum of the Teacup is sampled by the four negative velocity bins (see Figure \[fig7\]). We fitted a Gaussian model to the nuclear region detected in the corresponding flux maps (middle left and top panels in Figure \[fig6\]) and measured seeing-deconvolved FWHMs ranging from 1 to 1.1 kpc along the major axis, with PA=70–75. We measure a similar extension and orientation of the outflow when we sample the red wing of Pa$\alpha$ (middle right and bottom left panels in Figure \[fig6\]). This indicates that we can trace the nuclear outflow up to 1000 km s$^{-1}$ beyond the core of Pa$\alpha$. It is noteworthy that the elongation of the nuclear Pa$\alpha$ emission that we see at the highest velocities is less obvious at $\Delta$v=0 km s$^{-1}$ (i.e. at the core of the line), confirming that the nuclear outflow is extended. Therefore from the analysis of Figure \[fig6\] we conclude that the nuclear ionized outflow is resolved. The observed maximum extension, which corresponds to the -1000 km s$^{-1}$ velocity bin, is FWHM=1.37$\pm$0.06 along the 73.9$\pm$0.2 axis. As noted in Section \[observations\], the standard star that was used to characterize the seeing is slightly elongated along PA=88.3$\pm$0.1, with a maximum extension of 0.58. Even considering the maximum value of the seeing, the nuclear ionized outflow is resolved, with a seeing-deconvolved FWHM=0.68$\pm$0.09 (1.1$\pm$0.1 kpc). We cannot repeat the same exercise for the \[Si VI\] emission line because its blue wing is blended with H$_2$1-0S(3) and it has lower signal-to-noise than Pa$\alpha$. However, if we fit a Gaussian component to the broad \[Si VI\] flux map shown in Figure \[fig5\], we measure a FWHM=0.75$\pm$0.06, which corresponds to a seeing-deconvolved FWHM=0.54$\pm$0.09 (0.9$\pm$0.1 kpc) along the major axis (PA=72.4$\pm$0.3). Therefore, the coronal outflow is also resolved in the same direction as the ionized outflow. ### Large-scale Pa$\alpha$ emission {#bubbles} In Figure \[fig8\] we show the flux, velocity and dispersion maps of the narrow component of Pa$\alpha$ in a 9.5$\times$9 (15.1$\times$14.3 kpc$^2$) FOV. The flux map resembles the morphology of the \[O III\] and H$\alpha$ HST images [@Keel15]. The bulk of the Pa$\alpha$ emission is dominated by the nuclear component, and beyond we detect part of the NE bubble (radial size of $\sim$9.5 kpc), and the SW fan ($\sim$7 kpc). The velocity map (central panel of Figure \[fig8\]) is similar to the \[O III\] velocity map reported by @Harrison15, although it reveals that the kinematics of the gas in the bubble and fan are different from those of the NLR. Whilst the central $\sim$2 show a smooth velocity field, probably coincident with the rotation pattern of the galaxy, beyond this there is an abrupt change in velocity. The NE bubble is redshifted with maximum velocities of 300 km s$^{-1}$ relative to systemic, and the SW fan is blueshifted with V$_{max}$=-300 km s$^{-1}$. The dispersion map also shows a practically constant line width within the central 2 (FWHM$\sim$450-500 km s$^{-1}$) and smaller values in the bubble and fan (FWHM$\sim$200–300 km s$^{-1}$). We note that the FWHM$>$500 km s$^{-1}$ (i.e. $\sigma\ga 210$ km s$^{-1}$) that we measure around the nuclear region ($\sim$1 radius) corresponds to the transition zone where we no longer need to fit a broad component to reproduce the Pa$\alpha$ profiles. ![image](f8.eps){width="18cm"} To study the properties of the extended Pa$\alpha$ emission of the Teacup in more detail, we extracted spectra from different spatial locations in the bubble and fan. In Figure \[fig1\] we indicate the positions and apertures chosen for extracting the spectra in the galaxy nucleus, bubble (N1, N2, and N3) and fan (S1, S2, and S3). In Table \[tab4\] we report the different apertures chosen in each case and the radial distances from the nucleus. We show the six spectra extracted in the bubble and the fan in Figure \[fig9\]. We used [dipso]{} to fit the emission lines using Gaussian profiles, and in Table \[tab4\] we report the corresponding FWHMs, V$_s$ and line intesities. In the case of the bubble, single Gaussian components of FWHM$\sim$300 km s$^{-1}$ were sufficient to reproduce the narrow line profiles in the N1 and N2 spectra, extracted at $\sim$3 kpc from the nucleus. For the more distant N3 region ($\sim$8 kpc from the nucleus) the FWHM is narrower (FWHM$\sim$200 km s$^{-1}$). We note that, as we already knew from the Pa$\alpha$ velocity map shown in Figure \[fig8\], these narrow Pa$\alpha$ lines are significantly redshifted relative to the systemic velocity (V$_s$$\sim$120–270 km s$^{-1}$; see Figure \[fig9\]). We also extracted three spectra mapping two bright regions within the fan (S1 and S2) and a bright knot at $\sim$3 kpc towards the west (S3). Two Gaussians were necessary to reproduce the narrow Pa$\alpha$ profiles detected in the three spectra (see right panels of Figure \[fig9\]). First, the blueshifted (V$_s\sim$-250 km s$^{-1}$) narrow components with FWHM=100–200 km s$^{-1}$ already shown in the velocity map (see Figure \[fig8\]). Second, in the case of the S1 and S3 spectra we required an additional component of FWHM$\sim$200-300 km s$^{-1}$ redshifted by V$_s\sim$40–120 km s$^{-1}$ from systemic. For the S2 spectrum we fitted a slightly broader component of FWHM$\sim$500 km s$^{-1}$ whose V$_s$ is consistent with that of the narrow component (V$_s\sim$-300 km s$^{-1}$). Finally, we report tentative detection of a very broad Pa$\alpha$ component of FWHM$\sim$3000 km s$^{-1}$ in five of the regions considered here (shown as blue Gaussians in Figure \[fig9\]). These broad components are blueshifted relative to systemic in the bubble and fan, with V$_s$ ranging between -60 and -310 km s$^{-1}$. Unfortunately, the combination of sky residuals and low signal–to–noise of the spectra prevents confirmation of these very broad Pa$\alpha$ components. For this reason the results from the fits reported in Table \[tab4\] do not include the broad components. Discussion ========== The nuclear outflow ------------------- Based on the blueshifted broad lines detected in the nuclear K-band spectrum of the Teacup we confirm the presence of the nuclear ionized outflow previously reported by @Villar14 and @Harrison15 using optical spectra, and we report the existence of its coronal counterpart. Although coronal outflows are commonly detected in Seyfert galaxies (e.g. @Gelbord09 [@Davies14]), this is one of the first detections of coronal outflows in QSO2s. Another example is Mrk477, a QSO2 at z=0.037 with an \[O III\] luminosity of 3.3$\times$10$^{42}~erg~s^{-1}$ for which @Villar15 reported a blueshifted component of FWHM=2460$\pm$340 km s$^{-1}$ in the high-ionization line \[Fe VII\]$\lambda$6087 Å (IP=99.1 eV). The FWHM of this blueshifted component is larger than those measured in the optical for the central 3 of the Teacup using the SDSS spectrum (FWHM$\sim$1000 km s$^{-1}$; @Villar14) and GMOS/Gemini IFU data (maximum FWHM$\sim$1000 km s$^{-1}$; @Harrison14). This is consistent with the outflow being reddened, as first claimed by @Villar14 and confirmed in this work, supporting the advantage of using NIR observations to trace AGN outflows closer to their origin. We measure V$_s$=-234$\pm$35 km s$^{-1}$ for the nuclear broad Pa$\alpha$ component. This value is consistent with the velocities reported by @Harrison14 for the blueshifted component of the \[O III\] line measured from GMOS IFU data. They report a velocity offset of $\Delta$v=-150 km s$^{-1}$ for the broad component in the galaxy-integrated spectrum, which corresponds to the central 5$\times$3.5. However, if we look at the velocity maps (Figure A14 in @Harrison14), there is a strong velocity gradient around the position of the AGN, with a maximum velocity offset of -268 km s$^{-1}$. The broad Pa$\alpha$ velocity map that we report here resembles the \[O III\] velocity field, showing maximum velocities of -250 km s$^{-1}$. On the other hand, @Villar14 reported V$_s$=-70 km s$^{-1}$ for the broad component of \[O III\] relative to the narrow core, but measured in an aperture of 3 diameter ($\sim$5 kpc). The broad Pa$\alpha$ maps, shown in the top middle panel of Figure \[fig5\], demonstrate that the SDSS fiber includes both blueshifted and redshifted components, which explains the smaller V$_s$ reported in @Villar14. @Harrison15 extracted a spectrum at the position of the HR-B region detected in the VLA radio maps and measured a FWHM=720 km s$^{-1}$ with an observed velocity of -740 km s$^{-1}$ relative to the narrow component of \[O III\]. Our SINFONI data have very high S/N in the nuclear region and yet, we do not require such a high velocity component to reproduce any of the profiles (see top middle panel of Figure \[fig5\]). This could be due to the combined effect of reddening variation at different spatial locations, different aperture sizes and kinematic substructure within the outflow region. Using the Pa$\alpha$ and \[Si VI\] flux maps we confirm that both the ionized and coronal nuclear outflows are extended. We derived radial sizes of 1.1$\pm$0.1 kpc and 0.9$\pm$0.1 kpc respectively, with PA=72–74. This PA is almost identical to the radio jet orientation measured from the 1.4 GHz FIRST radio maps (PA$\sim$77; @Harrison14). This could be indicating, as first suggested by @Harrison15, that the interaction between the radio jet and the galaxy interstellar medium may have triggered and accelerated the nuclear outflow. The Teacup is then likely an example of radio jets driving outflows in a radio-quiet AGN. HST studies of Seyfert galaxies with linear radio structures showed strong evidence for interactions between the radio structures and the emission line gas in the NLR occurring on scales of $\la$1 kpc (e.g. @Axon98). In QSO2s, the nuclear outflow in Mrk477, which is thought to be triggered by the triple radio source present in this galaxy, has an estimated size of several hundreds of parsecs [@Heckman97; @Villar15], and recent observations of different samples of luminous QSO2s at z$<$0.6, some of them with relatively high radio luminosities, reveal compact ionized outflow sizes of $<$1–2 kpc [@Villar16; @Karouzos16; @Husemann16]. The case of powerful radio galaxies and quasars is different, with radio-induced outflows that can extend up to several kpc, even outside the galaxy boundaries [@Tadhunter94; @Villar99]. In the case of the H$_2$ lines we do not find evidence for broad components in the nuclear spectrum, and the velocity map shows a dominant rotation pattern. The only possible signatures of a molecular outflow in the Teacup are the nuclear narrow components blueshifted by -50 km s$^{-1}$ on average relative to the systemic velocity. In any case, the bulk of the H$_2$ emission comes from the rotating gas distribution shown in Figure \[fig5\], and only a small percentage would be outflowing. Deeper observations and a more accurate determination of the systemic velocity of the galaxy are needed to confirm the presence of a nuclear molecular outflow. The case of the Teacup is very different from that of its low-luminosity counterpart IC5063. This Seyfert 2 galaxy has a similar redshift and radio-power as the Teacup but its NIR spectrum shows wings in the H$_2$ profiles as broad as those of the Br$\gamma$ line. Furthermore, the H$_2$ emission shows a peak in brightness co-spatial with the radio lobe, which @Tadhunter14 interpreted as being due to gas cooling and forming molecules behind a jet-induced shock. The results presented here suggest a different scenario in the case of the Teacup, demonstrating that we do not always see broad molecular lines in the case of galaxies with strong jet-cloud interactions and ionized outflows. This non-detection of broad H$_2$ components in the Teacup could be explained by a two-stage quasar wind scenario [@Lapi05; @Menci08; @Zubovas12; @Faucher12]. Fast winds accelerated by the AGN interact with the galaxy ISM, reaching temperatures $\ge10^7$ K. Once the gas cools down to $\sim$10$^4$ K, it starts to emit warm ionized and coronal lines such as \[O III\], Pa$\alpha$, and \[Si VI\], but further cooling is necessary for the gas to emit in H$_2$. Emission-line structures ------------------------ The bulk of the Pa$\alpha$ emission is dominated by the nuclear component, and beyond, the NE bubble and SW fan expand in opposite directions with projected velocities $\pm$300 km s$^{-1}$ (see Figure \[fig8\]). This is entirely consistent with the \[O III\] velocity map reported by @Harrison15 in a much larger FOV ($\sim$25$\times$25). We extracted spectra at different locations within the emission line structures and we found double-peaked Pa$\alpha$ profiles in the SW fan with FWHMs=100–500 km s$^{-1}$ which are indicative of disturbed kinematics. In contrast, in the NE bubble only one Gaussian is needed to reproduce the Pa$\alpha$ lines, but we measure FWHMs=300 km s$^{-1}$ at two different positions. For comparison, the extended non-outflowing ionized gas detected in the most dynamically disturbed mergers with nuclear activity show FWHM$<$250 km s$^{-1}$ [@Bellocchi13]. According to this comparison with merger dynamics, the detection of turbulent gas in the Teacup is confirmed in apertures N1, N2, S1, and S2, located at distances between 2.8 and 5.6 kpc from the AGN, which correspond to $\sim$6.5 and $\sim$13.5 times half the seeing size. Furthermore, in both the bubble and the fan we find tentative detection of a a very broad Pa$\alpha$ component of FWHM$\sim$3000 km s$^{-1}$. These broad components are blueshifted relative to systemic, with Vs ranging between -60 and -310 km s$^{-1}$. The velocity shifts that we measure for the very broad components and also for the narrow components (see Table \[tab4\]) are consistent with those recently reported by @Keel17 using GMOS integral field spectroscopy. They claim that asymmetric \[O III\] profiles are present in different locations within the NE bubble, reaching maximum velocities of $\pm$1000 km s$^{-1}$. Unfortunately, @Keel17 did not characterize the lines profiles quantitatively, so we do not know if they are detecting the very broad components that we see in our SINFONI data. Our interpretation is that, if these broad components are real, we would be detecting highly turbulent gas in a huge outflow that has cooled sufficiently to emit hydrogen recombination lines and be observable in the NIR. Nonetheless, independently of whether or not these broad components are real, the large scale gas (up to 5.6 kpc from the nucleus) shows turbulent kinematics that can be explained by the action of the outflow. Finally, the H$_2$ maps presented here reveal a rotation pattern whose axis is misaligned with that of the narrow Pa$\alpha$ emission, suggesting that the molecular gas is not coupled with the velocity distribution shown by the ionized gas. The rotating distribution of molecular gas detected in the Teacup could be the quasar-luminosity counterpart of the 100 pc-scale circumnuclear disks (CNDs) observed in nearby Seyfert galaxies using both NIR [@Hicks13] and sub-mm observations [@Garcia14]. These CNDs rotate with similar velocities to those measured here $\sim$200–300 km s$^{-1}$ (e.g. @Helfer03 [@Garcia05]). Furthermore they are not present in matched samples of quiescent galaxies [@Hicks13], indicating that they constitute a key element in the feeding of active SMBHs. Conclusions =========== We have characterized the K-band emission-line spectrum of the radio-quiet QSO2 known as the Teacup galaxy. Thanks to the IFU capabilities of SINFONI we have not only studied the nuclear region of the galaxy, but also the Pa$\alpha$ emission of its kpc-scale emission-line structures (i.e. the NE bubble and the SW fan) within a $\sim$9$\times$9 FOV. Our major conclusions are as follows: - The nuclear K-band spectrum of the Teacup reveals the presence of a blueshifted component of FWHM$\sim$1600–1800 km s$^{-1}$ in the hydrogen recombination lines and also in the coronal line \[Si VI\]$\lambda$1.963. Therefore, we confirm the presence of the nuclear ionized outflow previously detected from optical spectra, and we reveal its coronal counterpart. - The FWHM of the NIR lines associated with the nuclear ionized outflow are larger than those of their optical counterparts. This is consistent with the idea that, because of the lower extinction in the NIR, we can trace the outflow closer to its origin. - Both the ionized and coronal nuclear outflows are spatially resolved, with seeing-deconvolved radial sizes of 1.1$\pm$0.1 and 0.9$\pm$0.1 kpc along the radio axis (PA=72–74). This suggests that the radio jet could have triggered the nuclear outflow. - We find kinematically disrupted ionized gas (FWHM$>$250 km s$^{-1}$) at up to 5.6 kpc from the AGN, which can be naturally explained by the action of the outflow. - The narrow component of \[Si VI\] is redshifted by V$_s$=54$\pm$11 km s$^{-1}$ with respect to the systemic velocity, which is not the case for any other emission line in the K-band spectrum of the Teacup. This indicates that the coronal region is not co-spatial with the NLR. - In the case of the H$_2$ lines, we do not require a broad component to reproduce the profiles seen in the nuclear spectrum, but the narrow lines are blueshifted by $\sim$50 km s$^{-1}$ on average from the galaxy systemic velocity. This could be an indication of the presence of a molecular outflow, although additional observations are required to confirm this. - The H$_2$ maps reveal a rotating structure oriented roughly perpendicular to the radio jet and the broad Pa$\alpha$ and \[Si VI\] maps. This molecular gas structure could be the quasar-luminosity equivalent of the 100 pc-scale CNDs detected in Seyfert galaxies. - We report tentative detection of very broad Pa$\alpha$ components (FWHM$\sim$3000 km s$^{-1}$) at different locations across the NE bubble and SW fan (at up to 5.6 kpc from the AGN). If confirmed, such extremely turbulent components could be hot shocked gas that has cooled sufficiently to be observable in Pa$\alpha$. Acknowledgments {#acknowledgments .unnumbered} =============== Based on observations made with ESO Telescopes at the Paranal Observatory under programme ID 094.B-0189(A). First, we would like to acknowledge the constructive feedback and suggestions of the referee. CRA acknowledges the Ramón y Cajal Program of the Spanish Ministry of Economy and Competitiveness through project RYC-2014-15779 and the Spanish Plan Nacional de Astronom' ia y Astrofis' ica under grant AYA2016-76682-C3-2-P. JPL acknowledges support from the Science and Technology Facilities Council (STFC) grant ST/N002717/1 and the Spanish Plan Nacional de Astronom' ia y Astrofis' ica under grant AYA2012-39408-C02-01. MVM acknowledges support from the Spanish Ministerio de Econom' ia y Competitividad through the grants AYA2012-32295 and AYA2015-64346-C2-2-P. PSB acknowledges support from FONDECYT through grant 3160374. The authors acknowledge the data analysis facilities provided by the Starlink Project, which is run by CCLRC on behalf of PPARC. We finally acknowledge José Antonio Acosta Pulido, Santiago Garc' ia Burillo, Richard Davies, Erin Hicks, and Clive Tadhunter for useful comments that have substantially contributed to improve this work. Appenzeller, I., Wagner, S. 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--- abstract: \| The ground state geometries of some small clusters have been obtained via ab initio molecular dynamical simulations by employing density based energy functionals. The approximate kinetic energy functionals that have been employed are the standard Thomas-Fermi $(T_{TF})$ along with the Weizsacker correction $T_W$ and a combination $F(N_e)T_{TF} + T_W$. It is shown that the functional involving $F(N_e)$ gives superior charge densities and bondlengths over the standard functional. Apart from dimers and trimers of Na, Mg, Al, Li, Si, equilibrium geometries for $Li_nAl, n=1,8$ and $Al_{13}$ clusters have also been reported. For all the clusters investigated, the method yields the ground state geometries with the correct symmetries with bondlengths within 5% when compared with the corresponding results obtained via full orbital based Kohn-Sham method. The method is fast and a promising one to study the ground state geometries of large clusters. PACS Numbers : 71.10, 31.20G, 02.70N, 36.40 address: 'Department of Physics, University of Poona, Pune 411 007, India' author: - 'Dinesh Nehete, Vaishali Shah and D. G. Kanhere [@email]' title: '[Ab initio]{} molecular dynamics using density based energy functionals: application to ground state geometries of some small clusters.' --- Introduction ============ During the last few years the technique of first principles molecular dynamics (MD), initiated by Car and Parrinello (CP) [@car; @rem], has emerged as a powerful tool for investigations of structural, electronic and thermodynamic properties of large scale systems. The standard implementation of this method which is based on density functional theory is via Kohn-Sham (KS) orbitals. Such orbital based algorithms scale as $N_a^3$, $N_a$ being the number of atoms in the system. Quite clearly, such methods turn out to be computationally expensive for system sizes over about 100 atoms [@footnote]. Recently, approaches based on total energy functionals, which depend on charge density only or orbital free density functionals have been proposed [@pear; @shah; @govind]. These methods are based on approximate representation of kinetic energy (KE) functionals and offer an attractive alternative for investigating large scale systems. Since the method is orbital free i.e there are no wavefunctions to handle, there is no computationally expensive orthogonality constraint and the methods scale linearly with system size. In addition, these methods are shown to yield stable dynamics even with large timesteps, a highly desirable feature for molecular dynamics simulations. It is clear that the utility of these methods is critically dependent on their ability to investigate the systems of interest with acceptable accuracy, at least for a class of physical properties. Madden and coworkers have investigated structural and thermodynamic properties of some simple metals with considerable success. For example, the dynamic structure factor of liquid Sodium and static structure factor, vacancy formation energy, free energies of point defects as well as phonon dispersion curves of Sodium [@pear; @smar] are well described by this method. The method has also been applied for ground state configurations of c-Si and H/Si (1 0 0) surface [@govind] and for geometries of some silicon clusters [@niranjan] and a good agreement has been found with experiments as well as with other calculations. However majority of the calculations reported so far have been performed on extended systems. In the present work, we focus our attention on studying the ground state and energetically low lying structures of clusters, a field of current interest. Obviously, due to the approximate nature of the KE functionals the bondlengths and binding energies will not be obtained with the same level of accuracy as the KS orbital based methods. However, it is of considerable interest to examine whether such a method is capable of yielding the correct shapes (i.e the right symmetries) of clusters by employing Car-Parrinello simulated annealing methods. If desired the KS method can then be used to search the local minimum around structures obtained by Orbital Free Method (OFM) in ‘quenching’ mode. This can be a computationally tractable way to avoid the long and costly simulated annealing runs of the orbital based KS molecular dynamics. Towards this end we have carried out a number of calculations on a variety of representative small clusters of simple metals. Specifically, we have investigated dimers and trimers of Na, Mg, Al, Li, Si, small clusters of $Na_n, (n=6,8)$, $Li_nAl, (n=1,8)$ and $Al_{13}$. These systems are representative of the small metal atom clusters of current interest and more accurate KS based results have been reported. Hence, it is possible to make an assessment of the present method by comparing the bondlengths and geometries with the reported ones. The question of appropriate choice of kinetic energy functionals has been addressed by Smargiassi and Madden [@smarg]. They have investigated a family of kinetic energy functionals which incorporate exact linear response properties. All such KE functionals are based on the Thomas Fermi (TF) and the Weizsacker correction term. Since our interest is in finite size systems, we have chosen to use simple KE functionals. These functionals have been previously used in the study of atoms and molecules. However, it must be mentioned that significant progress has been made towards improving the KE functionals notably by DePristo and Kress, Wang and Teter [@depristo; @wang]. In the next section we briefly discuss the method, the KE functionals used and give the relevant numerical details. This is then followed by the results and discussion. Formalism and Computational details =================================== Total Energy Calculation ------------------------- The total energy of a system of $N_e$ interacting electrons and $N_a$ atoms, according to the Hohenberg-Kohn theorem [@hoh; @ksh], can be uniquely expressed as a functional of the electron density $\rho({\bf r})$ under an external potential due to the nuclear charges at coordinates ${\bf R}_n$, $$E\Bigl[\rho,\{{\bf R}_n\}\Bigr] = T[\rho] + E_{xc}[\rho] + E_c[\rho] + E_{ext}\Bigl[\rho,\{{\bf R}_n\}\Bigr] + E_{ii}\Bigl(\{{\bf R}_n\}\Bigr),$$ where $E_{xc}$ is the exchange-correlation energy, $E_c$ is the electron-electron Coulomb interaction energy. The electron-ion interaction energy $E_{ext}$ is given by $$E_{ext}\Bigl[\rho,\{{\bf R}_n\}\Bigr] = \int{V({\bf r}) \rho({\bf r}) d^3r}$$ where $V({\bf r})$ is the external potential, usually taken to be a convenient pseudopotential [@bach]. The last term in Eq. (1), $E_{ii}$, denotes the ion-ion interaction energy. The first term in Eq. (1), the KE functional, is usually approximated as $$T[\rho] = T_{TF}[\rho] + T_W[\rho]$$ where $T_{TF}[\rho]$ is the Thomas-Fermi term, exact in the limit of homogeneous density, and has the form $$T_{TF}[\rho] = \frac{3}{10}(3\pi^2)^{\frac{2}{3}} \int{\rho({\bf r})^{5/3} d^3r}$$ and $T_W[\rho]$ is the gradient correction due to Weizsacker, given as $$T_W[\rho] = \frac{\lambda}{8} \int{\frac{\nabla\rho({\bf r}) \cdot\nabla\rho({\bf r}) d^3r}{\rho({\bf r})}}$$ which is believed to be the correct asymptotic behavior of T$[\rho]$ for rapidly varying densities. Instead of $\lambda = 1$, the original Weizsacker value, $\lambda = \frac{1}{9}$ and $\lambda = \frac{1}{5}$ [@parr] are also commonly used. It has been argued that for rapidly varying densities, which is the case for finite size clusters a more appropriate kinetic energy functional would be a following combination of these two terms[@acharya] $$T[\rho] = F(N_e) T_{TF}[\rho] + T_W[\rho]$$ where the factor $F(N_e)$ [@gaz] is $$F(N_e) = \bigglb(1-\frac{2}{N_e}\biggrb) \bigglb(1-\frac{A_1}{N_e^{\frac{1}{3}}} +\frac{A_2}{N_e^{\frac{2}{3}}}\biggrb)$$ with optimized parameter values $A_1 = 1.314$ and $A_2 = 0.0021$ [@ghosh]. This functional which includes the full contribution of the Weizsacker correction describes the response properties of the electron gas well. This functional has been used for investigating atoms and molecules with reasonable success. We briefly describe our procedure, details of which can be found in [@shah]. The total energy of the system (Eq. (1)), is minimized for fixed ionic positions using the conjugate gradient method [@press] which forms the starting point for molecular dynamics. The trajectories of ions and the fictitious electron dynamics are then simulated using Lagrange’s equations of motion which are solved by Verlet algorithm [@car]. The stability of CP dynamics has been discussed in [@pear] in the context of density based methods and timesteps of the order to 50 a.u. have been successfully used. We have verified that by appropriate adjustment of the fictitious electron mass the CP dynamics remains very stable for over 10000 iterations with a timestep of 40 a.u. in the present calculations of clusters. Typically, for free dynamics the grand total energy which is the sum of the kinetic energy of ions, kinetic energy of electrons and the potential energy of the system remains constant to within $10^{-5}$ a.u. For the calculations of the ground state structures for dimers and trimers of Na, Mg, Al, Li and Si a periodically repeated unit cell of length 26 a.u. with a 54 $\times$ 54 $\times$ 54 mesh and timestep $\Delta t \sim$ 10 to 20 a.u. was used. For the rest of the small clusters the calculations were done on a unit cell of length of 30 a.u with a 54 $\times$ 54 $\times$ 54 mesh. We have chosen to use the plane wave expansion on the entire fast fourier transform mesh without any truncation yielding the energy cutoff of 95 Rydberg. It must be mentioned that, due to the orbital free calculations the number of fast fourier transforms per iteration are constant irrespective of the number of electrons in the system. For clusters, the ground state configurations are obtained either by starting with different initial configurations and then quenching the structures or by dynamical simulated annealing where the cluster is heated to $300-350^{\circ} K$ and then cooled very slowly. In all the cases the stability of the final ground state configurations has been tested by reheating the clusters and allowing them to span the configuration space for a few thousand iterations and then cooling them to get the low energy configurations. Results and Discussion ====================== In this section, we first discuss the results for the equilibrium bondlengths and binding energies of dimers and trimers of Na, Mg, Al, Li, Si along with their KS results. All the results presented here are obtained with energy convergence up to $10^{-13}$ for total energy minimization. Table I shows the equilibrium bondlengths and binding energies for dimer and trimer systems using different kinetic energy functionals. These results have been compared with full nonlocal pseudopotential KS calculations. A few representative results using $\lambda = \frac{1}{5}$ have been given. It can be seen that for $\lambda = \frac{1}{5}$ the trend is similar to the $\lambda = \frac{1}{9}$ functional and there is no significant improvement in the results. Clearly, the results involving $F(N_e)$ functional show significant improvement over $\lambda = \frac{1}{9}$ (with the exception of Mg) and are in reasonable agreement with the bondlengths obtained by the KS method. The error in the bondlengths is around 10%. It is known that such methods based on approximate KE functionals are not expected to give accurate binding energies. One notable feature of the binding energy comparison is the considerable improvement by $F(N_e)$ over $\lambda = \frac{1}{9}$ (excepting again the case of Mg). The results for the Na, Li, Si trimer binding energies are not given because these are Jahn-Teller distorted isoscales triangles and the present method yields equilateral triangle geometries. Clearly, such density based methods are unable to reach the Jahn-Teller distorted geometries. The quality of the charge densities obtained by this method can be gauged by comparing them with the KS charge density. In Fig. 1 we have plotted the self-consistent charge densities obtained using the functionals involving $F(N_e)$ (curve a) and $\lambda = \frac{1}{9}$ (curve b) with the KS charge density (curve c) for Al dimer along the axis joining the atoms. The ionic positions are marked by arrows on the plot. The KS charge density has been obtained using the identical pseudopotentials and the same cell size as in the case of OFM. Three prominent features can be observed. 1. Overall the $F(N_e)$ functional densities compare very well with the KS densities except at the origin where both the $F(N_e)$ and $\lambda = \frac{1}{9}$ self-consistent densities show overestimation. 2. At the atomic sites the $F(N_e)$ and KS based densities are very close and nonzero, whereas the $\lambda = \frac{1}{9}$ shows a disturbing feature of almost zero density. 3. At the peaks on either side of the origin, the KS and $F(N_e)$ charge densities again are close, but the charge density by $\lambda = \frac{1}{9}$ shows considerable overestimation. In Fig. 2 we have plotted the superposed free atom charge density ($0^{th}$ iteration density) represented by the curve b and the self-consistent charge density for the functional involving $\lambda = \frac{1}{9}$ by the curve c and $F(N_e)$ by the curve a. The self consistent charge density obtained using $\lambda = \frac{1}{9}$ shows improvement only at the origin. Contrary to this the self consistent charge density using $F(N_e)$ shows a significant overall improvement, both at the origin and at the peaks on either side of the peak at the origin. To get an idea of the nature of the forces obtained by the OFM and KS dynamics, we have given the results for the vibrational frequencies for Na, Mg and Li dimers in Table 2. It is gratifying to note that the vibrational frequencies obtained by OFM method are in very good agreement with the KS ones. To assess the utility and performance of this method, it has been applied to calculate the ground state geometries of a range of small clusters. We report here our calculations on heteronuclear clusters of $Li_nAl, n=1,8$ and a highly symmetric homonuclear cluster of $Al_{13}$ and clusters of $Na_n, n = 6,8$ using the $F(N_e)$ functional. The results are compared with the ones reported by KS method. The geometries of the heteronuclear $Li_nAl$ clusters are shown in Fig. 3 and the bondlengths and symmetries in Table III. along with the KS results. Evidently, the present method not only reproduces the correct ground state geometry with bondlengths within 5% but also reproduces the two key features observed in the more accurate KS calculation [@cheng]. 1. The $Li_nAl$ clusters for $n < 3$ are two-dimensional whereas from $n \geq 3$ the clusters become three-dimensional. 2. The Al atom gets trapped inside the Li atoms at $n = 6$. It can also be noted that as the number of atoms in the cluster increases, the accuracy in the bondlengths appears to improve. However, for the case of $Li_3Al$ and $Li_8Al$ we get the ground structure configurations to be ideally symmetric rather than slightly Jahn Teller distorted geometries of the KS calculation. We have also investigated the $Al_{13}$ cluster since it shows an interesting icosahedral geometry. The calculations were performed in two different ways. First we started with a highly distorted icosahedron and applied the dynamical quenching to get the equilibrium geometry. In the second one, we started by placing the Al atoms at the fcc lattice points and heated the cluster to $300^{\circ} K$ and let the system span the configuration space for a few thousand iterations. This was then followed by a slow cooling schedule. It is very gratifying to note that in both the calculations the correct icosahedron is obtained with a bondlength of 4.88 a.u. as compared to the KS bondlength of 5.03 a.u. The error in the bondlengths being 3%. This strengthens our confidence in the ability of the method to reproduce the correct ground state geometries with acceptable bondlengths. In addition, we have also obtained the ground state geometries for Na$_6$, Na$_8$ and Na$_{20}$ and have verified that the geometries obtained are identical to those reported in [@urs], with the bondlengths differing by about 5%. Conclusion ========== In this work, we have presented the results obtained by using density based [ab initio]{} MD for a variety of small clusters and demonstrated that the method using approximate KE functionals is capable of yielding bondlengths within an accuracy of 5%. Our calculations indicate that the ground state geometries and symmetries of both homonuclear and heteronuclear clusters can be obtained within a reasonable accuracy and timesteps of the order of 40 a.u. can be used successfully for stable dynamics. The $F(N_e)$ functional is shown to give considerable improvement over standard $\lambda = \frac{1}{9}$ functional both in terms of charge densities and bondlengths and is thus recommended. We believe the method to be a promising tool in the study of finite temperature and dynamical properties of clusters. So far, all the reported OFM calculations have been performed using local pseudopotentials only and it would be interesting to implement the nonlocal pseudopotentials and study the effect of nonlocality on the bonding and binding properties of such clusters. More work is required in this direction and the implementation of nonlocality is under consideration. It may be possible to expand the applications of OFM by incorporating the nonlocal pseudopotentials and by employing more accurate KE functionals. It is hoped that the problems of current interest in the field of clusters like fragmentation, dissociation, interaction between clusters, which may involve large number of atoms as well as more than one atomic species will be amenable by the present technique. Acknowledgements ================ Partial financial assistance from the Department of Science and Technology (DST), Government of India and the Centre for Development of Advanced Computing (C-DAC), Pune is gratefully acknowledged. Two of us (V. S and D. N) acknowledge financial assistance from C-DAC. One of us (D. G. K.) acknowledges P. Madden for a number of fruitful discussions on the OFM. We also acknowledge K. Hermansson and L. Ojamoe for the MOVIEMOL animation program. Electronic Address: dinesh, vaishali, [email protected] R. Car, M. Parrinello, Phys. Rev. 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Comparison of the equilibrium bondlengths (in a.u) and binding energies (in eV/atom) using the different kinetic energy functionals with the KS self consistent method. = hspace[0.5in]{}= hspace[0.5in]{}= hspace[0.5in]{}= hspace[0.5in]{}= hspace[0.5in]{}= hspace[0.5in]{}= hspace[0.5in]{}= System Bondlengths Binding energies\ $\lambda = \frac{1}{9}$ $\lambda = \frac{1}{5}$ $F(N_e)$ KS $\lambda = \frac{1}{9}$ $\lambda = \frac{1}{5}$ $F(N_e)$ KS\ \ $Na_2$ 5.67 - 5.69 5.66$^a$ -0.116 - -0.867 -0.71$^a$\ $Na_3$ 5.81 5.99 5.75 6.00$^b$ -0.207 -0.281 -1.286 -\ $Mg_2$ 5.79 - 4.71 6.33$^c$ -0.195 - -1.432 -0.115$^c$\ $Mg_3$ 5.94 5.81 4.87 5.93$^c$ -0.355 -0.526 -2.096 -0.284$^c$\ $Al_2$ 5.74 - 4.14 4.66$^d$ -0.261 - -1.389 -1.06$^d$\ $Al_3$ 5.88 5.57 4.32 4.74$^d$ -0.483 -0.733 -2.074 -1.96$^d$\ $Li_2$ 5.87 - 5.51 5.15$^e$ -0.102 - -0.891 -\ $Li_3$ 6.03 6.11 5.58 5.3$^b$ -0.182 -0.256 -1.311 -\ $Si_2$ 5.35 - 3.74 4.29$^f$ -0.371 - -0.56 -0.6$^g$\ $Si_3$ 5.50 - 3.92 4.10$^f$ -0.651 - -0.938 -\ $^a$Reference[@urs] $^e$our own KS calculations $^b$Reference[@martin] $^f$Reference[@feuston] $^c$Reference[@kumar] $^g$Reference[@tomanek] $^d$Reference[@yang] TABLE II. The vibrational frequencies (in $cm^{-1}$) of Na, Mg, Li dimer using the OFM and KS self consistent method. == dimers OFM KS\ Na 167.4 168\ Mg 107.3 108.6\ Li 273.7 311\ TABLE III. The bondlengths between Li-AL of $Li_nAl, n=1,8$ using OFM compared with those obtained by KS method [@cheng]. All the bondlengths are in a.u. ==== system OFM KS % error Symmetry\ $LiAl $ 4.77 5.35 10.8 $C_{\infty v}$\ $Li_2Al$ 2 $\times$ 4.76 2 $\times$ 5.22 8.8 $C_{2v}$\ $Li_3Al$ 3 $\times$ 4.79 3 $\times$ 4.98 3.8 $C_{3v}$\ $Li_4Al$ 4 $\times$ 4.84 2 $\times$ 4.82 0.4 $C_{3v}$\ 2 $\times$ 4.89 1\ $Li_5Al$ 4 $\times$ 4.84 4 $\times$ 4.74 2 $C_{4v}$\ 4.95 5.13 3.3\ $Li_6Al$ 6 $\times$ 4.79 6 $\times$ 4.58 4.5 $O_h$\ $Li_7Al$ 4.97 4.70 5.7 $C_{1h}$\ 2 $\times$ 4.92 2 $\times$ 4.85 1.4\ 2 $\times$ 4.88 2 $\times$ 4.74 2.9\ 2 $\times$ 4.89 2 $\times$ 4.81 1.6\ $Li_8Al$ 8 $\times$ 4.99 8 $\times$ 4.82 3.5 $D_{4d}$\ Figure Captions 1. The self-consistent charge densities of Al dimer. Curve a represents the $F(N_e)$ functional charge density, curve b represents the $\lambda = \frac{1}{9}$ charge density and curve c denotes the charge density obtained using the KS method. 2. Comparison of self-consistent charge densities by the $F(N_e)$ (curve a) and $\lambda = \frac{1}{9}$ (curve c) functional for Al dimer with the superposed ( $0^{th}$ iteration) free Al atom charge density (curve b). 3. The ground state geometries of the $Li_nAl$ clusters for $n = 1, 8$. The large sphere represents the Li atom and small sphere represents the Al atom.	ArXiv
--- abstract: 'We report the first experimental demonstration of conditional preparation of a non-classical state of light in the continuous variable regime. Starting from a non-degenerate OPO which generates above threshold quantum intensity correlated signal and idler “twin beams”, we keep the recorded values of the signal intensity only when the idler intensity falls inside a band of values narrower than its standard deviation. By this very simple technique, we generate a sub-Poissonian state 4.4 dB (64%) below shot noise from twin beams exhibiting 7.5 dB (82%) of noise reduction in the intensity difference.' author: - 'J. Laurat' - 'T. Coudreau' - 'N. Treps' - 'A. Maître$^$' - 'C. Fabre' date: 'Received: 16 April 2003 / Revised version: 12 September 2003' title: \| Conditional preparation of a quantum state in the continuous variable regime :\ generation of a sub-Poissonian state from twin beams --- A well-known technique to generate a single photon state from quantum correlated photons (“twin photons”) is to use the method of conditional measurement: if one labels (1) and (2) the two modes in which the twin photons are emitted, it consists in retaining in the information collected on mode (1) only the counts occurring when a photon is detected in mode (2) (within a given time window $\Delta T$). This method has been widely and very successfully used over the past decades, firstly with twin photons generated by an atomic cascade [@Aspect], then by using the more efficient technique of parametric down conversion [@fock1]. Various protocols have been proposed to use conditional preparation in order to generate other kinds of non-classical states, for example Schrödinger cat states using a squeezed vacuum state transmitted through a beamsplitter and a measurement conditioned by the counts detected on the reflected port [@cat]. In a similar way, teleportation of a quantum state of light can be achieved by using conditional measurements [@teleport] and the degree of entanglement can be improved by photon subtractions [@improve]. In cavity QED, conditional measurements on the atomic state have also led to the experimental generation of non-classical photon states [@haroche]. State reduction is obviously not restricted to the case of photon counting, so that it may be interesting to extend this technique to the continuous variable regime, where a continuously varying photocurrent is measured instead of a series of photocounts. Continuous detection conditioned by a photon counting event has been implemented in various schemes [@OrozcoSchiller]. Closer to our proposal where continuous measurements are used both for triggering and characterizing the generated state, many theoretical protocols have been suggested relying on two-mode squeezed vacuum produced by a non-degenerate optical parametric amplifier [@reducetwins; @kumar]. For instance, homodyne measurements on the idler can be used to condition the detection of the signal and would reduce it to a squeezed state [@reducetwins]. In a conditional state preparation, the generation of the non-classical state can be seen as the collapse of the entangled wave-function induced by the measurement on one of its components. However, very frequently, the measurement is made by post-selection of the relevant events in the record of all the values measured on the two channels, which can be made after the end of the physical measurement, so that no wave-function collapse actually occurs during the experiment. Note that, in contrast to the methods of direct generation of a non-classical state, the exact time window when the state is produced is not controlled in a conditional measurement, and what we will call the “success rate”, i.e. the probability of generating the non-classical state in a given time interval, is an important parameter to characterize its efficiency. To the best of our knowledge, no scheme has been suggested so far to generate non-classical intense beams by the technique of conditional measurement performed on continuous variables. This is the purpose of the present letter, which proposes a very simple way of conditionally preparing a bright sub-Poissonian beam from twin beams, and reports on its experimental implementation. The theory of the presented technique will be detailed in a forthcoming publication [@theory]. It is well known that a non-degenerate optical parametric oscillator (ND-OPO) produces above threshold intense signal and idler “twin beams” [@traditional] : in the ideal case of a system without losses, the Fourier components of the signal and idler intensity quantum fluctuations which lie inside the cavity bandwidth are perfectly correlated. The correlations are characterized by the “gemellity”, which is the remaining noise on the intensity difference between the signal and idler intensities normalized to the corresponding quantum noise level [@Reynaud]. The instantaneous values of the signal and idler photocurrents play therefore the role of the occurrence of counts in the photon counting regime. The conditional measurement technique that we propose consists in selecting the signal photocurrent $I_s$ only during the time intervals when the idler intensity $I_i$ has a given value $I_0$ (within a band $\Delta I$ smaller than the photocurrent standard deviation). The measurements outside these time intervals are discarded. If the correlation is perfect and the interval $\Delta I$ close to zero, the recorded signal intensity is perfectly constant, and an intense number state is generated by this conditional measurement. In a real experiment, the correlation between the signal and idler photocurrents is not perfect, and the selection band $\Delta I$ is finite, so that the method will not prepare a perfect number state, but a sub-Poissonian state instead. The density matrix describing the state of light which is produced by such a state reduction technique can be determined within the approximation that the signal and idler photon distributions are Gaussian [@theory]. In the limit where $\Delta I$ is very small, one finds that, whatever the initial intensity noise of the beams are, the conditional measurement produces a sub-Poissonian signal beam, characterized by a Fano factor equal to the conditional variance of the intensity fluctuations of the signal beam knowing the intensity fluctuations of the idler beam, which plays an important role in the characterization of Quantum Non Demolition measurements [@grangier]. In other words, in the limit of large correlations, the intensity noise reduction on the conditionally prepared state is equal to the twin beam noise reduction minus 3dB. Obviously, if $\Delta I$ is very small, the probability that the idler intensity lies within the chosen band is also very small, and the success rate of the non-classical state production by such a conditional measurement is also very low. Computer simulations, as well as analytical calculations [@theory], show that the Fano factor of the generated state remains almost constant in a wide range of $\Delta I$ values, whereas the success rate of the method increases quickly. It is only when $\Delta I$ reaches values comparable to the shot noise standard deviation that the post-selection process becomes less efficient, and the Fano factor tends to its uncorrected value. The present conditional measurement technique has strong analogies with the method of active feed-forward correction of the signal beam intensity, by opto-electronics techniques, using the information obtained from the measurement of the idler intensity [@mertz1; @mertz2], which produces a sub-Poissonian state with a Fano factor also equal to the conditional variance. The present technique is much simpler to implement, whereas the active correction technique is non-conditional, and has the advantage of producing the non-classical state at all times. The experimental setup is shown in Figure \[setup\]. A continuous frequency-doubled Nd:YAG laser pumps a triply resonant ND-OPO above threshold, made of a semi-monolithic linear cavity : in order to increase the mechanical stability and reduce the reflection losses, the input flat mirror is directly coated on one face of the 10mm-long KTP crystal. The reflectivities for the input coupler are 95.5% for the pump (532nm) and almost 100% for the signal and idler beams (1064nm). The output coupler (R=38mm) is highly reflecting for the pump and its transmission is 5% for the infrared. At exact triple resonance, the oscillation threshold is less than 15mW. The OPO is actively locked on the pump resonance by the Pound-Drever-Hall technique: we detect by reflection the remaining 12MHz modulation used in the laser to lock the external doubling cavity. In order to stabilize the OPO infrared output intensity, the crystal temperature has to be drastically controlled (within a mK). The OPO can operate stably during more than one hour without mode-hopping. The signal and idler orthogonally polarized beams (in the 1-5mW range) generated by the OPO are then separated by a polarizing beam-splitter and detected on a pair of balanced high quantum efficiency InGaAs photodiodes (Epitaxx ETX300, quantum efficiency: 95%). An half-wave plate is inserted before the polarizing beam-splitter. When the polarization of the twin beams is turned by 45$^\circ$ with respect to its axes, it behaves as a 50% usual beam-splitter, which allows us to measure the shot noise level[@traditional]. ![\[setup\]Experimental layout. The 1064nm orthogonally polarized bright twin beams generated by the ND-OPO are separated by a polarizing beam splitter (PBS) and detected. The high frequency components of the photocurrents are amplified and demodulated at a given Fourier frequency. The two channels are digitized and simultaneously recorded on a computer.](Fig1.eps){width=".9\columnwidth"} In almost all the bright twin beams experiments to date [@traditional], the photocurrents are subtracted and the difference is sent onto a spectrum analyzer which gives the variance of the photocurrent distribution. We have implemented a different protocol to have access to the full photon-number quantum statistics of the signal and idler beams at a given Fourier frequency $\Omega$ (see also [@computer]): each photocurrent is amplified and multiplied by a sinusoidal current at frequency $\Omega$ produced by a signal generator, and filtered by a 22kHz low-pass filter in order to obtain the instantaneous value of the photocurrent Fourier component at frequency $\Omega$, which is then digitized at a sampling rate of 200 kHz by a 12-bit, 4-channel acquisition card (National Instruments PCI-6110E), which also simultaneously records the instantaneous values of the DC photocurrents. Two successive acquisitions (200 000 points for each channel) are required, one for calibrating the shot noise by rotating the half-wave plate and the other to record the quantum correlated signals. ![\[courbes\_exp\]Experimental results: (a) Idler intensity fluctuations: 200 000 acquired points at 3.5 MHz demodulation frequency (only 20 000 shown). (b) Corresponding probability distribution. The unit is the width $\sigma_0$ of the Poisson distribution of same mean intensity (shot noise). (c) Values of the signal intensity conditionally selected by the value of the idler intensity recorded at the same time (the selection bandwidth $\Delta I$ equal to 0.1 $\sigma_0$ around the mean), superimposed to the corresponding experimentally measured shot noise. (d) Corresponding probability distribution, compared to the Poisson distribution (grey line), displaying the sub-Poissonian character of the conditionally generated state. The black line is a gaussian fit of the intensity distribution](Fig2.eps){width=".9\columnwidth"} Figure \[courbes\_exp\] sums up the measurements obtained with a demodulation frequency $\Omega /2 \pi =3.5 MHz$. Figure \[courbes\_exp\]a shows the actual recording of the fluctuations of the idler beam during a time interval of 100ms. As the OPO is pumped close to threshold, the signal and idler beams have intensity fluctuations which are much larger than the standard quantum limit, as can be seen on curve \[courbes\_exp\]b, giving the probability distribution of the intensity fluctuations normalized to the shot noise. The corresponding Fano factor exceeds 100 (20 dB above the shot noise level). From the recorded data, one can calculate the noise variance on the difference between the signal and idler intensities. It reaches a value of 7.5 dB below the standard quantum limit (8.5 dB after correction of dark noise), in good agreement with the value of the noise variance measured on the spectrum analyzer. The dark noise, which is more than 6 dB below all measurements, is not subtracted in the following experimental results. The conditional measurement is performed in the following way : the signal intensity values are kept only if the idler intensity values recorded at the same time fall inside a narrow band around its mean value. The remaining ensemble of values of the signal intensity is given in figure \[courbes\_exp\]c, in which the shot noise is given at the same time : one indeed observes a significant narrowing of the probability distribution below the shot noise level. Let us stress that the shot noise is unchanged by this selection process: the beam exists only in the selection intervals so that the unselected intervals do not contribute to the average value. Figure \[courbes\_exp\]d gives the probability distribution of the intensity fluctuations of the conditionally prepared state normalized to the shot noise, together with the Poissonian distribution of photons for the same mean intensity. With a selection bandwidth $\Delta I$ equal to 0.1 times the standard deviation $\sigma_0$ of a coherent state having the same power (shot noise level), the conditionally prepared light state exhibits 4.4 dB of noise reduction below the Poisson distribution. This value is very close from the theoretical expectation in the case of vanishingly narrow intensity band. The success rate of the conditional preparation is around 0.85% (1700 points out of 200 000 are accepted). This value would be higher for an initial state with less excess noise above shot noise. ![\[limite\]Measured intensity noise on the post-selected signal (a) and success rate (proportion of selected points)(b) as a function of the selection bandwidth on the idler normalized to $\sigma_0$. Circles: experimental data. Squares: theoretical predictions.](Fig3.eps){width=".9\columnwidth"} The success rate can be improved by increasing the selection bandwidth, at the expense of a decreased non-classical character of the selected state. Figure \[limite\] shows the measured residual noise in the conditionally produced state, and the success rate of the state generation, as a function of the selection bandwidth normalized to the $\sigma_0$. The noise reduction turns out to be almost constant until the normalized selection bandwidth becomes of the order of 0.1, whereas the success rate steadily increases, in very good agreement with numerical simulations. However, one can see a slight increase in the noise when the selection bandwidth becomes very narrow. This artefact is due to the sampling process on a finite range of bits (12), which restricts the resolution of the acquisition. ![\[pertes\]Measured intensity noise reduction for different values of the signal-idler correlation. The selection bandwidth is taken equal to 0.1 times the shot noise. Circles: experimental data. Dotted line: theoretical prediction at the limit of a very small selection bandwidth (gemellity minus 3 dB).](Fig4.eps){width=".9\columnwidth"} In figure \[pertes\], we give the measured residual noise for different amounts of intensity correlations between the beams, which can be varied by inserting losses on the OPO beams. One checks on the figure the validity of the prediction that the noise reduction is equal to the gemellity minus 3dB. One also observes than, when the intensity difference noise is reduced by less than 3dB below the standard quantum level, the conditional state has reduced intensity noise fluctuations in comparison with the very noisy initial beam, but that it is not a non-classical sub-Poissonian state. The continuous variable regime offers a unique possibility to improve dramatically the efficiency of conditional strategy, by choosing multiple selection bands with different mean values on the idler intensity. Independent selection bands will correspond to independent sets of time windows. By using hundreds of independent intervals, one keeps most of the values of the signal, each of these intervals reducing the signal to a given sub-Poissonian state. Figure \[multiband\] shows that the noise reduction does not depend of the band center and that the success rate follows the initial gaussian noise distribution. ![\[multiband\] Measured intensity noise on the reduced state (a) and success rate (b) as a function of the band center normalized to $\sigma_0$. The selection bandwidth is taken equal to 0.1 times the shot noise.](Fig5.eps){width=".9\columnwidth"} To conclude, we have shown the first experimental demonstration of conditional preparation of a quantum state in the continuous variable regime. We have studied the influence of the selection bandwidth of the conditioning measurement on the obtained non-classical state and on the success rate of its preparation and shown that many sub-Poissonian states can be produced in parallel. This method to generate non-classical states of light in the continuous variable regime is equivalent to sending the signal beam through an intensity modulator which either totally transmits the beam when the idler beam has the right value, or blocks it when it is not the case. It therefore drastically changes the light state, and seems to be very different from the usual technique which consists of correcting the beam fluctuations by a feed-forward or feed-back opto-electronic loop which only slightly modifies the quantum fluctuations. This strongly non-linear character of the action on the light may lead to the generation of non-Gaussian states. One could also envision other criteria of conditioning the quantum state than simply imposing to the idler beam intensity to lie within a given band. This could also lead to the generation of new families of non-classical bright states of light. Laboratoire Kastler-Brossel, of the Ecole Normale Supérieure and the Université Pierre et Marie Curie, is associated with the Centre National de la Recherche Scientifique (UMR 8552). We acknowledge support from the European Commission project QUICOV (IST-1999-13071) and ACI Photonique (Ministère de la Recherche). P. Grangier, G. Roger, A. Aspect Europhys. Letters 1,173 (1986). C.K. Hong, L. Mandel, Phys. Rev. Lett. 56, 58 (1986). M. Dakna, T. Anhut, T. Opatrny, L. Knöll, D.G. Welsch, Phys. Rev. A 55, 3184 (1997). C.H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, W.K. Wootters, Phys. Rev. Lett. 70, 1895 (1993); D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eible, H. Weinfurter, A. Zeilinger, Nature 390, 575 (1997) T. Opatrny, G. Kurizki, D.G. Welsch, Phys. Rev. A 61, 032302 (2000); P.T. Cochrane, T.C. Ralph, G.J. Milburn, Phys. Rev. A 65, 062306 (2002). J.-M. Raimond, M. Brune, S. Haroche, Rev. Mod. Phys. 73, 565 (2001). G.T. Foster, L.A. Orozco, H.M. Castro-Beltran, H.J. 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--- abstract: \| A new measure of the crystal-field strength, complementary to the conventional one, is defined. It is based on the rotational invariants $\left\|B_{k0}\right\|_{\rm av}$ or $\left\|\sum_{k}B_{k0}\right\|_{\rm av}$, $k=2,4,6$, of the crystal-(ligand)-field (CF) Hamiltonian ${\cal H}_{\rm CF}$ parametrizations, i.e. on the axial CF parameters modules averaged over all reference frame orientations. They turn out to be equal to $\left\|{\cal H}_{\rm CF}^{(k)}\right\|_{\rm av}$ and $\left\|{\cal H}_{\rm CF}\right\|_{\rm av}$, respectively. While the traditional measure is established on the parametrization modules or on the second moment of the CF energy levels, the introduced scale employs rather the first moment of the energy modules and has better resolving power. The new scale is able to differentiate the strength of various iso-modular parametrizations according to the classes of rotationally equivalent parametrizations. Using both the compatible CF strength measures one may draw more accurate conclusions about the Stark levels arrays and particularly their total splitting magnitudes. author: - 'J. Mulak$^{1}$ and M. Mulak$^{2}$' date: \| [$^{1}$ W Trzebiatowski Institute of Low Temperature and Structure Research,\ Polish Academy of Sciences, 50–950, PO Box 1410, Wroclaw, Poland\ $^{2}$ Institute of Physics, Wroclaw University of Technology,\ Wyb. Wyspianskiego 27, 50–370 Wroclaw, Poland]{} title: 'On a complementary scale of crystal-field strength' --- [PACS]{}: 71.70.Ch\ [Key words]{}: crystal-field strength, crystal-field splitting\ 1. Introduction {#introduction .unnumbered} =============== Solid state experimentalists, especially spectroscopists, still need a reliable scale quantitatively characterizing the effect of crystal-field interaction, i.e. defining the so-called crystal-field strength. Such a parameter could directly verify and compare various parametrizations of the crystal-field Hamiltonian ${\cal H}_{\rm CF}$, which may come from different fittings experimental data when the orientations of reference frames associated with these parametrizations are unknown in the majority of cases. Although such a conventional scale for measuring the strength of the crystal-field has been already introduced over twenty years ago \[1,2\], in some cases it seems to be insufficiently precise. It employs the basic rotational invariants of the ${\cal H}_{\rm CF}$, i.e. the modules of its $2^{k}$-pole components ${\cal H}_{\rm CF}^{(k)}$, defined as $M_{k}=\left(\sum_{q}\|B_{kq}\|^{2}\right)^{1/2}$, as well as uses the global ${\cal H}_{\rm CF}$ modulus $M=\left(\sum_{k}\sum_{q}\|B_{kq}\|^{2}\right)^{1/2}$. In the first case the partial crystal-field strength is defined as $S_{k}\!=\!\left(\frac{1}{2k+1}\right)^{1/2}M_{k}$, while in the second case the global crystal-field strength is given by $S=\left(\sum_{k}S_{k}^{2}\right)^{1/2}$. Throughout the paper the tensor (Wybourne) notation for the crystal-field Hamiltonian and the crystal-field parameters (CFPs), ${\cal H}_{\rm CF}=\sum_{k}\sum_{q}B_{kq}C_{q}^{(k)}$, is consistently used \[3\]. The summations over $k$ and $q$ indices run, in each individual case, over strictly specified values according to the kind of central ion and its point symmetry. Both the parameters $S_{k}$ and $S$ themselves are not a direct measure of the real magnitude of the initial state splitting, since the crystal-field effect depends also on the properties of an object (a paramagnetic ion) upon which the ${\cal H}_{\rm CF}$ acts. Namely, the response of the system to the ${\cal H}_{\rm CF}$ perturbation reflects the symmetry of the electron density distribution of the central ion open-shell. For instance, an $S$-type ion like Gd$^{3+}$ feels no crystal field (in the first order of perturbation) no matter how strong is the surrounding field. The effect of splitting can be most simply expressed by the so-called second moments $\sigma_{k}^{2}$ or $\sigma^{2}$ of the CF sublevels within the initial state upon switching on the ${\cal H}^{(k)}_{\rm CF}$ (or ${\cal H}_{\rm CF}$) perturbation. In fact, the second moment is easily represented by the scalar crystal-field strength parameters, either $S_{k}$ or $S$ (section 2). However, although the effective ${\cal H}^{(k)}_{\rm CF}$ multipoles (for $k=2,4,6$) contribute to the energy of individual Stark levels independently (as an algebraic sum), the simple linear relations between $\sigma_{k}^{2}$ (or $\sigma^{2}$), and $S_{k}^{2}$ (or $S^{2}$) are always fulfilled. As it is proved these relations strongly confine both the maximal $(\Delta {\cal E}_{\rm max})$ and minimal $(\Delta {\cal E}_{\rm min})$ nominally allowed splittings of the initial state (section 3). Moreover the actual crystal field splittings $\Delta E$ can be additionally restricted (section 5). Naturally, all the iso-modular ${\cal H}_{\rm CF}$ parametrizations correspond to the same crystal-field strengths $S_{k}$ and $S$. However, apart from the modules $M_{k}$ and $M$, there exist also other rotational invariants of the ${\cal H}^{(k)}_{\rm CF}$ or ${\cal H}_{\rm CF}$ which distinguish the whole classes of the rotationally equivalent ${\cal H}_{\rm CF}$ parameterizations, in other words the parameterizations referring to the same real crystal-field potential, but expressed in variously oriented reference frame. Interestingly, the new invariants turn out to be the average values of the axial parameter modulus $\|B_{k0}\|_{\rm av}$, $k=2,4,6$, in the case of ${\cal H}^{(k)}_{\rm CF}$, or $\|\sum_{k}B_{k0}\|_{\rm av}$ for the global ${\cal H}_{\rm CF}$ obtained after the averaging over all orientations of the reference frame, i.e. over the solid angle $4\pi$. As it is shown in the paper the average value of the axial parameter modulus or the average of the modulus of their sum are just equal to $\|{\cal H}^{(k)}_{\rm CF}\|_{\rm av}$ and $\|{\cal H}_{\rm CF}\|_{\rm av}$, respectively (section 4). The new scale of the crystal-field strength based on the above invariants is in principle consistent with the conventional one but it reveals more resolving power. Applying the new measure to the iso-modular parametrizations may lead to different strength parameters what is exemplified below for several cases (section 5). The introduced more subtle strength gradation established rather on the first moment of the sublevel energy modules gives, comparing to the second moment, additional information about the Stark levels array for various iso-modular ${\cal H}_{\rm CF}$s, including the magnitude of the total splitting gap of the states. In this paper we confine ourselves to the pure model states of the zero-order approximation with a well defined angular momentum quantum number and the corresponding degeneration. These could be for instance Russell-Saunders coupled states $\|\alpha L S J \rangle$ coming from the $^{2S+1}$L terms, where $\alpha$ stands for the remaining quantum numbers needed for their complete determination. Such states have a well defined quantum number $J$ and the degeneration $2J+1$. The derivation of the analogical expressions including $J$-mixing effects \[4\] or a transformation to other functional bases of the zero-order approximation can be accomplished by using standard angular momentum re-coupling techniques \[4-8\]. In section 5 we analyse by way of example the crystal-field splitting of $p^{1}$, $d^{1}$ and $f^{1}$ one-electron configurations and a typical complex state $^{3}H_{4}$ for various iso-modular ${\cal H}_{\rm CF}^{(k)}$, $k=2,4,6$. In the first three cases we avoid complex states re-coupling procedure which is a side issue to the problem under consideration. Since we study the differentiation of the effects due to various iso-modular Hamiltonians ${\cal H}_{\rm CF}^{(k)}$, all CFPs values along with the Stark levels energies are given in $M_{k}$ units. 2. Conventional definition of the crystal-field strength parameter {#conventional-definition-of-the-crystal-field-strength-parameter .unnumbered} ------------------------------------------------------------------ The comparison and scaling of the crystal-field impact can be based upon the two types of scalar quantities, $M_{k}$ and/or $M$, since both of them are rotationally invariant. A scalar crystal-field strength parameter of this kind was given firstly by Auzel and Malta \[1,2\] as (in original notation): $$\begin{aligned} N_{v} &=& \left[\sum\limits_{k,q}\|B_{q}^{k}\|^2\left(\frac{2\pi}{2k+1}\right)\right]^{1/2},\end{aligned}$$ which is nothing more but $M$ in the space spanned by spherical harmonics $Y_{q}^{k}$. In other words, $N_{v}$ is a norm representing a distance in the space. Currently there are two definitions widely used in the literature \[9-12\]: $$\begin{aligned} S_{k} &=& \left(\frac{1}{2k+1}\sum\limits_{q}\|B_{kq}\|^2\right)^{1/2}= \left\{\frac{1}{2k+1}\left[B_{k0}^{2}+2\sum\limits_{q>0}({\rm Re}B_{kq})^2+({\rm Im}B_{kq})^2\right]\right\}^{1/2},\end{aligned}$$ for $k=2,4$ and $6$ in the case of $2^{k}$-pole ${\cal H}_{\rm CF}$ component and $$\begin{aligned} S &=& \left(S_{2}^{2}+S_{4}^{2}+S_{6}^{2}\right)^{1/2} \qquad {\rm or} \qquad S=\left[\frac{1}{3}\left(S_{2}^{2}+S_{4}^{2}+S_{6}^{2}\right)\right]^{1/2},\end{aligned}$$ for the global ${\cal H}_{\rm CF}=\sum_{k}{\cal H}_{\rm CF}^{(k)}$ \[4\]. A word of caution seems to be worthy at this point. Namely, the values of $S_{k}$ or $S$ can differ according to the type of the ${\cal H}_{\rm CF}$ parametrization (operators) applied. They can be compared with each other only after proper recalculation. Since both these quantities are independent of the assumed axis system they allow to check whether the original CFP data sets and the transformed ones are compatible. The strengths $S_{k}$ or $S$ enable also a broad comparison of CFP data sets when the axis systems have not been explicitly defined, and undoubtedly they play a central role in the CF theory. What is also important and useful they are linked to the second moment of the Stark levels within a particular initial state $\|\alpha S L J\rangle$ \[4,13\]. The second moment of the sublevels $\|n\rangle$ within $\|\alpha S L J\rangle$ state upon introduction of a ${\cal H}_{\rm CF}$ perturbation is defined by $$\begin{aligned} \sigma^{2}(\|\alpha S L J\rangle) &=& \frac{1}{2J+1}\sum\limits_{n}\left[E_{n}-\bar{E}(\|\alpha S L J\rangle)\right]^{2},\end{aligned}$$ where the center of gravity of the Stark levels belonging to the state $\|\alpha S L J\rangle$ is given by $$\begin{aligned} \bar{E}(\|\alpha S L J\rangle)&=& \frac{1}{2J+1}\sum\limits_{n}E_{n},\end{aligned}$$ and $E_{n}$ is the $\|n\rangle$ sublevel energy. Since ${\cal H}_{\rm CF}$ is diagonal in the $\|n\rangle$ basis and the second order effect of ${\cal H}_{\rm CF}$ interaction is neglected \[4\] $$\begin{aligned} \sigma^{2}(\|\alpha S L J\rangle) &=& \frac{1}{2J+1}Tr\left\{{\cal H}_{\rm CF}^{2}\right\}.\end{aligned}$$ Hence $$\begin{aligned} \sigma^{2}(\|\alpha S L J\rangle) &=& \frac{1}{2J+1}\sum\limits_{k}S_{k}^{2} \left(\langle\alpha SLJ\|\|C^{(k)}\|\|\alpha SLJ\rangle\right)^{2},\end{aligned}$$ what implies from the orthogonality of 3-$j$ symbols \[2,5,13\]. The symbols $\langle\alpha SLJ\|\|C^{(k)}\|\|\alpha SLJ\rangle$ are the double-bar or reduced matrix elements of the spherical tensor operators. According to Wigner-Eckart theorem \[14\] they are independent of the reference frame orientation. Their origin and physical meaning stem from the following relationships \[5-8,15\]: $$\begin{aligned} \langle\alpha SLJM_{J}\|C_{q}^{(k)}\|\alpha SL^{\prime}J^{\prime}M_{J}^{\prime}\rangle &=& (-1)^{J-M_{J}}\left(\begin{array}{ccc} J & k & J^{\prime} \\ -M_{J} & q & M_{J}^{\prime} \\ \end{array}\right) \langle\alpha SLJM_{J}\|\|C^{(k)}\|\|\alpha SL^{\prime}J^{\prime}\rangle,\end{aligned}$$ where the reduced matrix element follows the 3-$j$ factor. Further use of tensor formalism yields $$\begin{aligned} \langle\alpha SLJ\|\|C^{(k)}\|\|\alpha SL^{\prime}J^{\prime}\rangle &=& (-1)^{S+L^{\prime}+J+k} \left[(2J+1)(2J^{\prime}+1)\right]^{1/2}\left\{\begin{array}{ccc} J & J^{\prime} & k \\ L^{\prime} & L & S \\ \end{array}\right\} \langle\alpha SL\|\|C^{(k)}\|\|\alpha SL^{\prime}\rangle,\end{aligned}$$ where the double-reduced matrix element follows now the 6-$j$ symbol. We can also pass to the matrix elements of the unit operators $U^{(k)}$ \[5-8,15\], i.e. normalized equivalents of $C^{(k)}$, since $$\begin{aligned} \langle\alpha SL\|\|C^{(k)}\|\|\alpha SL^{\prime}\rangle &=& \langle\alpha SL\|\|U^{(k)}\|\|\alpha SL^{\prime}\rangle \langle\l\|\|C^{(k)}\|\|l\rangle,\end{aligned}$$ and $l$ is the angular momentum quantum number of the open-shell electrons. The reduced matrix elements of the $U^{(k)}$ operators have been compiled by Nielson and Koster \[16\], whereas the 3-$j$ and 6-$j$ symbols can be found in the tables by Rotenberg et al \[7\]. The simple relation between the $\sigma_{k}^{2}$ and $S_{k}^{2}$ (Eq.4) can be also proved employing Vieta’s formulas for roots of the ${\cal H}_{\rm CF}^{(k)}$ matrix characteristic polynomial $$\begin{aligned} E^{n}+a_{1}E^{n-1}+a_{2}E^{n-2}+\ldots a_{n}&=& 0,\end{aligned}$$ which is here of order of $n=2J+1$. All its coefficients and roots must be real what follows obviously from the ${\cal H}_{\rm CF}$ hermiticity. Interestingly, some characteristics of the sublevels spectrum may be described in terms of the elementary algebra. Firstly, as the energy center of gravity of the initial state is conserved, i.e. $\left(\sum_{i=1}^{n}E_{i}=0\right)^2$, the $a_{1}$ coefficient standing at $E^{n-1}$ must vanish. Next, since $0=\left(\sum_{i}^{n}E_{i}=0\right)^2=\sum_{i=1}^{n}E_{i}^{2}+ 2\sum_{i>j}E_{i}E_{j}$, the second moment, i.e. the sum of the root squares (divided by $2J+1$) is equal to $\frac{-2a_{2}}{2J+1}$. It can be also shown that $$\begin{aligned} -2a_{2} &=& \frac{1}{2k+1}\;M_{k}^{2}\;\langle J\|\|C^{(k)}\|\|J\rangle^{2},\end{aligned}$$ where the simplified notation for the reduced matrix element representing only the last quantum number has been introduced. Hence, between $\sigma_{k}^{2}$ and $S_{k}^{2}$ a simple formula holds (Eq.4) $$\begin{aligned} \sigma_{k}^{2} &=& \frac{1}{2J+1}\;S_{k}^{2}\; \langle J\|\|C^{(k)}\|\|J\rangle^{2}.\end{aligned}$$ In other words, $\sigma_{k}$ is proportional to $S_{k}$. Finally, a free term of the characteristic polynomial is given as $a_{n}=(-1)^{n}E_{1}E_{2}\ldots E_{n}$, what may be helpful analyzing the solutions. For instance, if one root equals zero then a free term vanishes. The problem becomes more complex for the global crystal-field strength $S$ (Eq.2), since then the components $S_{k}^{2}$ contribute to the sum with their weights $\langle J\|\|C^{(k)}\|\|J\rangle^{2}$ (Eq.4). This is why there is no straightforward proportionality between $\sigma^{2}$ and $S^{2}$ in this case. Nevertheless $\sigma$ is a positively defined quadratic form of $S_{k}$ and, in consequence, the inputs of particular ${\cal H}_{\rm CF}$ $2^{k}$-poles into $\sigma^{2}$ cannot compensate themselves. The condition that $\sigma^{2}$ is constant for various iso-modular ${\cal H}_{\rm CF}$ does not exclude, however, a possible differentiation of the CF sublevels sequence and structure, as well as the initial state total splitting. In fact, $\sigma$ and $S$ could be correlated similarly as $\sigma_{k}$ and $S_{k}$ in the previous case, only if the elements $\langle J\|\|C^{(k)}\|\|J\rangle$ were equal for all $k$. Nevertheless, the second moment of the Stark levels within a particular state $\|\alpha S L J\rangle$ is simply given in terms of $S_{2}$, $S_{4}$ and $S_{6}$. Auzel and Malta \[2\] made an attempt to average the $\sigma^{2}$ quadratic form by bringing down its respective ellipsoid $\sum_{k}S_{k}^{2}\langle J\|\|C^{(k)}\|\|J\rangle^{2}$ in the $k$-space to a sphere of the same volume $\sum_{k}S_{k}^{2}\left[\Pi_{k}\langle J\|\|C^{(k)}\|\|J\rangle^{2}\right]^{1/3}$ and having a radius equal to the geometric mean of the three ellipsoid axes. In practice, unfortunately, this elegant approach does not always lead to acceptable results. In the literature the overall effect of the crystal-field interaction is often characterized by a quantitative comparison of the crystal-field strength \[17-20\]. Additionally, a systematic correlation between the free ion parameters and the CF strength is observed, namely increase of the crystal-field interaction results in the reduction of the free-ion parameters \[17\]. The CF strength increases in the RE series with decreasing ionic radius of the RE$^{3+}$ host cation \[19\]. The physical meaning of the CF strength scalar parameter is also supported by the fact that it rises with pressure applied to a sample \[21,22\]. The CF strength parameter has also been used to compare the root mean square error obtained for crystal fields of different strength. However, its use in such a case is restricted only to comparisons of the identical site symmetries \[10,23\]. Furthermore, within the approximation to the second order in the crystal-field, the shift in the center of gravity of a particular $^{2S+1}L_{J}$ state due to $J$-mixing effects is a simple linear function of the $S_{k}^{2}$ \[4,17\]. The concept of the $S_{k}^{2}$ or $S$ can be extended to define the quantities $C_{k}$ and $C_{G}$ \[13\] as normalized “scalar products” of any two compared parametrizations. These quantities represent the “angles” between the two considered parametrizations and are a convenient measure of the closeness, i.e. the correlation of any two CFPs sets. 3. The correspondence of the Stark levels second moment of $\|J\rangle$ state to its nominally allowed splittings {#the-correspondence-of-the-stark-levels-second-moment-of-jrangle-state-to-its-nominally-allowed-splittings .unnumbered} ---------------------------------------------------------------------------------------------------------------- The second moment of CF levels, $\sigma^{2}$, essentially limits a formally allowed range of the initial state $\|J\rangle$ total splittings $\Delta {\cal E}$ for different but iso-modular ${\cal H}_{\rm CF}$s. Such energy splitting confinement differs for non-Kramers and Kramers ions what is specified in details below. Let us firstly study the case of any integer $J$, i.e. non-Kramers ions. Having to keep a constant $\sigma^{2}$ the minimal hypothetical splitting $\Delta {\cal E}_{\rm min}$ of the $(2J+1)$-fold degenerate state takes place when $J$ levels assume identical energy of $\frac{J+1}{2J+1}\Delta{\cal E}_{\rm min}$, and the remaining $J+1$ levels take the energy $\frac{-J}{2J+1}\Delta {\cal E}_{\rm min}$, or vice versa. Further this is referred as Type I splitting. Then $$\begin{aligned} \sigma^{2} &=& \frac{J(J+1)(\Delta {\cal E}_{\rm min})^{2}}{(2J+1)^{2}}, \qquad \qquad \Delta {\cal E}_{\rm min}=\sigma\frac{2J+1}{\sqrt{J(J+1)}}.\end{aligned}$$ In turn, the maximal hypothetical splitting $\Delta {\cal E}_{\rm max}$ occurs for one level of $\Delta {\cal E}_{\rm max}/2$ energy, one of $-\Delta {\cal E}_{\rm max}/2$, and the rest $(2J-1)$ levels with zero energy. Further this is referred as Type II splitting. Then $$\begin{aligned} \sigma^{2} &=& \frac{2\left(\Delta {\cal E}_{\rm max}/2\right)^2}{2J+1}, \qquad \qquad \Delta {\cal E}_{\rm max}=\sigma \sqrt{2(2J+1)},\end{aligned}$$ and hence $$\begin{aligned} \frac{\Delta {\cal E}_{\rm max}}{\Delta {\cal E}_{\rm min}} &=& \sqrt{\frac{2J(J+1)}{2J+1}}.\end{aligned}$$ Let us also consider, following Auzel and Malta \[2\], the case of the homogenous splitting $\Delta {\cal E}_{\rm hom}$, when $$\begin{aligned} \sigma^{2} &=& \frac{2(1+4+\ldots+J^{2})\left(\Delta {\cal E}_{\rm hom}/2J\right)^2}{2J+1} \qquad {\rm and} \qquad \Delta {\cal E}_{\rm hom}=2\sigma \sqrt{\frac{3J}{J+1}}.\end{aligned}$$ Below it will be referred as Type III splitting. For example, if $J=4$ then $\Delta {\cal E}_{\rm min}=\sigma\frac{9}{2\sqrt{5}}=2.01\sigma$, $\Delta {\cal E}_{\rm max}=\sigma3\sqrt{2}\sigma=4.24\sigma$, $\Delta {\cal E}_{\rm hom}=4\sigma\sqrt{3/5}=3.10\sigma$, and finally the ratio $\frac{\Delta {\cal E}_{\rm max}}{\Delta {\cal E}_{\rm min}} = 2.11$. For $J=1$, $\Delta {\cal E}_{\rm min}=\sigma\frac{3}{\sqrt{2}}$, $\Delta {\cal E}_{\rm max}=\Delta {\cal E}_{\rm hom}=\sigma\sqrt{6}$. The ratio $\frac{\Delta {\cal E}_{\rm max}}{\Delta {\cal E}_{\rm min}} = 2/\sqrt{3}=1.16$ and this narrow interval strictly limits the $\Delta {\cal E}$ variation. It has a simple graphical interpretation. As is known, three real roots of a third order equation must fulfill the conditions (Cardan’s formulas) presented in Fig.1a, where the angle $\varphi$ is a function of the equation coefficients. The maximal and minimal splittings $\Delta {\cal E}$ correspond to the solutions shown in Figs 1b and 1c, respectively. Let us now pass to the Kramers ions with a half integer $J$. Here, two cases should be analyzed. Firstly, if an even number of doublets $(2J+1)/2$ occurs the minimal $\|J\rangle$ state splitting takes place when $(2J+1)/4$ doublets have the energy $\Delta {\cal E}_{\rm min}/2$, and the next $(2J+1)/4$ doublets the energy $-\Delta {\cal E}_{\rm min}/2$. Then, $$\begin{aligned} \sigma^{2} &=& \frac{4[(2J+1)/4]\left(\Delta {\cal E}_{\rm min}/2\right)^2}{2J+1}, \qquad \qquad \Delta {\cal E}_{\rm min}=2\sigma.\end{aligned}$$ In turn, the maximal splitting, $\Delta {\cal E}_{\rm max}$, will appear if one of the doublets will be of energy $\Delta {\cal E}_{\rm max}/2$, and the second one of energy $-\Delta {\cal E}_{\rm max}/2$ with all the rest of levels with zero energy. This time $$\begin{aligned} \sigma^{2} &=& \frac{4\left(\Delta {\cal E}_{\rm max}/2\right)^2}{2J+1}, \qquad \qquad \Delta {\cal E}_{\rm max}=\sigma \sqrt{2J+1},\end{aligned}$$ and therefore now $$\begin{aligned} \frac{\Delta {\cal E}_{\rm max}}{\Delta{\cal E}_{\rm min}} &=& \frac{\sqrt{2J+1}}{2}.\end{aligned}$$ Secondly, for an odd number of Kramers doublets $\Delta {\cal E}_{\rm min}=\frac{2\sigma(2J+1)}{\sqrt{(2J+3)(2J-1)}}\;$ and $\;\Delta {\cal E}_{\rm max}=\sigma\sqrt{2J+1}$, with $\frac{\Delta {\cal E}_{\rm max}}{\Delta {\cal E}_{\rm min}} = \frac{1}{2}\sqrt{\frac{(2J+3)(2J-1)}{2J+1}}$. The homogenous splitting $\Delta {\cal E}_{\rm hom}$ for an even number of doublets $(J=(4k+3)/2)$ and for an odd number of doublets $(J=(4k+1)/2)$, where $k=0,1,\ldots$, amounts correspondingly to $\sigma\sqrt{\frac{3(2J-1)}{J}}$ and $2\sigma\sqrt{\frac{3(2J-1)}{2J+3}}$. By way of example, if $J=9/2$ (i.e. for five doublets), then $\Delta {\cal E}_{\rm min}=\sigma\frac{5}{\sqrt{6}}=2.04\sigma$, $\Delta {\cal E}_{\rm max}=\sigma\sqrt{10}=3.16\sigma$, and $\Delta {\cal E}_{\rm hom}=2.83\sigma$. In the case of Kramers ions the $\Delta {\cal E}$ variation range turns out to be smaller than that for non-Kramers ions, which is seen comparing the $\Delta {\cal E}_{\rm min}$ and $\Delta {\cal E}_{\rm max}$ for $J=4$ and $J=9/2$. Finally, taking the most extreme case of $J=15/2$ for $f$-electron configurations (e.g. for Dy$^{3+}$, Er$^{3+}$) with eight doublets, we would obtain $\Delta {\cal E}_{\rm min}=\sigma\frac{16}{3\sqrt{7}}=2.02\sigma$, $\Delta {\cal E}_{\rm max}=4\sigma$, and $\Delta {\cal E}_{\rm hom}=2.37\sigma$. 4. The new scale of the crystal-field strength. Comparison of both the scales $S_{k}=\frac{1}{2k+1}M_{k}$ and $S_{k}^{\prime}=\|{\cal H}_{\rm CF}^{(k)}\|_{\rm av}$ {#the-new-scale-of-the-crystal-field-strength.-comparison-of-both-the-scales-s_kfrac12k1m_k-and-s_kprimecal-h_rm-cfk_rm-av .unnumbered} ================================================================================================================================================================= 4.1. Average values of the axial parameter modules $\|B_{k0}^{\prime}\|_{\rm av}$ and $\|\sum_{k}B_{k0}^{\prime}\|_{\rm av}$, where $k=2,4,6$ – the rotational invariants of the equivalent ${\cal H}_{\rm CF}$ parametrizations {#average-values-of-the-axial-parameter-modules-b_k0prime_rm-av-and-sum_kb_k0prime_rm-av-where-k246-the-rotational-invariants-of-the-equivalent-calh_rm-cf-parametrizations .unnumbered} ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Rotating the reference frame by the two Euler angles $\alpha$ and $\beta$ we obtain all the equivalent ${\cal H}_{\rm CF}$ parametrizations (with the accuracy to the third Euler angle $\gamma$ about the $z$ axis) \[5,24\]. Their axial parameters for a $2^{k}$-pole component are given as: $$\begin{aligned} B_{k0}^{\prime} &=& \sum\limits_{q=-k}^{k}{\cal D}_{0q}^{(k)} (\alpha,\beta,0)B_{kq}= \sum\limits_{q=-k}^{k}C_{q}^{(k)}(\beta,\alpha)B_{kq} \nonumber\\ &=& C_{0}^{(k)}(\beta)B_{k0} + 2\sum\limits_{q=1}^{k}C_{q}^{(k)}(\beta)\cos q(\alpha+\varphi_{q}) \|B_{kq}\|,\end{aligned}$$ where ${\cal D}_{0q}^{(k)} (\alpha,\beta,\gamma)$ are the middle row rotation matrix elements, $C_{q}^{(k)}(\beta,\alpha)=\left(\frac{4\pi}{2k+1}\right)^{1/2}Y_{q}^{k}(\beta,\alpha)$ are the spherical tensors, whereas $C_{q}^{(k)}(\beta)=(-1)^{q}\left[\frac{(k-q)!}{(k+q)!}\right]^{1/2}P_{k}^{q}(\cos\beta)$, and $P_{k}^{q}(\cos\beta)$ are the associated Legendre functions, $B_{kq}=\|B_{kq}\|e^{{\rm i}q\varphi_{q}}$, and $B_{k-q}=(-1)^{q}\|B_{kq}\|e^{-{\rm i}q\varphi_{q}}$. The primed parameters correspond to the transformed parametrization while the unprimed to the initial one. It can be directly proved that $\left(B_{k0}^{\prime}\right)_{\rm av}=0$ and $\left(\sum_{k}B_{k0}^{\prime}\right)_{\rm av}=0$, while the average absolute values $$\begin{aligned} \left\|B_{k0}^{\prime}\right\|_{\rm av}&=& \frac{1}{4\pi}\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi} \left\|B_{k0}^{\prime}\|(\alpha,\beta)\right\|\sin\beta d\beta d\alpha \;,\nonumber\\ \left\|\sum_{k}B_{k0}^{\prime}\right\|_{\rm av}&=& \frac{1}{4\pi}\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi} \left\|\sum_{k}B_{k0}^{\prime}(\alpha,\beta)\right\|\sin\beta d\beta d\alpha \;,\end{aligned}$$ as the rotational group invariants are discriminants of the equivalent parametrizations classes \[24\]. By the mean value we understand the magnitude averaged over all possible orientations of the reference frame, i.e. over the solid angle $4\pi$. Interestingly, they can be used to estimate the CF strength independently of the parametrization modulus. 4.2. Average values of the modules $\left\|{\cal H}_{\rm CF}^{(k)}\right\|_{\rm av}$ and $\left\|{\cal H}_{\rm CF}\right\|_{\rm av}$ {#average-values-of-the-modules-leftcal-h_rm-cfkright_rm-av-and-leftcal-h_rm-cfright_rm-av .unnumbered} -------------------------------------------------------------------------------------------------------------------------------- Since the expression $$\begin{aligned} {\cal H}_{\rm CF}^{(k)} &=& \sum\limits_{q=-k}^{k}B_{kq}C_{q}^{(k)}(\beta,\alpha),\end{aligned}$$ where $\beta$ and $\alpha$ are the spherical angle coordinates in the central-ion reference system, is identical with that for $B_{k0}^{\prime}$ (Eq.10), the following important identity holds $$\begin{aligned} \left\|{\cal H}_{\rm CF}^{(k)}\right\|_{\rm av}&=& \frac{1}{4\pi}\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi} \left\|{\cal H}_{\rm CF}^{(k)}\|(\alpha,\beta)\right\|\sin\beta d\beta d\alpha= \left\|B_{k0}^{\prime}\right\|_{\rm av}.\end{aligned}$$ The average value of the modulus of the $2^{k}$-pole ${\cal H}_{\rm CF}^{(k)}$ component turns out to be equal to the average value of the modulus of the relevant axial parameter $B_{k0}$. This identity, Eq.12, obvious when we properly interpret the rotation angles in both cases of averaging, associates $\left\|B_{k0}^{\prime}\right\|_{\rm av}$ with the complementary measure of the CF strength $S_{k}^{\prime}$: $$\begin{aligned} S_{k}^{\prime} &=& \left\|{\cal H}_{\rm CF}^{(k)}\right\|_{\rm av}= \left\|B_{k0}^{\prime}\right\|_{\rm av}\end{aligned}$$ Although the expression for $S_{k}^{\prime}$ in the above form is limited to a given $2^{k}$-pole ${\cal H}_{\rm CF}^{(k)}$ component, it may be generalized for the global ${\cal H}_{\rm CF}$ $$\begin{aligned} S^{\prime} &=& \left\|\sum\limits_{k}{\cal H}_{\rm CF}^{(k)}\right\|_{\rm av}= \left\|\sum\limits_{k}B_{k0}^{\prime}\right\|_{\rm av}.\end{aligned}$$ Contrary to the conventional CF strengths $S_{k}$ and $S$ (Eqs 1,2), which are constant for all the iso-modular parametrizations, the new strengths $S_{k}^{\prime}$ and $S^{\prime}$ calculated for the constant modulus (and modules) change their magnitudes within certain ranges discussed in the next section. To compare both the measures it is convenient to express $S_{k}^{\prime}$ in the product form $f_{k}\cdot M_{k}$, where $f_{k}$ is a specified factor. Now, these two measures will be compatible if the factor $f_{k}$ is close to $\sqrt{\frac{1}{2k+1}}$, Eq.1, i.e. to $0.447$, $0.333$ and $0.277$ for $k=2,4$ and $6$, respectively. This compatibility is demonstrated in the next section, where a thorough discussion of the relation between both the CF strength scales is provided, by way of example of the CF splitting of $p^{1}$, $d^{1}$ and $f^{1}$ electron configurations with the spin-orbit coupling deliberately neglected , and the $^{3}H_{4}$ state for various iso-modular ${\cal H}_{\rm CF}^{(k)}$s. 5. Computational results and discussion {#computational-results-and-discussion .unnumbered} --------------------------------------- 5.1. Crystal-field splitting of $p^{1}$, $d^{1}$ and $f^{1}$ electron configurations for various iso-modular ${\cal H}_{\rm CF}^{(k)}$s, $k=2,4,6$. {#crystal-field-splitting-of-p1-d1-and-f1-electron-configurations-for-various-iso-modular-cal-h_rm-cfks-k246. .unnumbered} --------------------------------------------------------------------------------------------------------------------------------------------------- We consider the model results of interaction of any iso-modular ${\cal H}_{\rm CF}^{(k)}$ ($k=2,4,6$) with $M_{k}=1$, on the initial states with well defined angular momentum quantum numbers. The magnitudes of all quantities under discussion, i.e. the new CF strength parameters $S_{k}^{\prime}$, the total splittings $\Delta E^{(k)}$, the second moments $\sigma_{k}^{2}$ of CF levels and the averages of the absolute values of the Stark level energies $\|E_{n}^{(k)}\|_{\rm av}$ are given in $M_{k}$ units. Tables 1, 2 and 3 present a comprehensive review of $S_{k}^{\prime}$ values for various iso-modular ${\cal H}_{\rm CF}^{(k)}$s, with $k=2,4,6$, respectively. Correspondingly, these five, ten and eleven ${\cal H}_{\rm CF}^{(k)}$s compiled in Tables are the representative ones including those with the highest and lowest $S_{k}^{\prime}$ values found during the survey. No other ${\cal H}_{\rm CF}^{(k)}$s seem to yield $S_{k}^{\prime}$ out of these ranges. The strength parameters $S_{k}^{\prime}$ change themselves within the rather narrow intervals: $0.368 - 0.385$, $0.251 - 0.287$ and $0.195 - 0.239$, while the relevant $S_{k}$ are constant and equal to $0.447$, $0.333$ and $0.277$ for $k=2,4$ and $6$, respectively. The maximal $S_{k}^{\prime}$ parameters refer to the purely axial ${\cal H}_{\rm CF}^{(k)}$s when $B_{k0}$s achieve 1. For other parametrizations this maximal value of 1 is not achieved in any reference frame. As implies from Tables 4, 5 and 6 there is a certain mapping between the above $S_{k}^{\prime}$ ranges and the referring to them intervals of $\Delta E^{(k)}$ and $\|E_{n}^{(k)}\|_{\rm av}$. As is shown in the paper this quantitative mapping is determined by the roots of the ${\cal H}_{\rm CF}^{(k)}$ matrix characteristic polynomial, and the key part of the matrix elements is the product $(-1)^{M_{J}}B_{kq}\left(\begin{array}{ccc} J & k & J^{\prime} \\ -M_{J} & q & M_{J}^{\prime} \\ \end{array}\right)$. The remaining factors coming into the matrix elements are common and play the role of a scaling factor. In the below examples concerning the CF splitting of one-electron states with $J=J^{\prime}=l$ for $l=1,2$ and $3$, the role of such a scaling factor play the double-bar matrix elements $\langle l\|\|C^{(k)}\|\|l \rangle$. It should be pointed out, however, that the mappings $S_{k}^{\prime}\leftarrow\!\!\rightarrow \Delta E^{(k)}$, $S_{k}^{\prime}\leftarrow\!\!\rightarrow \|E_{n}^{(k)}\|_{\rm av}$, $\Delta E^{(k)}\leftarrow\!\!\rightarrow \|E_{n}^{(k)}\|_{\rm av}$ are neither straightforward nor explicit. With the increase of the initial state degeneration $2J+1$ they become less clear due to a big variety of possible splitting schemes. Nevertheless, one may presume a dominant tendency: the greater $S_{k}^{\prime}$ the greater $\|E_{n}^{(k)}\|_{\rm av}$ and the lesser $\Delta E^{(k)}$ (Type I splittings). In the reverse case, i.e. for a small $S_{k}^{\prime}$, Type II splittings are expected. However, such reasoning does not take into account the unique characteristics of the Hamiltonian averages $\left\|{\cal H}_{\rm CF}^{(k)}\right\|_{\rm av}$, and the space density distribution of unpaired electrons in the states of various $J$. From this point of view the analysis of Tables 4, 5 and 6 seems to be instructive, indeed. On the other hand, the allowed spans of the $\Delta {\cal E}^{(k)}$ for a fix $M_{k}$, i.e. $\sigma_{k}$ are known. In the light of the above mapping it turns out that not all of these values $\Delta {\cal E}^{(k)}$, and corresponding to them splitting schemes, can actually occur. Namely, depending on the initial state quantum number $J$ and the multipole’s rank $k$ some specified limitations of the $\Delta E^{(k)}$ are observed (Tables 4, 5 and 6). They are listed briefly below. For $l=1$ ($p$-electron) and $k=2$ the full nominal range of the $\Delta {\cal E}^{(2)}$ and all splittings of Types I, II and III are admitted. More particularly, $\Delta E^{(2)}$ can vary from $0.600\;M_{2}$ to $0.693\;M_{2}$ (Table 4). For $l=2$ ($d$-electron) and $k=2$ the magnitude of $\Delta E^{(2)}$ is constant and equals $0.572\;M_{2}$ for each iso-modular ${\cal H}_{\rm CF}^{(2)}$ what corresponds to splittings similar to those of Type I. Other splittings, including e.g. $\Delta {\cal E}^{(2)}_{\rm hom}$ are impossible in this case (Table 5). For $l=2$ ($p$-electron) and $k=4$ again the full nominal range of the $\Delta {\cal E}^{(4)}$ is allowed beginning from the smallest $0.363M_{4}$ for the cubic ${\cal H}_{\rm CF}^{(4)}$, up to the biggest $0.564 M_{4}$ for ${\cal H}_{\rm CF}^{(4)}=\frac{1}{\sqrt{2}}B_{44}C_{4}^{(4)}+\frac{1}{\sqrt{2}}B_{4-4}C_{-4}^{(4)}$ (Table 5). For $l=3$ ($f$-electron) and the value of $k=2$ $\Delta E^{(2)}$ weakly depends on $S_{2}^{\prime}$, varying in all its range merely from $0.600M_{2}$ to $0.608M_{2}$, i.e. somewhat below the $\Delta {\cal E}^{(2)}_{\rm hom}$ (Type III splittings) (Table 6). Next, for $l=3$ and $k=4$, the possible $\Delta E^{(4)}$ varies within the range from $0.358M_{4}$ to $0.482M_{4}$, i.e. around the $\Delta {\cal E}^{(4)}_{\rm hom}$ (Table 6). Finally, for $l=3$ and $k=6$ the allowed $\Delta E^{(6)}$ varies from $0.326M_{6}$ to $0.501M_{6}$ covering the majority of the nominal range together with its upper limit, but excluding the smallest splittings (Table 6). The obtained results may be generalized for states with $J$ or $L$ equal to 1, 2 or 3, multiplying $\Delta E^{(k)}$ and $\|E_{n}^{(k)}\|_{\rm av}$ by the scaling factors $\langle J\|\|C^{(k)}\|\|J\rangle$ or $\langle L\|\|C^{(k)}\|\|L\rangle$. 5.2. Crystal-field splitting of $^{3}H_{4}$ state in various iso-modular ${\cal H}_{\rm CF}^{(k)}$s, $k=2,4,6$. {#crystal-field-splitting-of-3h_4-state-in-various-iso-modular-cal-h_rm-cfks-k246. .unnumbered} --------------------------------------------------------------------------------------------------------------- Let us end up with the analysis of splitting of nine-fold degenerate $^{3}H_{4}$ state subjected to the iso-modular ${\cal H}_{\rm CF}^{(k)}$s enclosed in Tables 1, 2 and 3. Table 7 shows the correlation between $S_{k}^{\prime}$, $\Delta E^{(k)}$ and $\|E_{n}^{(k)}\|_{\rm av}$. The scaling factors $\langle J\!=\!4\|\|C^{(k)}\|\|J\!=\!4\rangle=\langle J\!=\!4\|\|U^{(k)}\|\|J\!=\!4\rangle\langle f\|\|C^{(k)}\|\|f\rangle$, required here due to the coupled initial state $(L=5, S=1, J=4)$, are equal to $-1.2365$, $-0.7389$ and $0.7706$ for $k=2,4,6$, respectively. Hence $\sigma_{2}=0.184M_{2}$, $\sigma_{4}=0.082M_{4}$ and $\sigma_{6}=0.071M_{6}$, while the global second moment of the Stark levels takes the form $$\begin{aligned} \sigma^{2}&=& \frac{1}{9}\left[\frac{1}{5}(-1.2365)^{2}M_{2}^{2}+\frac{1}{9}(-0.7389)^{2}M_{4}^{2} +\frac{1}{13}(0.7706)^{2}M_{6}^{2}\right].\end{aligned}$$ The ranges of the formally allowed $\Delta {\cal E}^{(k)}$ corresponding to the above second moments $\sigma_{2},\sigma_{4},\sigma_{6}$ are marked in Fig.2 by the solid lines. We can see in Fig.2 that from the set of all potentially allowed total splittings $\Delta {\cal E}^{(k)}$ only certain $\Delta E^{(k)}$ may be realized (those between the dashed lines), and consequently, only certain splitting schemes (roughly between Types I and III) may occur. For instance, in the case of all the three effective multipoles neither $\Delta {\cal E}^{(k)}_{\rm max}$ nor $\Delta {\cal E}^{(k)}_{\rm min}$ are possible, while $\Delta {\cal E}^{(k)}_{\rm hom}$ can appear solely in the case of $2^{6}$-pole. Based on Table 7 it is seen also that for all the three effective ${\cal H}_{\rm CF}^{(k)}$s the biggest $\Delta E^{(k)}$ are achieved for intermediate $S_{k}^{\prime}$ values. 6. Conclusions {#conclusions .unnumbered} -------------- The conventional scales $S_{k}$ or $S$ with the associated second moments of the CF levels, $\sigma_{k}$ or $\sigma$, do not distinguish the iso-modular ${\cal H}_{\rm CF}^{(k)}$ or ${\cal H}_{\rm CF}$ parametrizations, which, however, can be differentiated by an another scale – the spherically averaged $S_{k}^{\prime}=\left\|{\cal H}_{\rm CF}^{(k)}\right\|_{\rm av}$ and $S^{\prime}=\left\|{\cal H}_{\rm CF}\right\|_{\rm av}$. It is proved that the $S_{k}^{\prime}$ variation ranges for all the iso-modular parametrizations are limited and lie slightly below the relevant $S_{k}$ magnitudes. The span of these ranges amounts to 5, 10 and 20% of their values for $k=2,4$ and $6$, respectively. There exists a direct mapping of $S_{k}^{\prime}$ ranges into the total splitting $\Delta E^{(k)}$ ranges and $\|E_{n}^{(k)}\|_{\rm av}$ intervals, which may be interpreted more clearly for the initial states with low degeneration. Such mapping allows to estimate the total splittings $\Delta E^{(k)}$ or $\Delta E$ to be expected and characterize their spectrum. It is shown that not all the nominally admitted total $\Delta {\cal E}^{(k)}$ or $\Delta {\cal E}$ splittings determined by the modules $M_{k}$ or $M$, i.e. the second moments $\sigma_{k}$ or $\sigma$, can actually occur. This essentially confines the set of the allowed splitting schemes. [10]{} Auzel F 1979 [Matt. Res. 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Gen.]{} [38]{} 6081 ----- -- ---------------------- ------------------------------------------------ ----------------------------------------------------- -- ------- No. $S_{2}^{\prime}=\|{\cal H}_{\rm CF}^{(2)}\|_{\rm av}$ $B_{20}$ $B_{21}$ $B_{22}$ 1 1 0 0 0.385 2 $\frac{1}{\sqrt{5}}$ $\frac{1}{\sqrt{5}}$ $-\frac{1}{\sqrt{5}}$ 0.381 3 $\frac{1}{\sqrt{5}}$ $\frac{1}{\sqrt{5}}$ $-\frac{1}{\sqrt{5}}$ 0.374 4 $\frac{1}{\sqrt{5}}$ $\frac{1}{\sqrt{5}}\;\;{\rm e}^{{\rm i}\pi/4}$ $\frac{1}{\sqrt{5}}$ 0.369 5 0 0 $\frac{1}{\sqrt{2}}$ 0.368 ----- -- ---------------------- ------------------------------------------------ ----------------------------------------------------- -- ------- : The spherical averages of five representative iso-modular ${\cal H}_{\rm CF}^{(2)}$s, $\;S_{2}^{\prime}=\|{\cal H}_{\rm CF}^{(2)}\|_{av}$, acc. to Eqs 11-13, expressed in $M_{2}$ units. Only $B_{2q}$ CFPs are given, $B_{2-q}=(-1)^{q}B_{2q}^{\ast}$[]{data-label="tab"} ----- -- --------------------------------- --------------------------------------- ----------------------------------------------------- ---------------------------------------- --------------------------------- -- ------- No. $S_{4}^{\prime}=\|{\cal H}_{\rm CF}^{(4)}\|_{\rm av}$ $B_{40}$ $B_{41}$ $B_{42}$ $B_{43}$ $B_{44}$ 1 1 0 0 0 0 0.287 2 $\frac{1}{2}\sqrt{\frac{7}{3}}$ 0 0 0 $\frac{1}{2}\sqrt{\frac{5}{6}}$ 0.280 3 $\frac{1}{3}$ $\frac{1}{3}$ $\frac{1}{3}\;{\rm e}^{{\rm i}\pi/2}$ $\frac{1}{3}$ $\frac{1}{3}$ 0.277 4 $\frac{1}{3}$ $-\frac{1}{3}$ $\frac{1}{3}$ $\frac{1}{3}$ $\frac{1}{3}$ 0.276 5 $\frac{1}{3}$ $\frac{1}{3}$ $\frac{1}{3}$ $-\frac{1}{3}\;{\rm e}^{{\rm i}\pi/2}$ $\frac{1}{3}$ 0.273 6 0 0 $\frac{1}{\sqrt{2}}$ 0 0 0.269 7 0 0 0 $\frac{1}{\sqrt{2}}$ 0 0.266 8 $\frac{1}{3}$ $\frac{1}{3}$ $\frac{1}{3}$ $\frac{1}{3}$ $\frac{1}{3}$ 0.265 9 $\frac{1}{3}$ $\frac{1}{3}\;{\rm e}^{{\rm i}\pi/4}$ $\frac{1}{3}$ $\frac{1}{3}$ $\frac{1}{3}$ 0.261 10 0 0 0 0 $\frac{1}{\sqrt{2}}$ 0.251 ----- -- --------------------------------- --------------------------------------- ----------------------------------------------------- ---------------------------------------- --------------------------------- -- ------- : The spherical averages of ten representative iso-modular ${\cal H}_{\rm CF}^{(4)}$s, $\;S_{4}^{\prime}=\|{\cal H}_{\rm CF}^{(4)}\|_{av}$, acc. to Eqs 11-13, expressed in $M_{4}$ units. Only $B_{4q}$ CFPs are given, $B_{4-q}=(-1)^{q}B_{4q}^{\ast}$[]{data-label="tab"} ----- -- ------------------------ ------------------------ ----------------------------------------------------- ----------------------- ------------------------- ----------------------- ------------------------ -- ------- No. $S_{6}^{\prime}=\|{\cal H}_{\rm CF}^{(6)}\|_{\rm av}$ $B_{60}$ $B_{61}$ $B_{62}$ $B_{63}$ $B_{64}$ $B_{65}$ $B_{66}$ 1 1 0 0 0 0 0 0 0.239 2 $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $-\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ 0.231 3 $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $-\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ 0.228 4 $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $-\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $-\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ 0.227 5 $-\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ 0.225 6 $\frac{1}{\sqrt{13}}$ $-\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ 0.223 7 $\frac{1}{2\sqrt{2}}$ 0 0 0 $\pm\frac{\sqrt{7}}{4}$ 0 0 0.223 8 $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $-\frac{1}{\sqrt{13}}$ 0.222 9 0 $\frac{1}{\sqrt{2}}$ 0 0 0 0 0 0.219 10 $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ $\frac{1}{\sqrt{13}}$ 0.213 11 0 0 0 0 0 0 $\frac{1}{\sqrt{2}}$ 0.195 ----- -- ------------------------ ------------------------ ----------------------------------------------------- ----------------------- ------------------------- ----------------------- ------------------------ -- ------- : The spherical averages of eleven representative iso-modular ${\cal H}_{\rm CF}^{(6)}$s, $\;S_{6}^{\prime}=\|{\cal H}_{\rm CF}^{(6)}\|_{av}$, acc. to Eqs 11-13, expressed in $M_{6}$ units. Only $B_{6q}$ CFPs are given, $B_{6-q}=(-1)^{q}B_{6q}^{\ast}$[]{data-label="tab"} [lcccc]{} No.& &$\left\|{\cal H}_{\rm CF}^{(2)}\right\|_{\rm av}$&$\Delta E^{(2)}$& $\left\|E_{n}^{(2)}\right\|_{\rm av}$\ 1& &0.385& 0.600 &0.267\ 2& &0.381& 0.656 &0.262\ 3& &0.374& 0.683 &0.250\ 4& &0.369& 0.692 &0.239\ 5& &0.368& 0.693 &0.231\ [lcccc]{} No.& &$\left\|{\cal H}_{\rm CF}^{(2)}\right\|_{\rm av}$&$\Delta E^{(2)}$& $\left\|E_{n}^{(2)}\right\|_{\rm av}$\ 1& &0.385& 0.572 &0.229\ 2& &0.381& 0.572 &0.227\ 3& &0.374& 0.572 &0.221\ 4& &0.369& 0.572 &0.217\ 5& &0.368& 0.572 &0.213\ \ \ & &$\left\|{\cal H}_{\rm CF}^{(4)}\right\|_{\rm av}$&$\Delta E^{(4)}$& $\left\|E_{n}^{(4)}\right\|_{\rm av}$\ 6& &0.287& 0.476 &0.152\ 7& &0.280& 0.363 &0.174\ 8& &0.277& 0.449 &0.169\ 9& &0.276& 0.437 &0.169\ 10& &0.273& 0.463 &0.164\ 11& &0.269& 0.426 &0.159\ 12& &0.266& 0.398 &0.159\ 13& &0.265& 0.549 &0.137\ 14& &0.261& 0.555 &0.132\ 15& &0.251& 0.564 &0.113\ [lcccc]{} No.& &$\left\|{\cal H}_{\rm CF}^{(2)}\right\|_{\rm av}$&$\Delta E^{(2)}$& $\left\|E_{n}^{(2)}\right\|_{\rm av}$\ 1& &0.385& 0.600 &0.190\ 2& &0.381& 0.603 &0.191\ 3& &0.374& 0.607 &0.192\ 4& &0.369& 0.608 &0.193\ 5& &0.368& 0.608 &0.193\ \ \ & &$\left\|{\cal H}_{\rm CF}^{(4)}\right\|_{\rm av}$&$\Delta E^{(4)}$& $\left\|E_{n}^{(4)}\right\|_{\rm av}$\ 6& &0.287& 0.394 &0.121\ 7& &0.280& 0.417 &0.119\ 8& &0.277& 0.449 &0.116\ 9& &0.276& 0.458 &0.117\ 10& &0.273& 0.464 &0.115\ 11& &0.269& 0.478 &0.113\ 12& &0.266& 0.482 &0.115\ 13& &0.265& 0.399 &0.129\ 14& &0.261& 0.363 &0.129\ 15& &0.251& 0.358 &0.131\ \ \ & &$\left\|{\cal H}_{\rm CF}^{(6)}\right\|_{\rm av}$&$\Delta E^{(6)}$& $\left\|E_{n}^{(6)}\right\|_{\rm av}$\ 16& &0.239& 0.408 &0.107\ 17& &0.231& 0.326 &0.130\ 18& &0.228& 0.365 &0.120\ 19& &0.227& 0.379 &0.123\ 20& &0.225& 0.445 &0.106\ 21& &0.223& 0.420 &0.112\ 22& &0.223& 0.346 &0.127\ 23& &0.222& 0.422 &0.112\ 24& &0.219& 0.468 &0.097\ 25& &0.213& 0.481 &0.097\ 26& &0.195& 0.501 &0.072\ [lcccc]{} No.& &$\left\|{\cal H}_{\rm CF}^{(2)}\right\|_{\rm av}$&$\Delta E^{(2)}$& $\left\|E_{n}^{(2)}\right\|_{\rm av}$\ 1& &0.385& 0.504 &0.163\ 2& &0.381& 0.543 &0.170\ 3& &0.374& 0.562 &0.164\ 4& &0.369& 0.560 &0.164\ 5& &0.368& 0.524 &0.155\ \ \ & &$\left\|{\cal H}_{\rm CF}^{(4)}\right\|_{\rm av}$&$\Delta E^{(4)}$& $\left\|E_{n}^{(4)}\right\|_{\rm av}$\ 6& &0.287& 0.215 &0.079\ 7& &0.280& 0.227 &0.073\ 8& &0.277& 0.249 &0.070\ 9& &0.276& 0.241 &0.071\ 10& &0.273& 0.232 &0.073\ 11& &0.269& 0.230 &0.064\ 12& &0.266& 0.232 &0.067\ 13& &0.265& 0.231 &0.077\ 14& &0.261& 0.229 &0.076\ 15& &0.251& 0.196 &0.077\ \ \ & &$\left\|{\cal H}_{\rm CF}^{(6)}\right\|_{\rm av}$&$\Delta E^{(6)}$& $\left\|E_{n}^{(6)}\right\|_{\rm av}$\ 16& &0.239& 0.202 &0.058\ 17& &0.231& 0.192 &0.069\ 18& &0.228& 0.249 &0.051\ 19& &0.227& 0.212 &0.060\ 20& &0.225& 0.224 &0.062\ 21& &0.223& 0.233 &0.059\ 22& &0.223& 0.245 &0.053\ 23& &0.222& 0.233 &0.061\ 24& &0.219& 0.258 &0.052\ 25& &0.213& 0.208 &0.059\ 26& &0.195& 0.206 &0.058\ FIGURE CAPTIONS: Fig.1. Crystal-field splitting of $\|J\!=\!1\rangle$ state – geometrical interpretation ($x$ is the energy) a) general case: $\Delta {\cal E}^{(2)}=x_{1}-x_{2}$, b) $\Delta {\cal E}^{(2)}_{\rm min}=x_{1}-x_{2}$, and c) $\Delta {\cal E}^{(2)}_{\rm hom}=\Delta {\cal E}^{(2)}_{\rm max}=x_{2}-x_{3}$. Fig.2. Nominally allowed $\Delta {\cal E}^{(k)}$ (bold solid borders) and the actual $\Delta E^{(k)}$ (dashed borders) ranges of the total splittings of the $^{3}H_{4}$ state subjected to the iso-modular ${\cal H}_{CF}^{(k)}$. The $\Delta {\cal E}^{(k)}_{\rm hom}$ are also given (thin solid lines). ![image](fig1.eps){width="17cm"} Fig.1 ![image](fig2.eps){width="15cm"} Fig.2	ArXiv
--- abstract: 'We study the convexity of the entropy functional along particular interpolating curves defined on the space of finitely supported probability measures on a graph.' author: - 'Erwan Hillion [^1]' title: 'Entropy along $W_{1,+}$-geodesics on graphs' --- Introduction ============ The Wasserstein distance $W_p(\mu_0,\mu_1)$ between two finitely supported probability measures on a metric space $(X,d)$ with its Borel $\sigma$-algebra is defined for $p \geq 1$ by $$\label{eq:WassersteinDef} W_p(\mu_0,\mu_1)^p := \inf_{\pi \in \Pi(\mu_0,\mu_1)} \int_{X \times X} d(x_0,x_1)^p d\pi(x_0,x_1),$$ where $\Pi(\mu_0,\mu_1)$ is the (non-empty) set of couplings between $\mu_0$ and $\mu_1$, i.e. the set of probability measures on $X \times X$ having $\mu_0$ and $\mu_1$ as marginals. The optimization problem defined by equation is called the Monge-Kantorovitch problem and any minimizer for is called optimal coupling between $\mu_0$ and $\mu_1$. For a comprehensive study of optimal transportation theory, the reader is referred to the textbooks [@VillaniBook1] and [@VillaniBook2] by Villani. Under mild conditions, it is possible to show that the set $\Pi_p(\mu_0,\mu_1)$ of optimal couplings between $\mu_0$ and $\mu_1$ is non-empty. Furthermore, under the additional assumptions that $p>1$, $(X,d)$ is the Euclidean space $(\mathbb{R}^d,\|.\|)$ and $\mu_0$ is absolutely continuous with respect to the Lebesgue measure, one can prove the existence of a measurable map $T : \mathbb{R}^d \rightarrow \mathbb{R}^d$ such that the coupling $\pi := (Id \times T)_\mu_0$ is a minimizer for . In particular, $\mu_1$ is the pushforward of $\mu_0$ by the application $T$: $\mu_1 := T_\mu_0$ and equation can be rewritten $$W_p(\mu_0,\mu_1)^p = \int_{\mathbb{R}^d} \|x-T(x)\|^p d\mu_0(x).$$ It is possible to go further by considering, for $0 \leq t \leq 1$, the measure $\mu_t := (T_t)_\mu_0$, where the application $T_t : \mathbb{R}^d \rightarrow \mathbb{R}^d$ is defined as the barycenter $T_t(x) := (1-t)x + t T(x)$. One can then show that the family $(\mu_t)_{t \in [0,1]}$ is a geodesic for the Wasserstein distance $W_p$, in the sense that $$W_p(\mu_0,\mu_1) = \sup_{0 = t_0 \leq t_1 \cdots \leq t_{n} = 1} \sum_{i=0}^{n-1} W_p(\mu_{t_i},\mu_{t_{i+1}}).$$ Moreover, a fundamental property of optimal couplings asserts that $T_t$ is injective, which allows us to define unambiguously a velocity field $(v_t)_{t \in [0,1]}$ by $$v_t(T_t(x)) := T(x)-x.$$ The terminology ’velocity field’ comes from the fact that, if we write $d\mu_t(x) = f_t(x) dx$, then the density $f_t(x)$ satisfy, at least formally, the transport equation $$\label{eq:velocitytransport} \frac{\partial}{\partial t} f_t(x) + \operatorname{div}(v_t(x) f_t(x)) = 0.$$ Moreover, the velocity field $v_t(x)$ satisfies the Hamilton-Jacobi-type equation $$\label{eq:velocityoptimalHJ} \frac{\partial}{\partial t} v_t(x) + \frac{1}{2} \operatorname{grad}\|v_t(x)\|^2 = 0,$$ which can be simplified into $$\label{eq:velocityoptimal} \frac{\partial}{\partial t} v_t(x) = - \operatorname{div}(v_t(x)) v_t(x).$$ In [@BenamouBrenier], Benamou and Brenier proved that both equations and can be used to give a characterization of $W_p$-geodesics, more precisely we have: \[th:BenamouBrenier\] Given two finitely supported probability measures $d\mu_0(x) := f_0(x) dx$ and $d\mu_1(x) := f_1(x) dx$, we have $$\label{eq:MKBenamou} W_p(\mu_0,\mu_1)^p = \inf \int_0^1 \int_{\mathbb{R}^d} \|v_t(x)\|^p d\mu_t(x),$$ where the infimum is taken over the set of curves $(\mu_t)_{t \in [0,1]} = (f_t(x) dx)_{t \in [0,1]}$ joining the prescribed measures $\mu_0$ and $\mu_1$, and where $(v_t(x))_{t \in [0,1]}$ is a velocity field such that equation holds. Moreover, the formal optimality condition for the optimization problem is given by equation . Theorem \[th:BenamouBrenier\] is also true for families of probability measures defined on a Riemannian manifold, having smooth enough densities with respect to the Riemannian volume measure. However, in this framework, equations and are no longer equivalent. The optimality condition is the starting point of the article [@HillionGeodesic] by the author. The main idea is the following: given two distinct probability measures $f_0,f_1$ on a graph $G$, there is no interpolating curve $(f_t)_{t \in [0,1]}$ with a finite length for the Wasserstein $W_p$, for any $p>1$. However, in generic cases there are infinitely many geodesics $(f_t)_{t \in [0,1]}$ for the $W_1$ distance. The aim of [@HillionGeodesic] is to choose among this set a particular $W_1$-geodesic satisfying a discrete version of equation . These interpolating curves are called $W_{1,+}$-geodesics on $G$; we recall their basic properties in Section 2. The purpose of this article is to study the behaviour of the entropy functional along a $W_{1,+}$-geodesic $(f_t(x))_{t \in [0,1], x \in G}$ on a graph $G$. More precisely, we will study the convexity of the function $t \mapsto H(t)$ defined by $$H(t) := \sum_{x \in G} f_t(x) \log(f_t(x)),$$ where by convention $0 \log 0 = 0$. The methods used to prove such convexity properties are adapted from the previous article [@HillionContraction] by the author, and use the first-order-calculus formalism introduced in [@HillionGeodesic]. The motivation behind this research work comes from Sturm-Lott-Villani theory, developed in the articles [@SturmRicci01], [@SturmRicci02] and [@LottVillani]. The main idea of this theory is the following: it is possible to obtain some information about the geometry of a measured length space $(X,d,\nu)$ by studying the behaviour of entropy functionals along $W_2$-geodesics on the space of probability measures over $(X,d)$. A major result asserts that a compact Riemannian manifold $(M,g)$ satisfies the Ricci curvature bound $\operatorname{Ric}\geq K g$ if and only if each pair of absolutely continuous probability measures $\mu_0, \mu_1$ can be joined by a Wasserstein $W_2$-geodesic $(\mu_t)_{t \in [0,1]}$ such that $$\label{eq:LVSconvexity} H(\mu_t) \leq (1-t) H(\mu_0) + t H(\mu_1) - K \frac{t(1-t)}{2} W_2(\mu_0,\mu_1)^2,$$ where the relative entropy $H(\mu)$ is defined by $H(\mu): = \int_{M} \rho \log(\rho) d\operatorname{vol}$ if $d\mu = \rho . d\operatorname{vol}$ and by $H(\mu)=\infty$ if $\mu$ is not absolutely continuous with respect to the Riemannian volume measure. It is then possible to define the curvature condition ’$\operatorname{Ric}\geq K$’ on a measured length space $(X,d,\nu)$ if Equation is satisfied for any $W_2$-Wasserstein geodesic on $\mathcal{P}_2(X)$. Several geometric theorems and functional inequalities holding on Riemannian manifolds satisfying a Ricci curvature bound are still valid in the framework of measured length spaces with a curvature condition ’$\operatorname{Ric}\geq K$’. The generalization of Sturm-Lott-Villani theory to discrete setting has been the subject of many research works, each leading to its own definition of Ricci curvature bounds on graphs, among which we can cite papers by Ollivier [@OllivierRicci] and Erbar-Maas [@ErbarMaas]. The latter is based on the study of a discrete version of the minimization problem for $p=2$, whereas our approach is based on a discrete version of equation characterizing the solutions of . Another important work in discrete Sturm-Lott-Villani theory is [@GRST] which, like this present work, is based on the study of the behaviour of the entropy functional along mixtures of binomial measures. The results proven in our paper show that the convexity properties of the entropy along $W_{1,+}$-geodesics are linked with some intuitive notion of curvature bounds on graphs. However, it seems that our study of the convexity of the entropy does not lead to a definition of Ricci curvature bounds strong enough to imply important functional inequalities, such as the modified logarithmic Sobolev inequality introduced in [@BobkovLedoux]. Our article is outlined as follows: in Section \[sec:Rappel\], we recall the definition and basic properties of the $W_1$-orientation and $W_{1,+}$-geodesics, which are developed in the previous article [@HillionGeodesic]. We also introduce the notion of canonical $W_{1,+}$-geodesic, see Theorem \[th:CanonicalGeodesic\], which will be used in Section \[sec:ProductSpaces\]. In Section \[sec:GeneralFormula\], we begin the study of the entropy function $H(t)$ along a $W_{1,+}$-geodesic on a graph; we use the Benamou-Brenier equation , which is at the heart of definition of $W_{1,+}$-geodesics, to obtain bounds on the second derivative $H''(t)$. The calculations done in this section are inspired by those done in the previous article [@HillionContraction] by the author; the results obtained are also linked with the more general theory of entropic interpolations, developed by Léonard in a recent series of articles, including [@LeoConvex], [@LeoLazy] and [@LeonardSurvey]. In Section \[sec:ProductSpaces\], we refine the calculations done in Section \[sec:GeneralFormula\] to prove a tensorization property. This property allows us to give bounds on the second derivative $H''(t)$ when the underlying graph is a product graph. Interesting examples are given by $\mathbb{Z}^n$, the cube $\{0,1\}^n$, or more generally by the Cayley graph of a finitely generated abelian group. In the Appendix, we present two additional results on families of probability measures on $\mathbb{Z}$. We first prove that, along a $W_{1,+}$-geodesic on $\mathbb{Z}$, other types of functionals are convex, belonging to the family of Renyi entropies functionals. The second part of the Appendix is devoted to another type of interpolation of probability measures on $\mathbb{Z}$, defined as a mixture of binomial distributions with respect to a $W_2$-optimal coupling. $W_{1,+}$-geodesics on graphs {#sec:Rappel} ============================= In this section, we first recall the main definitions and properties of [@HillionGeodesic]. The reader is referred to this paper for detailed proofs and additional explanations. We then introduce the new notion of canonical $W_{1,+}$-geodesic, which will be used in the study of product spaces in Section \[sec:ProductSpaces\]. Definition and construction --------------------------- Let $G$ be a locally finite, connected graph. We denote by $d$ the usual graph distance on $G$ and by $x \sim y$ the adjacency relation on $G$, meaning that $(x,y)$ is an edge of $G$. A curve of length $n$ on $G$ is an application $\gamma : \{0,\ldots n\} \rightarrow G$ satisfying $\gamma(i) \sim \gamma(i+1)$. A geodesic between two vertices $x$ and $y$ is a curve of minimal length joining $x$ to $y$. The set of geodesics between $x$ and $y$ is denoted by $\Gamma_{x,y}$ and its cardinality by $\|\Gamma_{x,y}\|$. The set of all geodesic curves of $G$ is denoted by $\Gamma(G)$. Let $f_0,f_1$ be two finitely supported probability distributions on $G$. We denote by $\Pi_1(f_0,f_1)$ the set of $W_1$-optimal couplings between $f_0$ and $f_1$, i.e. the set of couplings between $f_0$ and $f_1$ which minimize the functional $$I_1(\pi) := \sum_{x,y \in G} d(x,y) \pi(x,y).$$ Using properties of supports of optimal couplings, one can prove that the following definition in unambiguous: Let $f_0,f_1$ be two finitely supported probability measures on $G$. - The $W_1$-orientation on $G$ with respect to $f_0,f_1$ is constructed in the following way: a couple $(x,y)$ of adjacent vertices is oriented by $x \rightarrow y$ if there exists an optimal coupling $\pi \in \Pi_1(f_0,f_1)$ and a geodesic $\gamma \in \Gamma(G)$ of length $n$ such that $(\gamma(0),\gamma(n)) \in \operatorname{Supp}(\pi)$ and such that there exists $i \in \{0,\ldots n-1\}$ with $\gamma(i)=x$ and $\gamma(i+1)=y$. - Let $x_1 \in G$. The set $\mathcal{E}(x_1)$, resp. $\mathcal{F}(x_1)$, is the (possibly empty) set of vertices $x_0 \in G$, resp. $x_2 \in G$, such that $x_0 \rightarrow x_1$, resp. $x_1 \rightarrow x_2$. - An oriented path on $G$ is a mapping $\gamma : \{0,\ldots n\}$ with $\gamma(i) \rightarrow \gamma(i+1)$. - The $W_1$-orientation w.r.t. $f_0,f_1$ induces a partial order on the vertices of $G$: we denote $x \leq y$ if there exists an oriented path $x = \gamma(0) \rightarrow \cdots \rightarrow \gamma(n)=y$. One important property of this orientation is the fact that every oriented path is a geodesic: \[prop:OrientedGeodesic\] If we have $\gamma(0) \rightarrow \cdots \rightarrow \gamma(n)$ then $d(\gamma(0),\gamma(n)) = n$. A particular subset of geodesics on the oriented $G$ is given by extremal geodesics: Let $\gamma : \gamma(0) \rightarrow \cdots \rightarrow \gamma(n)$ be a geodesic on the oriented $G$. We say that $\gamma$ is an extremal geodesic, and we write $\gamma \in \operatorname{E\Gamma}$ if it cannot be extended in a longer geodesic, i.e. if the sets $\mathcal{E}(\gamma(0))$ and $\mathcal{F}(\gamma(n))$ are empty. The introduction of an orientation makes possible the introduction of a first-order calculus on $G$. We first define: The oriented edge graph $(E(G),\rightarrow)$ associated to $(G,\rightarrow)$ is defined as follows: its vertices are denoted by $(x_0x_1)$, where $x_0 \rightarrow x_1 \in G$ and its oriented edges join each couple $(x_0x_1) \rightarrow (y_0y_1)$ such that $x_1=y_0$. The oriented graph of oriented triples $(T(G),\rightarrow)$ is the graph $(E(E(G)),\rightarrow)$: its vertices are the triples $(x_0x_1x_2)$ with $x_0 \rightarrow x_1 \rightarrow x_2$ and its edges are defined between each couple $(x_0x_1x_2) \rightarrow (x_1x_2x_3)$. When the choice of the orientation on $G$ is unambiguous, we will often write $E(G), T(G)$ instead of $(E(G),\rightarrow), (T(G),\rightarrow)$. The divergence of a function $g : E(G) \rightarrow \mathbb{R}$ defined on the oriented edges of $G$ is the function $\nabla \cdot g : G \rightarrow \mathbb{R}$ defined by: $$\nabla \cdot g (x_1)= \sum_{x_2 \in \mathcal{F}(x_1)} g(x_1x_2) - \sum_{x_0 \in \mathcal{E}(x_1)} g(x_0x_1).$$ We define similarly the divergence $\nabla \cdot h : E(G) \rightarrow \mathbb{R}$ of a function $h : T(G) \rightarrow \mathbb{R}$ defined on the oriented triples of $G$. We denote $\nabla_2 \cdot h := \nabla \cdot (\nabla \cdot h)$. This first-order differential operator on the oriented graph allows us to introduce a discrete version of the formal optimality condition , on which is based the definition of $W_{1,+}$-geodesics: Let $G$ be a graph, $W_1$-oriented with respect to a couple of probability measures $f_0,f_1$. A family $(f_t)=(f_t)_{t \in [0,1]}$ is said to be a $W_{1,+}$-geodesic if: 1. The curve $(f_t)$ is a $W_1$-geodesic. 2. There exist two families $(g_t)$ and $(h_t)$ defined respectively on $E(G)$ and $T(G)$, such that: $$\frac{\partial}{\partial t} f_t = - \nabla \cdot g_t \ , \ \frac{\partial}{\partial t} g_t = - \nabla \cdot h_t.$$ 3. For every $(xy) \in E(G)$ we have $g_t(xy)>0$. 4. The triple $(f_t,g_t,h_t)$ satisfies the Benamou-Brenier equation $$\label{eq:BBcondition} \forall (x_0x_1x_2) \in T(G) \ , \ f_t(x_1)h_t(x_0x_1x_2) = g_t(x_0x_1) g_t(x_1x_2).$$ Let us fix a couple $f_0$, $f_1$ of probability measures on $G$ and endow $G$ with the $W_1$-orientation with respect to $f_0,f_1$. The existence of a $W_{1,+}$-interpolation $(f_t)$ joining $f_0$ to $f_1$ is the main result of [@HillionGeodesic]. Moreover, any such curve $(f_t)$ can be seen as a mixture of binomial families of distributions with respect to a coupling which is solution of a certain minimization problem. Canonical $W_{1,+}$-geodesics ----------------------------- In this paper we are mostly interested in particular $W_{1,+}$-geodesics, called canonical $W_{1,+}$-geodesics, which correspond to the case where $\forall \gamma \in \operatorname{E\Gamma}\ , \ C(\gamma)=1$, with the notations of [@HillionGeodesic]. The existence, uniqueness, and construction of such curves can be summed up by the following: \[th:CanonicalGeodesic\] Let $x_0 \leq \cdots \leq x_n \in G$ be an oriented $n+1$-uples of vertices of $G$. We define: $$m(x_0,\ldots, x_n) := \frac{\|\Gamma_{x_0,\cdots, x_n}\|}{\|\operatorname{E\Gamma}\|},$$ where $\Gamma_{x_0,\cdots, x_n}$ is the set of extremal geodesics visiting $x_0,\ldots, x_n$: $$\Gamma_{x_0,\cdots x_n} := \{\gamma \in \operatorname{E\Gamma}\ : \ \exists i_0 \leq \cdots \leq i_n \ ,\ \gamma(i_k)= x_k\}.$$ There exists a unique couple of families of functions $P_t(x),Q_t(x)$, defined for $x \in G$ and $t \in [0,1]$, such that each $t \mapsto P_t(x)$ and $t \mapsto Q_t(x)$ is positive and polynomial in $t$, and satisfying the following property: let us consider the families of functions $(f_t)$, $(g_t)$, $(h_t)$ respectively defined on $G$, $E(G)$ and $T(G)$ by $$\begin{aligned} f_t(x_0) &:=& m(x_0)P_t(x_0)Q_t(x_0), \\ g_t(x_0x_1) &:=& m(x_0,x_1) P_t(x_0) Q_t(x_1), \\ h_t(x_0x_1x_2) &:=& m(x_0,x_1,x_2) P_t(x_0)Q_t(x_2). \end{aligned}$$ Then the triple $(f_t,g_t,h_t)$ satisfies all the items of the definition of a $W_{1,+}$-geodesic. Such a curve will be called canonical $W_{1,+}$-geodesic joining $f_0$ to $f_1$. The reason why we introduce these particular geodesics comes from the following property, which will be used in Section \[sec:ProductSpaces\]: \[prop:CanonicalTriple\] If the triple $(f_t,g_t,h_t)$ defines a canonical $W_{1,+}$-geodesic, then for any oriented triple $(x_0x_1x_2) \in T(G)$, the quantity $h(x_0x_1x_2)$ does not depend on $x_1$, and therefore can be written $h(x_0x_2)$. Proof:* It suffices to show that the cardinality $\|\Gamma_{x_0,x_1,x_2}\|$ does not depend on $x_1$. This comes from the fact that every $\gamma \in \Gamma_{x_0,x_1,x_2}$ can be written $$\gamma \ : \ \gamma(0) \rightarrow \cdots \rightarrow \gamma(i)=x_0 \rightarrow x_1 \rightarrow x_2=\gamma(i+2) \rightarrow \cdots \rightarrow \gamma(n).$$ We thus have $\|\Gamma_{x_0,x_1,x_2}\| = A(x_0) B(x_2)$, where $A(x_0)$ is the number of oriented paths joining some $\gamma(0)$ such that $\mathcal{E}(\gamma(0))=\emptyset$ to $x_0$ and where $B(x_2)$ is defined similarly. $\square$ General bounds on $H''(t)$ {#sec:GeneralFormula} ========================== In this section, we adapt the method used in [@HillionContraction] to prove the convexity of the entropy along the contraction of a probability measure on $\mathbb{Z}$ to the more general framework of $W_{1,+}$-geodesics on a graph. We then apply this method in the cases where $G$ is the graph $\mathbb{Z}$ or a complete graph. We finally study the behaviour, along a $W_{1,+}$-geodesic, of the relative entropy with respect to a log-concave reference probability measure and discuss why the hypothesis of a uniform bound on the second derivative $H''(t)$ may not be by itself a sufficient condition for interseting functional inequalities to hold. Benamou-Brenier triples ----------------------- Let $G$ be a graph, endowed with the $W_1$-orientation with respect to a couple of probability distributions $f_0,f_1$ on $G$. A Benamou-Brenier triple, or BB-triple, on $(G,\rightarrow)$, is a triple of positive functions $f,g,h$ defined respectively on $G$, $E(G)$ and $T(G)$ such that $$\forall (x_0x_1x_2) \in T(G) \ , \ h(x_0x_1x_2)f(x_1) = g(x_0x_1)g(x_1x_2).$$ It is clear that, if a triple $(f_t,g_t,h_t)$ defines a $W_{1,+}$-geodesic, then for each $t \in [0,1]$, $(f_t,g_t,h_t)$ is a BB-triple. Other types of BB-triples will be considered in Section \[sec:ProductSpaces\]. The functional $\mathcal{I}$ is defined for every BB-triple on $(G,\rightarrow)$ by $$\label{eq:defI} \mathcal{I}(f,g,h) := \sum_{x \in G} \nabla_2 \cdot h(x) \log(f(x)) + \frac{(\nabla \cdot g(x))^2}{f(x)}.$$ Let us consider $(f_t,g_t,h_t)$ defining a $W_{1,+}$-geodesic on $G$. The entropy $H(t)$ of $f_t$ satisfies $$H''(t) = \mathcal{I}(f_t,g_t,h_t).$$ Proof: This simply comes from the definition of the families $(g_t)_{t \in [0,1]}$ and $(h_t)_{t \in [0,1]}$: $$\frac{\partial}{\partial t}f_t(x) = - \nabla \cdot g_t(x) \ , \ \frac{\partial^2}{\partial t^2} f_t(x) = \nabla_2 \cdot h_t(x). \ \square$$ Integration by parts on $G$ --------------------------- In order to obtain bounds on $H''(t)$, we first use integration by parts to transform the sum in : \[prop:Itelescopic\] For any BB-triple $(f,g,h)$ we have $$\begin{aligned} \mathcal{I}(f,g,h) &=& \sum_{x \in G} \left[ \sum_{x_1 \in \mathcal{F}(x)} \sum_{x_2 \in \mathcal{F}(x_1)} h(xx_1x_2) \log\left(\frac{f(x) h_t(x x_1 x_2)}{g(x x_1)^2}\right) \right] \\ && + \sum_{x \in G} \left[ \sum_{x_{-1} \in \mathcal{E}(x)} \sum_{x_{-2} \in \mathcal{E}(x_{-1})} h(x_{-2}x_{-1}x) \log\left(\frac{f(x) h_t(x_{-2}x_{-1}x)}{g(x_{-1}x)^2}\right) \right] \\ && + \sum_{x \in G} \frac{(\nabla \cdot g(x))^2}{f(x)}.\end{aligned}$$ Proof: We add to the sum defining $\mathcal{I}(f,g,h)$ (see equation ) the following telescopic sums $$0 = \sum_{x \in G} \nabla_2 \cdot (h \log(h))(x) \ , \ 0= -2 \sum_{x \in G} \nabla \cdot \left(g \nabla \cdot h\right)(x).$$ The proposition is then proven by noticing that, $(f,g,h)$ being a BB-triple, we have $$\forall x \in G \ , \ -2 \sum_{x_{-1} \in \mathcal{E}(x)} \sum_{x_1 \in \mathcal{F}(x)} h(x_{-1}xx_1) \log\left(\frac{h(x_{-1}xx_1) f(x)}{g(x_{-1}x)g(xx_1)} \right) =0. \ \square$$ Combining Proposition \[prop:Itelescopic\] with the elementary inequality $\log(x) \geq 1-1/x$ allows us to obtain bounds on $\mathcal{I}(f,g,h)$: \[prop:Iboundnaif\] For any triple $(f,g,h)$ we have $$\label{eq:Iboundnaif} \mathcal{I}(f,g,h) \geq \sum_{(x_0x_1) \in E(G)} \frac{g(x_0x_1)^2}{f(x_0)}\left(1-\left\|\mathcal{F}(x_1)\right\| \right) + \frac{g(x_0x_1)^2}{f(x_1)}\left(1-\left\|\mathcal{E}(x_0)\right\| \right).$$ Proof: The inequality $\log(x) \geq 1-1/x$ implies $$\begin{aligned} \mathcal{I}(f,g,h) &\geq& \sum_{x \in G} \left[ \sum_{x_1 \in \mathcal{F}(x)} \sum_{x_2 \in \mathcal{F}(x_1)} h(xx_1x_2) - \frac{g(x x_1)^2}{f(x)} \right] \\ && + \sum_{x \in G} \left[ \sum_{x_{-1} \in \mathcal{E}(x)} \sum_{x_{-2} \in \mathcal{E}(x_{-1})} h(x_{-2}x_{-1}x) - \frac{g(x_{-1}x)^2}{f(x)} \right] \\ && + \sum_{x \in G} \frac{(\nabla \cdot g(x))^2}{f(x)}.\end{aligned}$$ The following are obvious: $$\sum_{x_2 \in \mathcal{F}(x_1)} \frac{g(x x_1)^2}{f(x)} = \|\mathcal{F}(x_1)\| \frac{g(x x_1)^2}{f(x)} \ , \ \sum_{x_{-2} \in \mathcal{E}(x_{-1})} \frac{g(x_{-1}x)^2}{f(x)} = \|\mathcal{E}(x_{-1})\| \frac{g(x_{-1}x)^2}{f(x)}.$$ Moreover, we have: $$\begin{aligned} \sum_{x \in G} \sum_{x_1 \in \mathcal{F}(x)} \sum_{x_2 \in \mathcal{F}(x_1)} h(xx_1x_2) &=& \sum_{(x_0x_1x_2) \in T(G)} h(x_0x_1x_2) \\ &=& \sum_{x \in G} \sum_{x_{-1} \in \mathcal{E}(x)} \sum_{x_1 \in \mathcal{F}(x)} \frac{g(x_{-1}x)g(xx_1)}{f(x)},\end{aligned}$$ and similarly: $$\sum_{x \in G} \sum_{x_1 \in \mathcal{F}(x)} \sum_{x_2 \in \mathcal{F}(x_1)} h(xx_1x_2) = \sum_{x \in G} \sum_{x_{-1} \in \mathcal{E}(x)} \sum_{x_1 \in \mathcal{F}(x)} \frac{g(x_{-1}x)g(xx_1)}{f(x)}.$$ Expanding $\sum_{x \in G} \frac{(\nabla \cdot g(x))^2}{f(x)}$ allows us to find similar terms: $$\begin{aligned} \sum_{x \in G} \frac{(\nabla \cdot g(x))^2}{f(x)} &=& \sum_{x \in G} \frac{\left(\sum_{x_1 \in \mathcal{F}(x)} g(xx_1) - \sum_{x_{-1} \in \mathcal{E}(x)} g(x_{-1}x)\right)^2}{f(x)} \\ &\geq & \sum_{x \in G} \sum_{x_1 \in \mathcal{F}(x)} \frac{g(xx_1)^2}{f(x)}+\sum_{x \in G} \sum_{x_{-1} \in \mathcal{E}(x)} \frac{g(x_{-1}x)^2}{f(x)} \\ && -2 \sum_{x \in G} \sum_{x_{-1} \in \mathcal{E}(x)} \sum_{x_1 \in \mathcal{F}(x)} \frac{g(x_{-1}x)g(xx_1)}{f(x)}.\end{aligned}$$ We used the fact that $g$ is non-negative to apply the inequality $$\left(\sum_{x_1 \in \mathcal{F}(x)} g(xx_1) \right)^2 \geq \sum_{x_1 \in \mathcal{F}(x)} g(xx_1)^2,$$ which is far from being optimal, unless $\|\mathcal{F}(x)\|=0$ or $1$. Combining these estimations leads to $$\begin{aligned} \mathcal{I}(f,g,h) &\geq& \sum_{x \in G} \sum_{x_1 \in \mathcal{F}(x)} \left(1-\|\mathcal{F}(x_1)\|\right) \frac{g(x x_1)^2}{f(x)}\\&&+\sum_{x \in G} \sum_{x_{-1} \in \mathcal{E}(x)} \left(1-\|\mathcal{E}(x_{-1})\|\right)\frac{g(x_{-1}x)^2}{f(x)},\end{aligned}$$ which, up to a change of indices, is exactly inequality . $\square$ The bound obtained in Proposition \[prop:Iboundnaif\] is interesting in two fundamental cases: \[cor:IboundCube\] Let $(f,g,h)$ be a BB-triple of functions on $(G,\rightarrow)$ where $G$ is the complete graph with $n$ points, $W_1$-oriented with respect to some couple $(f_0,f_1)$. We have $$\mathcal{I}(f,g,h) \geq \sum_{(x_0x_1) \in E(G)} g(x_0x_1)^2 \left(\frac{1}{f(x_0)}+\frac{1}{f(x_1)} \right).$$ Proof: We apply Proposition \[prop:Iboundnaif\], using the fact that, if $(x_0x_1) \in E(G)$, then the sets $\mathcal{E}(x_0)$ and $\mathcal{F}(x_1)$ are empty, or the equivalent fact that the set of oriented triple $T(G)$ is empty: indeed if there exists $(x_0x_1x_2) \in T(G)$ then, by Proposition \[prop:OrientedGeodesic\], we have $d(x_0,x_2)=2$, which is a contradiction. $\square$ \[cor:IboundZ\] Let $(f,g,h)$ be a (finitely supported) BB-triple of functions on $(G,\rightarrow)$ where $G$ is the graph $\mathbb{Z}$, $W_1$-oriented with respect to some couple $(f_0,f_1)$. T $\mathcal{I}(f,g,h) \geq 0.$ Proof: We use this time the fact that each vertex of $\mathbb{Z}$ has two neighbours, which implies that, for every $x \in \mathbb{Z}$, $\|\mathcal{E}(x)\|+\|\mathcal{F}(x)\| \leq 2$. In particular, if $(x_0x_1) \in E(G)$, then $\mathcal{E}(x_1)$ is non-empty (as it contains $x_0$), so $\|\mathcal{F}(x_1)\| \leq 1$. Similarly we have $\|\mathcal{E}(x_0)\| \leq 1$. Applying Proposition \[prop:Iboundnaif\] leads to the result. $\square$ Remark. Corollary \[cor:IboundZ\] can be extended to the framework of cyclic graphs $\mathbb{Z}_r$ for $r \geq 2$, because in this case every vertex has also two neighbours. About the convexity of the relative entropy ------------------------------------------- We have been so far interested in the behaviour of the Shannon entropy functional $H(f) := \sum_{x \in G} f(x) \log(f(x))$ along $W_{1,+}$ geodesics on $G$. However, the functional which is considered in Sturm-Lott-Villani theory are the relative entropy $H_\nu$ with respect to some reference probability measure $\nu$. In this paragraph, we present some results about the behaviour of $H_\nu$ along $W_{1,+}$-geodesics on graphs. Let $\nu$ be a probability measure fully supported on $G$. The relative entropy $H_\nu(f)$ of a probability measure $f$ on $G$ is defined by $$H_\nu(f) := \sum_{x \in G} f(x) \log\left(\frac{f(x)}{\nu(x)}\right).$$ Remark. Let $(f_t)_{t \in [0,1]}$ be a $W_{1,+}$-geodesic supported on a finite subset of vertices $A \subset G$. Let $\nu$ be the uniform probability distribution on $A$. Then the Shannon and relative entropies are linked by $$H_\nu(f_t) = H(f_t) + \log(\|A\|)$$ so the convexity of $t \mapsto H_\nu(f_t)$ is equivalent to the convexity of $t \mapsto H(f_t)$. As in the Riemannian case, it is interesting to consider log-concave reference measures: We endow $G$ with a reference measure $\nu(x) := \exp(-V(x))$. We suppose that there exists $K>0$ such that, for every geodesic path of length $2$ $\gamma_0,\gamma_1,\gamma_2$ we have $$V(\gamma_0)-2 V(\gamma_1) + V(\gamma_2) \geq K.$$ Let $(f_t)$ be a $W_{1,+}$-geodesic, $H(t)$ be the Shannon entropy of $f_t$ and $H_\nu(t)$ its relative entropy. Then $$\label{eq:EntroRelative} H_\nu''(f_t) \geq H''(t) + K W^2(f_0,f_1),$$ where $$W^2(f_0,f_1) := \sum_{(x_0x_1x_2) \in T(G)} h_t(x_0x_1x_2)$$ does not depend on $t$. Proof: We have $$H_\nu(t) - H(t) = - \sum_{x \in G} f_t(x) \log(\nu(x)) = \sum_{x \in G} f_t(x) V(x),$$ and by differentiating twice with respect to $t$ we have $$H_\nu''(t) = H''(t) + \sum_{x \in G} (\nabla_2 \cdot h_t)(x) V(x) = \sum_{(x_0x_1x_2) \in T(G)} h_t(x) (V(x_2)-2V(x_1)+V(x_0)),$$ which, by the convexity assumption made on $V$, proves equation . In order to prove that $\sum_{(x_0x_1x_2) \in T(G)} h_t(x_0x_1x_2)$ does not depend on $t$, we first use the Benamou-Brenier condition to write $$\frac{\partial}{\partial t} h_t(x_0x_1x_2) = - \sum_{x_{-1} \in \mathcal{E}(x_0)} \frac{g_t(x_{-1}x_0) g_t(x_0x_1) g_t(x_1x_2)}{f_t(x_0)f_t(x_1)}+ \sum_{x_3 \in \mathcal{F}(x_2)} \frac{g_t(x_0x_1)g_t(x_1x_2)g_t(x_2x_3)}{f_t(x_1)f_t(x_2)}.$$ A simple change of indices then show that $$\frac{\partial}{\partial t} \sum_{(x_0x_1x_2) \in T(G)} h_t(x_0x_1x_2) = \sum_{x_0 \rightarrow \cdots \rightarrow x_3 \in G} \frac{g_t(x_0x_1)g_t(x_1x_2)g_t(x_2x_3)}{f_t(x_1)f_t(x_2)}-\frac{g_t(x_0x_1)g_t(x_1x_2)g_t(x_2x_3)}{f_t(x_1)f_t(x_2)}=0,$$ so $\sum_{(x_0x_1x_2) \in T(G)} h_t(x_0x_1x_2)$ does not depend on $t$. $\square$ Remark. One major difference with the continuous case is the fact that, although acting as the Wasserstein distance $W_2$, the quantity $W(f_0,f_1)$ does not define a distance on $\mathcal{P}(G)$. For instance, if $f_0$ and $f_1$ are two Dirac distributions at two adjacent vertices, we have $W(f_0,f_1)=0$. A different perspective on the same issue consists in writing $$W^2(f_0,f_1) = \sum_{x_1 \in G} f_t(x_1) V_{+,t}(x_1) V_{-,t}(x_1),$$ where $V_{+,t}(x_1) := \sum_{x_2 \in \mathcal{F}(x_1)} \frac{g_t(x_1x_2)}{f_t(x_1)}$ and $V_{-,t}(x_1) := \sum_{x_0 \in \mathcal{E}(x_1)} \frac{g_t(x_0x_1)}{f_t(x_1)}$ are the two velocity functions, which can be written $W^2 = \langle V_{+,t} , V_{-,t}\rangle$ for the scalar product with respect to $f_t$. This formula is the discrete analogue of the Benamou-Brenier formula for $p=2$, but in the continuous setting we have $W_2^2 = <v_t,v_t>= \|\|v_t\|\|^2$ for the scalar product with respect to $f_t$. The fact that $V_{+,t} \neq V_{-,t}$ is a major obstacle to a generalization of the HWI inequality which holds for instance in the measured length space $(\mathbb{R}^d,\exp(-V(x))dx)$ (see [@LottVillani] for a proof of this fact). Product of graphs {#sec:ProductSpaces} ================= Let $G_1$ and $G_2$ be two locally finite and connected graphs. In this section we study the behaviour of the entropy along $W_{1,+}$-geodesics defined on the product graph $G := G_1 \times G_2$ endowed with the usual product metric $$d_G((x_1,x_2),(y_1,y_2)) := d_{G_1}(x_1,y_1)+d_{G_2}(x_2,y_2).$$ The $W_1$-orientation on a product graph ---------------------------------------- The neighbours of a vertex $(x_1,x_2)$ in $G$ are the vertices $(x_1,y_2)$, where $d_{G_2}(x_2,y_2)=1$ and $(y_1,x_2)$ where $d_{G_1}(x_1,y_1)=1$. From this fact we easily deduce the following description of geodesic curves in $G$: \[prop:GeodProduct\] Let $\gamma \in \Gamma(x,y)$ be a geodesic on $G$, where $(x,y)=((x_1,x_2),(y_1,y_2))$. There exist two geodesics $\gamma_1 \in \Gamma(x_1,y_1)$, $\gamma_2 \in \Gamma(x_2,y_2)$ defined respectively on $G_1$ and $G_2$, and an application $$\phi : \{0,\ldots, d(x,y)\} \rightarrow \{0,\ldots, d_1(x_1,y_1)\}$$ with $\phi(0)=0$, $\phi(d(x,y))=d_1(x_1,y_1)$ and $\phi(k+1)-\phi(k) \in \{0,1\}$, such that $$\gamma(k) = (\gamma_1(\phi(k)),\gamma_2(k-\phi(k))).$$ In particular, the cardinality of $\Gamma(x,y)$ satisfies $$\|\Gamma(x,y)\| = \binom{d(x,y)}{d(x_1,y_1)}\|\Gamma(x_1,y_1)\|\|\Gamma(x_2,y_2)\|.$$ If $f$ is a probability distribution on $G$, we denote by $f^{(1)}$, $f^{(2)}$ its marginals on $G_1$ and $G_2$. To a coupling $\pi$ between two distributions $f_0,f_1$, which can be seen as a probability measure on $$G \times G = (G_1 \times G_2) \times (G_1 \times G_2) = (G_1 \times G_1) \times (G_2 \times G_2),$$ we associate the marginal couplings $\pi^{(1)}$ on $G_1 \times G_1$ between $f_0^{(1)}$ and $f_1^{(1)}$ and $\pi_2$ on $G_2 \times G_2$ between $f_0^{(2)}$ and $f_1^{(2)}$. We then describe the $W_{1,+}$-orientation on $G$ with respect to a couple of measures $f_0,f_1$. \[prop:W1OrientProduct\] Let $f_0,f_1 \in \mathcal{P}(G)$. For $i=1,2$ we define $$\mathcal{E}_i(x^{(i)}) := \{y^{(i)} \in G_i \ : \ y^{(i)} \rightarrow x^{(i)} \} \ , \ \mathcal{F}_i(x^{(i)}) := \{z^{(i)} \in G_i \ : \ x^{(i)} \rightarrow z^{(i)} \}$$ for the $W_1$ orientation on $G_i$ between $f_0^{(i)}$ and $f_1^{(i)}$. The $W_1$-orientation between $f_0$ and $f_1$ is then described by $$\begin{aligned} \mathcal{E}(x) &=& \left( \bigcup_{y^{(2)} \in \mathcal{E}_2(x^{(2)})} (x^{(1)},y^{(2)}) \right) \bigcup \left( \bigcup_{y^{(1)} \in \mathcal{E}_2(x^{(1)})} (y^{(1)},x^{(2)}) \right)\\ &=:& \mathcal{E}_1(x) \cup \mathcal{E}_2(x), \\ \mathcal{F}(x) &=& \left( \bigcup_{y^{(2)} \in \mathcal{F}_2(x^{(2)})} (x^{(1)},y^{(2)}) \right) \bigcup \left( \bigcup_{y^{(1)} \in \mathcal{F}_2(x^{(1)})} (y^{(1)},x^{(2)}) \right)\\ &=:& \mathcal{F}_1(x) \cup \mathcal{F}_2(x).\end{aligned}$$ Proof: Let $\pi \in \Pi(f_0,f_1)$ be a coupling between $f_0$ and $f_1$. We have $$\begin{aligned} I_1(\pi) &=& \sum_{(x_1,x_2),(y_1,y_2) \in G \times G} d((x_1,x_2),(y_1,y_2)) \pi((x_1,x_2),(y_1,y_2)) \\ &=& \sum_{(x_1,y_1) \in G_1 \times G_1} \sum_{(x_2,y_2) \in G_2\times G_2} d_1(x_1,y_1)+d_2(x_2,y_2) \pi((x_1,x_2),(y_1,y_2)) \\ &=& \sum_{(x_1,y_1) \in G_1 \times G_1} d_1(x_1,y_1) \pi^{(1)}(x_1,y_1) + \sum_{(x_2,y_2) \in G_2\times G_2}d_2(x_2,y_2) \pi^ {(2)}(x_2,y_2) \\ &=& I_1(\pi^{(1)})+ I_1(\pi^{(2)}),\end{aligned}$$ which proves that $\pi$ is $W_1$-optimal between $f_0$ and $f_1$ (for the distance $d_G$) if and only if its marginals $\pi^{(1)}$, $\pi^{(2)}$ are $W_1$-optimal between $f_0^{(1)}$ and $f_1^{(1)}$, resp $f_0^{(2)}$ and $f_1^{(2)}$ for the distance $d_{G_1}$, resp. $d_{G_2}$. We now fix a $W_1$-optimal coupling $\pi \in \Pi_1(f_0,f_1)$. Let $x=(x_1,x_2)$ and $y=(y_1,y_2)$ be two vertices of $G$ such that $\pi(x,y)>0$. We then have $\pi^{(1)}(x_1,y_1)>0$ and $\pi^{(2)}(x_2,y_2)>0$ for the marginal couplings, which are also $W_1$-optimal. Let $\gamma \in \Gamma_G(x,y)$, and $\gamma_1 \in \Gamma_{G_1}(x_1,y_1)$, $\gamma_2 \in \Gamma_{G_2}(x_2,y_2)$, $\phi : \{0,\ldots, d(x,y)\} \rightarrow \{0,\ldots, d_1(x_1,y_1)\}$ be associated to $\gamma$ by Proposition \[prop:GeodProduct\]. For $k \in \{0,\ldots, d(x,y)-1\}$, we have $$\gamma(k) = (\gamma_1(\phi(k)),\gamma_2(k-\phi(k))) \ , \ \gamma(k+1) = (\gamma_1(\phi(k)),\gamma_2(k-\phi(k+1)+1)).$$ If $\phi(k+1)=\phi(k)+1$, resp. $\phi(k+1)=\phi(k)$, then $\gamma(k+1) \in \mathcal{E}_1(\gamma(k))$, resp. $\gamma(k+1) \in \mathcal{E}_2(\gamma(k))$. Conversely, let us consider a vertex $x=(x_1,x_2) \in G$. We suppose that $x=\gamma(k)$ for some geodesic $\gamma$ of length $n$ such that $\pi(\gamma(0),\gamma(n))>0$ for a $W_1$-optimal coupling $\pi \in \Pi_1(f_0,f_1)$. We denote by $\gamma^{(1)}, \gamma^{(2)}$ the projections of $\gamma$, as defined in Proposition \[prop:GeodProduct\], $n_1$ and $n_2$ their respective lengths, and $\pi^{(1)}$, $\pi^{(2)}$ the marginals of $\pi$. Let $y \in \mathcal{E}_1(x)$. We have $y=(y_1,x_2)$ with $y_1 \in \mathcal{E}_1(x_1)$. There exists a $W_1$-optimal coupling $\tilde{\pi}^{(1)} \in \Pi_1(f_0,f_1)$ and a geodesic $\tilde{\gamma}_1$ on $G_1$, of length $\tilde{n}_1$, such that $\tilde{\gamma}_1(k_1)=x_1$, $\tilde{\gamma}_1(k_1+1)=y_1$ and $\tilde{\pi}^{(1)}(\tilde{\gamma}_1(0),\tilde{\gamma}_1(\tilde{n}_1)) > 0$. Let $\tilde{\pi}$ be any coupling between $f_0$ and $f_1$ having $\tilde{\pi}^{(1)}$ and $\pi^{(2)}$ as marginals and $\gamma$ be a geodesic of $G$ having $\tilde{\gamma_1}$ and $\gamma_2$ as projections. Then there exists some $k$ for which $\tilde{\gamma}(k)=x$, $\tilde{\gamma}(k+1)=y$. Furthermore $\tilde{\pi}$ is $W_1$-optimal between $f_0$ and $f_1$ and $\tilde{\pi}(\gamma(0),\gamma(\tilde{n}_1+n_2)) > 0$, which proves that $y \in \mathcal{E}(x)$. We can prove similarly that, if $y \in \mathcal{E}_2(x)$ then $y \in \mathcal{E}(x)$, which finishes the proof. $\square$ An immediate consequence of Proposition \[prop:W1OrientProduct\] is a decomposition of the divergence operator: \[prop:divdecomp\] The divergence $\nabla \cdot g$ of a function $g : E(G) \rightarrow \mathbb{R}$ can be written $\nabla \cdot g= \nabla^{(1)} \cdot g+\nabla^{(2)} \cdot g$ where $$\nabla^{(i)} \cdot g(x_1) := \sum_{x_2 \in \mathcal{F}_i(x_1)} g(x_1 x_2) - \sum_{x_0 \in \mathcal{E}_i(x_1)} g(x_0x_1).$$ Similarly, the second order divergence operator of a function $h : T(G) \rightarrow \mathbb{R}$ can be written $$\nabla_2 \cdot h = \nabla_2^{(11)} \cdot h +\nabla_2^{(12)} \cdot h+\nabla_2^{(21)} \cdot h+\nabla_2^{(22)} \cdot h,$$ with $\nabla_2^{(ij)} := \nabla^{(i)} \circ \nabla^{(j)}$. The structure of the oriented graph $(G_1 \times G_2)$ is better understood by introducing oriented product squares: An oriented product square of $G$ is a 4-uple of vertices $(x_0,x_1,x_1',x_2) \in G^4$ such that $x_1 \in \mathcal{F}_1(x_0)$, $x_1' \in \mathcal{F}_2(x_0)$, $x_2 \in \mathcal{F}_2(x_1)$ and $x_2 \in \mathcal{F}_1(x_1')$. We denote by $S(G)$ the set of oriented product squares of $G$. \[prop:OrientedSquares\] Let $x_0 \in G$. The following sets are all in bijection: - $\mathcal{A}_1$:= $\mathcal{F}_1(x_0) \times \mathcal{F}_2(x_0)$. - $\mathcal{A}_2$:= $\{ x_2 \in G \ : \ \exists x_1,x_1' \in G\times G \ , \ (x_0,x_1,x_1',x_2) \in S(G) \}$. - $\mathcal{A}_3$:= $\{ (x_0x_1x_2) \in T(G) \ : x_1 \in \mathcal{F}_1(x_0) , x_2 \in \mathcal{F}_2(x_1) \ \}$. - $\mathcal{A}_4$:= $\{ (x_0x_1'x_2) \in T(G) \ : x_1' \in \mathcal{F}_2(x_0) , x_2 \in \mathcal{F}_1(x_1') \ \}$. Proof: Let us fix $x_1 \in \mathcal{F}_1(x_0)$ and $x_1' \in \mathcal{F}_2(x_0)$. We write $x_0=(x_0^{(1)},x_0^{(2)})$ in $G_1 \times G_2$. There exist a unique $x_1^{(1)} \in \mathcal{F}_{G_1}(x_0^{(1)})$ and a unique $x_1'^{(2)} \in \mathcal{F}_{G_{2}}(x_0^{(2)})$ such that $x_1 = (x_1^{(1)},x_0^{(2)})$ and $x_1'=(x_0^{(1)},x_1'^{(2)})$ in $G_1 \times G_2$. We then set $x_2 := (x_1^{(1)},x_1'^{(2)})$ and it is easy to see that $(x_0,x_1,x_1',x_2) \in S(G)$. $\square$ Proposition \[prop:OrientedSquares\] shows that an oriented square $(x_0x_1x_1'x_2)$ is uniquely determined by the couple $x_0,x_2$. We will use the notation $(x_0x_2) \in S(G)$ to denote such squares. We will also denote the two midpoints $x_1,x_1'$ respectively by $m_1(x_0x_2)$ and $m_2(x_0x_2)$. Let $(f_t)$ be a $W_{1,+}$-geodesic on $G$. There exist two families of functions $(g_t)$ and $(h_t)$, defined respectively on $E(G)$ and $T(G)$, such that $\frac{\partial}{\partial t}f = - \nabla \cdot g$, $\frac{\partial}{\partial t}g = - \nabla \cdot h$ and satisfying $$\forall (x_0x_1x_2) \in T(G) \ , \ f_t(x_1)h_t(x_0x_1x_2) = g_t(x_0x_1)g_t(x_1x_2).$$ Given a vertex $x^{(2)}$, we now define, for $(x_0^{(1)}x_1^{(1)}x_2^{(1)}) \in T(G_1)$, the functions $$\begin{aligned} \label{eq:BBprojection} f_{t,x^{(2)}}(x_0^{(1)}) &:=& f_t(x_0^{(1)},x^{(2)}), \\ g_{t,x^{(2)}}(x_0^{(1)}x_1^{(1)}) &:=& g_t((x_0^{(1)},x^{(2)})(x_1^{(1)},x^{(2)})), \\ h_{t,x^{(2)}}(x_0^{(1)}x_1^{(1)}x_2^{(1)}) &:= & h_t((x_0^{(1)},x^{(2)})(x_1^{(1)},x^{(2)})(x_2^{(1)},x^{(2)})).\end{aligned}$$ The triple of functions $(f_{t,x^{(2)}},g_{t,x^{(2)}},h_{t,x^{(2)}})$ is then a BB-triple on $G_1$. Given $x^{(1)} \in G_1$, we define similarly the BB-triples of functions $(f_{t,x^{(1)}},g_{t,x_{(1)}},h_{t,x_{(1)}})$ on $G_2$. The divergence of $g_{t,x^{(2)}} : E(G_1) \rightarrow \mathbb{R}$ satisfies the relation $$(\nabla \cdot g_{t,x^{(2)}})(x^{(1)}) = (\nabla^{(1)} \cdot g_t)(x^{(1)},x^{(2)}).$$ The second order divergence $h_{t,x^{(2)}} : T(G_1) \rightarrow \mathbb{R}$ satisfies $$(\nabla_2 \cdot h_{t,x^{(2)}})(x^{(1)}) = (\nabla^{(11)} \cdot h_t)(x^{(1)},x^{(2)}).$$ A tensorization result ---------------------- We are now able to state the tensorization theorem: \[th:EntroTensorization\] Let $(f_t,g_t,h_t)$ be a canonical $W_{1,+}$-geodesic on $G$ and $H(t)$ denote the entropy of $f_t$. Then: $$H''(t) \geq \sum_{x^{(2)} \in G_2} \mathcal{I}(f_{t,x^{(2)}},g_{t,x^{(2)}},h_{t,x^{(2)}}) + \sum_{x^{(1)} \in G_1} \mathcal{I}(f_{t,x^{(1)}},g_{t,x^{(1)}},h_{t,x^{(1)}}).$$ Proof: We apply Proposition \[prop:divdecomp\]: $$\begin{aligned} \sum_{x^{(2)} \in G_2} \mathcal{I}(f_{t,x^{(2)}},g_{t,x^{(2)}},h_{t,x^{(2)}}) &=& \sum_{x^{(2)} \in G_2} \left( \sum_{x^{(1)} \in G_1} \nabla_2 \cdot h_{t,x^{(2)}}(x^{(1)}) \log(f_{t,x^{(2)}}(x^{(1)})) \right) \\ && + \sum_{x^{(2)} \in G_2} \left( \sum_{x^{(1)} \in G_1} \frac{(\nabla \cdot g_{t,x^{(2)}}(x^{(1)}))^2}{f_{t,x^{(2)}}(x^{(1)})} \right) \\ &=& \sum_{x \in G} \nabla_2^{(11)} \cdot h_t(x) \log(f_t(x)) + \frac{(\nabla^{(1)} \cdot g_t(x))^2}{f_t(x)}.\end{aligned}$$ Similarly, $$\sum_{x^{(1)} \in G_1} \mathcal{I}(f_{t,x^{(1)}},g_{t,x^{(1)}},h_{t,x^{(1)}}) = \sum_{x \in G} \nabla_2^{(22)} \cdot h_t(x) \log(f_t(x)) + \frac{(\nabla^{(2)} \cdot g_t(x))^2}{f_t(x)}.$$ To prove Theorem \[th:EntroTensorization\], it thus suffices to show the inequality $$\label{eq:CrossTerms} \sum_{x \in G} (\nabla_2^{(12)}+\nabla_2^{(21)}) \cdot h_t(x) \log(f_t(x)) + 2 \frac{\nabla^{(1)} g_t(x) \nabla^{(2)} \cdot g_t(x)}{f_t(x)} \geq 0.$$ By considering the telescopic sums $$\sum_{x \in G} \nabla^{(12)} \cdot h_t \log(h_t)(x)=0 \ , \ -2\sum_{x\in G} \nabla^{(1)} \cdot (\nabla^{(2)} \cdot h_t \log(g_t))=0,$$ we prove, as in Proposition \[prop:Itelescopic\], that $$\begin{aligned} \sum_{x \in G} \nabla_2^{(12)} \cdot h_t(x) \log(f_t(x)) &=& \sum_{(x_0x_1x_2) \in T^{(12)}(G)} h(x_0x_1x_2) \log\left(\frac{f_t(x_0)h_t(x_0x_1x_2)}{g_t(x_0x_1)^2}\right)\\&&+\sum_{(x_0x_1x_2) \in T^{(12)}(G)} h(x_0x_1x_2)\log\left(\frac{f_t(x_2)h_t(x_0x_1x_2)}{g_t(x_1x_2)^2}\right),\end{aligned}$$ where $T^{(12)}(G)$ is the set of oriented triples $(x_0x_1x_2) \in T(G)$ such that $x_0 \in \mathcal{E}_1(x_1)$ and $x_1 \in \mathcal{E}_2(x_2)$. We now use the bijection between $T^{(12)}(G)$ and $S(G)$, proven in Proposition \[prop:OrientedSquares\], and the fact that $h(x_0x_1x_2)$ does not depend on $x_1$, which comes from the assumption that $(f_t)$ is canonical and from Proposition \[prop:CanonicalTriple\], to write: $$\sum_{x \in G} \nabla_2^{(12)} \cdot h_t(x) \log(f_t(x)) = \sum_{(x_0x_2) \in S(G)} h(x_0x_2) \left( \log \left(\frac{f(x_0)h(x_0x_2)}{g(x_0m_1(x_0,x_2))^2 } \right)+\log \left(\frac{f(x_2)h(x_0x_2)}{g(m_1(x_0,x_2)x_2)^2 } \right)\right).$$ Similarly, we have: $$\sum_{x \in G} \nabla_2^{(21)} \cdot h_t(x) \log(f_t(x)) = \sum_{(x_0x_2) \in S(G)} h(x_0x_2) \left( \log \left(\frac{f(x_0)h(x_0x_2)}{g(x_0m_2(x_0,x_2))^2 } \right)+\log \left(\frac{f(x_2)h(x_0x_2)}{g(m_2(x_0,x_2)x_2)^2 } \right)\right).$$ Adding both equations and using the inequality $\log(x) \geq 1-1/x$ gives: $$\begin{aligned} \sum_{x \in G} (\nabla_2^{(12)}+\nabla_2^{(21)}) \cdot h_t(x) \log(f_t(x)) &=& 2 \sum_{(x_0x_2) \in S(G)} h(x_0x_2) \log \left(\frac{f(x_0)h(x_0x_2)}{g(x_0m_1(x_0,x_2))g(x_0m_2(x_0,x_2)) } \right) \\&&+ 2 \sum_{(x_0x_2) \in S(G)} h(x_0x_2) \log \left(\frac{f(x_2)h(x_0x_2)}{g(m_1(x_0,x_2)x_2)g(m_2(x_0,x_2)x_2) } \right) \\ &\geq& 4 \sum_{(x_0x_2) \in S(G)} h(x_0x_2) \\ && - 2 \sum_{(x_0x_2) \in S(G)} \frac{g(x_0m_1(x_0,x_2))g(x_0m_2(x_0,x_2))}{f_t(x_0)} \\ && - 2 \sum_{(x_0x_2) \in S(G)} \frac{g(m_1(x_0,x_2)x_2)g(m_2(x_0,x_2)x_2)}{f_t(x_2)}.\end{aligned}$$ We use again the bijection in Proposition \[prop:OrientedSquares\] to write $$\begin{aligned} \sum_{(x_0x_2) \in S(G)} \frac{g(x_0m_1(x_0,x_2))g(x_0m_2(x_0,x_2)}{f_t(x_0)} &=& \sum_{x_0 \in G} \left( \sum_{(x_1,x_1') \in \mathcal{F}_1(x_0) \times \mathcal{F}_2(x_0)} \frac{g(x_0x_1)g(x_0x_1')}{f(x_0)} \right)\\ &=& \sum_{x_0 \in G} \frac{\sum_{x_1 \in \mathcal{F}_1(x_0)} g(x_0x_1) \cdot \sum_{x_1' \in \mathcal{F}_2(x_0)} g(x_0x_1')}{f(x_0)} \end{aligned}$$ and $$\begin{aligned} \sum_{(x_0x_2) \in S(G)} \frac{g(m_1(x_0,x_2)x_2)g(m_2(x_0,x_2)x_2)}{f_t(x_2)} = \sum_{x_0 \in G} \frac{\sum_{x_{-1} \in \mathcal{E}_1(x_0)} g(x_{-1}x_0) \cdot \sum_{x_{-1}' \in \mathcal{E}_2(x_0)} g(x_{-1}'x_0)}{f(x_0)}. \end{aligned}$$ We also have: $$\begin{aligned} \sum_{(x_0x_2) \in S(G)} h(x_0x_2) &=& \sum_{(x_{-1}x_0x_1) \in T^{(12)}(G)} h(x_{-1}x_0x_1) \\ &=& \sum_{x_0 \in G} \sum_{x_{-1} \in \mathcal{E}_1(x_0) \ , \ x_1 \in \mathcal{F}_2(x_0)} \frac{g(x_{-1}x_0) g(x_0x_1)}{f(x_0)} \\ &=& \sum_{x_0 \in G} \frac{\sum_{x_{-1} \in \mathcal{E}_1(x_0)} g(x_{-1}x_0) \cdot \sum_{x_{1} \in \mathcal{F}_2(x_0)} g(x_{1}x_0)}{f(x_0)},\end{aligned}$$ and: $$\begin{aligned} \sum_{(x_0x_2) \in S(G)} h(x_0x_2) &=& \sum_{(x_{-1}x_0x_1) \in T^{(21)}(G)} h(x_{-1}x_0x_1) \\ &=& \sum_{x_0 \in G} \frac{\sum_{x_{-1} \in \mathcal{E}_2(x_0)} g(x_{-1}x_0) \cdot \sum_{x_{1} \in \mathcal{F}_1(x_0)} g(x_{1}x_0)}{f(x_0)}.\end{aligned}$$ Adding the last four identities gives: $$\begin{aligned} \sum_{x \in G} (\nabla_2^{(12)}+\nabla_2^{(21)}) \cdot h_t(x) \log(f_t(x)) & \geq & -\sum_{x_0 \in G} \frac{\nabla^{(1)} \cdot g(x_0) \nabla^{(2)} \cdot g(x_0)}{f(x_0)},\end{aligned}$$ which is exactly the inequality we wanted to obtain. $\square$ Examples -------- The tensorization Theorem \[th:EntroTensorization\] is generalized to products of more than two graphs: let $G = G_1 \times \cdots \times G_p$. For $i = 1,\ldots,p$, we denote by $\hat{G}_i$ the product $G_1 \times \cdots G_p$, where $G_i$ is omitted. Given some vertex $\hat{x} \in \hat{G}_i$ and a BB-triple $(f_t,g_t,h_t)$ on $G$, we define a BB-triple $(f_{t,\hat{x}} ,g_{t,\hat{x}},h_{t,\hat{x}})$ as in equation . We then have: \[cor:EntroTensoMultiple\] Let $(f_t,g_t,h_t)$ be a BB-triple on $G$. The entropy $H(t)$ of $f_t$ satisfies $$H''(t) \geq \sum_{i=1}^p \sum_{\hat{x} \in \hat{G}_i} \mathcal{I}(f_{t,\hat{x}} ,g_{t,\hat{x}},h_{t,\hat{x}}).$$ Applying Corollary \[cor:EntroTensoMultiple\] to the examples studied in Section 3 allows us to obtain interesting bounds on the second derivative $H''(t)$ in other important cases: \[prop:EntroTensoExemples\] Theorem \[th:EntroTensorization\] can be applied in the following fundamental examples: - The entropy $H(t)$ along a $W_{1,+}$-geodesic $(f_t)_{t \in [0,1]}$ on $\mathbb{Z}^n$ is a convex function of $t$. - Let $(f_t,g_t,h_t)$ be a $W_{1,+}$-geodesic on the cube $\mathbb{Z}_2^n$. Then $$\mathcal{I}(f_t,g_t,h_t) = \sum_{(x_0x_1) \in E(G)} g_t(x_0x_1)^2 \left(\frac{1}{f_t(x_0)} + \frac{1}{f_t(x_1)} \right) \geq 0.$$ Proof: The first point follows directly from Corollary \[cor:IboundZ\]. To prove the second point, we notice that the cube is described by the product $G_1 \times \cdots \times G_n$ where each $G_i$ is the two-point graph $\mathbb{Z}_2$. Each $\hat{G}_i$ is isometric to the $n-1$-dimensional cube. To each $\hat{x} \in \hat{G}_i$, we associate two vertices $\hat{x}_0,\hat{x}_1 \in G$ by setting the $i$-th coordiante to $0$ or $1$. If $\hat{x}_0 \rightarrow \hat{x}_1$ in $G$, we define $g_t(\hat{x}) := g_t(\hat{x}_0 \hat{x}_1)$. If $\hat{x}_1 \rightarrow \hat{x}_0$ we define $g_t(\hat{x}) := g_t(\hat{x}_1 \hat{x}_0)$. Finally if the edge $(\hat{x}_0 \hat{x}_1)$ is not oriented in $G$ we set $g_t(\hat{x}):=0$. In any case, we have, by Corollary \[cor:IboundCube\], $$\mathcal{I}(f_{t,\hat{x}} ,g_{t,\hat{x}},h_{t,\hat{x}}) = g_t(\hat{x})^2 \left(\frac{1}{f_t(\hat{x}_0)}+\frac{1}{f_t(\hat{x}_1)} \right).$$ A (non-ordered) edge $(x_0 x_1)$ of $G$ is described in the following way: $x_0$ and $x_1$ differ by exactly one coordinate. In other terms, there is a bijection between the set of edges of $G$ and the disjoint union $\bigcup_{i=1}^p \hat{G}_i$. We can then write $$\sum_{i=1}^n \sum_{\hat{x} \in \hat{G}_i} \mathcal{I}(f_{t,\hat{x}} ,g_{t,\hat{x}},h_{t,\hat{x}}) = \sum_{(x_0x_1) \in (E(G),\rightarrow)} g_t(x_0x_1)^2 \left(\frac{1}{f_t(x_0)}+\frac{1}{f(x_1)} \right),$$ which is what we wanted. $\square$ These two examples can be seen as particular cases of a more general theorem: \[th:GroupCurvature\] Let $G$ be the Cayley graph of a finitely generated abelian group, with a set of generators $T=(\tau_1,\ldots, \tau_q)$. Let $(f_t)$ be a $W_{1,+}$-interpolation on $G$ and $H(t)$ the entropy of $f_t$. Then : $$\label{eq:IboundGroup} H''(t) \geq \sum_{(x_0 x_1) \in \tilde{E}(G)} g_t(x_0x_1)^2 \left(\frac{1}{f_t(x_0)}+\frac{1}{f(x_1)} \right),$$ where $\tilde{E}(G)$ is the subset of oriented edges $(x_0 \rightarrow x_1) \in E(G)$ such that $x_1 = \tau_i x_0$ for some generator $\tau_i \in T$ such that $\tau_i^2 = id$. Proof: Theorem \[th:GroupCurvature\] can be proven with the help of Theorem \[th:EntroTensorization\]. Indeed, any finitely generated abelian group is isomorphic to the direct product $$\mathbb{Z}^n \times \mathbb{Z}_2^{n_2} \times \cdots \times \mathbb{Z}_p^{n_p} \times \cdots,$$ where all but a finite number of coefficients $n_p$ are equal to $0$. As we have proven that $\mathcal{I}(f,g,h) \geq 0$ for any BB-triple on $\mathbb{Z}_p$ or on $\mathbb{Z}$, a direct application of Theorem \[th:EntroTensorization\] gives that $H''(t) \geq 0$. The more precise bound given in equation is proven as in the second point of Proposition \[prop:EntroTensoExemples\]. $\square$ Appendix: further results on $W_{1,+}$-geodesics on $\mathbb{Z}$. ================================================================= Renyi entropy along $W_{1,+}$-geodesics on $\mathbb{Z}$. -------------------------------------------------------- In this appendix we prove that, along a $W_{1,+}$-geodesic on $\mathbb{Z}$, not only the relative entropy is convex, but also a larger class of functionals belonging to the family of Renyi entropies: given a probability distribution $(f(k))_{k \in \mathbb{Z}}$ and a parameter $0<p<1$, we set $$H_p(f) := -\sum_{k \in \mathbb{Z}} f(k)^p .$$ The relative entropy $H(f) := \sum_k f(k) \log(f(k))$ can be seen as a limit case of Renyi entropy as the parameter $p \rightarrow 1$, in the sense that $$H_p(f) = -1 + (1-p) H(f) + o((1-p)^2).$$ We then have: \[th:RenyiZ\] Let $(f_t)_{t \in [0,1]}$ be a $W_{1,+}$ geodesic on $\mathbb{Z}$. Then $t \mapsto H_p(f_t)$ is convex. In order to have simpler notations, we are going to prove Theorem \[th:RenyiZ\] under the additional assumption that $f_0$ is stochastically dominated by $f_1$ (see also Theorem \[th:EntroBinoW2\]). Under this assumption, the $W_1$-orientation on $\mathbb{Z}$ is simply described by orienting the edge $(k,k+1)$ by $k \rightarrow k+1$. If $g : E(G) \rightarrow \mathbb{R}$ is a function defined on oriented edges, we can then simply write $g(k)$ instead of $g(k,k+1)$ and the divergence operator $(\nabla \cdot g)(k) = (g(k)-g(k-1)$ can be seen as the left derivative of $g$. We will denote $\nabla g(k) := (\nabla \cdot g)(k)$. Similarly, if $k \rightarrow k+1 \rightarrow k+2$ is an oriented triple, we will write $h(k)$ instead of $h(k,k+1,k+2)$ and $(\nabla_2 \cdot h)(k)$ will be the twice left derivative $\nabla_2 h(k) := h(k)-2h(k-1)+h(k-2)$. With these notations, the Benamou-Brenier condition is written $$\label{eq:BBCondZ} h_t(k-1) f_t(k) = g_t(k-1) vg_t(k) .$$ The proof of Theorem \[th:RenyiZ\] is based on two technical lemmas: For every triple of non negative numbers $f,g,h$ we have $$\label{eq:HolderLike} h f^{p-1} \leq \frac{1}{2-p} h^{2-p}g^{2p-2}+ \frac{1-p}{2-p} g^2 f^{p-2}.$$ Proof: The convexity of the exponential function implies the inequality: $$\forall a,b >0 \ , \ \forall \alpha,\beta >0 \ s.t. \ \frac{1}{\alpha}+\frac{1}{\beta}=1 \ , \ ab \leq \frac{a^\alpha}{\alpha}+ \frac{b^\beta}{\beta},$$ Setting $$\alpha := 2-p \ , \ \beta := \frac{2-p}{1-p} \ , \ a:= h g^{-2\frac{1-p}{2-p}} \ , \ b:= g^{2\frac{1-p}{2-p}} f^{p-1},$$ we obtain as wanted. $\square$ For every $x \geq 0$ we have $$\psi(x) := (1-p)(x-1)^2 - \frac{x^p-1}{2-p}-\frac{1}{2-p}x^{2-p}-\frac{1-p}{2-p}x^2+2x-1 \geq 0.$$ Proof: Let us compute the first derivatives of $h$: $$\psi'(x) = 2(1-p)(x-1) -\frac{p}{2-p} x^{p-1} -x^{1-p}-2\frac{1-p}{2-p}x+2,$$ $$\psi''(x) = 2(1-p) + \frac{p(1-p)}{(2-p)} x^{p-2}-(1-p)x^{-p}-2 \frac{1-p}{2-p},$$ $$\psi^{(3)}(x) = p(1-p) (x^{-p-1}-x^{p-3}).$$ As $p<1$, we have $-p-1>p-3$, so $\psi^{(3)}(x)$ is negative for $0 \leq x \leq 1$ and positive for $x\geq 1$. This means that $\psi''(x) \geq \psi''(1)=0$, so $\psi$ is convex. As we have $\psi(1)=0$ and $\psi'(1)=0$, we deduce that $\psi(x) \geq 0$ for every $x \geq 0$. $\square$ Proof of Theorem \[th:RenyiZ\]: The second derivative of $H_p(f_t)$ satisfies: $$-H_p''(t) = p \sum_{k \in \mathbb{Z}} \nabla_2 h_t(k) f_t(k)^{p-1} + p(p-1) \sum_{k \in \mathbb{Z}} (\nabla g_t(k))^2 f_t(k)^{p-2}.$$ As $p \in (0,1)$, we have $p(1-p) < 0$. To prove the convexity of $t \mapsto H_p(t)$, we first apply twice the inequality with $(f,g,h)=(f_t(k),g_t(k),h_t(k))$ and $(f,g,h)=(f_t(k),g_t(k-1),h_t(k-2))$ and then change indices to write $$\begin{aligned} \sum_{k \geq 0} \nabla_2(h_t(k)) f_t(k)^{p-1} &\leq& \sum_{k \geq 0} \frac{1}{2-p} h_t(k)^{2-p} g_t(k)^{2p-2}+ \frac{1-p}{2-p} g_t(k)^2 f_t(k)^{p-2}\\ && -2h_t(k-1) f_t(k)^{p-1} \\ &&+ \frac{1}{2-p} h_t(k-2)^{2-p} g_t(k-1)^{2p-2} + \frac{1-p}{2-p} g_t(k-1)^2 f_t(k)^{p-2} \\ &=& \sum_{k \geq 0} \frac{1}{2-p} h_t(k-1)^{2-p} g_t(k-1)^{2p-2}+ \frac{1-p}{2-p} g_t(k)^2 f_t(k)^{p-2}\\ && -2h_t(k-1) f_t(k)^{p-1} \\ &&+ \frac{1}{2-p} h_t(k-1)^{2-p} g_t(k)^{2p-2} + \frac{1-p}{2-p} g_t(k-1)^2 f_t(k)^{p-2} .\end{aligned}$$ We denote $v_{+,t}(k):= \frac{g_t(k)}{f_t(k)}$ and $v_{-,t}(k) :=\frac{g_t(k-1)}{f_t(k)}$. The Benamou-Brenier equation is then written $h_t(k-1)=v_{+,t}(k)v_{-,t}(k) f_t(k)$ and we have $$\begin{aligned} \sum_{k \geq 0} \nabla_2(h_t(k)) f_t(k)^{p-1} &\leq& \sum_{k \geq 0} \frac{1}{2-p} v_{+,t}(k)^{2-p} v_{-,t}(k)^p f_t(k)^p + \frac{1-p}{2-p} v_{+,t}(k)^2 f_t(k)^p\\ && - 2 v_{+,t}(k)v_{-,t}(k) f_t(k)^p + \frac{1}{2-p} v_{+,t}(k)^p v_{-,t}(k)^{2-p} f_t(k)^p + \frac{1-p}{2-p} v_{-,t}(k)^2 f_t(k)^p.\end{aligned}$$ With the same notations, we have $$(g_t(k)-g_t(k-1))^2 f_t(k)^{p-2} = (v_{+,t}(k)^2-2v_{+,t}(k)v_{-,t}(k)+v_{-,t}(k)^2) f_t(k)^{p},$$ and $$0 = \sum_{k \geq 0} g_t(k)^{p}-g_t(k-1)^{p} = \sum_{k \geq 0} (v_{+,t}(k)^p-v_{-,t}(k)^p)f_t(k)^p.$$ We use these estimations and the positivity of $\psi$ to write $$\begin{aligned} -\frac{1}{p} H_p''(t) &=& \sum_{k \geq 0} \nabla_2(h_t(k)) f_t(k)^{p-1}-(1-p) \sum_{k\geq 0} \nabla (g_t(k))^2 f_t(k)^p \\ && -\frac{1}{2-p} \sum_{k \geq 0} g_t(k)^{p}-g_t(k-1)^{p} \\ &\leq& \sum_{k \geq 0} v_{+,t}(k)^2 f_t(k)^p \psi\left(\frac{v_{-,t}(k)}{v_{+,t}(k)}\right) \\ &\leq& 0,\end{aligned}$$ which finishes the proof of the theorem. $\square$ Binomial mixtures and $W_2$-optimal couplings. ---------------------------------------------- In this appendix we consider two finitely supported probability measures $f_0$ and $f_1$ on $\mathbb{Z}$. Through this article and the previous one (see [@HillionGeodesic]), we have seen that, by considering a mixture of binomial measures with respect to a proper coupling between $f_0$ and $f_1$, it is possible to construct a $W_1$-geodesic $(f_t)_{t \in [0,1]}$ which satisfies a Benamou-Brenier condition which is a discrete analogue of a characterization of $W_2$-geodesics on the real line. Another natural way to generalize the notion of $W_2$-geodesic from the continuous setting to the discrete setting is the following: Let $\pi$ be the unique $W_2$-optimal coupling between $f_0$ and $f_1$. The binomial/$W_2$ interpolation $(f_t)_{t \in [0,1]}$ is defined by: $$f_t(k) : =\sum_{(i,j)} \pi_{i,j} \operatorname{bin}_{(i,j),t}(k),$$ where $\operatorname{bin}_{(i,j),t}$ is the binomial family between $i$ and $j$. Basic theorems on optimal transportation give the existence and uniqueness of a $W_2$-optimal coupling $\pi$ between $f_0$ and $f_1$. Thus the binomial/$W_2$ interpolation $(f_t)_{t \in [0,1]}$ exists and is unique. The question of the convexity of the entropy along $(f_t)$ is still open. The particular case where $f_1$ is a translation of $f_0$ has been studied by the author in [@HillionTranslation]. In this appendix we prove the more general: \[th:EntroBinoW2\] We make the following assumptions: 1. The measure $f_0$ is stochastically dominated by $f_1$ : $f_0 << f_1$, which means that for each $l \in \mathbb{Z}$, $\sum_{l \leq k} f_0(l) \geq \sum_{l \leq k} f_1(l)$. 2. Each $f_t$ is log-concave, i.e. that the inequality $f_t(k+1)^2 \geq f_t(k)f_t(k+2)$ holds for any $t \in [0,1]$ and $k \in \mathbb{Z}$. Then the entropy $H(t)$ of $f_t$ is a convex function of $t$. The stochastic domination assumption is not necessary but allows us to give a simpler proof. We will use it through the following: \[lem:W2coupling\] We suppose that $f_0 << f_1$. Then the $W_2$-optimal coupling $\pi$ between $f_0$ and $f_1$ satisfies the following: - If $\pi(i,j)>0$ then $i \leq j$. - If $\pi(i_1,j_1)>0$ and $\pi(i_2,j_2)>0$ then $(i_2-i_1)(j_2-j_1) \geq 0$. Remark. In particular the stochastic domination assumption allows us to use the same notations $g(k):=g(k,k+1)$, $h(k) := h(k,k+1,k+2)$ as in the first part of the Appendix. Proof of Theorem \[th:EntroBinoW2\]: Using the first point of Lemma \[lem:W2coupling\], we can write $$f_t(k) = \sum_{i \leq j} \pi(i,j) \operatorname{bin}_{(j-i),t}(k-i).$$ We now define the families of functions $(g_t)_{t \in [0,1]}$ and $(h_t)_{t \in [0,1]}$ by: $$\begin{aligned} g_t(k) &:=& \sum_{i \leq j} \pi(i,j) (j-i) \operatorname{bin}_{(j-i),t}(k-i) \\ h_t(k) &:= & \sum_{i \leq j} (j-i)(j-i-1) \pi(i,j) \operatorname{bin}_{(j-i-2),t}(k-i),\end{aligned}$$ so we have $$\frac{\partial}{\partial t} f_t(k) = -\nabla g_t(k) \ , \ \frac{\partial}{\partial t} g_t(k) = -\nabla h_t(k).$$ The study of the entropy of $f_t$ is similar to case of $W_{1,+}$-interpolations. We have: $$\begin{aligned} H''(t) &=& \sum_{k \in \mathbb{Z}} \nabla_2h_t(k) \log(f_t(k)) + \sum_{k \in \mathbb{Z}} \frac{(\nabla g_t(k))^2}{f_t(k)}.\end{aligned}$$ The major difference with $W_{1,+}$-interpolations comes from the fact that $f_t(k)h_t(k-1)$ is a priori not equal to $g_t(k)g_t(k-1)$. Let us introduce the family of functions $\tilde{h_t}(k) := \frac{g_t(k)g_t(k+1)}{f_t(k+1)}$. The triple $(f_t,g_t,\tilde{h}_t)$ is a BB-triple on $\mathbb{Z}$, so we have: $$\begin{aligned} H''(t) &=& \sum_{k \in \mathbb{Z}} \nabla_2(h_t-\tilde{h}_t)(k) \log(f_t(k))+\sum_{k \in \mathbb{Z}} \nabla_2\tilde{h}_t(k) \log(f_t(k)) + \sum_{k \in \mathbb{Z}} \frac{(\nabla g_t(k))^2}{f_t(k)} \\ &\geq& \sum_{k \in \mathbb{Z}} \nabla_2(h_t-\tilde{h}_t)(k) \log(f_t(k)) \\ &=& \sum_{k \in \mathbb{Z}} (h_t-\tilde{h}_t)(k) \nabla_2 \log(f_t(k+2)).\end{aligned}$$ By the assumption on the log-concavity of $f_t$, it thus suffices to show that $h_t \leq \tilde{h}_t$. to prove this fact, we notice that we can write $g_t(k)$, $g_t(k-1)$ and $h_t(k-1)$ under the form: $$g_t(k) = \sum_{i \leq j} \pi(i,j) \operatorname{bin}_{(j-i),t}(k-i) \frac{j-k}{1-t} \ , \ g_t(k) = \sum_{i \leq j} \pi(i,j) \operatorname{bin}_{(j-i),t}(k-i) \frac{k-i}{t},$$ $$h_t(k-1) = \sum_{i \leq j} \pi(i,j) \operatorname{bin}_{(j-i),t}(k-i) \frac{(j-k)(k-i)}{t(1-t)}.$$ Let us denote, for $i \leq j$, $a(i,j) := \pi(i,j) \operatorname{bin}_{(j-i),t}(k-i)$. Then $g_t(k)g_t(k-1)-f_t(k)h_t(k-1)$ can be seen as a quadratic form in the variables $(a(i,j))_{(i,j) \in \operatorname{Supp}(\pi)}$. The coefficient associated to $a(i,j)^2$ is $$\frac{j-k}{1-t} \frac{k-i}{t} - \frac{(j-k)(k-i)}{t(1-t)} =0 .$$ If $(i_1,j_1) \neq (i_2,j_2)$ are in $\operatorname{Supp}(\pi)$ then the coefficient associated to $a(i_1,j_1) a(i_2,j_2)$ is $$\begin{aligned} \frac{j_1-k}{1-t}\frac{k-i_2}{t}+\frac{j_2-k}{1-t}\frac{k-i_1}{t}-\frac{(j_1-k)(k-i_1)}{t(1-t)}-\frac{(j_2-k)(k-i_2)}{t(1-t)} &=& \frac{(j_2-j_1)(i_2-i_1)}{t(1-t)} \\&\geq& 0.\end{aligned}$$ This shows that $h_t \leq \tilde{h}_t$, and finishes the proof of Theorem \[th:EntroBinoW2\]. $\square$ [10]{} Jean-David Benamou and Yann Brenier. A numerical method for the optimal time-continuous mass transport problem and related problems. In [Monge [A]{}mpère equation: applications to geometry and optimization ([D]{}eerfield [B]{}each, [FL]{}, 1997)]{}, volume 226 of [Contemp. Math.]{}, pages 1–11. Amer. Math. 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A survey of the Schrödinger problem and some of its connections with optimal transport. . 34 (2014), no. 4, 1533–1574. John Lott and C[é]{}dric Villani. Ricci curvature for metric-measure spaces via optimal transport. , 169(3):903–991, 2009. Yann Ollivier. Ricci curvature of Markov chains on metric spaces. , 256(3), 810-864. Karl-Theodor Sturm. On the geometry of metric measure spaces. [I]{}. , 196(1):65–131, 2006. Karl-Theodor Sturm. On the geometry of metric measure spaces. [II]{}. , 196(1):133–177, 2006. Cédric Villani. Topics in optimal transportation. , vol. 58. Cédric Villani. Optimal transport: old and new. , vol. 338. [^1]: Department of Mathematics, University of Luxembourg, [email protected]	ArXiv
--- abstract: 'In this contribution, the evaluation of the diversity of the MIMO MMSE receiver is addressed for finite rates in both flat fading channels and frequency selective fading channels with cyclic prefix. It has been observed recently that in contrast with the other MIMO receivers, the MMSE receiver has a diversity depending on the aimed finite rate, and that for sufficiently low rates the MMSE receiver reaches the full diversity - that is, the diversity of the ML receiver. This behavior has so far only been partially explained. The purpose of this paper is to provide complete proofs for flat fading MIMO channels, and to improve the partial existing results in frequency selective MIMO channels with cyclic prefix.' author: - \| Florian Dupuy and Philippe Loubaton\ Thales Communication EDS/SPM, 92704 Colombes (France)\ Université Paris Est, IGM LabInfo, UMR-CNRS 8049, 77454 Marne-la-Vallée (France),\ Telephone: +33 146 132 109, Fax: +33 146 132 555, Email: [email protected]\ Telephone: +33 160 957 293, Fax: +33 160 957 755, Email: [email protected] bibliography: - 'IEEEabrv.bib' - 'bibMarne.bib' title: Diversity of the MMSE receiver in flat fading and frequency selective MIMO channels at fixed rate --- Diversity, Flat fading MIMO channels, Frequency selective MIMO channels, Outage probability, MMSE receiver Introduction ============ The diversity-multiplexing trade-off (DMT) introduced by [@zheng2003diversity] studies the diversity function of the multiplexing gain in the high SNR regime. [@kumar2009asymptotic] showed that the MMSE linear receivers, widely used for their simplicity, exhibit a largely suboptimal DMT in flat fading MIMO channels. Nonetheless, for a finite data rate (i.e. when the rate does not increase with the signal to noise ratio), the MMSE receivers take several diversity values, depending on the aimed rate, as noticed earlier in [@hedayat2005linear], and also in [@hedayat2004outage; @tajer2007diversity] for frequency-selective MIMO channels. In particular they achieve full diversity for sufficiently low data rates, hence their great interest. This behavior was partially explained in [@kumar2009asymptotic; @mehana2010diversity] for flat fading MIMO channels and in [@mehana2011diversity] for frequency-selective MIMO channels. Indeed the proof of the upper bound on the diversity order for the flat fading case given in [@mehana2010diversity] contains a gap, and the approach of [@mehana2010diversity] based on the Specht bound seems to be unsuccessfull. As for MIMO frequency selective channels with cyclic prefix, [@mehana2011diversity] only derives the diversity in the particular case of a number of channel taps equal to the transmission data block length, and claims that this value provides an upper bound in more realistic cases, whose expression is however not explicitly given. In this paper we provide a rigorous proof of the diversity for MMSE receivers in flat fading MIMO channels for finite data rates. We also derive the diversity in MIMO frequency selective channels with cyclic prefix for finite data rates if the transmission data block length is large enough. Simulations corroborate our derived diversity in the frequency selective channels case. Problem statement ================= We consider a MIMO system with $M$ transmitting, $N \geq M$ receiving antennas, with coding and ideal interleaving at the transmitter, and with a MMSE linear equalizer at the receiver, followed by a de-interleaver and a decoder (see Fig. \[fig:scheme\]). We evaluate in the following sections the achieved diversity by studying the outage probability, that is the probability that the capacity does not support the target data rate, at high SNR regimes. We denote $\rho$ the SNR, $I$ the capacity and $R$ the target data rate. We use the notation $\doteq$ for [exponential equality]{} [@zheng2003diversity], i.e. $$f(\rho) \doteq \rho^d \Leftrightarrow \lim_{\rho \to \infty} \frac{\log f(\rho)}{\log \rho}= d, \label{eq:exp-equ}$$ and the notations $\dot\leq$ and $\dot\geq$ for exponential inequalities, which are similarly defined. We note $\log$ the logarithm to base $2$. ![image](div_sch3){width="6.5in"} Flat fading MIMO channels ========================= In this section we consider a flat fading MIMO channel. The output of the MIMO channel is given by $$\y= \sqrt{\frac{\rho}{M}} \H \x + \n,$$ where $\n \sim \mathcal{CN}({\boldsymbol}{0},\I_N)$ is the additive white Gaussian noise and $\x$ the channel input vector, $\H$ the $N \times M$ channel matrix with i.i.d. entries $\sim \mathcal{CN}(0,1)$. For a rate $R$ such that $\log \frac{M}{m} < \frac{R}{M} < \log \frac{M}{m-1}$, with $m \in \{ 1, \ldots, M \}$, the outage probability verifies $$\PP(I<R) \doteq \rho^{-m(N-M+m)},$$ that is, a diversity of $m(N-M+m)$. Note that for a rate $R < M \log \frac{M}{M-1}$ (i.e. $m=M$) full diversity $MN$ is attained, while for a rate $R > M \log M$ the diversity corresponds to the one derived by DMT approach. This result was stated by [@mehana2010diversity]. Nevertheless the proof of the outage lower bound in [@mehana2010diversity] omits that the event noted $\mathcal{B}_a$ is not independent from the eigenvalues of $\H^H\H$, hence questioning the validity of the given proof. We thus provide an alternative proof based on an approach suggested by the analysis of [@kumar2009asymptotic] in the case where $R = r \log \rho$ with $r > 0$. The capacity $I$ of the MIMO MMSE considered system is given by $$I = \sum_{j=1}^M \log ( 1 + \beta_j),$$ where $\beta_j$ is the SINR for the $j$th stream: $$\beta_j= \frac{1}{\left( \left[ \I + \frac{\rho}{M} \H^\H \right]^{-1} \right)_{jj} } - 1.$$ We lower bound in the first place $\PP(I<R)$ and prove in the second place that the bound is tight by upper bounding $\PP(I<R)$ with the same bound. Lower bound of the outage probability {#sec:lowB_flat} ------------------------------------- We here assume that $R/M>\log (M/m$). In order to lower bound $\PP(I<R)$ we need to upper bound the capacity $I$. Using Jensen’s inequality on function $x \mapsto \log x$ yields $$\begin{aligned} I &\leq M \log \Bigg[ \frac{1}{M} \sum_{j=1}^M \left( 1+\beta_j \right) \Bigg] \label{ineq:jensen1} \\ &= M \log \Bigg[ \frac{1}{M} \sum_{j=1}^M \bigg( \left[ \left( \I + \frac{\rho}{M} \H^\H \right)^{-1} \right]_{jj} \bigg)^{-1} \Bigg]. \label{ineq:logconcave} \end{aligned}$$ We note $\H^\H= \U^\Lambda\U$ the SVD of $\H^\H$ with $\Lambda=\mathrm{diag}(\lambda_1,\ldots,\lambda_M)$, $\lambda_1 \leq \lambda_2 \ldots \leq \lambda_M$. We recall that the $(\lambda_k)_{k=1, \ldots, M}$ are independent from the entries of matrix $\U$ and that $\U$ is a Haar distributed unitary random matrix, i.e. the probability distribution of $\U$ is invariant by left (or right) multiplication by deterministic matrices. Using this SVD we can write $$\frac{1}{M} \sum_{j=1}^M \bigg( \left[ \left( \I + \frac{\rho}{M} \H^\H \right)^{-1} \right]_{jj} \bigg)^{-1} = \frac{1}{M} \sum_{j=1}^M \bigg( \sum_{k=1}^M \frac{\|\U_{kj}\|^2}{1+ \frac{\rho}{M} \lambda_k } \bigg)^{-1}. \label{eq:sumSINR} $$ ### Case m=1 In order to better understand the outage probability behavior, we first consider the case $m=1$. In this case $R/M>\log M$. We review the approach of [@kumar2009asymptotic III], which consists in upper bounding by $\left( 1+ \frac{\rho}{M} \lambda_1 \right) \frac{1}{M} \sum_{j=1}^M \frac{1}{\|\U_{1j}\|^2}$, as $\sum_{k=1}^M \frac{\|\U_{kj}\|^2}{1+ \frac{\rho}{M} \lambda_k } \geq \frac{\|\U_{1j}\|^2}{1+\frac{\rho}{M}\lambda_1}$. Using this bound in gives $$I \leq M \log \bigg[ \left( 1 + \frac{\rho}{M} \lambda_1 \right) \frac{1}{M} \sum_{j=1}^M \frac{1}{\|\U_{1j}\|^2} \bigg].$$ Therefore $$\Big( \left( 1 + \frac{\rho}{M} \lambda_1 \right) \frac{1}{M} \sum_{j=1}^M \frac{1}{\|\U_{1j}\|^2} < 2^{R/M} \Big) \subset (I<R).$$ In order to lower bound $\PP(I<R)$, [@kumar2009asymptotic] introduced the set $$\mathcal{A}_1 = \bigg\{ \frac{1}{M} \sum_{j=1}^M \frac{1}{\|\U_{1j}\|^2} < M + \eps \bigg\}$$ for $\eps > 0$. Then, $$\begin{aligned} \PP(I <R) & \geq \PP\left( (I<R) \cap \mathcal{A}_1 \right) \\ & \geq \PP \bigg[ \bigg( \left( 1 + \frac{\rho}{M} \lambda_1 \right) \frac{1}{M} \sum_{j=1}^M \frac{1}{\|\U_{1j}\|^2} < 2^{R/M} \bigg) \cap \mathcal{A}_1\bigg] \\ & \geq \PP \left[ \left( 1 + \frac{\rho}{M} \lambda_1 < \frac{2^{R/M}}{M+\eps} \right) \cap \mathcal{A}_1\right] \\ & = \PP(\mathcal{A}_1) \cdot \PP\left[ 1 + \frac{\rho}{M} \lambda_1 < \frac{2^{R/M}}{M+\eps} \right], \end{aligned}$$ where the last equality comes from the independence between eigenvectors and eigenvalues of Gaussian matrix $\H^\H$. It is shown in [@kumar2009asymptotic Appendix A] that $\PP(\mathcal{A}_1) \neq 0$. Besides, as we supposed $2^{R/M}>M$, we can take $\eps$ such that $\frac{2^{R/M}}{M+\eps}>1$, ensuring that $\PP\Big[ \left( 1 + \frac{\rho}{M} \lambda_1 \right) < \frac{2^{R/M}}{M+\eps} \Big] \neq 0$. Hence there exists $\kappa>0$ such that $$\PP(I<R) \ \dot\geq \ \PP\left( \lambda_1 < \frac{\kappa}{\rho} \right),$$ which is asymptotically equivalent to $\rho^{-(N-M+1)}$ in the sense of (see, e.g., [@jiang2011performance Th. II.3]). ### General case 1<=m<= M By the same token as for $m=1$ we now consider the general case – we recall that we assumed that $\log (M/m) < R/M$. We first lower bound $\sum_k\frac{\|\U_{kj}\|^2}{1+ \frac{\rho}{M} \lambda_k}$ which appears in by the $m$ first terms of the sum and then use Jensen’s inequality applied on $x \mapsto x^{-1}$, yielding $$\begin{aligned} \sum_{k=1}^M \frac{\|\U_{kj}\|^2}{1+ \frac{\rho}{M} \lambda_k} &\geq \sum_{k=1}^m \frac{\|\U_{kj}\|^2}{1+ \frac{\rho}{M} \lambda_k} \\ &\geq \frac{\left( \sum_{l=1}^m \|\U_{lj}\|^2 \right)^2}{\sum_{k=1}^m \|\U_{kj}\|^2 \left( 1+ \frac{\rho}{M} \lambda_k\right)}. \end{aligned}$$ Using this inequality in , we obtain that $$\begin{aligned} \frac{1}{M} \sum_{j=1}^M \bigg( \left[ \left( \I + \frac{\rho}{M} \H^\H \right)^{-1} \right]_{jj} \bigg)^{-1} \leq \frac{1}{M} & \sum_{j=1}^M \frac{\sum_{k=1}^m \|\U_{kj}\|^2\left( 1+ \frac{\rho}{M} \lambda_k\right)}{\left( \sum_{l=1}^m \|\U_{lj}\|^2 \right)^2} \notag \\ & = \sum_{k=1}^m \left( 1+ \frac{\rho}{M} \lambda_k\right) \delta_k(\U), \label{ineq:intr_delta} \end{aligned}$$ where $\delta_k(\U) = \frac{1}{M} \sum_{j=1}^M \frac{\|\U_{kj}\|^2}{\left( \sum_{l=1}^m \|\U_{lj}\|^2 \right)^2}$. Equation , together with , yields the following inclusion: $$\begin{aligned} \Bigg( \sum_{k=1}^m \delta_k(\U) \left( 1+ \frac{\rho}{M} \lambda_k\right) < 2^{R/M} \Bigg) \subset (I<R). \end{aligned}$$ Similarly to the case $m=1$, we introduce the set $\mathcal{A}_m$ defined by $$\mathcal{A}_m = \left\{ \delta_k(\U) < \frac{M}{m^2} + \eps, \ k=1,\ldots,m \right\}$$ for $\eps > 0$. We now use this set to lower bound $\PP(I<R)$. $$\begin{aligned} \PP(I<R) & \geq \PP\left( (I<R) \cap \mathcal{A}_m \right) \\ & \geq \PP \bigg[ \Bigg( \sum_{k=1}^m \delta_k(\U) \left( 1+ \frac{\rho}{M} \lambda_k\right) < 2^{R/M} \Bigg) \cap \mathcal{A}_m\bigg] \\ & \geq \PP \left[ \Bigg( \sum_{k=1}^m \left( 1+ \frac{\rho}{M} \lambda_k\right) < \frac{2^{R/M}}{\frac{M}{m^2} + \eps} \Bigg) \cap \mathcal{A}_m\right] \\ &= \PP(\mathcal{A}_m) \cdot \PP\left[ \sum_{k=1}^m \left( 1 + \frac{\rho}{M} \lambda_k \right) < \frac{2^{R/M}}{\frac{M}{m^2}+\eps} \right]. \end{aligned}$$ The independence between eigenvectors and eigenvalues of Gaussian matrix $\H^\H$ justifies the last equality. As we assumed that $\log(M/m) < R/M$, that is $m < \frac{2^{R/M}}{M/m^2}$, we can choose $\eps$ such that $m < \frac{2^{R/M}}{M/m^2 +\eps}$. That ensures that $\PP\left[ \sum_{k=1}^m \left( 1 + \frac{\rho}{M} \lambda_k \right) < \frac{2^{R/M}}{ M/m^2 +\eps} \right] \neq 0$. We show in Appendix \[apx:prob\_sum\_first\_ev\] that this probability is asymptotically equivalent to $\rho^{-m(N-M+m)}$ in the sense of , leading to $$\PP(I<R) \ \dot\geq \ \frac{\PP(\mathcal{A}_m)}{\rho^{m(N-M+m)}}. \label{ineq:P_I<R}$$ We still need to prove that $\PP(\mathcal{A}_m) \neq 0$. Any Haar distributed random unitary matrix can be parameterized by $M^2$ independent angular random variables $(\alpha_1, \ldots, \alpha_{M^2})={\boldsymbol}\alpha$ whose probability distributions are almost surely positive (see [@dita2003factorization; @lundberg2004haar] and Appendix \[apx:ang\_par\]). We note $\Phi_m$ the functions such that $\U=\Phi_m({\boldsymbol}\alpha)$. Consider a deterministic unitary matrix $\U_$ such that $\|(\U_)_{ij}\|^2 = \frac{1}{M} \ \forall i,j$, and denote by ${\boldsymbol}\alpha_$ a corresponding $M^2$ dimensional vector. It is straightforward to check that $\delta_k \circ \Phi_m (\alpha_) = M / m^2$. Functions ${\boldsymbol}\alpha \mapsto (\delta_k \circ \Phi_m)({\boldsymbol}\alpha)$ are continuous at point ${\boldsymbol}\alpha_$ for $1 \leq k \leq m$ and therefore there exists $\eta>0$ such that the ball $\mathcal{B} \left( {\boldsymbol}\alpha_, \eta \right)$ is included in the set $\left\{ {\boldsymbol}\alpha, \ (\delta_k \circ \Phi_m)(\alpha) < \frac{M}{m^2}+\eps, \ k=1, \ldots, m \right\}$. We have therefore $\PP(\mathcal{A}_m) \neq 0$ as $$\begin{aligned} \PP(\mathcal{A}_m) & = \int_{\left\{ (\delta_k \,\circ\, \Phi_m) ({\boldsymbol}\alpha) < \frac{M}{m^2} + \eps, \, k=1, \ldots, m \right\}} p({\boldsymbol}\alpha) d{\boldsymbol}\alpha \\ & > \int_{\mathcal{B} \left( {\boldsymbol}\alpha_, \eta \right)} p({\boldsymbol}\alpha) d{\boldsymbol}\alpha > 0 \end{aligned}$$ Coming back to , we eventually have $$\PP(I<R) \ \dot\geq \ \frac{1}{\rho^{m(N-M+m)}},$$ that is the diversity of the MMSE receiver is upper bounded by $m(N-M+m)$. Upper bound of the outage probability {#sec:upper-flat} ------------------------------------- We now conclude by studying the upper bound of the outage probability, showing that $m(N-M+m)$ is also a lower bound for the diversity. Note that this lower bound has been derived in [@kumar2009asymptotic; @mehana2010diversity] using however rather informal arguments; we provide a more rigorous proof here for the sake of completeness. We now assume that $R/M < \log (M/(m-1))$, i.e. $m-1 < M 2^{-R/M}$. Using Jensen inequality on function $y \mapsto \log(1/y)$, the capacity $I$ can be lower bounded: $$\begin{aligned} I &= - \sum_{j=1}^M \log \left( \left[ \left( \I + \frac{\rho}{M} \H^\H \right)^{-1} \right]_{jj} \right) \\ &\geq -M \log \left( \frac{1}{M} \Tr \left[ \left( \I + \frac{\rho}{M} \H^\H \right)^{-1} \right] \right), \end{aligned}$$ which leads to an upper bound for the outage probability: $$\PP(I<R) \leq \PP \left[ \Tr \left[ \left( \I + \frac{\rho}{M} \H^\H \right)^{-1} \right] > M 2^{-R/M} \right]. \label{ineq:PP-B0}$$ We need to derive the probability in the right-hand side of the above inequality. Noting $\mathcal{B}_0= \left\{ \lambda_1 \leq \lambda_2 \ldots \leq \lambda_M, \ \sum_{k=1}^M \left( 1 + \frac{\rho}{M} \lambda_k \right)^{-1} > M 2^{-R/M} \right\}$, $$\begin{aligned} \PP \bigg[ \Tr \Big[ \Big( \I + & \frac{\rho}{M} \H^\H \Big)^{-1} \Big] > M 2^{-R/M} \bigg] \notag \\ & = \int_{\mathcal{B}_0} p(\lambda_1, \ldots, \lambda_M) d\lambda_1 \ldots d\lambda_M. \label{eq:int_B0_PP} \end{aligned}$$ We now introduce $\mu_m = \sup_{(\lambda_1, \ldots, \lambda_M) \in \mathcal{B}_0}\{ \rho \,\lambda_m \}$ and prove by contradiction that $\mu_m < +\infty$. If $\mu_m = +\infty$, there exists a sequence $(\lambda_1^{(n)}, \lambda_2^{(n)}, \ldots, \lambda_M^{(n)})_{n\in\mathbb{N}}$ such that $\lambda_k^{(n)} \rightarrow +\infty$ for any $k \geq m$. Besides, $$M 2^{-R/M} \mhs < \sum_{k=1}^M \mhs \Big( 1+\frac{\rho}{M}\lambda^{(n)}_k \Big)^{-1} \mhs \leq (m-1) + \sum_{k=m}^M \mhs \Big(1+\frac{\rho}{M}\lambda^{(n)}_k\Big)^{-1}.$$ In particular $M 2^{-R/M} < (m-1) + \sum_{k=m}^M \mhs \big(1+\frac{\rho}{M}\lambda^{(n)}_k\big)^{-1} $, which, taking the limit when $n \rightarrow +\infty$, leads to $m-1 \geq M 2^{-R/M}$, a contradiction with the assumption $m-1 < M 2^{-R/M}$. Hence, $\mu_m < +\infty$. We introduce the set $\mathcal{B}_1= \{ \lambda_1 \leq \lambda_2 \ldots \leq \lambda_M, \, 0 < \lambda_k \leq \frac{\mu_m}{\rho}, \, k=1,\ldots,m \}$, which verifies $\mathcal{B}_0 \subset \mathcal{B}_1$. Using and , this implies that $$\PP(I<R) \leq \int_{\mathcal{B}_1} p(\lambda_1, \ldots, \lambda_M) d\lambda_1 \ldots d\lambda_M,$$ which is shown to be asymptotically smaller than $\rho^{-m(N-M+m)}$ in the sense of in Appendix \[apx:prob\_first\_ev\_bounded\]. The diversity is thus lower bounded by $m(N-M+m)$, ending the proof. Frequency selective MIMO channels with cyclic prefix {#sec:freqsel} ==================================================== We consider a frequency selective MIMO channel with $L$ independent taps. We consider a block transmission cyclic prefix scheme, with a block length of $K$. The output of the MIMO channel at time $t$ is given by $$\begin{aligned} \y_t =\sqrt{\frac{\rho}{ML}}\,\sum_{l=0}^{L-1} \H_l \x_{t-l} +\n_t = \sqrt{\frac{\rho}{ML}}\,[\H(z)] \x_t +\n_t\end{aligned}$$ where $\x_t$ is the channel input vector at time $t$, $\n_t \sim \mathcal{CN}({\boldsymbol}{0}, \I_N)$ the additive white Gaussian noise, $\H_l$ is the $N \times M$ channel matrix associated to $l^{\mathrm{th}}$ channel tap, for $l \in \{0,\ldots, L-1\}$, and $\H(z)$ denotes the transfer function of the discrete-time equivalent channel defined by $$\H(z) = \sum_{l=0}^{L-1} \H_l \, z^{-l}.$$ We make the common assumption that the entries of $\H_l$ are i.i.d and $\mathcal{CN}(0,1)$ distributed. We can now state the second diversity theorem of the paper. Assume that the non restrictive condition $K > {M^{2}(L-1)}$ holds, ensuring that $\log \frac{M}{m} < -\log\big(\frac{m-1}{M}+\frac{(L-1)(M-(m-1))}{K}\big)$ for any $m=1, \ldots, M$. Then, for a rate $R$ verifying $$\hspace{-11pt} \textstyle \log \frac{M}{m} < \frac{R}{M} < -\log \left(\frac{m-1}{M} + \frac{(L-1)(M-(m-1))}{K} \right), \label{eq:bounds_R_fsel}$$ $m \in \{ 1, \ldots, M \}$, the outage probability verifies $$\PP(I<R) \doteq \rho^{-m(LN-M+m)},$$ that is a diversity of $m(LN-M+m)$. The diversity of the MMSE receiver is thus $m(LN-M+m)$, corresponding to a flat fading MIMO channel with $M$ transmit antennas and $LN$ receive antennas. For a large block length $K$, the upper bound for rate $R$ is close to the bound of the previous flat fading case $\log \frac{M}{m-1}$. Concerning data rates verifying $ -\log\big(\frac{m-1}{M} + \frac{L-1}{K} (M-(m-1))\big) < \frac{R}{M} < \log \frac{M}{m-1}, $ the $m(LN-M+m)$ diversity is only an upper bound; nevertheless the diversity is also lower bounded by $(m-1)(LN-M+(m-1))$. Similarly to previous section the capacity of the MIMO MMSE system is written $ I = \sum_{j=1}^M \log ( 1 + \beta_j), $ where $\beta_j$ is the SINR for the $j$th stream of $\x_t$. It is standard material that in MIMO frequency selective channel with cyclic prefix the SINR of the MMSE receiver is given by $$\beta_j = \frac{1}{ \frac{1}{K}\sum_{k=1}^K \left[ \left( \S\left(\frac{k-1}{K}\right) \right)^{-1} \right]_{jj} } -1, \label{eq:SINR_freqsel}$$ where $\S(\nu)= \I_N + \frac{\rho}{M} \H(e^{2 i \pi \nu})^\H(e^{2 i \pi \nu}) $. Lower bound for the outage probability -------------------------------------- We assume that $R/M > \log(M/m)$. One can show that function $\A \mapsto (\A^{-1})_{jj}$, defined over the set of positive-definite matrices, is convex. Using Jensen’s inequality then yields $$\begin{aligned} \frac{1}{K}\sum_{k=1}^K {\textstyle \left[ \left( \S\left(\frac{k-1}{K}\right) \right)^{-1} \right]_{jj} } & \geq \bigg( \bigg[ \frac{1}{K} \sum_{k=1}^K {\textstyle \S\left(\frac{k-1}{K}\right) }\bigg]^{-1} \bigg)_{jj} \\ & = \bigg( \bigg[ \I_N + \sum_{l=0}^{L-1} \frac{\rho}{M} \H_l^\H_l \bigg]^{-1} \bigg)_{jj}. \end{aligned}$$ The last equality follows from the fact that $\frac{1}{K}\sum_{k=1}^K e^{2 i\pi \frac{k-1}{K}(l-n)}=\delta_{ln}$. Using this inequality in the SINR expression gives $$1+\beta_j \leq \bigg( \bigg( \bigg[ \I_N + \sum_{l=0}^{L-1} \frac{\rho}{M} \H_l^\H_l \bigg]^{-1} \bigg)_{jj} \bigg)^{-1}.$$ We now come back to the capacity $I$ of the system; similarly to , using Jensen’s inequality yields $$\begin{aligned} I & \leq M \log \Bigg[ \frac{1}{M} \sum_{j=1}^M \left( 1+\beta_j \right) \Bigg] \\ & \leq M \log \Bigg[ \frac{1}{M} \sum_{j=1}^M \bigg( \bigg( \bigg[ \I_N + \frac{\rho}{M} \sum_{l=0}^{L-1} \H_l^\H_l \bigg]^{-1} \bigg)_{jj} \bigg)^{-1} \,\Bigg]. \end{aligned}$$ We can now use the results of section \[sec:lowB\_flat\] by simply replacing $N \times M$ matrix $\H$ in by $LN \times M$ matrix $\Hb=[\H_0^T, \H_1^T, \ldots, \H_{L-1}^T]^T$. They lead to the following lower bound for the outage capacity, for a rate $R$ verifying $R/M > \log (M/m)$: $$\PP(I<R) \ \dot\geq \ \frac{1}{\rho^{m(LN-M+m)}}.$$ Upper bound for the outage probability -------------------------------------- We assume that $\frac{R}{M} < -\log\big(\frac{m-1}{M}+\frac{(L-1)(M-(m-1))}{K}\big)$, that is $2^{-R/M} < \frac{m-1}{M} \mhs + \mhs \frac{L-1}{K} (M \mhs - \mhs (m-1))$. We first derive a lower bound for the capacity $I$. $$\begin{aligned} I & = - \sum_{j=1}^M \log \left( \frac{1}{K}\sum_{k=1}^K \left( \left[ {\textstyle \S\left(\frac{k-1}{K}\right)} \right]^{-1} \right)_{jj} \right) \\ & \geq - M \log \left( \frac{1}{KM} \sum_{k=1}^K \Tr\left( \left[ {\textstyle \S\left(\frac{k-1}{K}\right)} \right]^{-1} \right) \right) \end{aligned}$$ The latter inequality follows once again from Jensen’s inequality on function $x \mapsto \log x$. We now analyze $\Tr\left(\S(\nu)^{-1}\right)$. To that end, we write $LN \times M$ matrix $\Hb=[\H_0^T, \ldots, \H_{L-1}^T]^T$ under the form $\Hb={\boldsymbol}\Theta (\Hb^\Hb)^{1/2}$, where ${\boldsymbol}\Theta=[{\boldsymbol}\Theta_0^T, \ldots, {\boldsymbol}\Theta_{L-1}^T]^T$ and ${\boldsymbol}\Theta^{\boldsymbol}\Theta=\I_M$. Besides, we note $\U^\Lambda\U$ the SVD of $\Hb^\Hb$ with $\Lambda=\mathrm{diag}(\lambda_1,\ldots,\lambda_M)$, $\lambda_1 \leq \ldots \leq \lambda_M$. Hence, $$\H(e^{2 i\pi \nu})={\boldsymbol}\Theta(e^{2 i\pi \nu})\U^\Lambda^{1/2}\U,$$ where ${\boldsymbol}\Theta(z)=\sum_{l=0}^{L-1}{\boldsymbol}\Theta_l z^{-l}$. Using this parametrization, $$\begin{aligned} \Tr\left(\S(\nu)^{-1}\right) &= \Tr \left[ \left( \I + \frac{\rho}{M} \U {\boldsymbol}\Theta^(e^{2 i \pi \nu}){\boldsymbol}\Theta(e^{2 i \pi \nu}) \U^* \Lambda \right)^{-1} \right] \\ &\leq \Tr \left[ \left( \I + \frac{\rho}{M} \gamma(e^{2 i \pi \nu}) \Lambda \right)^{-1} \right], \end{aligned}$$ where $\gamma(\nu)=\lambda_{\mathrm{min}}({\boldsymbol}\Theta^(e^{2 i \pi \nu}){\boldsymbol}\Theta(e^{2 i \pi \nu}))$. Coming back to the outage probability, $$\begin{aligned} \PP(I<R) \leq & \PP \mhs \left[ \frac{1}{K} \sum_{k=0}^{K-1} \sum_{j=1}^M \mhs \left( \mhs 1 + \frac{\rho \lambda_j }{M} \gamma \mhs\left(\frac{k}{K}\right) \mhs \right)^{\mhs-1} \mhs > M2^{-R/M}\right] \notag \\ &= \PP \mhs \left[ \Hb \in \mathcal{B}_0 \right], \label{ineq:PP-fs} \end{aligned}$$ where $\mathcal{B}_0 = \big\{ \Hb, \frac{1}{K} \sum_{k=0}^{K-1} \sum_{j=1}^M \mhs \big( 1 + \frac{\rho \lambda_j }{M} \gamma \mhs\left(\frac{k}{K}\right) \big)^{-1} \mhs > M2^{-R/M} \big\}$. We now prove by contradiction that $\mu_m < +\infty$, where $\mu_m =\sup_{\Hb \in \mathcal{B}_0} \{ \rho \lambda_m \}$. If $\mu_m = +\infty$ there exists a sequence of matrices $\Hb^{(n)} \mhs \in \mhs \mathcal{B}_0$ such that $\rho \lambda_m^{(n)} \rightarrow +\infty$. Besides, $$\begin{aligned} M 2^{-\frac{R}{M}} & < \frac{1}{K} \sum_{k=0}^{K-1} \sum_{j=1}^M \bigg( 1 + \frac{\rho \lambda_j^{(n)} }{M} \gamma^{(n)} \mhs\left(\frac{k}{K}\right) \bigg)^{\mhs-1} \notag \\ & \leq (m-1) + \frac{1}{K} \sum_{k=0}^{K-1} \sum_{j=m}^M \bigg( 1 + \frac{\rho \lambda_j^{(n)} }{M} \gamma^{(n)} \mhs\left(\frac{k}{K}\right) \bigg)^{\mhs-1} \label{ineq:mat-seq} \end{aligned}$$ As ${\boldsymbol}\Theta^{(n)}$ belongs to a compact we can extract a subsequence ${\boldsymbol}\Theta^{(\psi(n))}$ which converges towards a matrix ${\boldsymbol}\Theta_\infty$. For this subsequence, inequality becomes $$M 2^{-\frac{R}{M}} \leq (m-1) + \frac{1}{K} \sum_{k=0}^{K-1} \sum_{j=m}^M \bigg( 1 + \frac{\rho \lambda_j^{(\psi(n))} }{M} \gamma^{(\psi(n))} \mhs\left(\frac{k}{K}\right) \bigg)^{-1}. \label{ineq:sumsum}$$ Let $\gamma_\infty$ be the function defined by $\gamma_\infty(\nu)=\lambda_{\mathrm{min}}({\boldsymbol}\Theta^_\infty(e^{2 i \pi \nu}){\boldsymbol}\Theta_\infty(e^{2 i \pi \nu}))$ and $k_1, \ldots, k_p$ be the integers for which $\gamma_{\infty}(k_j/K) = 0$. Then $\det {\boldsymbol}\Theta_\infty(z) = \det \big( \sum_{l=0}^{L-1} {\boldsymbol}\Theta_{\infty,l} z^{-l} \big)=0$ for all $z \in \big\{ e^{2 i \pi k_j/K}, j=1,\ldots,p \big\}$. Nevertheless, polynomial $z \mapsto \sum_{l=0}^{L-1} {\boldsymbol}\Theta_{\infty,l} z^{-l}$ has a maximum degree of $M(L-1)$, therefore $p \leq M(L-1)$. Inequality then leads to $$M 2^{-\frac{R}{M}} \leq (m-1) + \frac{M(L-1)}{K} +\frac{1}{K} \sum_{k\notin\{k_1, \ldots, k_p\}} \sum_{j=m}^M \bigg( 1 + \frac{\rho \lambda_j^{(\psi(n))} }{M} \gamma^{(\psi(n))} \left( \frac{k}{K} \right) \bigg)^{\mhs-1} \label{ineq:M2-RMp}$$ Moreover, if $k \notin \{k_1, \ldots, k_p\}$, $\lambda_j^{(\psi(n))} \gamma^{(\psi(n))}(\frac{k}{K}) \rightarrow +\infty$ for $j \geq m$, as $\gamma^{(\psi(n))} \mhs\left(\frac{k}{K}\right) \rightarrow \gamma_\infty \mhs\left(\frac{k}{K}\right) \neq 0$ for $k \notin \left\{ k_1, \ldots, k_p \right\}$. Therefore taking the limit of when $n \rightarrow +\infty$ gives $$M 2^{-\frac{R}{M}} \leq (m-1) + \frac{M(L-1)}{K},$$ which is in contradiction with the original assumption $2^{-R/M} < \mhs \frac{m-1}{M} \mhs + \mhs \frac{L-1}{K} (M \mhs - \mhs (m-1))$. Hence $\mu_m < +\infty$, and $\mathcal{B}_0 \subset \mathcal{B}_1= \{ \Hb, \rho \lambda_m(\Hb^\Hb) < \mu_m \}$. Using , we thus have $$\PP(I<R) \leq \PP(\Hb \in \mathcal{B}_1),$$ which, by Appendix \[apx:prob\_first\_ev\_bounded\], is asymptotically smaller than $\rho^{-m(NL-M+m)}$ in the sense of , therefore ending the proof. Numerical Results ================= We here illustrate the derived diversity in the frequency selective case. In the conducted simulation we took a block length of $K=64$, a number of transmitting and receiving antennas $M=N=2$, $L=2$ channel taps and an aimed data rate $R=3$ bits/s/Hz. Rate $R$ then verifies with $m=1$, therefore the expected diversity is $LN-M+1=3$. The outage probability is displayed on Fig. \[fig:Pout\] as a function of SNR. We observe a slope of $-10^{-3}$ per decade, hence a diversity of $3$, confirming the result stated in part \[sec:freqsel\]. ![Outage probability of the MMSE receiver, L=2, K=64, M=N=2[]{data-label="fig:Pout"}](div_40M){width="3in"} Conclusion ========== In this paper we provided rigorous proofs regarding the diversity of the MMSE receiver at fixed rate, in both flat fading and frequency selective MIMO channels. The higher the aimed rate the less diversity is achieved; in particular, for sufficiently low rates, the MMSE receiver achieves full diversity in both MIMO channel cases, hence its great interest. Nonetheless, in frequency selective channels, the diversity bounds are not tight for some specific rates; this could probably be improved. Simulations corroborated our results. {#apx:prob_sum_first_ev} We prove in this appendix that, for $b>0$, $\PP(\sum_{k=1}^m \rho \lambda_k < b) \ \dot\geq \ \rho^{-m(N-M+m)}$. We note $\mathcal{C}_m$ the set defined by $\mathcal{C}_m = \{ \lambda_1, \ldots, \lambda_m: \ 0 < \lambda_1 \leq \ldots \leq \lambda_m, \ \sum_{k=1}^m \rho \lambda_k < b \}$. As the $\lambda_i$ verify $0 < \lambda_1 \leq \ldots \leq \lambda_M$, we can write $$\PP \Bigg( \sum_{k=1}^m \rho \lambda_k < b \Bigg) = \int_{(\lambda_1, \ldots, \lambda_m) \in \mathcal{C}_m} \int_{\lambda_m}^{+\infty} \bhs\ldots \int_{\lambda_{M-1}}^{+\infty} p_{M,N}(\lambda_1, \ldots, \lambda_M) \ d\lambda_1 \ldots d\lambda_M, \label{eq:prob_sum_first_ev}$$ where $p_{M,N}: \mathbb{R}^M \rightarrow \mathbb{R}$ is the joint probability density function of the ordered eigenvalues of a $M \times M$ Wishart matrix with scale matrix $\I_M$ and $N$ degrees of freedom, given by (see, e.g., [@zheng2003diversity]): $$p_{M,N} = K_{M,N}^{-1} \prod_{i=1}^M \left( \lambda_i^{N-M} e^{-\lambda_i} \right) \prod_{i<j}(\lambda_i-\lambda_j)^2, \label{eq:p-wish}$$ where $K_{M,N}$ is a normalizing constant. We now try to separate the integral in in two integrals, one over $\lambda_1, \ldots, \lambda_m$, the other over $\lambda_{m+1}, \ldots, \lambda_M$. As we have $(\lambda_1, \ldots, \lambda_m) \in \mathcal{C}_m$ in , $\lambda_m < b / \rho$ and thus $$\begin{split} \int_{\lambda_m \leq \lambda_{m+1} \leq \ldots \leq \lambda_M} & p_{M,N}(\lambda_1, \ldots, \lambda_M) \ d\lambda_{m+1} \ldots d\lambda_M \\ & \geq \int_{(\lambda_{m+1}, \ldots, \lambda_M) \in \mathcal{D}} p_{M,N}(\lambda_1, \ldots, \lambda_M) \ d\lambda_{m+1} \ldots d\lambda_M \end{split} \label{ineq:2intg}$$ where $\mathcal{D}=\{(\lambda_{m+1}, \ldots, \lambda_M) \in \mathbb{R}_+^{M-m}; \ b/\rho \leq \lambda_{m+1}\leq \ldots \leq \lambda_M \}$. This integral can be simplified by noticing that $p_{M,N}(\lambda_1, \ldots, \lambda_M)$ explicit expression is invariant by permutation of its parameters $\lambda_1, \ldots, \lambda_M$, in particular by permutation of its parameters $\lambda_{m+1}, \ldots, \lambda_M$. Therefore, noting $\mathcal{S}=\mathrm{Sym}(\{\lambda_{m+1}, \ldots, \lambda_M\})$ the group of permutations over the finite set $\{\lambda_{m+1}, \ldots, \lambda_M\}$, we get $$\begin{aligned} \int_{b/\rho}^{+\infty} \bhs\ldots \int_{b/\rho}^{+\infty} & p_{M,N}(\lambda_1, \ldots, \lambda_M) \ d\lambda_{m+1} \ldots d\lambda_M \notag \\ & = \sum_{s \in \mathcal{S}} \int_{s(\lambda_{m+1}, \ldots, \lambda_M) \in \mathcal{D}} p_{M,N}(\lambda_1, \ldots, \lambda_M) \ d\lambda_{m+1} \ldots d\lambda_M \notag \\ & = \mathrm{Card}(\mathcal{S}) \int_{ (\lambda_{m+1}, \ldots, \lambda_M) \in \mathcal{D}} p_{M,N}(\lambda_1, \ldots, \lambda_M) \ d\lambda_{m+1} \ldots d\lambda_M \notag \\ & = (M-m)! \int_{ (\lambda_{m+1}, \ldots, \lambda_M) \in \mathcal{D}} p_{M,N}(\lambda_1, \ldots, \lambda_M) \ d\lambda_{m+1} \ldots d\lambda_M. \label{eq:intg-symm}\end{aligned}$$ Using and in , we obtain $$\PP \Bigg( \sum_{k=1}^m \rho \lambda_k < b \Bigg) \geq \frac{1}{(M-m)!} \int_{\mathcal{C}_m} \int_{b/\rho}^{+\infty} \bhs\ldots \int_{b/\rho}^{+\infty} p_{M,N}(\lambda_1, \ldots, \lambda_M) \ d\lambda_1 \ldots d\lambda_M.$$ We now replace $p_{M,N}$ by its explicit expression and then try to separate the $m$ first eigenvalues from the others. Note that we can drop the constants $(M-m)!$ and $K_{M,N}$ as we only need an asymptotic lower bound. $$\begin{aligned} \PP \Bigg( \sum_{k=1}^m \rho \lambda_k < b \mhs \Bigg) \mhs \ \dot\geq & \int_{\mathcal{C}_m} \int_{b/\rho}^{+\infty} \bhs \ldots \int_{b/\rho}^{+\infty} \prod_{i=1}^M \left( \lambda_i^{N-M} e^{-\lambda_i} \right) \prod_{i<j}(\lambda_i-\lambda_j)^2 \, d\lambda_1 \ldots d\lambda_M \\ = & \ \int_{\mathcal{C}_m} \int_{b/\rho}^{+\infty} \bhs \ldots \int_{b/\rho}^{+\infty} \Bigg( \prod_{i=1}^m \left( \lambda_i^{N-M} e^{-\lambda_i} \right) \prod_{i<j\leq m}(\lambda_i-\lambda_j)^2 \, \Bigg) \\ & \cdot \Bigg( \prod_{i=m+1}^M \left( \lambda_i^{N-M} e^{-\lambda_i} \right) \prod_{i\leq m<j}(\lambda_i-\lambda_j)^2 \prod_{m<i<j}(\lambda_i-\lambda_j)^2 \, \Bigg) \ d\lambda_1 \ldots d\lambda_M \end{aligned}$$ For $i\leq m<j$, we have that $\lambda_i \leq b/\rho$ and thus $(\lambda_i-\lambda_j)^2 \geq \big(\lambda_j-\frac{b}{\rho}\big)^2$. Hence, $$\begin{aligned} \PP \Bigg( \sum_{k=1}^m \rho \lambda_k < b \mhs \Bigg) \dot\geq & \ \bigg( \int_{\mathcal{C}_m} \prod_{i=1}^m \left( \lambda_i^{N-M} e^{-\lambda_i} \right) \prod_{i<j\leq m}(\lambda_i-\lambda_j)^2 \ d\lambda_1 \ldots d\lambda_m \bigg) \label{eq:2sep-intg} \\ & \cdot \mhs \bigg( \mhs \int_{b/\rho}^{+\infty} \bhs\mhs\mhs \ldots \int_{b/\rho}^{+\infty} \mhs\mhs \prod_{i=m+1}^M \mhs\mhs\mhs \left( \lambda_i^{N-M}e^{-\lambda_i} \right) \mhs \prod_{j=m+1}^M \mhs\mhs\mhs \left(\lambda_j-\frac{b}{\rho}\right)^{\,\mhs\mhs 2m} \mhs\mhs \prod_{m<i<j} \mhs\mhs\mhs (\lambda_i-\lambda_j)^2 \ d\lambda_{m+1} \ldots d\lambda_M \mhs\mhs\bigg) \notag \end{aligned}$$ We now have two separate integrals. We first consider the second one, in which we make the substitution $\beta_i= \lambda_i - b/\rho$ for $i=m+1, \ldots, M$. $$\begin{aligned} \int_{b/\rho}^{+\infty} & \bhs \ldots \int_{b/\rho}^{+\infty} \prod_{i=m+1}^M \mhs\mhs\mhs \left( \lambda_i^{N-M}e^{-\lambda_i} \right) \mhs \prod_{j=m+1}^M \mhs\mhs\mhs \left(\lambda_j-\frac{b}{\rho}\right)^{\,\mhs\mhs 2m} \mhs\mhs \prod_{m<i<j} \mhs\mhs\mhs (\lambda_i-\lambda_j)^2 \ d\lambda_{m+1} \ldots d\lambda_M \notag \\ & = e^{- {(M-m)b/\rho}} \int_0^{+\infty} \bhs \ldots \int_0^{+\infty} \mhs\mhs \prod_{i=m+1}^M \mhs\mhs \left( \mhs \left( {\textstyle \beta_i + \frac{b}{\rho}} \right)^{N-M} e^{- \beta_i} \beta_i ^{2m} \mhs \right) \mhs \prod_{m<i<j}(\beta_i-\beta_j)^2 \ d\beta_{m+1} \ldots d\beta_M \notag \\ & \geq \frac{1}{2} \int_0^{+\infty} \bhs \ldots \int_0^{+\infty} \prod_{i=m+1}^M \left( \beta_i ^{N-M+2m} e^{- \beta_i} \right) \prod_{m<i<j}(\beta_i-\beta_j)^2 \ d\beta_{m+1} \ldots d\beta_M \label{ineq:2nd-intg} \end{aligned}$$ for $\rho$ large enough, i.e. such that $e^{- (M-m) b/\rho} > 1/2$. It is straightforward to see that the integral in is nonzero, finite, independent from $\rho$ and therefore asymptotically equivalent to $1$ in the sense of . Hence, we can drop the second integral in , leading to: $$\PP \Bigg( \sum_{k=1}^m \rho \lambda_k < b \mhs \Bigg) \dot\geq \ \int_{\mathcal{C}_m} \prod_{i=1}^m \left( \lambda_i^{N-M} e^{-\lambda_i} \right) \prod_{i<j\leq m}(\lambda_i-\lambda_j)^2 \ d\lambda_1 \ldots d\lambda_m. \label{ineq:only1-intg}$$ Making the substitution $\alpha_i= \rho \lambda_i$ for $i=1, \ldots, m$ in and noting $\mathcal{C}'_m=\{ \alpha_1, \ldots, \alpha_m: \ 0 < \alpha_1 \leq \ldots \leq \alpha_m, \ \sum_{k=1}^m \alpha_k < b \}$ we then have $$\begin{aligned} \PP \Bigg(\sum_{k=1}^m \rho \lambda_k < b \Bigg) & \, \dot\geq \, \bigg( \rho^{-m-m(N-M)-m(m-1)} \int_{\mathcal{C}'_m} \prod_{i=1}^m \left( \alpha_i^{N-M} e^{-\alpha_i/\rho} \right) \prod_{i<j\leq m}(\alpha_i-\alpha_j)^2 \ d\alpha_1 \ldots d\alpha_m \bigg) \notag \\ & \geq \rho^{-m(N-M+m)} \int_{\mathcal{C}'_m} \prod_{i=1}^m \left( \alpha_i^{N-M} e^{-\alpha_i} \right) \prod_{i<j\leq m}(\alpha_i-\alpha_j)^2 \ d\alpha_1 \ldots d\alpha_m \label{ineq:no-intg} \end{aligned}$$ for $\rho \geq 1$, as we have then $e^{- \alpha_i/\rho} \geq e^{- \alpha_i}$ for $i=1, \ldots, m$. As $b>0$ it is straightforward to see that the integral in is nonzero but also finite and independent from $\rho$; it is therefore asymptotically equivalent to $1$ in the sense of , yielding $$\PP \Bigg( \sum_{k=1}^m \rho \lambda_k < b \Bigg) \dot\geq \ \rho^{-m(N-M+m)},$$ which concludes the proof. {#apx:prob_first_ev_bounded} We prove in this section that $\PP\left( \mathcal{B}_1 \right) \dot\leq \, \rho^{-m(M-N+m)}$, where the set $\mathcal{B}_1$ is defined by $$\mathcal{B}_1= \{ \lambda_1 \leq \lambda_2 \ldots \leq \lambda_M, \, 0 < \lambda_k \leq b, \, k=1,\ldots,m \},$$ with $b>0$ and $\lambda_1, \ldots, \lambda_M$ the ordered eigenvalues of the Wishart matrix $\H^\H$. We use the same approach as in Appendix \[apx:prob\_sum\_first\_ev\]. For we note $p_{M,N}$ the joint probability density function of the ordered eigenvalues of a $M \times M$ Wishart matrix with scale matrix $\I_M$ and $N$ degrees of freedom, the probability $\PP(\mathcal{B}_1)$ can be written as $$\PP(\mathcal{B}_1) = \int_{(\lambda_1, \ldots, \lambda_M) \in \mathcal{B}_1} p_{M,N}(\lambda_1, \ldots, \lambda_M) \ d\lambda_1 \ldots d\lambda_M.$$ Similarly to Appendix \[apx:prob\_sum\_first\_ev\] we try to upper bound $\PP(\mathcal{B}_1)$ by the product of two integrals, one containing the $m$ first eigenvalues and the other the $M-m$ remaining eigenvalues. We first replace $p_{M,N}$ by it explicit expression : $$\begin{aligned} \PP(\mathcal{B}_1) & = K_{M,N}^{-1} \int_{(\lambda_1, \ldots, \lambda_M) \in \mathcal{B}_1} \prod_{i=1}^M \lambda_i^{N-M} e^{-\lambda_i} \prod_{i<j}(\lambda_i-\lambda_j)^2 \ d\lambda_1 \ldots d\lambda_M \\ & \doteq \int_{(\lambda_1, \ldots, \lambda_M) \in \mathcal{B}_1} \Bigg( \prod_{i=1}^m \left( \lambda_i^{N-M} e^{-\lambda_i} \right) \prod_{i<j\leq m}(\lambda_i-\lambda_j)^2 \Bigg) \\ & \fhs \cdot \Bigg( \prod_{i=m+1}^M \left( \lambda_i^{N-M} e^{-\lambda_i} \right) \prod_{i\leq m<j}(\lambda_i-\lambda_j)^2 \prod_{m<i<j}(\lambda_i-\lambda_j)^2 \Bigg) \ d\lambda_1 \ldots d\lambda_M. \end{aligned}$$ Note that we dropped the normalizing constant $K_{M,N}$, as $K_{M,N}^{-1} \doteq 1$. For $i \leq m < j$, we have $\|\lambda_i - \lambda_j\| \leq \lambda_j$ and thus $\prod_{i\leq m<j}(\lambda_i-\lambda_j)^2 \leq \prod_{j=m+1}^M \lambda_j^{2m}$, yielding $$\begin{aligned} \PP(\mathcal{B}_1) & \,\dot\leq\, \int_0^{b/\rho} \int_{\lambda_1}^{b/\rho} \bhs \ldots \int_{\lambda_{m-1}}^{b/\rho} \int_{\lambda_m}^{+\infty} \bhs \ldots \int_{\lambda_{M-1}}^{+\infty} \Bigg( \prod_{i=1}^m \left( \lambda_i^{N-M} e^{-\lambda_i} \right) \prod_{i<j\leq m}(\lambda_i-\lambda_j)^2 \Bigg) \\ & \fhs \cdot \Bigg( \prod_{i=m+1}^M \left( \lambda_i^{N+2m-M} e^{-\lambda_i} \right) \prod_{m<i<j}(\lambda_i-\lambda_j)^2 \Bigg) \ d\lambda_1 \ldots d\lambda_M \end{aligned}$$ In order to obtain two separate integrals we discard the $\lambda_m$ in the integral bound simply by noticing that $\lambda_m>0$, therefore $$\begin{aligned} \PP(\mathcal{B}_1) & \,\dot\leq\, \Bigg( \int_0^{b/\rho} \int_{\lambda_1}^{b/\rho} \bhs \ldots \int_{\lambda_{m-1}}^{b/\rho} \prod_{i=1}^m \left( \lambda_i^{N-M} e^{-\lambda_i} \right) \prod_{i<j\leq m}(\lambda_i-\lambda_j)^2 \ d\lambda_1 \ldots d\lambda_m \Bigg) \\ & \fhs \cdot \Bigg( \int_0^{+\infty} \int_{\lambda_{m+1}}^{+\infty} \bhs \ldots \int_{\lambda_{M-1}}^{+\infty} \prod_{i=m+1}^M \left( \lambda_i^{N+2m-M} e^{-\lambda_i} \right) \prod_{m<i<j}(\lambda_i-\lambda_j)^2 \ d\lambda_{m+1} \ldots d\lambda_M \Bigg) \end{aligned}$$ As the second integral (in $\lambda_{m+1}$, …, $\lambda_M$) is nonzero, finite and independent of $\rho$ it is asymptotically equivalent to $1$ in the sense of . Hence, $$\PP(\mathcal{B}_1) \,\dot\leq\, \int_0^{b/\rho} \int_{\lambda_1}^{b/\rho} \bhs \ldots \int_{\lambda_{m-1}}^{b/\rho} \prod_{i=1}^m \left( \lambda_i^{N-M} e^{-\lambda_i} \right) \prod_{i<j\leq m}(\lambda_i-\lambda_j)^2 \ d\lambda_1 \ldots d\lambda_m. \label{ineq:PB1-1intg}$$ We now make the substitutions $\alpha_i=\rho\lambda_i$ for $i=1, \ldots, m$ inside the remaining integral. $$\begin{aligned} & \int_0^{b/\rho} \int_{\lambda_1}^{b/\rho} \bhs \ldots \int_{\lambda_{m-1}}^{b/\rho} \prod_{i=1}^m \left( \lambda_i^{N-M} e^{-\lambda_i} \right) \prod_{i<j\leq m}(\lambda_i-\lambda_j)^2 \ d\lambda_1 \ldots d\lambda_m \notag \\ & \hspace{15pt} = \ \rho^{-m(N-M+m)} \int_0^{b} \int_{\alpha_1}^{b} \bhs \ldots \int_{\alpha_{m-1}}^{b} \prod_{i=1}^m \left( \alpha_i^{N-M} e^{-\alpha_i/\rho} \right) \prod_{i<j\leq m}(\alpha_i-\alpha_j)^2 \ d\alpha_1 \ldots d\alpha_m \notag \\ & \hspace{15pt} \leq \rho^{-m(N-M+m)} \mhs \int_0^{b} \int_{\alpha_1}^{b} \bhs \ldots \int_{\alpha_{m-1}}^{b} \prod_{i=1}^m \alpha_i^{N-M} \mhs \prod_{i<j\leq m} \mhs\mhs (\alpha_i-\alpha_j)^2 \ d\alpha_1 \ldots d\alpha_m , \label{ineq:rem_intg} \end{aligned}$$ as $e^{-\alpha_i/\rho} \leq 1$. The remaining integral in is nonzero ($b > 0$), finite and does not depend on $\rho$; therefore, is asymptotically equivalent to $\rho^{-m(N-M+m)}$ in the sense of . Coming back to we obtain $$\PP(\mathcal{B}_1) \ \dot\leq \ \rho^{-m(N-M+m)}.$$ {#apx:ang_par} In this appendix, we review the results of [@dita2003factorization; @lundberg2004haar] for the reader’s convenience. It has been shown in [@dita2003factorization] that any $n \times n$ unitary matrix $A_n$ can be written as $$A_n = d_n {\mathcal{O}}_n \begin{bmatrix} 1 & 0 \\ 0 & A_{n-1} \end{bmatrix}, \label{eq:unitary_dec}$$ with $A_{n-1}$ a $(n-1) \times (n-1)$ unitary matrix, $d_n$ a diagonal phases matrix, that is $d_n = \mathrm{diag}(e^{i\varphi_1}, \ldots, e^{i\varphi_n})$ with $\varphi_1, \ldots, \varphi_n \in [0, 2\pi]$, and $\mathcal{O}_n$ an orthogonal matrix (the angles matrix). Matrix $\mathcal{O}_n$ can be written in terms of parameters $\theta_1, \ldots, \theta_n \in [0, \frac{\pi}{2}]$ thanks to the following decomposition: $\mathcal{O}_n = J_{n-1,n} J_{n-2,n-1} \ldots J_{1,2}$, where $$J_{i,i+1}= \begin{bmatrix} \I_{i-1} & 0 & 0 & 0 \\ 0 & \cos\theta_i & -\sin\theta_i & 0 \\ 0 & \cos\theta_i & -\sin\theta_i & 0 \\ 0 & 0 & 0 & \I_{n-i-1} \end{bmatrix}.$$ Let $\U_M$ be a $M \times M$ unitary Haar distributed matrix. Then, using decomposition , $$\U_M= \D_M({\boldsymbol}\varphi_1) \V_M({\boldsymbol}\theta_1) \begin{bmatrix} 1 & 0 \\ 0 & \U_{M-1} \end{bmatrix},$$ with ${\boldsymbol}\varphi_1=(\varphi_{1,1},\ldots,\varphi_{1,M}) \in [0,2\pi]^M$, ${\boldsymbol}\theta_1=(\theta_{1,1},\ldots,\theta_{1,M-1}) \in [0,\frac{\pi}{2}]^{M-1}$, $\D_M({\boldsymbol}\varphi_1)$ the diagonal matrix defined by $\D_M({\boldsymbol}\varphi_1)=\mathrm{diag}(e^{i\varphi_{1,1}},\ldots,e^{i\varphi_{1,M}})$, $\V_M({\boldsymbol}\theta_1)$ the orthogonal matrix defined by $\V_M({\boldsymbol}\theta_1)=J_{M-1,M} J_{M-2,M-1} \ldots J_{1,2}$ and $\U_M$ a $M-1 \times M-1$ unitary matrix. Matrix $\U_{M-1}$ can naturally be similarly factorized. Similarly to [@lundberg2004haar], we can show that, in order $\U_M$ to be a Haar matrix it is sufficient that $(\varphi_{1,i})_{i=1,\ldots,M}$ are i.i.d. random variables uniformly distributed over interval $[0, 2\pi[$, that $\theta_{1,1},\ldots,\theta_{1,M-1}$ are independent with densities respectively equal to $(\sin \theta_1)^{M-2}, (\sin \theta_2)^{M-3}, \ldots, (\sin \theta_{M-2}), 1$ and independent from ${\boldsymbol}\varphi_1$ and that $\U_{M-1}$ is Haar distributed and independent from ${\boldsymbol}\varphi_1$ and ${\boldsymbol}\theta_1$. The proof consists in first showing, by a simple variable change, that if the $(\varphi_{1,i})_{i=1,\ldots,M}$ and the $\theta_{1,1},\ldots,\theta_{1,M-1}$ follow the mentioned distributions then $\D_M({\boldsymbol}\varphi_1) \V_M({\boldsymbol}\theta_1)$ is uniformly distributed over the unity sphere of $\mathbb{C}^M$. The proof is then completed by showing that if $\U_{M-1}$ is a Haar matrix independent from ${\boldsymbol}\varphi_1$ and $\theta_1$ then $\U_M$ is Haar distributed. Finally one can parameterize a Haar matrix $\U_M$ by ${\boldsymbol}\varphi_1$, $\theta_1$ and $\U_{M-1}$. Repeating the same parametrization for $\U_{M-1}$ we obtain that $\U_M$ can be parameterized by the $M^2$ following independent variables $$\begin{aligned} &(\varphi_{1,1}, \ldots, \varphi_{1,M}), (\theta_{1,1}, \ldots, \theta_{1,M-1}), (\varphi_{2,1}, \ldots, \varphi_{2,M-1}), (\theta_{2,1}, \ldots, \theta_{2,M-2}), \ldots, \\ &(\varphi_{M-2,1}, \varphi_{M-2,2}), \theta_{M-2,1}, \varphi_{M-1,1}, \end{aligned}$$ whose probability laws are almost surely positive.	ArXiv
--- abstract: \| We represent a theory of polymer gelation as an analogue of liquid-glass transition in which elastic fields of stress and strain shear components appear spontaneously as a consequence of the cross-linking of macromolecules. This circumstance is explained on the basis of obvious combinatoric arguments as well as a synergetic Lorenz system, where the strain acts as an order parameter, a conjugate field is reduced to the elastic stress, and the number of cross-links is a control parameter. Both the combinatoric and synergetic approaches show that an anomalous slow dependence of the shear modulus on the number of cross-links is obtained. [PACS:]{} 05.70Ln, 47.17+e, 61.43Fs [Keywords:]{} [Polymer gelation; sol-gel transition; Shear stress/strain; Shear modulus]{} address: - \| Sumy State University\ Rimskii-Korsakov St. 2, Sumy, 40007 Ukraine\ - \| Department of Macromolecular Physics, Faculty of Mathematics & Physics\ Charles University\ V Holešovičkách 2, 180 00 Prague 8, Czech Republic\ author: - 'Alexander I. Olemskoi' - Ivan Krakovský title: 'Two Simple Approaches to Sol-Gel Transition' --- INTRODUCTION ============ Within the phenomenological framework the basic distinction between a liquid and a glass consists in character of the relaxation law of shear components of the elastic stress: if in an ideal glass they are kept infinitely long, in a liquid such relaxation proceeds in the finite time $\tau = \eta /G$, where $\eta$ is the dynamical shear viscosity and $G$ is the shear modulus [@1]. In naive manner it is possible to assume, that a glass transition is caused purely by kinetic effect of a liquid freezing, for which viscosity $\eta$ gets infinite value for a finite value of the shear modulus $G$ [@2]. However in the course of usual second-order phase transition, where infinite increase of the $\tau$ at critical point is also observed the situation is reverse. Really, proceeding from the viscoelastic liquid to a general case, one has $\tau = \chi / \gamma$, where $\chi$ is the generalized susceptibility and $\gamma$ is the kinetic coefficient (in the case under consideration they are reduced to the quantities $G^{-1},~\eta^ {-1}$, respectively) [@3]. The infinite increase of a susceptibility $\chi$ occurs and a kinetic coefficient $\gamma$ does not manifest any peculiarity at the phase transition. In our case, this is equivalent to the situation when the shear modulus $G$ is approaching zero value at a fixed viscosity $\eta$. This situation corresponds to a viscoelastic transition [@4]. Usually under a glass transition, the transition into a glassy state which occurs during fast cooling of a liquid is understood. However, an analogous type of transition occurs during polymer network formation ([polymer gelation]{}) from monomers or linear macromolecules ([sol]{}) by means of a chemical reaction at constant temperature. This transition is referred to as the [sol-gel transition]{}. In such a case, thermodynamic peculiarities are observed as, e.g., appearance of the shear modulus when the system reaches a critical point.[^1] The first theoretical representations of such type transitions have been elaborated in classical works by Flory [@flory1] – [@flory3]. Modern description of the glassy state of the macromolecule networks (see [@G] – [@E]) is based on pioneering contributions to the theory of soft condensed matter: the Deam-Edwards theory [@DE] of a cross-linked tangled macromolecule, and the Edwards-Anderson theory [@EA] of spin glass. The aim of this paper is to elaborate the simplest theoretical scheme so that the transition occurring during polymer gelation can be explained on the base of obvious combinatoric arguments (Section 2) as well as within framework of a synergetic theory (Section 3). In the former case, the critical point and behaviour of the system behind the critical point are obtained using information about the cross-linking properties of the constituent parts of the system ([monomers]{}) and extent of the cross-linking process, only. In Section 3, the starting point is that a sol-gel transition is ensured by self-organization of the elastic fields of stress and strain shear components and the number of cross-links. Both the combinatoric approach and the synergetic one allow to obtain anomalous dependence of the shear modulus on the number of cross-links. Combinatoric approach ===================== The gelation process has been described first by Flory[@flory1] and Stockmayer[@stock]. They used a laborious approach based on combinatoric considerations of the most probable composition of the system. A much more effective variant of this approach, exploiting theory of branching processes, has been developed later by Dobson and Gordon [@dg1; @dg2]. This approach allows also an analysis of structural details of the system behind the critical point, see, e.g. [@dusek]. As the simplest case let us consider the gelation process in the system which at the beginning consists of a large number of monomers $\cal N$, wearing $f$ functional terminal groups of independent reactivity. Monomers react mutually and irreversibly via their terminal groups and if $f>2$, branched molecules of increasing size and complexity are formed progressively in the system. Extent of the reaction is described by the conversion of terminal groups $\alpha$, which is defined as the ratio of the number of groups consumed by the reaction at given time and the starting number of groups. Eventually, when a critical conversion $\alpha_{\rm c}$ is achieved a molecule of macroscopic dimensions ([gel]{}) appears in the system [[@flory1], [@flory2]]{}. Gel contains cycles of long sequences of linked monomers and, consequently, attains elastic properties. Flory [@flory3] has shown that shear modulus of gel $G$ is proportional to its “cycle rank” $\xi$ defined as the number of “superfluous” links formed in gel which can be cut without breaking integrity of gel. In other words, cycle rank is the number of cuts needed for elimination of all cycles from gel. The key role in the approach is played by the extinction probability $v$, which is a probability that a link formed in the process has just a finite continuation. It can be shown easily [@dg1] that for the system considered here the extinction probability can be obtained as a root of the equation $$v=(1-\alpha+\alpha v)^{f-1} \label{v1}$$ satisfying the condition $0\le v\le 1$. Eq.(\[v1\]) expresses fact that a given link has a finite continuation in a direction only if $f-1$ functional groups remaining on the monomer connected by the link are either unreacted (with probability $1-\alpha$) or reacted giving rise to links with finite continuation only (with probability $\alpha v$). Below the critical conversion, the only solution of Eq.(\[v1\]) is $v=1$, i.e., only molecules of finite size are formed in the system. However, behind the critical conversion monomers can be found either in sol or in gel: $v<1$. As a measure of the “distance” from the critical conversion let us introduce the parameter $\epsilon$ defined as $$\epsilon\equiv\alpha-\alpha_{\rm c} \label{eps}$$ and expand the extinction probability in a series: $$v=1+A_{1}\epsilon+A_{2}\epsilon^{2}+\ldots \label{v3}$$ Substituting Eqs.(\[eps\]) and (\[v3\]) into Eq.(\[v1\]), expressions for the critical conversion and the parameters $A_{i}$ are obtained as $$\begin{aligned} \alpha_{\rm c}&=&\frac{1}{f-1}\label{alpha},\\ A_{1}&=&-2~\frac{(f-1)^{2}}{f-2}\label{a1},\\ A_{2}&=&\frac{4}{3}~\frac{(2f-3)(f-1)^{3}}{(f-2)^{2}}. \label{a2}\end{aligned}$$ The expression for the cycle rank can be found in the following way. Obviously, to join $\cal N$ monomers into a cycle-free structure, ${\cal N}-1$ links are necessary. By definition, the cycle rank $\xi$ is the number of “superfluous” links formed in gel, i.e., the difference between the number ${\rm N}_{\rm G}$ of all links formed in gel and the number ${\cal N}_{\rm G}-1$ of links sufficient to join together ${\cal N}_{\rm G}$ monomers of gel: $$\xi\equiv {\rm N}_{\rm G}-({\cal N}_{\rm G}-1)\simeq {\rm N}_{\rm G}-{\cal N}_{\rm G} \label{xi1}$$ as ${\rm N}_{\rm G}$, ${\cal N}\gg 1$. Correspondingly, the number of links formed in gel is the difference between the numbers of links formed in the total system and in sol, i.e., $${\rm N}_{\rm G}=\frac{1}{2}{\cal N}\alpha f(1-v^2) \label{ng}$$ as any link in sol has to have finite continuations in two directions. On the other hand, the number ${\cal N}_{\rm G}$ of monomers incorporated in gel is the difference between the total number of monomers and the number of monomers in sol which is made of monomers with links of only finite continuation: $${\cal N}_{\rm G}={\cal N}-{\cal N}(1-\alpha+\alpha v)^{f}. \label{Ng}$$ By virtue of Eqs.(\[xi1\]) – (\[Ng\]), one gets for cycle rank of the system considered $$\frac{\xi}{\cal N} = (1-\alpha+\alpha v)^{f}+\frac{1}{2}\alpha f(1-v^{2})-1. \label{xi2}$$ Finally, substituting Eqs.(\[v3\]),(\[a1\]) and (\[a2\]) into the formula (\[xi2\]), one gets the necessary expansion: $$\frac{\xi}{\cal N} = \frac{2}{3}~\frac{(f-1)^{4}}{(f-2)^2}f~\epsilon^3+O(\epsilon^{4}). \label{xi3}$$ Respectively, the weight fraction of gel $w_{\rm G}\equiv{\cal N}_{\rm G}/{\cal N}$ is determined by Eq.(\[Ng\]) to read $$w_{\rm G}=2~\frac{(f-1)^{2}}{f-2}~\epsilon+O(\epsilon^{2}). \label{wg2}$$ So, if the critical exponent of the gel weight fraction is equal 1 as usual, this for the cycle rank is anomalous large being equal 3. It is worthwhile to note that combinatoric approach is based exclusively on the information about the functionality of the monomers and extent of the chemical reaction between the monomers in the system considered. Synergetic approach =================== Now, let us consider the polymer network as a viscoelastic continuum matter that is characterized by the shear modulus $G$ and the shear viscosity $\eta$. Process of polymer gelation is determined by the number of the cross-links $N$, which value is different from a stationary magnitude $N_0$ at a time $t$. Therein, an elastic state of the polymer is defined by the shear component of the proper (internal) values of deformation $\varepsilon(t)$ and stress $\sigma(t)$. The keypoint is that these values are not reduced to the external elastic deformation $e\ll 1$ and stress $\sigma_e\ll G$, in particular they can get large magnitudes $\varepsilon\sim 1$, $\sigma\sim G$. Our consideration of evolution of the elastic continuum with the internal structure is stated on the phenomenological equations by Maxwell-Kelvin [@1] $${d\varepsilon\over dt}=-{\varepsilon\over \tau}+{\sigma\over\eta}, \label{a}$$ $${d\sigma\over dt}=-{\sigma\over \tau_{\sigma}}+ g_{\sigma}\varepsilon N. \label{b}$$ Here we introduce a macroscopic relaxation time $\tau$ for the strain and a microscopic one $\tau_{\sigma}$ for the stress, as well as a constant $g_{\sigma} > 0$ of the positive feedback between the deformation $\varepsilon$ and the number of cross-links $N$. Within the microscopic interval $t\gg\tau_{\sigma}$, steady-state condition $d\sigma/dt=0$ in Eq.(\[b\]) leads to the Hooke law with the microscopic shear modulus $$G_{\sigma}\equiv\tau_{\sigma} g_{\sigma}N \label{c}$$ being determined by the number of cross-links $N$. Respectively, within a macroscopic interval $t\gg\tau$ Eq.(\[a\]) gives the magnitude $G\equiv\eta/\tau$ that is characteristic for the usual modulus of the viscoelastic matter. Lastly, a variation rate $dN/dt$ of the internal degree of freedom is supposed to be determined by the equation $${dN\over dt}={N_0-N\over \tau_N}-g_N\sigma\varepsilon \label{e}$$ where $\tau_N$ is a mesoscopic relaxation time, $g_N > 0$ is constant of negative feedback between the deformation $\varepsilon$ and the stress $\sigma$. Within a mesoscopic interval $\tau_N\ll t\ll\tau$, Eq.(\[e\]) determines a steady-state value $$N=N_0-\tau_N g_N\sigma\varepsilon \label{f}$$ that is smaller than the magnitude $N_0$ fixed by external conditions due to the fact that the elastic energy is proportional to the product $\sigma\varepsilon$. System of Eqs.(\[a\]), (\[b\]) and (\[e\]) is known in synergetics [@Haken] as the Lorenz system where the deformation $\varepsilon$, the stress $\sigma$ and the number of cross-links $N$ play roles of an order parameter, a conjugate field and a control parameter, respectively. It is very important for following considerations that the relation between micro-, meso- and macroscopic values of the relaxation times $$\tau_{\sigma}, \tau_N \ll \tau \label{g}$$ is satisfied. Due to this condition the evolution of the quantities $\sigma$, $N$ turns out to be subordinated to the long-time variation of $\varepsilon$. A peculiarity of the Lorenz system consists in linear character of the equation (\[a\]) for the order parameter $\varepsilon$ and in non-linearity of equations (\[b\]), (\[e\]) for the conjugate field $\sigma$ and the control parameter $N$. The negative nature of non-linearity in Eq.(\[e\]) means a decrease of the number $N$ of cross-links. Evidently, this fact reflects Le Chatelier principle. A non-linear term in Eq.(\[b\]) for a field $\sigma$ describes the positive feedback causing the system self-organization. Expressions (\[a\]), (\[b\]) and (\[e\]) form the complete system of equations determining the polymer cross-linking behaviour. Because of a slow evolution, the order parameter $\varepsilon(t)$ subordinates variations of quantities $\sigma (t)$, $N(t)$, so that one can take $d\sigma/dt =dN/dt =0$ within the framework of the adiabatic approximation [@Haken]. Then $N$, $\sigma$ are expressed in terms of $\varepsilon$ by the equations: $$N={N_0\over 1+ \varepsilon^2/ \varepsilon^2_m},\quad \varepsilon_m^{-2}\equiv \tau_{\sigma} \tau_N g_{\sigma} g_N; \label{h}$$ $$\sigma=G_0{\varepsilon \over 1+ \varepsilon^2/ \varepsilon^2_m},\quad G_0 \equiv \tau_{\sigma} g_{\sigma} N_0. \label{i}$$ In accordance with Eq.(\[h\]) the number of cross-links $N$ decreases monotoneously with increase of the strain $\varepsilon$ from the value $N_0$ at $\varepsilon = 0$ to $N_0/2$ at $\varepsilon =\varepsilon_m$.[^2] In Eq.(\[i\]) the elastic stress in terms of the strain has the linear form of the Hooke law at $\varepsilon\ll \varepsilon_m$ with the effective shear modulus $G_0$. Then, at $\varepsilon = \varepsilon_m$ the function $\sigma (\varepsilon) $ has a maximum and at $\varepsilon > \varepsilon_m$ it decreases which is physically meaningless. Thus, the constant $\varepsilon_m$, defined by the second equation (\[h\]), has the meaning of the maximum achievable strain. Substituting Eq.(\[i\]) in Eq.(\[a\]) we find the equation describing evolution of a system in the course of the sol-gel transition: $${d\varepsilon\over dt}=-\gamma {\partial E \over \partial\varepsilon},\qquad \gamma\equiv {\varepsilon_m^2\over\tau T~N_0} \label{j}$$ where constant $\gamma$ plays a role of the kinetic coefficient. Behaviour of the system under consideration is determined by the dependence $E (\varepsilon) $ of the elastic energy on the strain: $$E \equiv {T~N_0\over 2} \left [{\varepsilon^2 \over \varepsilon^2_m} - {N_0\over N_c} \ln\left(1+{\varepsilon^2\over \varepsilon^2_m}\right)\right] \label{k}$$ where the characteristic value of the number of cross-links is introduced $$N_c \equiv {\eta\over\tau_{\sigma} g_{\sigma}}. \label{l}$$ At $N_0\le N_c$ dependence (\[k\]) is monotoneously increasing with a minimum at the point $\varepsilon = 0$. It means that in the stationary state ($\dot\varepsilon {=} 0$) the elastic strain is not spontaneous. Thus, a liquid state is realized, in which the strain caused by the external stress relaxes during the time $$\tau_ {ef} = \tau \left (1 - N_0/N_c\right)^{-1}. \label{m}$$ The relaxation time increases infinitely when the number of cross-links $N_0$ reaches the critical value $N_c$ and at $N_0> N_c$ the system undergoes a sol-gel transition. In the gel state the multiplier 1/2 appears in the dependence (\[m\]), and the minimum of the elastic energy corresponds to the elastic strain $${\varepsilon^2 \over \varepsilon^2_m} = {N_c \over N_0}~{N_0-N_c\over N_c}\equiv {\epsilon \over 1 + \epsilon} \label{n}$$ where we introduce the distance from the critical value $N_c$ $$\epsilon\equiv {N_0-N_c\over N_c} \label{o}$$ being equivalent to the definition (\[eps\]). Inserting Eq.(\[m\]) into the dependence (\[k\]), we obtain the elastic energy of the steady-state: $$E_0\equiv E(\varepsilon_0)=-~{TN_0\over 4}~\epsilon^2 + O(\epsilon^3). \label{p}$$ As would be expected, this energy is proportional to the second power of the parameter (\[o\]) and is negative in nature (the latter means the energy benefit of the gel state in comparison with the liquid state). Taking into account that the glassy state is determined by density of the localized monomers, let us find now the shear modulus of the appeared gel state. It is principally important in our considerations that the gel state is determined by the value of the Deam-Edwards parameter of localization $\omega$ [@DE] which is supposed to be proportional to the square of the proper strain $\varepsilon_0$ of the matter. Under the condition of the appearance of the elastic strains $e$, a generalized Deam-Edwards parameter $\omega(e)$ has to be considered which is related to the condition $\omega(e=0)\equiv \omega$. Then, expanding the function $\omega(e)$ into a series and keeping the first two terms only, one obtains $$\omega(e)\simeq \omega(1+e^2), \quad \omega\propto\varepsilon^2_0,\quad e\ll 1. \label{r1}$$ By virtue of the parity condition $\omega(e)=\omega(-e)$, this expansion does not contain a linear term.[^3] Because of that the total strain $\varepsilon\propto\sqrt{\omega(e)}$, internal one $\varepsilon_0\propto\sqrt{\omega}$ and elastic one $e$ are connected by the following relation: $$\varepsilon^2 = \varepsilon_0^2 (1+e^2), \qquad e\ll 1. \label{r}$$ The key point is that Eq.(\[r\]) supposes the additivity rule holds not for quantities $\varepsilon$, $\varepsilon_0$, $e$ themselves, but for their squares. The physical reason for such a situation is that the system under consideration is random in character and described by a symmetrical distribution function. Therefore, all odd-power moments vanish identically and making use of the additivity rule (\[r\]) for variances follows. It is worthwhile to note the seeming contradiction between above relations for the localization parameter $\omega$ and the Deam-Edwards results [@DE]. Obvious reason consists in that the formers are obtained within framework of the mean-field theory, whereas the latters suppose fluctuation effects. According to [@DE] the cross-linking process is not sensible to strain $\varepsilon$ and corresponding localization parameter $\omega$ is proportional to the cross-link number $N$ but not to the difference $N-N_c$, as in Eqs.(\[n\]), (\[o\]). In our opinion, this is caused by non-self-consistency of the approach [@DE] in sense that the stress field $\sigma$ is switched off. On the other hand, making use of the statistical scheme [@DE] arrives at the strain-dependence for the localization parameter $\omega$ due to appearance of the polymer network entanglement, whereas within the above phenomenological approach this dependence has to be postulated. A contribution to the elastic energy caused by the external strain is determined by equality $$\Delta E(e)\equiv \| E\left(\varepsilon(e)\right)- E(\varepsilon_0)\|. \label{s}$$ With use of Eqs. (\[k\]), (\[p\]), (\[r\]) and expansion $\ln(1+x)\approx x-x^2/2+x^3/3$, $ x\ll 1$ the expression $$\Delta E(e)\approx N_0~{T\epsilon^3\over 2} e^2 \label{u}$$ is easily obtained where only the first non-zero term is kept. Comparing this relation with the usual expression for the elastic energy [@1] $$\Delta E(e)\equiv V~{G\over 2} e^2 \label{v}$$ where $V$ is volume, we arrive at the final expression for the shear modulus of the gel state of the polymer network: $$G={T\epsilon^3\over\Omega}, \quad \epsilon\equiv {N_0-N_c\over N_c}\ll 1, \quad \Omega\equiv {V\over N_0}. \label{w}$$ The notable peculiarity of this result consists in that, in accordance with previous result (\[xi3\]), the shear modulus is proportional to the third power of the distance $\epsilon$. It is worthwhile to note that an expression of this kind can be obtained only within framework of the synergetic approach used, but not on the basis of the phase transition theory by Landau. Finally, comparing Eqs. (\[xi3\]) and (\[w\]), we obtain the relation between the micro- and macroscopic parameters of the gel under consideration in the proximity of the critical point ($\epsilon\ll 1$): $$G=g~\frac{\xi}{\cal N}~\frac{T}{\Omega},\quad g\equiv\frac{3}{2}~f^{-1}~\frac{(f-2)^2}{(f-1)^{4}}. \label{xy}$$ [00]{} L. D. Landau, E. M. Lifshitz, [Elasticity Theory]{} (Nauka, Moscow, 1987). A. Havranek, M. Marvan, Ferroelectrics [176]{}, 25 (1996). A. I. Olemskoi, [Theory of Structure Transformations in Non-equilibrium Condensed Matter]{} (NOVA Science, N.-Y., 1999). V. G. Bar’yakhtar, A. I. Olemskoi, Sov. Phys. Solid State [33]{}, 2705 (1991). P. J. Flory, J.Am.Chem.Soc. [63]{}, 3083 (1941). P. J. Flory, [Principles of Polymer Chemistry]{} (Cornell University Press, 1953). P. J. Flory, Proc.Roy.Soc.London A[351]{}, 351 (1976). P. M. Goldbart, H. E. Castillo, A. Zippelius, Adv. Phys. [45]{}, 393 (1996). S. Panyukov, Y. Rabin, Phys. Rep. [269]{}, 1 (1996). S. F. Edwards, H. Takano, E. M. Terentjev, cond-mat/0007270. R. T. Deam, S. F. Edwards, Phil.Trans.R.Soc. A[280]{}, 317 (1976). S. F. Edwards, P. W. Anderson, J.Phys. F[5]{}, 965 (1975). W. H. Stockmayer, J.Chem.Phys. [11]{}, 45 (1943). G. R. Dobson, M. Gordon, J.Chem.Phys. [41]{}, 2389 (1964). G. R. Dobson, M. Gordon, J.Chem.Phys. [42]{}, 705 (1965). K. Dušek, Adv.Polym.Sci. [78]{}, 1 (1986). G. Haken, [Synergetics]{} (Springer-Verlag, Berlin, Heidelberg, N.Y., 1983). [^1]: Of course, when the temperature of the polymer network formed is decreased the kinetic glass transition in above sense occurs, as well. [^2]: Obviously this decrease is caused by the negative feedback in Eq.(\[e\]), that is reflection of Le Chatelier principle for analyzes problem. Actually, a liquid self-organization, resulting in a sol-gel transition, is caused by the positive feedback between the strain $\varepsilon$ and the number of cross-links $N$ in Eq.(\[b\]). Hence, the increase of the number of cross-links should intensify the self-organization effect. However, according to Eq.(\[h\]) system behaves in such way that the consequence of self-organization, i.e., growth of the elastic strain, leads to decrease of its origin – the number of cross-links. [^3]: Otherwise, the usual squared dependence (\[u\]) will not be obtained.	ArXiv
--- author: - 'F. Arrigoni Battaia' - 'Chian-Chou Chen' - 'M. Fumagalli' - Zheng Cai - 'G. Calistro Rivera' - Jiachuan Xu - 'I. Smail' - 'J. X. Prochaska' - Yujin Yang - 'C. De Breuck' bibliography: - 'allrefs.bib' date: 'Received —; accepted —' subtitle: 'The environment and the powering of an Enormous Lyman-Alpha Nebula' title: 'Overdensity of submillimeter galaxies around the $z\simeq 2.3$ MAMMOTH-1 nebula' --- Introduction ============ In the present-day Universe, giant elliptical galaxies are found at the centers of massive clusters. Being characterized by old, and coeval stellar populations, these central galaxies must have formed the bulk of their stars in exceptional star-forming events at early epochs, or must have accreted several coeval galaxies (e.g., @Kauffmann96). Indeed, the current hierarchical structure formation model predicts that these central galaxies merge with several nearby satellite galaxies to build up their stellar mass (e.g., @West1994). This violent merging process is thought to take place in the highest density peaks in the early Universe, in the so-called protoclusters. Despite that a lot of effort has been put in characterizing overdensities of galaxies at high-redshift, there is still an open debate on which is the best technique to map protoclusters and on which systems represent the nurseries of present-day massive clusters, and thus the site of formation of elliptical galaxies (e.g., @steidel00 [@Venemans2007; @Dannerbauer2014; @Orsi2016; @Cai2017a; @Miller2018; @Oteo2018]). To date, high-redshift radio galaxies (HzRGs) are one of the best candidates for pinpointing the location of these extremely dense environments (@mileyd08). This result is supported by the rarity of these systems, by Ly$\alpha$ emitter (LAEs) overdensities near them, and in some cases by overdensities in submillimeter observations (@Stevens2003 [@Humphrey2011; @Rigby2014; @Zeballos2018]). Being the host of an active galactic nucleus (AGN) and characterized by intense radio emission, HzRGs are also known for their associated giant Ly$\alpha$ nebulae on hundreds of kpc scales, suggesting the presence of a large amount of gas in these systems (e.g., @rvr+03). This Ly$\alpha$ emission is a complex result of AGN ionization, jet-ambient gas interaction, and intense star formation (@VillarMartin2003 [@Vernet2017]). Despite these pieces of evidence for protoclusters around HzRGs, we have recently discovered enormous Ly$\alpha$ nebulae (ELANe; @Cai2016), more extended than those around HzRGs, and in even more extreme environments at $z\sim2-3$ (@hennawi+15 [@Cai2016; @FAB2018]). The ELANe – with observed Ly$\alpha$ surface brightness SB$_{\rm Ly\alpha}\gtrsim10^{-17}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$ on $\gtrsim 100$ kpc, maximum extents of $>250$ kpc and Ly$\alpha$ luminosities of $L_{\rm Ly\alpha}>10^{44}$ erg s$^{-1}$ – represent the extrema of known radio-quiet Ly$\alpha$ nebulosities. Indeed, previously well studied radio-quiet Ly$\alpha$ nebulae at $z\sim2-6$, [a.k.a]{} Ly$\alpha$ blobs (LABs; e.g., @steidel00 [@matsuda04; @Yang2010; @Matsuda2011; @Prescott2015; @Geach2016; @Umehata2017]), are characterized by smaller luminosities $L_{\rm Ly\alpha}\sim10^{43-44}$ erg s$^{-1}$, and smaller extents (50-120 kpc) down to similar surface brightness levels (SB$_{\rm Ly\alpha}\sim10^{-18}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$). While most of the LABs have a powering mechanism that is still debated (e.g., @Mori2004 [@Dijkstra2009_hd; @Rosdahl12; @Overzier2013; @fab+15a; @Prescott2015b; @Geach2016]), ELANe are usually explained by photoionization and/or feedback activity of the associated quasars and companions (@cantalupo14 [@hennawi+15; @Cai2016; @FAB2018]). The current sample of ELANe still comprises only a handful of objects (@hennawi+15 [@Cai2016; @FAB2018; @Cai2018]). All these ELANe are associated with local overdensities of AGN, with up to 4 known quasars sitting at the same redshift of the extended Ly$\alpha$ emission for the ELAN Jackpot (@hennawi+15). Given the current clustering estimates for AGN, the probability of finding a multiple AGN system is very low, $\approx 10^{-7}$ for a quadrupole AGN system (@hennawi+15). This occurrence makes a compelling case that these nebulosities are sitting in very dense environments. This working hypothesis is further strengthen by the detection of a large number of associated LAEs on small (@FAB2018) and on large scales (@hennawi+15 [@Cai2016]). Such overdensities of LAEs are comparable or even higher than in the case of HzRGs and LABs (@hennawi+15 [@Cai2016; @FAB2018]). Most of the known ELANe (@cantalupo14 [@hennawi+15; @fab+15b; @FAB2018]) show (i) at least one bright type-1 quasar embedded in the extended emission, (ii) non-detections in $\lambda1640$Å and $\lambda1549$Å down to sensitive SB limits ($\sim10^{-18}-10^{-19}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$), and (iii) relatively quiescent kinematics for the Ly$\alpha$ emission (FWHM$\simeq600$ km s$^{-1}$) with a single peaked Ly$\alpha$ line down to the current resolution of the instrument used. Notwithstanding these results, the ELANe and their environment have been currently studied only in unobscured tracers, possibly resulting in a biased vision of the phenomenon. A complete view of these systems requires a multiwavelength dataset. In particular, submillimeter galaxies (SMGs; @Smail1997) have been shown to be linked to merger events (e.g., @Engel2010 [@Ivison2012; @Alaghband-Zadeh2012; @Fu2013; @TC2015; @Oteo2016]) and to be good tracers of protoclusters (e.g., @Smail2014 [@Casey2016; @Hung2016; @Wang2016; @Oteo2018; @Miller2018]). For these reasons, and to directly test whether our newly discovered ELANe could be powered by intense obscured star-formation, we have initiated a submillimeter campaign with the James Clerk Maxwell Telescope (JCMT) and the Atacama Pathfinder EXperiment (APEX) telescopes to map the obscured star-forming activity (if any) associated with these rare systems and their environment. Here we report the results of our observations of the ELAN MAMMOTH-1 at $z=2.319$ (@Cai2016) using the Submillimetre Common-User Bolometer Array 2 (SCUBA-2; @Holland2013) on JCMT. This ELAN has been discovered close to the density peak of the large-scale structure BOSS1441 (@Cai2017a). BOSS1441 has been identified thanks to a group of strong IGM Ly$\alpha$ absorption systems (@Cai2017a). Follow-up narrow-band imaging, together with spectroscopic observations have constrained the Lyman-$\alpha$ Emitters (LAEs, i.e. sources with rest-frame equivalent width EW$_0^{Ly\alpha}>20$Å) in this field (@Cai2017a). With a LAE density of $\approx12\times$ that in random fields in a (15 cMpc)$^3$ volume, BOSS1441 is one of the most overdense fields discovered to date. The ELAN MAMMOTH-1 is unique compared to the other few ELAN so far discovered, showing (i) only a relatively faint source ($i=24.2$) embedded in it, (ii) extended emission ($\gtrsim30$ kpc) in $\lambda1640$Å and $\lambda1549$Å and (iii) double-peaked line profiles with velocity offsets of $\approx700$ km s$^{-1}$ for Ly$\alpha$, , and . In light of these evidences, @Cai2016 explained this ELAN as circumgalactic/intergalactic gas powered by photoionization or shocks due to a galactic outflow, most likely powered by an enshrouded AGN. With the SCUBA-2 data we can start to better constrain the nature of this powering source. This work is structured as follows. In Section \[sec:obs\], we describe our observations and data reduction. In Section \[catalogs\] we present the catalogs at 450 and 850 $\mu$m, along with reliability and completeness tests. In Section \[NC\], we describe how we determined the pure source number counts, estimated the underlying counts model through Monte Carlo simulations, and how we used this models to get the true counts. The same Monte Carlo simulations allowed us to assess the flux boosting (Section \[fluxBoost\]) and the positional uncertainties (Section \[pos\_err\]) inherent to our observations. In Section \[sec:results\] we show (i) the true number counts and compare them to number counts in blank fields, and (ii) the location of the discovered submillimeter sources in comparison to known LAEs. We then discuss our overall detections and the counterpart of the ELAN MAMMOTH-1 in Section \[sec:disc\], and we summarize our results in Section \[sec:summ\]. Throughout this paper, we adopt the cosmological parameters $H_0=70$ km s$^{-1}$ Mpc$^{-1}$, $\Omega_M =0.3$ and $\Omega_{\Lambda}=0.7$. In this cosmology, 1 corresponds to about 8.2 physical kpc at $z=2.319$. All distances reported in this work are proper. Observations and Data Reduction {#sec:obs} =============================== The SCUBA-2 observations for the MAMMOTH-1 field were conducted at JCMT during flexible observing in 2018 January 16, 17, and 18 (program ID: M17BP024) under good weather conditions (band 1 and 2, $\tau_{225{\rm GHz}}\leq 0.07$). The observations were performed with a Daisy pattern covering $\simeq13.7\arcmin$ in diameter, and were centered at the location of the ELAN MAMMOTH-1 as indicated in @Cai2016. Note however that the exact coordinate of the ELAN MAMMOTH-1 have been refined to be R.A. $=$ 14:41:24.456, and Dec. $=$ +40:03:09.45. To facilitate the scheduling we divided the observations in 6 scans/cycles of about 30 minutes, for a total of 3 hours.\ The data reduction follows closely the procedures detailed in @TC2013a. In short, the data were reduced using the Dynamic Iterative Map Maker (DIMM) included in the SMURF package from the STARLINK software (@Jenness2011 [@Chapin2013]). The standard configuration file dimmconfig\_blank\_field.lis was adopted for our science purposes. Data were reduced for each scan and the MOSAIC\_JCMT\_IMAGES recipe in PICARD, the Pipeline for Combining and Analyzing Reduced Data (@Jenness2008), was used to coadd the reduced scans into the final maps. The final maps were applied standard matched filter to increase the point source detectability, using the PICARD recipe SCUBA2\_MATCHED\_FILTER. Standard flux conversion factors (FCFs; 491 Jy pW$^{-1}$ for 450 $\mu$m and 537 Jy pW$^{-1}$ for 850 $\mu$m) with 10% upward corrections were adopted for flux calibration. The relative calibration accuracy is shown to be stable and good to 10% at 450 $\mu$m and 5% at 850 $\mu$m (@Dempsey2013). The final central noise level for our data is 0.88 mJy/beam and 5.4 mJy/beam, respectively at 850 $\mu$m and 450 $\mu$m. In the reminder of this work we focused on the regions of the data characterized by a noise level less than three times the central noise. We refer to this area as effective area. In Figure \[BOSS1441\] we overlay the field-of-view (corresponding to the effective area) of our [SCUBA-2]{} observations (dashed red) on the overdensity of LAEs known from the work of @Cai2017a (green contours). In Table \[obs\] we summarize the center, the effective area, and the central noise ($\sigma_{\rm CN}$) of our observations. ![The galaxy overdensity BOSS1441 at $z=2.32\pm0.02$ (@Cai2017a). The density contours (green) for LAEs are shown in steps of 0.1 galaxies per arcmin$^2$, with the inner density peak of 1.0 per arcmin$^2$. The density contours are shown with increasing thickness for increasing galaxy number density. The brown crosses indicate the positions of known QSOs in the redshift range $2.30\leq z < 2.34$, and thus likely within the overdensity. We also highlight the position of the ELAN MAMMOTH-1 (dotted crosshair), and the effective area of our SCUBA-2 observations (red dashed contour).[]{data-label="BOSS1441"}](Fig_1.pdf){width="1.0\columnwidth"} Source Extraction and Catalogues {#catalogs} ================================ To extract the detections from both maps, we proceed following @TC2013a. We first extracted sources with a peak S/N $>2$ within the effective area of our observations (see Table \[obs\]). Specifically, our algorithm for source extraction finds the maximum pixel within the selected region, takes the position and the information of the peak, and subtracts a scaled PSF centered at such position[^1]. The process has been iterated until the peak S/N went below 2.0. These sources constituted the preliminary catalogs at 850 and 450 $\mu$m. We then cross-checked the two catalogs to find counterparts in the other band. We considered a source as a counterpart if its position at 450 $\mu$m lays within the 850 $\mu$m beam. The final catalogs were built by keeping every $> 4\sigma$ source in the preliminary catalogs, but also every $>3\sigma$ source characterized by a $>3\sigma$ counterpart in the other band. Overall, we discovered 27 sources at 850 $\mu$m and 14 sources at 450 $\mu$m. In Tables \[850\] and \[450\] we list the information for these sources. Figure \[Maps\] shows the final S/N maps at 850 and 450 $\mu$m for the targeted field with the discovered sources over-plotted. ![image](Fig_2.pdf){width="18.5cm"} Reliability of Source Extraction {#jackknife} -------------------------------- To determine the number of spurious sources that could affect our catalogues, we proceeded as follows. First, we applied the source extraction algorithm to the inverted maps. We found 2 and 1 detections at $>4\sigma$ at 850 and 450 $\mu$m, respectively. Second, we constructed true noise maps, we applied the source extraction algorithm, and checked the number of detections with $>4\sigma$. To obtain true noise maps we used the jackknife resampling technique. Specifically, we subtracted two maps obtained by co-adding roughly half of the data for each band. In this way, any real source in the maps is subtracted irrespective of its significance. The residual maps are thus source-free noise maps. To account for the difference in exposure time, we scaled these true noise maps by a factor of $\sqrt{t1\times t2}/(t1+t2)$, with t1 and t2 being the exposure time of each pixel from the two maps. These jackknife maps are characterized by a central noise of 0.88 and 5.39 mJy/beam, respectively at 850 $\mu$m and 450 $\mu$m, in agreement with the noise in the science data. By applying the source extraction algorithm to these maps, we found 1 and 4 detections at $>4\sigma$ at $850$ and $450$ $\mu$m, respectively. We thus expect a similar number of spurious sources in our $>4\sigma$ source catalogs. Further, we tested the number of spurious detections for the $3\sigma$ sources identified as having a counterpart in the other bandpass (lower portion of Table \[850\] and \[450\]) by using once again the jackknife maps. Specifically, from this maps we extracted sources between $3$ and $4\sigma$, and cross-correlated them with the detections in the real data at the other wavelength. We found that none of such spurious sources matched a detection in the real maps. We then repeated the test $1000$ times by randomizing the position of the spurious sources within the effective area of our observations. On average we found $0.3$ and $0.7$ spurious sources at $850$ and $450$ $\mu$m, indicating that sources selected at the $>3\sigma$ level in both bandpasses are even more reliable than $>4\sigma$ sources selected in only one bandpass. Overall these tests suggest that – most likely – the sources at $450$ $\mu$m without a detection at $850$ $\mu$m are spurious for our observations. For the sake of completeness, we decided to list all the sources in our catalogs. As it will be clear from our analysis, our conclusions are not affected. Completeness Tests ------------------ We tested at which flux our data can be considered complete. We proceeded as follows. We took the true noise maps introduced in section \[jackknife\], and populated them with mock sources of a given flux and placed at random positions. We have then extracted the sources, considering them as recovered if the detection is above 4$\sigma$ and within the beam area. Specifically, we injected sources with flux from 0.1 to 25.1 mJy (0.1 to 80.1 mJy) with a step of 0.5 mJy (1.0 mJy) for 850 (450) $\mu$m. For each step in flux we iterate the extraction by introducing 1000 sources. To fully characterize the completeness in the whole extent of the maps, we repeated this procedure for areas of the images characterized by different depths, i.e. $<3\sigma_{\rm CN}$, $2\sigma_{\rm CN}<\sigma<3\sigma_{\rm CN}$, $1.5\sigma_{\rm CN}<\sigma<2\sigma_{\rm CN}$, and $\sigma<1.5\sigma_{\rm CN}$ (see Fig. \[Maps\]). Figure \[completeness\] shows the results of the tests. For the whole area with $<3\sigma_{\rm CN}$, the $50\%$ completeness is at 5.8 and 37 mJy at 850 and 450 $\mu$m, respectively, and the $80\%$ is at 6.8 and 44 mJy, respectively. As expected the central portion of the maps with $\sigma<1.5\sigma_{\rm CN}$ have a better sensitivity, with the $50\%$ completeness being around 4.8 and 30 mJy at 850 and 450 $\mu$m, respectively, and the $80\%$ being around 5.3 and 32 mJy, respectively. ![Top: completeness at $850$ $\mu$m versus flux for different portions of the map, i.e. $2\sigma_{\rm CN}<\sigma<3\sigma_{\rm CN}$, $1.5\sigma_{\rm CN}<\sigma<2\sigma_{\rm CN}$, $\sigma<1.5\sigma_{\rm CN}$, and $<3\sigma_{\rm CN}$ (see Fig. \[Maps\]). Bottom: same as above, but fot the $450$ $\mu$m dataset.[]{data-label="completeness"}](Fig_3a.pdf "fig:"){width="0.98\columnwidth"}\ ![Top: completeness at $850$ $\mu$m versus flux for different portions of the map, i.e. $2\sigma_{\rm CN}<\sigma<3\sigma_{\rm CN}$, $1.5\sigma_{\rm CN}<\sigma<2\sigma_{\rm CN}$, $\sigma<1.5\sigma_{\rm CN}$, and $<3\sigma_{\rm CN}$ (see Fig. \[Maps\]). Bottom: same as above, but fot the $450$ $\mu$m dataset.[]{data-label="completeness"}](Fig_3b.pdf "fig:"){width="0.98\columnwidth"} Number Counts {#NC} ============= In this section we determine the pure source number counts at 850 and 450 $\mu$m around the ELAN MAMMOTH-1, and estimate the underlying counts models for each wavelength. A precise measurement of the galaxy number counts needs an accurate estimate of the number of spurious sources contaminating the counts. For this purpose, we followed the procedure in @TC2013a [@TC2013b], and use the jackknife maps produced in Section \[jackknife\] to assess how many spourious sources affect the counts. As a first step, in Figure \[histos\] we show the S/N histograms of the true noise maps (orange shading) and the signal maps (blue shading with black edge). The excess signals with respect to the pure noise distribution are from real astronomical sources. On the other hand, the negative excesses are due to the negative throughs of the matched-filter PSF. From these histograms it is well evident that the 450 $\mu$m data are less sensitive and more affected by the presence of spurious sources (as already noted in Section \[jackknife\]). ![Normalized histograms of the S/N values for the pixels within the portions of the 850 and 450 $\mu$m maps characterized by less than three times the central noise. The orange and blue histograms indicate the distributions of the jackknife maps, and of the data, respectively. The jackknife maps represents the pixel noise distributions which dominates at low S/N (see Section \[jackknife\] for details). The data (especially at 850 $\mu$m) shows excesses at high S/N where sources contribute to the distributions. The matched filter technique introduces residual troughs around bright detections, which are visible here as negative excesses.[]{data-label="histos"}](Fig_4.pdf){width="0.95\columnwidth"} In contrast to what done with the catalogs in Section \[catalogs\] – where we have selected only detections with S/N$>4$ or with S/N$>3$ at both wavelengths – we lowered our detection threshold to $2\sigma$. Indeed, as the positional information is not relevant for number counts analyses, the detection threshold can be lowered to explore statistically significant positive excesses (e.g., @TC2013b). We thus use the preliminary catalogues produced in Section \[catalogs\], and additionally ran the source extraction algorithm on the true noise maps down to S/N$=2$. The pure source differential number counts are then obtained as follows. First, for each extracted source in the signal map we calculated the number density by inverting the detectable area, which is the portion of the field-of-view with noise level low enough to allow the detection of the source. Second, we obtained the raw number counts by summing up the number densities of the sources within each flux bin. Finally, to get to the pure source differential number counts, we subtracted the number counts similarly obtained from the true noise maps, if any, from the counts obtained from the signal maps. Figure \[counts\_MCMC\] shows the obtained pure source differential number counts (black data points) for both 850 (top panel) and 450 $\mu$m (lower panel). ![Pure source differential number counts (black data-point) at 850 and 450 $\mu$m around the ELAN MAMMOTH-1 compared to the simulated mean counts (red dashed line). The yellow shadings represent the 90% confidence range obtained from 500 realizations of the blue dot-dashed curves. The blue dot-dashed curves are the final adopted underlying models for the Monte Carlo simulations (see section \[NC\]), and represent the true number counts. The dashed vertical lines indicate the mean $4\sigma$ within the effective area. The horizontal errorbars for the data-points indicate the width of each flux bin.[]{data-label="counts_MCMC"}](Fig_5.pdf){width="0.98\columnwidth"} To obtain the underlying counts models we ran Monte Carlo simulations following e.g., @TC2013a [@TC2013b]. First, we create a simulated image by randomly injecting mock sources onto the jackknife maps. The mock sources are drawn from an assumed model and convolved with the PSFs. For the counts models we use a broken power-law of the form $$\label{eqn:diff_counts} \frac{dN}{dS} = \left\{ \begin{array}{l l} {N_0}\left(\frac{S}{S_0}\right)^{-\alpha} & \quad \text{if $S \leq S_0$}\\ {N_0}\left(\frac{S}{S_0}\right)^{-\beta} & \quad \text{if $S > S_0$}\\ \end{array} \right.,$$ and started from a fit to the observed counts. As faintest fluxes for our models, we use the fluxes at which the integrated flux density agrees with the values for the extragalactic background light (EBL; e.g., @Puget1996). After obtaining a mock map, we ran the source extraction algorithm and computed the number counts in exactly the same way as done with the real data. We then calculated the ratio between the recovered counts and the input model, which reflects the Eddington bias (@Eddington1913), and then applied this ratio to the observed counts to correct for this bias. A $\chi^2$ fit is performed to the corrected observed counts using the broken power-law to get the normalization and power-law indices, which are then used in the next iteration. This iterative process was terminated once the input model agreed with the corrected counts at the $1\sigma$ level. Given the low number of data-points at 450 $\mu$m, we only fitted the normalization and the bright-end slope at this wavelength. We fixed $S_0$ and $\alpha$ to the values in @TC2013b. To test the reliability of our results, we have then created 500 realizations of simulated maps using as input the model curves determined through the Monte Carlo simulations, and calculated the pure source number counts for each of them. In Figure \[counts\_MCMC\] we show the results of the Monte Carlo simulations and compare them to the data. We give the derived underlying counts models (blue dot-dashed lines), the mean counts (red dashed lines), and the 90% confidence range of the 500 realization (yellow). The 500 realizations well match the pure source number counts within the uncertainties. We can then apply the ratio between the mean number counts and the input model to correct our data, and thus obtain the true differential number counts (see Section \[true\_number\_counts\]). Table \[MC\_input\] summarizes the parameters of the obtained underlying count models at both 850 and 450 $\mu$m. Flux boosting estimates {#fluxBoost} ======================= With the Monte Carlo simulations we found a systematic flux/count boost, which we characterized by comparing the flux of the injected mock sources with the detections. In particular, we selected the brightest input source located within the beam area of each of the $>3\sigma$ recovered sources, and computed the flux ratio. In Figure \[Flux\_boosting\] we show this test as a function of S/N for both wavelengths. We plot the mean (red) and the median (yellow) values of the flux boosting, together with the $1\sigma$ ranges (blue) relative to the mean values. At S/N$=4$, the estimated median flux boosting is 2.0 and 1.5 at 450 and 850 $\mu$m, respectively. This values are in agreement within the uncertainties with similar previous studies conducted with SCUBA-2 (e.g., @Casey2013 [@TC2013a]). We then corrected the observed fluxes for the catalogs obtained in Section \[catalogs\] using the median curves, and listed the de-boosted fluxes in Tables \[450\] and \[850\]. This flux boost is usually found in previous SCUBA studies (e.g., @Wang2017) and it is ascribed to the so-called Eddington bias (@Eddington1913). ![Ratio between the fluxes of the detected sources and the injected sources from the 500 realizations of the estimated underlying counts models (section \[NC\]) as a function of the S/N of the detections. The gray dots are $\sim100,000$ simulated data-points. We show the mean (red) and median (yellow) values of the flux ratio in different S/N bins. The blue dashed curves enclose the $1\sigma$ range relative to the mean curve. The test is shown for both 850 (top) and 450 $\mu$m (bottom).[]{data-label="Flux_boosting"}](Fig_6.pdf){width="0.98\columnwidth"} Positional uncertainties {#pos_err} ======================== Using the same Monte Carlo simulations and the same algorithm to find counterparts in the injected and recovered catalogs, we can estimate the positional offset between the location of the injected and the recovered sources. Figure \[pos\_off\] shows this test at both 450 and 850 $\mu$m. At S/N$\lesssim5$, there is a large scatter, suggesting positional uncertainties of the order of $\gtrsim1.7$ and $\gtrsim2.2$ arcsec, respectively for 450 and 850 $\mu$m. At larger S/N the uncertainty is lower, down to 1 arcsec for sources as strong as the brightest objects in our 850 $\mu$m catalog (S/N$\approx16$). At 450 $\mu$m – characterized by a $\approx 1.5\times$ smaller beam – the positional uncertainties are slightly smaller. These results well agrees – within the uncertainties – with the predicted positional offset based on the LESS sample (dashed black line; equation B22 in @Ivison2007). Based on this test, we can then assign the mean value of the offsets as positional uncertainty to the detections listed in Tables \[450\] and \[850\]. ![Positional offset between the detected sources and the injected sources from the 500 realizations of the estimated underlying counts models (section \[NC\]) as a function of the S/N of the detections. The gray dots are $\sim100,000$ simulated data-points. We show the mean (red) and median (yellow) values of the positional offsets in different S/N bins. The blue dashed curves enclose the $1\sigma$ range relative to the mean curve. The test is shown for both 850 (top) and 450 $\mu$m (bottom). The dot-dashed black lines indicate the predictions from @Ivison2007 based on the LESS sample.[]{data-label="pos_off"}](Fig_7.pdf){width="0.98\columnwidth"} Results {#sec:results} ======= True Number Counts {#true_number_counts} ------------------ ![image](Fig_8.pdf){width="95.00000%"} Figure \[true\_counts\] presents the corrected differential and cumulative number counts at both 450 and 850 $\mu$m for the effective area of our observations, together with the derived underlying broken power-law (bpl) models (blue dot-dashed lines). As explained in Section \[NC\], these true number counts have been obtained by dividing the pure source counts by the ratio between the mean number counts and the input models. We list the values of our corrected data-points in Table \[table:counts\]. We then compare our data-points with the most comprehensive literature studies for blank fields at both 450 and 850 $\mu$m (@TC2013b [@Casey2013; @Geach2013; @Wang2017; @Zavala2017; @Geach2017]). In Figure \[true\_counts\] we plot the fit – Schechter (Sc.) or broken power-law (bpl)[^2] – from those works. Our 450 $\mu$m data well agree with these literature curves[^3], while the 850 $\mu$m data-points are above these current expectations for blank fields. Especially the more robust data at about 5 and 7 mJy are clearly suggesting the presence of higher number counts with respect to the literature values. To quantify this overdensity of counts at 850 $\mu$m, we fit our corrected differential number counts with the functions from each of the literature works allowing only the normalization to vary. The difference in counts is then estimated through the ratio between the normalizations. Specifically, - @TC2013b quoted a best fit with a broken power-law function of the form shown in eqn. \[eqn:diff\_counts\], with[^4] $N_0=120^{+65}_{-45}$ mJy$^{-1}$ deg$^{-2}$, $S_0=6.21$ mJy, $\alpha=2.27$, $\beta=3.71$; - @Casey2013 and @Geach2017 reported a Schechter function of the form $$\label{eqn:Schechter1} \frac{dN}{dS} = \frac{N_0}{S_0}\left(\frac{S}{S_0}\right)^{\gamma}{\rm exp}\left(-\frac{S}{S_0}\right),$$ with $N_0=(3.3\pm1.4)\times10^3$ deg$^{-2}$, $S_0=3.7$ mJy, $\gamma=1.4$, and[^5] $N_0=4550\pm546$ deg$^{-2}$, $S_0=3.40\pm0.21$ mJy, $\gamma=1.97\pm0.08$, respectively; - @Zavala2017 preferred a Schechter function of the form $$\label{eqn:Schechter1} \frac{dN}{dS} = \frac{N_0}{S_0}\left(\frac{S}{S_0}\right)^{1-\gamma}{\rm exp}\left(-\frac{S}{S_0}\right),$$ with $N_0=8300\pm300$ deg$^{-2}$, $S_0=2.3$ mJy, $\gamma=2.6$ for all their data. We show the results of the fit with free normalizations $N_0$ in Figure \[fit\_with\_Lit\], and we list in Table \[table:overdensity\] the ratio between the derived normalizations needed to match our data and the literature values. From these ratios it is clear that the probed effective area is indeed overdense with respect to blank fields. On average, around the ELAN MAMMOTH-1 there are $4.0\pm1.3$ times more counts than in blank fields. In this mean estimate we do not include the ratio with respect to @Zavala2017 because this work does not cover effectively the sources bright-end (their last bin is at 4.9 mJy), probably biasing their fit. We however report the comparison with this work for completeness. ![Top: fit of the SCUBA-2 MAMMOTH-1 true differential number counts at 850 $\mu$m using the functions given in literature works for blank fields (@TC2013b [@Casey2013; @Zavala2017; @Geach2017]) with $N_0$ free to vary. Bottom: the SCUBA-2 MAMMOTH-1 true cumulative number counts at 850 $\mu$m compared to the fit models obtained in the top panel. In both panels, the blue dot-dashed curve is our true number counts curve from Table \[MC\_input\]. All the literature models need a significant increase of their normalization parameter $N_0$ to fit our data at $850~\mu$m, revealing that the covered effective area is overdense with respect to blank fields. We list the values in Table \[table:overdensity\].[]{data-label="fit_with_Lit"}](Fig_9.pdf){width="0.95\columnwidth"} Position of the catalog sources within the LAE overdensity {#Pos} ---------------------------------------------------------- Even though the association to the BOSS1441 overdensity of the sources listed in the SCUBA-2 catalogs has to be confirmed spectroscopically, we can still search for LAE counterparts to our submillimeter detections, if any. In Figure \[Comp\_with\_LAEs\], we show the location of the 450 (yellow squares) and 850 $\mu$m (blue circles, with fluxes) catalog sources along with (i) the position of known LAEs (black circles; @Cai2017a), (ii) the position of known QSOs at $2.30\leq z<2.34$ (brown crosses; @Cai2017a), and (iii) the LAEs density contours (green; @Cai2017a). From this figure it is clear that only 2 sources out of the 27 850 $\mu$m detections could be considered to be possibly associated with a LAE from the catalogue of @Cai2017a. This 2 sources (highlighted in orange) are (i) MAM-850.14 close to the ELAN MAMMOTH-1 (see Section \[MAMMOTH-1\] for a discussion), and (ii) MAM-850.16 close to a LAE at RA=220.3906 and Dec=40.0286, with rest-frame equivalent width $EW_0=25.16\pm0.01\AA$, which is actually a $z\simeq2.3$ QSO. The other 25 LAEs lay at a separation greater than the 850 $\mu$m beam from any of our detections. Given the large offsets, the positional uncertainties presented in Section \[pos\_err\] are not affecting the lack of association between LAEs and our submillimeter detections. If future follow-up studies confirm the association of most of the SCUBA-2 sources with the BOSS1441 overdensity, the lack of submillimeter flux at the location of LAEs is consistent with the usual finding that most of the strongly Ly$\alpha$ emitting galaxies are relatively devoid of dust (e.g., @Ono2010 [@Hayes2013; @Sobral2018]). In addition, the brightest detections at 850 $\mu$m, MAM-850.1 and MAM-850.2 ($f_{850}^{\rm Deboosted}=18.3\pm2.8$ mJy and $f_{850}^{\rm Deboosted}=16.3\pm2.7$ mJy) lay intriguingly close to the peak of the LAEs overdensity. Their observed ratios between 450 and 850 $\mu$m suggest that these two bright detections are unlikely to be low redshift sources. Therefore, they probably are associated with the protocluster given the rare alignment with the peak of the LAEs overdensity. ![image](Fig_10.pdf){width="95.00000%"} Discussion {#sec:disc} ========== BOSS1441: a rich and diverse protocluster {#disc:rich} ----------------------------------------- In the previous sections we have demonstrated the presence of a $\sim 4$ times higher density fluctuation compared to blank fields at 850 $\mu$m. In addition, we found that the brightest of our detections are located at the peak of the LAEs overdensity. This unique alignments suggest that most of the SCUBA-2 detections are likely associated with the BOSS1441 overdensity (and the ELAN MAMMOTH-1), rather than being intervening. To test this, we search the available multiwavelength catalogs and build the spectral energy distributions (SEDs) for all the sources in our sample. We thus look for counterparts in the [AllWISE]{} Source Catalog[^6] (@Wright2010) at 3.4, 4.6, 12.1, 22.2 $\mu$m (W1, W2, W3, W4), and in the Faint Images of the Radio Sky at Twenty-cm (FIRST) Survey at 1.4GHz (@Becker1994). This portion of the sky has not been covered by the [Herschel]{} telescope and thus our SCUBA-2 observations are key in covering the far-infrared portion of the SED. To match the different catalogs, we look for counterparts within a 850 $\mu$m beam, and select the closest source. We found that 8 of our detections have a counterpart in [AllWISE]{}, while none has been detected in FIRST down to the catalog detection limit at each source position ($\approx0.95$ mJy). To estimate the likelihood of false match for the WISE counterparts, we use the $p$-value as defined in @Downes1986 $$p = 1 - \exp(-\pi n \theta^{2}),$$ where $n$ is the [AllWISE]{} source density within the effective area, $n\simeq0.00134$ sources/arcsec$^2$, and $\theta$ is the angular separation between the [AllWISE]{} source and the SCUBA-2 detection[^7]. A value of $p<0.05$ usually makes a counterpart reliable (e.g., @Ivison2002 [@TC2016]), while $0.05<p<0.1$ makes it tentative (e.g., @Chapin2009). Of the eight counterparts we found that only four are robust, i.e. MAM-850.8, MAM-850.18, MAM-850.26 and MAM-850.27, while the others are tentative. However, we have shown in Section \[Pos\] that MAM-850.16 is likely associated with a quasar at $z=2.30$. As the quasar is detected by WISE and no other close WISE detections are present, we consider this match secure. Further, the lack of available radio and/or high resolution submm data prevents us to perform a robust identification of counterparts in our recently obtained LBT/LBC $U$, $V$ and $i$ band images (@Cai2017a) and in our UKIRT/WIRCAM $J$, $H$, and $K$ band images (Xu et al. in prep.). We summarize the sources with multiwavelength detections in Table \[SED\_values\_ALL\] , and display for illustration purposes the SEDs of the five sources with robust counterparts in [AllWISE]{} in Figure \[SEDs\_ALL\]. We leave a detailed classification of our detections to future studies encompassing the whole protocluster extent, and better covering the electromagnetic spectrum. However, we used the average SED template for SMGs obtained from 99 sources in the ALESS survey (@daCunha2015)[^8] to compute a rough estimate for the far-infrared (FIR) luminosity $L_{\rm FIR}$ for each source assuming they are all SMGs at $z=2.32$ (redshift of BOSS1441). After normalizing the SED template to our SCUBA-2 observations, we found $L_{\rm FIR}\geq4.8\times10^{12}$ L$_{\odot}$ for each of the sources[^9]. Next, we compared the volume density implied by our observations with expectations from the current luminosity function of SMGs. Assuming the effective area of our observations ($\sim336.9$ Mpc$^2$) and the distance interval spanned in redshift by the protocluster ($z=2.3-2.34$; $\sim34.9$ Mpc$^{-3}$), the comoving volume targeted by our observations is about $11800$ Mpc$^3$. If all (75%) of the detections at 850$\mu$m belongs to the protocluster, their volume density would then be $2.3\times10^{-3}$ Mpc$^{-3}$ ($1.7\times10^{-3}$ Mpc$^{-3}$). These values are a factor of $>30$ above the volume density expected from the current luminosity function of SMGs with $L_{\rm FIR}\geq4.8\times10^{12}$ L$_{\odot}$ ($\sim5\times10^{-5}$ Mpc$^{-3}$; @Casey2014). Therefore, BOSS1441 is a potentially SMG-rich volume. We further noticed that the two brightest sources, MAM-850.1 and MAM-850.2, seem to depart from the templates at 850$\mu$m, showing higher fluxes than expected. This deviation could be explained by allowing a different (higher) redshift, or most probably by the fact that single-dish submillimeter sources as bright as MAM-850.1 and MAM-850.2 are usually a blend of $\geq2$ SMGs once observed with interferometers (@Karim2013 [@Simpson2015]). This explanation is compelling as the two sources sit at the peak of the overdensity. They could thus be groups of interacting galaxies, pinpointing the core of the protocluster. Extremely compact ($20\arcsec$-$40\arcsec$) protocluster cores made of several ($>10$) starbursting galaxies at $z\sim4$ have been recently discovered by @Miller2018 and @Oteo2018. These structures have a global $L_{\rm FIR}\approx10^{14}$ L$_{\odot}$ (@Miller2018). The BOSS1441 protocluster might thus have similar, but scaled down central structures. We can compare this central portion of the overdensity with other known protoclusters at $z\sim2$ (@Dannerbauer2014 [@Casey2015; @Kato2016]). These studies found that spheres with 1 Mpc radius centered at the protocluster core enclose a star-formation rate density of ${\rm SFRD}\sim1000-1500$ M$_{\odot}$ yr$^{-1}$ Mpc$^{-3}$. Following the approach in those works, we centered a 1 Mpc sphere at the position of MAM-850.2, which is the closest SMG to the peak of the LAE overdensity. Within this sphere we potentially found six detections (MAM-850.1,MAM-850.2,MAM-850.12,MAM-850.14,MAM-850.16,MAM-850.21), which add up to a total star formation rate of ${\rm SFR}=9100$ M$_{\odot}$ yr$^{-1}$, translating to ${\rm SFRD}\approx2200$ M$_{\odot}$ yr$^{-1}$ Mpc$^{-3}$ after dividing by the sphere volume[^10]. As our detections are candidate SMGs, this value represents an upper limit. Subtracting the field average value as done in @Kato2016 for $z=2.3$, and assuming that only 75% of our detections are within the sphere, we obtained ${\rm SFRD}\approx1200$ M$_{\odot}$ yr$^{-1}$ Mpc$^{-3}$. We conservatively conclude that ${\rm SFRD}=1200^{+1000}_{-1100}$ M$_{\odot}$ yr$^{-1}$ Mpc$^{-3}$, with the lower limit given by the very unlike case that none of the detections (apart the ELAN MAMMOTH-1 counterpart) are within the sphere. This SFRD value is in agreement with values for protoclusters in the literature (e.g. see Fig. 5 in @Kato2016). Spectroscopic and interferometric follow-ups are needed to confirm the redshift of our sources, and to unveil their nature. Our analysis however suggests that BOSS1441 is a rich protocluster hosting several LAEs, one ELAN, and likely several SMGs. The Counterpart of the ELAN MAMMOTH-1 {#MAMMOTH-1} ------------------------------------- @Cai2016 reported the presence of a continuum source at $z=2.319\pm0.004$, named source B, and of a $z=0.16$ AGN, both at the location of the peak of the ELAN MAMMOTH-1. Source B has been invoked as the powering source of the extended Ly$\alpha$ emission. At a separation of $4.58\arcsec$ from source B and at $2.75\arcsec$ from the $z=0.16$ AGN, our SCUBA-2 observations resulted in a bright detection at 850 $\mu$m, MAM-850.14, with flux density of $f_{850}=4.57\pm0.93$ mJy ($f_{850}^{Deboosted}=2.83\pm1.03$ mJy), and a $3\sigma$ upper limit of $f_{450}<16.65$ mJy at 450 $\mu$m. The non-detection at 450 $\mu$m suggests that the emission at 850 $\mu$m is associated with the $z=2.319$ source. Indeed, a $z=0.16$ AGN with such a detection at 850 $\mu$m should have a much brighter dust thermal emission at 450 $\mu$m. Specifically, if we assume a modified black-body for optically thin thermal dust emission, a dust temperature of $T_{\rm dust}=45$ K, and the largely used emissivity index $\beta=1.5$ (e.g., @Casey2012), we find that a $z=0.16$ AGN should have $f_{450}=35.3$ mJy (or $21.8$ mJy for the deboosted flux). These fluxes would be detected at high significance ($\gtrsim4\sigma$) even in our shallow 450 $\mu$m data at the location of the ELAN MAMMOTH-1. To constrain the nature of source B, we compiled all its data available from the literature, and compared them to known SED. We summarize all the available observations in Table \[SED\_values\] [^11], while we plot them in Figure \[SEDsourceB\]. To compare these data-points to known SEDs, we fixed the redshift of the source to the redshift $z=2.319$ determined from the line emission (@Cai2016), and we fitted the data leaving the normalization free. We first take in consideration the average SED for SMGs obtained from the 99 sources in ALESS (@daCunha2015), and all the available average SEDs from that publication. The left panel of Figure \[SEDsourceB\] shows this test, highlighting the shortage of emission at the WISE bands for these SED templates (we plot only two to avoid confusion) in comparison to the source B’s data-points. None of the average templates in @daCunha2015 match the W1,W2,W3 data-points, with the SED with $A_V<1$ (yellow) giving the closer values, though differing still significantly. We then follow the same procedure with the template SED of the local starburst galaxy M82 (@Silva1998; solid black line in Figure \[SEDsourceB\]). This template match significantly better the observations, with only the W3 data-point underestimated. Most likely a hotter dust component powered by an AGN (e.g., @Silva2004 [@Fritz2006]) would allow a better match of the data of source B. Indeed @Cai2016 demonstrate that only hard-ioninzing sources – most likely an AGN or a wind – could power the and emission in this object. To test this interpretation further, we used the publicly available SED fitting code, [AGNfitter]{} (@CalistroRivera2016), which adopts a fully Bayesian Markov Chain Monte Carlo method to model the SEDs of galaxies and AGN. [AGNfitter]{} fits simultaneously the sub-mm to UV photometry decomposing the SED into four physically motivated components: the AGN accretion disk emission (Big Blue Bump), the hot dust emission from the obscuring structure around the accretion disk (torus), the cold dust emission from star-forming regions and the stellar populations of the host galaxy[^12]. Details on the specific models are presented by @CalistroRivera2016 and references therein. The right panel of Figure \[SEDsourceB\] shows the best fit (in gray) produced by [AGNfitter]{}, with each component highlighted by a different color. The W3 data-point is now well covered by a composition of the AGN-powered hot dust, star emission and starformation-powered cold dust. This analysis thus suggests that source B is an enshrouded strong starbursting galaxy, likely hosting an obscured AGN. Using the output from [AGNfitter]{}, we can separate the far-infrared (FIR) luminosity $L_{\rm FIR}$ (rest-frame 8-1000 $\mu$m) due to the AGN and to star-formation (SF). We find $L_{\rm FIR}^{\rm AGN}=8.0^{+1.2}_{-6.3}\times10^{11}$ L$_{\odot}$ and $L_{\rm FIR}^{\rm SF}=2.4^{+7.4}_{-2.1}\times10^{12}$ L$_{\odot}$, respectively. Source B thus meets the generally used criteria to define an UltraLuminous InfraRed Galaxy (ULIRG; L$_{\rm 8-1000\mu m}>10^{12}$ L$_{\odot}$; e.g., @SandersMirabel1996). Following the classical conversion in @kennicutt98 and considering only the star-formation powered emission, one would then obtain a star formation rate of SFR $=400^{+1300}_{-400}$ M$_{\odot}$ yr$^{-1}$. In agreement with the observations and analysis in @Cai2016, our analysis thus suggests a strong similarity between the source B embedded within the ELAN MAMMOTH-1 and the ULIRG sample hosting AGN activity in @Harrison2012. Source B – with its obscured AGN and starburst – can thus easily power the surrounding ELAN, and thus the outflow resulting in the velocity offset of 700 km s$^{-1}$ between the two spectral components in Ly$\alpha$, , and (@Cai2016). The very broad \[\] emission presented for the targets in @Harrison2012 however extends to lower distances (15 kpc) with respect to the rest-frame UV lines seen in the ELAN MAMMOTH-1 ($\gtrsim30$ kpc; @Cai2016). As the ELAN MAMMOTH-1 hosts an ULIRG we thus expect to see broad \[\] emission in its central portion down to similar depths. Finally, [AGNfitter]{} estimated the stellar mass of source B to be log$(M_{\rm star}/{\rm M_{\odot}})=11.4^{+0.3}_{-0.2}$. By inverting the halo mass $M_{\rm halo}$ - $M_{\rm star}$ relation in @Moster2013, we derived that – if source B is a central galaxy – the ELAN MAMMOTH-1 is hosted by a very massive halo of log$(M_{\rm halo}/{\rm M_{\odot}})=15.2^{+1.4}_{-1.6}$. Given the large uncertainties this result has to be confirmed. However, it certainly highlights the peculiarity of the halo hosting the ELAN MAMMOTH-1, indicating that it sits at the high-mass end of the halo population at this redshift. We further note that the stellar mass of source B is intriguingly close to current estimates for the stellar mass of host galaxies of HzRGs (log$(M_{\rm star}/{\rm M_{\odot}})\simeq 11 - 11.5$; @Seymour2007 [@DeBreuck2010]). HzRGs are currently thought to reside in massive halos of mass log$(M_{\rm halo}/{\rm M_{\odot}})\approx 13$ (e.g., @Stevens2003). The halo hosting the ELAN MAMMOTH-1 could thus be similarly massive or exceed such halos. ![image](Fig_11.pdf){width="100.00000%"} Summary {#sec:summ} ======= We are conducting a survey of all the known ELANe (@cantalupo14 [@hennawi+15; @Cai2016; @FAB2018]) with the JCMT and APEX telescopes to assess the presence of starburst activity in these systems and their environments. In this work we focused on the SCUBA-2/JCMT data at 450 and 850 $\mu$m obtained for an effective area of $\sim127$ arcmin$^2$ around the ELAN MAMMOTH-1 at $z=2.319$ (@Cai2016), and thus targeting the known peak area of the LAE overdensity BOSS1441 (@Cai2017a). Thanks to this dataset we found that 1. the 850 $\mu$m source counts are $4.0\pm1.3$ times higher than in blank fields (@TC2013b [@Casey2013; @Geach2017]), confirming also in obscured tracers the presence of an overdensity. Intriguingly, the two brightest submillimeter detections, MAM-850.1 and MAM-850.2, are located at the peak of the LAE overdensity, possibly pinpointing the core of the protocluster and multiple mergers/interactions (e.g., @Miller2018). The association of the discovered submillimeter sources with BOSS1441 needs however a spectroscopic confirmation. 2. the continuum source at the center of the ELAN MAMMOTH-1, source B (@Cai2016), is associated to a strong detection at 850 $\mu$m, MAM-850.14, with flux density of $f_{850}=4.6\pm0.9$ mJy ($f_{850}^{\rm Deboosted}=2.8\pm1.0$ mJy) and a $3\sigma$ upper limit of $f_{450}<16.6$ mJy at 450 $\mu$m. Together with the data from the literature, the SED of source B agrees with a strongly starbursting galaxy hosting an obscured AGN, and having a FIR luminosity of $L_{\rm FIR}^{\rm SF}=2.4^{+7.4}_{-2.1}\times10^{12}$ L$_{\odot}$. Source B is thus an ULIRG with a star-formation rate of SFR $=400^{+1300}_{-400}$ M$_{\odot}$ yr$^{-1}$ assuming the classical @kennicutt98 calibration. Such a source – containing both an AGN and a violent starburst – is able to power the hard photoionization plus outflow scenario depicted in @Cai2016. The acquisition of wide-field multiwavelength data (X-ray, UV, optical, submillimeter, radio) is key in painting a coherent and detailed picture of a protocluster, and ultimately to understand the assembly of massive galaxies within the cosmic nurseries of the soon-to-be large clusters. The results of this pilot project are encouraging and reflect the importance of such a multi-wavelength approach in fully comprehending the ELAN phenomenon and the environment in which they reside. We thank the referee Yuichi Matsuda for his careful read of the manuscript. The James Clerk Maxwell Telescope is operated by the East Asian Observatory on behalf of The National Astronomical Observatory of Japan; Academia Sinica Institute of Astronomy and Astrophysics; the Korea Astronomy and Space Science Institute; the Operation, Maintenance and Upgrading Fund for Astronomical Telescopes and Facility Instruments, budgeted from the Ministry of Finance (MOF) of China and administrated by the Chinese Academy of Sciences (CAS), as well as the National Key R&D Program of China (No. 2017YFA0402700). Additional funding support is provided by the Science and Technology Facilities Council of the United Kingdom and participating universities in the United Kingdom and Canada. This publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, and NEOWISE, which is a project of the Jet Propulsion Laboratory/California Institute of Technology. WISE and NEOWISE are funded by the National Aeronautics and Space Administration. M.F. acknowledges support by the Science and Technology Facilities Council \[grant number ST/P000541/1\]. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 757535). I.R.S. acknowledges support from the ERC Advanced Grant [DUSTYGAL]{} (321334) and STFC (ST/P000541/1). Y.Y.’s research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (NRF-2016R1C1B2007782). The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain. [AllWISE]{} counterparts to our SCUBA-2 detections ==================================================== In this appendix we show the multiwavelength dataset for [AllWISE]{} counterparts to our SCUBA-2 detections. Specifically, in Table \[SED\_values\_ALL\] we list (i) the likelihood of false match for each [AllWISE]{} counterpart, i.e. the $p$-value (see Section \[disc:rich\]), (ii) the magnitudes from the [AllWISE]{} catalog, (iii) the flux from the FIRST survey, and (iv) the rough estimate of $L_{\rm FIR}$ for each source. The four sources with $p<0.05$ are considered robust, while the others are tentative. We further consider robust the match with MAM-850.16 has it is clearly associated with a quasar at $z=2.30$ (see Section \[Pos\] for more details). Finally, for illustration purposes, in Figure \[SEDs\_ALL\] we show the SED of the five sources with robust [AllWISE]{} counterparts along with the template SED of M82 (@Silva1998; black line), of the average ALESS SMGs (dashed brown line), and the average ALESS SMGs with $A_V<1$, and $A_V \geq 3$ (@daCunha2015; yellow). All the template SEDs have been normalized to the SCUBA-2 data assuming $z=2.32$. We will perform a more detailed analysis of the SED in future works encompassing the full extent of the protocluster, and covering a broader range of the electromagnetic spectrum. ![image](Fig_A1.pdf){width="90.00000%"} [^1]: As PSFs for our observations, we adopt the PSFs at 850 and 450 $\mu$m generated by @TC2013b (see their Figure 2). [^2]: If a work presented both functions for their fits, we selected their Schechter fit. Our results do not depend on this choice. [^3]: We remind the reader that – as already noted in @Casey2013 – the equation (1) of @Geach2013 should be written as $dN/dS=(N'/S')(S/S')^{1-\alpha}\exp(-S/S')$, and the best-fitting parameter $N'$ for this 450 $\mu$m data should be quoted as $N'=4900\pm1040$ deg$^{-2}$ mJy$^{-1}$ rather than $N'=490\pm1040$ deg$^{-2}$ mJy$^{-1}$. [^4]: For all the fits in the literature, we report only the errors on the parameter $N_0$. [^5]: @Geach2017 only showed a Schechter fit to their data in their Figure 15. Here we thus report the values for a Schechter fit to their data. [^6]: <http://wise2.ipac.caltech.edu/docs/release/allwise/> [^7]: As the positional uncertainty is small for the 450 $\mu$m band, for our sources we adopt the coordinates at 450 $\mu$m if available. [^8]: @daCunha2015 provides average SEDs made in bins of redshifts, observed ALMA 870 $\mu$m flux, average V-band attenuation $A_V$, total dust luminosity (<http://astronomy.swinburne.edu.au/~ecunha/ecunha/SED_Templates.html>). [^9]: We estimated the FIR luminosity $L_{\rm FIR}$ for each source as frequently done by integrating the luminosity in the rest-frame range 8-1000 $\mu$m. [^10]: We convert $L_{\rm FIR}$ to SFR using the classical conversion in @kennicutt98. [^11]: The $U$, $V$, $i$-band photometry here reported is slightly (within errors) different from @Cai2016 because of the image degradation applied to the data to match the UKIRT observations (Xu et al. in prep.). [^12]: [AGNfitter]{} does not currently cover the radio portion of the spectrum. This does not affect our results as we do not have stringent limits at the radio wavelengths.	ArXiv
--- abstract: \| We consider the dynamical Gross-Pitaevskii (GP) hierarchy on $\R^d$, $d\geq1$, for cubic, quintic, focusing and defocusing interactions. For both the focusing and defocusing case, and any $d\geq1$, we prove local existence and uniqueness of solutions in certain Sobolev type spaces $\cH_\xi^\alpha$ of sequences of marginal density matrices. The regularity is accounted for by $$\alpha \, \left\{ \begin{array}{rcl} > &\frac12& {\rm if} \; d=1 \\ >&\frac d2-\frac{1}{2(p-1)} & {\rm if} \; d\geq2 \; {\rm and} \; (d,p)\neq(3,2)\\ \geq & 1 & {\rm if} \; (d,p)=(3,2) \,, \end{array} \right.$$ where $p=2$ for the cubic, and $p=4$ for the quintic GP hierarchy; the parameter $\xi>0$ is arbitrary and determines the energy scale of the problem. This result includes the proof of an a priori spacetime bound conjectured by Klainerman and Machedon for the cubic GP hierarchy in $d=3$. In the defocusing case, we prove the existence and uniqueness of solutions globally in time for the cubic GP hierarchy for $1\leq d\leq3$, and of the quintic GP hierarchy for $1\leq d\leq 2$, in an appropriate space of Sobolev type, and under the assumption of an a priori energy bound. For the focusing GP hierarchies, we prove lower bounds on the blowup rate. Also pseudoconformal invariance is established in the cases corresponding to $L^2$ criticality, both in the focusing and defocusing context. All of these results hold without the assumption of factorized initial conditions. address: - 'T. Chen, Department of Mathematics, University of Texas at Austin.' - 'N. Pavlović, Department of Mathematics, University of Texas at Austin.' author: - Thomas Chen - Nataša Pavlović title: 'On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies' --- Introduction ============ The derivation of the nonlinear Schrödinger equation as the dynamical mean field limit of the manybody quantum dynamics of interacting Bose gases is a research area that is recently experiencing remarkable progress, see [@esy1; @esy2; @ey; @kiscst; @klma; @rosc] and the references therein, and also [@adgote; @eesy; @frgrsc; @frknpi; @frknsc; @he; @sp]. A main motivation to investigate this problem is to understand the dynamical behavior of Bose-Einstein condensates. For recent developments in the mathematical analysis of Bose gases and their condensation, we refer to the fundamental work of Lieb, Seiringer, Yngvason, et al.; see [@ailisesoyn; @lise; @lisesoyn; @liseyn] and the references therein. The procedure developed in the landmark works of Erdös, Schlein, and Yau, [@esy1; @esy2; @ey], to obtain the dynamical mean field limit of an interacting Bose gas, comprises the following main ingredients. One determines the BBGKY hierarchy of marginal density matrices for particle number $N$, and derives the Gross-Pitaevskii (GP) hierarchy in the limit $N\rightarrow\infty$, for a scaling where the particle interaction potential tends to a delta distribution; see also [@kiscst; @sc]. For factorized initial data, the solutions of the GP hierarchy are governed by a cubic NLS for systems with 2-body interactions, [@esy1; @esy2; @ey; @kiscst], and quintic NLS for systems with 3-body interactions, [@chpa]. The proof of the uniqueness of solutions of the GP hierarchy is the most difficult part of this analysis, and is obtained in [@esy1; @esy2; @ey] by use of highly sophisticated Feynman graph expansion methods inspired by quantum field theory. Recently, an alternative method to prove the uniqueness of solutions in the $d=3$ case has been developed by Klainerman and Machedon in [@klma], using spacetime bounds on the density matrices in the GP hierarchy; this result makes the assumption of a particular a priori spacetime bound on the density matrices which has so far remained conjectural. In the work [@kiscst] of Kirkpatrick, Schlein, and Staffilani, the corresponding problem in $d=2$ is solved, and the assumption made in [@klma] is replaced by a spatial a priori bound which is proven in [@kiscst]. Alternative methods to obtain dynamical mean field limits of interacting Bose gases using operator-theoretic methods are developed by Fröhlich et al in [@frgrsc; @frknpi; @frknsc]. All of the above mentioned works discuss Bose gases with [repulsive]{} interactions; it is currently not known how to obtain a GP hierarchy from the $N\rightarrow\infty$ limit of a BBGKY hierarchy with attractive interactions. In the work at hand, we have nothing to add to this issue. Instead, we start here directly from the level of the GP hierarchy, and are thus free to also consider [attractive]{} interactions within this context. Accordingly, we will refer to the corresponding GP hierarchies as [cubic]{}, [quintic]{}, [focusing]{}, or [defocusing GP hierarchies]{}, depending on the type of the NLS governing the solutions obtained from factorized initial conditions. In the present work, we investigate the Cauchy problem for the cubic and quintic GP hierarchy with focusing and defocusing interactions. Our results do not assume any factorization of the initial data. As a crucial ingredient of our arguments, we introduce Banach spaces $\cH_\xi^\alpha=\{ \, \Gamma\in\Gspace \, \| \, \\| \, \Gamma \, \\|_{\cH_\xi^\alpha} <\infty \, \}$ where \[bigG\] = { = ( \^[(k)]{}(x\_1,…,x\_k;x\_1’,…,x\_k’) )\_[k]{} \| \^[(k)]{} < } is the space of sequences of $k$-particle density matrices, and \_[\_\^]{} := \_[k]{} \^k \^[(k)]{} \_[H\^(\^[dk]{}\^[dk]{})]{} . The parameter $\xi>0$ is determined by the initial condition, and it sets the energy scale of a given Cauchy problem. If $\Gamma\in\cH_\xi^\alpha$, then $\xi^{-1}$ is the typical $H^\alpha$-energy per particle. The parameter $\alpha$ determines the regularity of the solution, and our results hold for $\alpha\in\alphaset(d,p)$ where \[eq-alphaset-def-0\] (d,p) := { [cc]{} (12,) & [if]{} d=1\ (d2-, ) & [if]{} d2 (d,p)(3,2)\ \[1,) & [if]{} (d,p)=(3,2) , . in dimensions $d\geq1$, and where $p=2$ for the cubic, and $p=4$ for the quintic GP hierarchy. The parameter $\xi>0$ determines the energy scale of the problem. The main results proven in this paper are: 1. We prove local existence and uniqueness of solutions for the cubic and quintic GP hierarchy with focusing or defocusing interactions, in $\cH_\xi^\alpha$, for $\alpha\in \alphaset(d,p)$, which satisfy a spacetime bound $\\|\opB\Gamma\\|_{L^1_{t\in I}\cH^{\alpha}_{\xi}}<\infty$ for some $\xi>0$ (the operator $\opB$ is defined in Section \[sec-defandresults-1\] below). This spacetime bound has been conjectured by Klainerman and Machedon in [@klma]. It is of Strichartz-type, and is proven in Section \[sec-locwp-1\] using a Picard-type fixed point argument on the space $L^1_{t\in [0,T]}\cH_\xi^\alpha$; see inequality (\[eqn-BGamma-spacetime-bd-0\]) and Remark \[rem-Strichartz-1\] below. Accordingly, we conclude that a solution of the GP hierarchy in $\cH_\xi^\alpha$ is unique if and only if this spacetime bound holds.\ 2. We prove the global existence and uniqueness of solutions in $\cH_\xi^1$ satisfying the above noted spacetime bound, for the defocusing cubic GP hierarchy for $1\leq d\leq3$, and the defocusing quintic GP hierarchy for $1\leq d\leq 2$, provided that an a priori bound $\\|\Gamma(t)\\|_{\cH_\xi^1}<c$ holds for $\xi>0$ sufficiently small.\ 3. We indroduce generalized pseudoconformal transformations, and prove the invariance of the cubic GP hierarchy in $d=2$, and of the quintic GP hierarchy in $d=1$, under their application. Because the NLS obtained from factorized initial data in these cases are $L^2$-critical, we will, for brevity, refer to these GP hierarchies as being $L^2$-critical.\ 4. For the focusing cubic or quintic GP hierarchy, we prove lower bounds on the blowup rate in $\cH^\alpha_\xi$ and $\cL^r_\xi$, where both spaces are defined in Section \[sec-defandresults-1\] below. An important ingredient of our proof of the local existence and uniqueness of solutions is the use of certain spacetime bounds for the non-interacting GP hierarchy established in [@klma] for the cubic GP hierarchy in $d=3$ (which were generalized to cubic in $d=2$ in [@kiscst], and to the quintic GP hierarchy in [@chpa]), and the “boardgame estimates" developed in [@klma] (and generalized to the quintic case in [@chpa]), which were motivated by the Feynman graph expansion techniques of [@esy1; @esy2]. For our discussion of blowup solutions of the focusing (cubic or quintic) GP hierarchy, we make extensive use of a quantity that controls the average $H^\alpha$-energy per particle, and, in a different form, the average $L^r$-norm per particle. It is introduced in Definition \[def-AvHLp-1\] below, and turns out to be the key observable for our discussion of blowup solutions.\ Organization of the paper {#organization-of-the-paper .unnumbered} ------------------------- In Section \[sec-defandresults-1\], we introduce the cubic and quintic GP hierarchy, and state our main theorems. In Section \[sec-locwp-1\], we prove the local wellposedness of the Cauchy problem for the cubic and quintic GP hierarchy, for both focusing and defocusing interactions. In Section \[sec-gwp-GP-1\], the local wellposedness is enhanced to global wellposedness for the cubic and quintic defocusing GP hierarchies, using energy conservation. In Section \[sec-Proof-blowuprate-1\], we prove lower bounds on the blowup rate of blowup solutions in the spaces $\cH_\xi^\alpha$ and $\cL^r_\xi$ (see below for their definitions). In Section \[sec-confinv-1\], we prove the pseudoconformal invariance of the $L^2$-critical cubic (in $d=2$) and quintic (in $d=1$) GP hierarchies. In the Appendix, we reformulate the Klainerman-Machedon spacetime bounds in a form convenient for our work. Definition of the model and statement of the main results {#sec-defandresults-1} ========================================================= We introduce the space := \_[k=1]{}\^L\^2(\^[dk]{}\^[dk]{}) of sequences of density matrices := ( \^[(k)]{} )\_[k]{} where $\gamma^{(k)}\geq0$, $\tr\gamma^{(k)} =1$, and where every $\gamma^{(k)}(\ux_k,\ux_k')$ is symmetric in all components of $\ux_k$, and in all components of $\ux_k'$, respectively. We call $\Gamma=(\gamma^{(k)})_{k\in\N}$ admissible if\ && = dx\_[k+1]{} dx\_[k+p2]{} \^[(k+)]{}(\_[k]{},x\_[k+1]{},…,x\_[k+p2]{};\_k’,x\_[k+1]{},…,x\_[k+p2]{}) for all $k\in\N$. Let $0<\xi<1$ and $r>1$. We define \_\^r := { \| \_[\_\^r]{} < } where \_[\_\^r]{} := \_[k=1]{}\^\^[ k]{} \^[(k)]{} \_[L\^r(\^[dk]{}\^[dk]{})]{} . Furthermore, we define \_\^ := { \| \_[\_\^]{} < } where \_[\_\^]{} = \_[k=1]{}\^\^[ k]{} \^[(k)]{} \_[H\^(\^[dk]{}\^[dk]{})]{} , with \^[(k)]{} \_[H\^(\^[dk]{}\^[dk]{})]{} = S\^[(k,)]{} \^[(k)]{} \_[L\^2(\^[dk]{}\^[dk]{})]{} , and $S^{(k,\alpha)}:=\prod_{j=1}^k\langle\nabla_{x_j}\rangle^\alpha\langle\nabla_{x_j'}\rangle^\alpha$. Clearly, $\cL_\xi^r$, $\cH_\xi^\alpha$ are Banach spaces. We note that Banach spaces of integral kernels of a similar type as those introduced above are, for instance, used for operator-theoretic renormalization group methods in the spectral analysis of quantum electrodynamics, [@bcfs]. Let $p\in\{2,4\}$. We consider the $p$-GP (Gross-Pitaevskii) hierarchy given by \[eq-def-b0-2\] i\_t \^[(k)]{} = \_[j=1]{}\^k \[-\_[x\_j]{},\^[(k)]{}\] + B\_[k+p2]{} \^[(k+p2)]{} in $d$ dimensions, for $k\in\N$. Here,\ && := \_[j=1]{}\^k (B\_[j;k+1,…,k+p2]{}\^[(k+)]{})(t,x\_1,…,x\_k;x\_1’,…,x\_k’)\ && := \_[j=1]{}\^kdx\_[k+1]{}dx\_[k+p2]{} dx\_[k+1]{}’dx\_[k+p2]{}’\ &&\ && \^[(k+p2)]{}(t,x\_1,…,x\_[k+p2]{};x\_1’,…,x\_[k+p2]{}’) accounts for $\frac p2+1$-body interactions between the Bose particles. For a factorized initial condition \[eq-initcond-fact-1\] \^[(k)]{}(0) = \| \_0 \_0 \|\^[k]{} with $\phi_0\in H^\alpha$, one obtains that \^[(k)]{}(t) = \| (t) (t) \|\^[k]{} is a solution of if $\phi_t$ satisfies the NLS \[eq-NLS-p-1\] i\_t \_t + \_x \_t - \|\_t\|\^p \_t = 0 with initial condition $\phi(0)=\phi_0$, where $\mu\in\{1,-1\}$. For $p=2$, this is the cubic NLS, and for $p=4$, this is the quintic NLS. The NLS is defocusing for $\mu=1$, and focusing for $\mu=-1$. Accordingly, we refer to (\[eq-def-b0-2\]) as the [cubic GP hierarchy]{} if $p=2$, and as the [quintic GP hierarchy]{} if $p=4$. Moreover, for $\mu=1$ or $\mu=-1$ we refer to the GP hierarchies as being defocusing or focusing, respectively. We recall the definition of the set $\alphaset(d,p)$, for $p=2,4$ and $d\geq1$, \[eq-alphaset-def-1\] (d,p) = { [cc]{} (12,) & [if]{} d=1\ (d2-, ) & [if]{} d2 (d,p)(3,2)\ \[1,) & [if]{} (d,p)=(3,2) . Our main result in this paper is the following theorem. \[thm-main-0\] Let $0<\xi_2=\eta\xi_1\leq\xi_1<1$. Assume that $\alpha\in\alphaset(d,p)$ where $d\geq1$ and $p\in\{2,4\}$, and $0<\eta<1$ sufficiently small. Then, the following hold. - For every $\Gamma_0\in\cH_{{\xi_1}}^\alpha$, there exist constants $T>0$ and $0<\xi_2\leq\xi_1$ such that there exists a unique solution $\Gamma(t)\in\cH_{\xi_2}^\alpha$ for $t\in[0,T]$ with $\\|\opB\Gamma\\|_{L^1_{t\in[0,T]}\cH_{\xi_2}^\alpha}<\infty$.\ - Assume that given $\Gamma_0\in\cH_{{\xi_1}}^\alpha$, there are constants $T>0$ and $0<\xi_2\leq\xi_1$ such that for $t\in I=[0,T]$, there exists a solution $\Gamma(t)$ of the $p$-GP hierarchy (\[seclwp-pGP\]) in the space $L^\infty_{t\in I}\cH_{\xi_2}^\alpha$.\ \ Then, the solution $\Gamma(t)\in L^\infty_{t\in I}\cH_{\xi_2}^\alpha$ is [unique]{} if and only if $\\| \, \opB \Gamma \, \\|_{L^1_{t\in I}\cH_{\xi}^\alpha}<\infty$ holds for some $\xi>0$.\ \ If the latter is satisfied, then in fact, the Strichartz-type bound \[eqn-BGamma-spacetime-bd-0\] \_[L\^1\_[tI]{}\_[\_2]{}\^]{} &&C(d,p,\_1,\_2) \_0\_[\_[\_1]{}\^]{} holds.\ \[rem-BBGKYlim-1\] An immediate implication of part (ii) of Theorem \[thm-main-0\] is that every solution $\Gamma(t)$ extracted by a diagonal argument from the $N\rightarrow\infty$ limit of the $N$-particle BBGKY hierarchies (with repulsive interactions) studied by Erdös-Schlein-Yau in [@esy1; @esy2], Kirkpatrick-Schlein-Staffilani in [@kiscst], and Chen-Pavlović in [@chpa], satisfies $\\| \, \opB \Gamma \, \\|_{L^1_{t\in I}\cH_{\xi}^\alpha}<\infty$ for some $\xi>0$. This is true because uniqueness of those solutions has been established in these works with independent methods. The role of the parameters $\xi_1,\xi_2$ is as follows: Given initial data $\Gamma_0=(\gamma^{(k)})_{k\in\N}$ with $\\|\gamma^{(k)}\\|_{H^\alpha(\R^{dk}\times\R^{dk})}<\infty$ for all $k$, we determine $\xi_1>0$ sufficiently small such that $\Gamma_0\in\cH_{\xi_1}^\alpha$. This means that the energy per particle in $\Gamma_0$ is bounded by $\xi_1^{-1}$. In cases of physical interest, $\xi_1>0$; the notion of an energy per particle will be quantified below. Then, we find a suitable $\xi_2=\eta\xi_1\ll\xi_1$ such that the Cauchy problem for the the GP hierarchy can be solved in a sufficiently large space $\cH^\alpha_{\xi_2}$. The requirement $\xi_2 \ll \xi_1$ is used to ensure that a solution $\Gamma(t)$ does not drift out of $\cH^\alpha_{\xi_2}$ for $t\in I=[0,T]$ with $T=T(\xi_2)>0$; we thereby impose the assumption that the energy per particle does not exceed $\xi_2^{-1}$ while $t\in I$, but once this assumption is violated, we may choose $0<\xi_2'<\xi_2$ to continue the solution to $T(\xi_2')>T(\xi_2)$. In particular, there is no implication of the size of $\xi_2$ on the regularity accounted for by $\alpha$. For factorized initial data, the statement that the solution of the NLS remains in $H^\alpha$ for $t\in I$ is equivalent to the statement that the solution of the GP hierarchy remains in $\cH^\alpha_{\xi}$ for [an arbitrary]{} nonzero $\xi>0$. \[rem-Strichartz-1\] We note that the estimate (\[eqn-BGamma-spacetime-bd-0\]), for the cubic GP hierarchy with $d=3$ and $\alpha=1$, proves the a priori spacetime bound conjectured in [@klma]. For factorized initial data $\Gamma=(\|\phi_0\rangle\langle\phi_0\|^{\otimes k})_{k\in\N}$ in the cubic case, so that $\Gamma=(\|\phi(t)\rangle\langle\phi(t)\|^{\otimes k})_{k\in\N}$ where $i\partial_t\phi+\Delta\phi-\mu\|\phi\|^2\phi=0$, it corresponds to the inequality \|\|\^2 \_[L\^1\_[tI]{}H\^]{}\^[13]{} C(T) \_0 \_[H\^]{} which is of Strichartz type. The example of factorized solutions with $\phi(t)\in H^1$, $t\in I$, is discussed in detail in [@klma]. We say that a solution $\Gamma(t)$ of the GP hierarchy blows up in finite time with respect to $H^\alpha$ if there exists $T^<\infty$ such that for every $\xi>0$ there exists $T_{\xi,\Gamma}^<T^$ such that $\\|\Gamma(t)\\|_{\cH_{\xi}^\alpha}\rightarrow\infty$ as $t\nearrow T^{}_{\xi,\Gamma}$, and $T_{\xi,\Gamma}^\nearrow T^$ as $\xi\rightarrow0$. For the study of blowup solutions, it is convenient to introduce the following quantity. \[def-AvHLp-1\] We refer to \_[H\^]{}() := \^[-1]{} , \_[L\^r]{}() := \^[-1]{} , respectively, as the typical (or average) $H^\alpha$-energy and the typical $L^r$-norm per particle. We note that = ( \| \|\^[k]{} )\_[k]{} \_[H\^]{}() = \_[H\^]{}\^2 \_[L\^r]{}() = \_[L\^r]{}\^2 in the factorized case. The fact that $\Gamma\in\cH_\xi^\alpha$ means that the typical energy per particle is bounded by $\Av_{H^\alpha}(\Gamma )<\xi^{-1}$. Therefore, the parameter $\xi$ determines the $H^\alpha$-energy scale in the problem. While solutions with a bounded $H^\alpha$-energy remain in the same $\cH_\xi^\alpha$, blowup solutions undergo transitions $\cH_{\xi_1}^\alpha\rightarrow\cH_{\xi_2}^\alpha\rightarrow\cH_{\xi_3}^\alpha\rightarrow\cdots$ where the sequence $\xi_1>\xi_2>\cdots$ converges to zero as $t\rightarrow T^$. It is easy to see that blowup in finite time of $\Gamma(t)$ with respect to $H^\alpha$ is equivalent to the statement that $\Av_{H^\alpha}(\Gamma(t))\rightarrow\infty$ as $t\nearrow T^$. Clearly, $(\Av_{N}(\Gamma ))^{-1}$ is the convergence radius of $\\|\Gamma\\|_{\cN_\xi}$ as a power series in $\xi$, for the norms $N=H^\alpha,L^r$ and $\cN_\xi=\cH^\alpha_\xi ,\cL^r_\xi$, respectively. \[thm-blowuprate-L2crit-1\] Assume that $\Gamma(t)$ is a solution of the (cubic $p=2$ or $p=4$ quintic) $p$-GP hierarchy with initial condition $\Gamma(t_0)=\Gamma_{0}\in\cH_{\xi}^\alpha$, for some $\xi>0$, which blows up in finite time. Then, the following lower bounds on the blowup rate hold: 1. Assume that $\frac4d\leq p < \frac{4}{d-2\alpha}$. Then, ( \_[H\^]{}((t)) )\^[12]{} > . Thus specifically, for the cubic GP hierarchy in $d=2$, and for the quintic GP hierarchy in $d=1$, ( \_[H\^1]{}((t )) )\^[12]{} , with respect to the Sobolev spaces $H^\alpha$, $\cH_\xi^\alpha$. 2. ( \_[L\^r]{}((t )) )\^[12]{} , < r. We note that in the factorized case, the above lower bounds on the blow-up rate coincide with the known lower bounds on the blow-up rate for solutions to the NLS (see, for example, [@ca]). The cubic GP hierarchy in $d=2$, and the quintic GP hierarchy in $d=1$ are distinguished by being invariant under a class of generalized pseudoconformal transformations, as presented below. Let us first recall pseudoconformal invariance on the level of the NLS (\[eq-NLS-p-1\]). If the NLS (\[eq-NLS-p-1\]) is $L^2$-critical, that is, $p=\frac 4d$, it is invariant under the pseudoconformal transformations \[eq-NLSpc-def-1\] [P]{}\_t (x) := e\^[-i]{} \_( ) , for $b\in\R\setminus\{0\}$. That is, i\_t[P]{}\_t + \_t - \|[P]{}\_t\|\^p [P]{}\_t = 0 ; see for instance [@ca]. There are two cases of $L^2$-critical NLS with $p\in\N$: The cubic ($p=2$) NLS in $d=2$, and the quintic ($p=4$) NLS in $d=1$. For the GP hierarchy, one can likewise introduce pseudoconformal transformations, and as we prove in this paper, the GP hierarchy is pseudoconformally invariant when $p=2$ and $d=2$ (cubic), or $p=4$ and $d=1$ (quintic). This property is independent of whether the GP hierarchy is defocusing, $\mu=1$, or focusing, $\mu=-1$. \[thm-pseudoconf-gamma-1\] For $d=2$ and $p=2$ (cubic), or $d=1$ and $p=4$ (quintic), the focusing or defocusing ($\mu\in\{1,-1\}$) GP hierarchy is invariant under the pseudoconformal transformations \[eq-pseudoconf-gamma-1\]\ && := e\^[-i]{} \^[(k)]{}( , ; ) , for $b\in\R\setminus\{0\}$. That is, \[eq-infhierarch-pseudoconf-1\] i\_t \^[(k)]{} + \_\^[(k)]{}\^[(k)]{} - B\_[k+p2]{} \^[(k+2)]{} = 0 , for all $k\geq1$. The proof is given in Section \[sec-confinv-1\]. For a survey of related matters for the NLS, see for instance [@ca; @ra; @ta]. Of course, the following is immediately clear. Assume that $\alpha\in\alphaset(d,p)$ where $d\geq1$ and $p\in\{2,4\}$. Moreover, assume that $\Gamma(t)\in\cH_{\xi_2}^\alpha$ solves the (cubic or quintic) focusing ($\mu=-1$) GP hierarchy with factorized initial condition $\Gamma_0=(\|\phi_0\rangle\langle\phi_0\|^{\otimes k})_{k\in\N}\in\cH_{\xi }^\alpha$ for some $\xi>0$, where $\phi_0\in H^\alpha $.\ \ Then, if there exists $T^<\infty$ such that $\\|\phi(t)\\|_{H^\alpha}\rightarrow\infty$ as $t\nearrow T^$, it follows that also $\Av_{H^\alpha}(\Gamma(t))\rightarrow\infty$ as $t\nearrow T^$. This follows from $\Av_{H^\alpha}(\Gamma(t))=\\|\phi(t)\\|_{H^\alpha}^2$ for product states. For various scenarios in which blowup occurs for solutions of the cubic or quintic NLS, we refer to the literature; see for instance [@ca; @ra] for surveys. Local existence and uniqueness of solutions for the focusing and defocusing GP hierarchy {#sec-locwp-1} ======================================================================================== In this section, we prove a local existence and uniqueness result for the cubic and quintic GP hierarchy for both focusing and defocusing interactions. We formulate all arguments for the cubic hierarchy ($p=2$). For the quintic hierarchy ($p=4$), the generalizations are straightforward, and will only be briefly described. We introduce the notation \_\^[(k)]{} = \_[\_k]{} - \_[’\_k]{} with \_[\_k]{} = \_[j=1]{}\^k\_[x\_j]{} and \_[, x\_j]{} = \_[x\_j]{} - \_[x\_j’]{} . Moreover, we write \_ := ( \^[(k)]{}\_\^[(k)]{} )\_[k]{} and := ( B\_[k+1]{} \^[(k+1)]{} )\_[k]{} . Then, the $p$-GP hierarchy can be written as \[seclwp-pGP\] i\_t + \_ = . In integral formulation, it is formally given by \[int-pGP\] (t) = e\^[it\_]{}\_0 - i \_0\^t ds e\^[i(t-s)\_]{} (s) . We will prove the local existence and uniqueness of such a solution in the following manner. We note that (\[int-pGP\]) can be formally written as a system of integral equations (t) & = & e\^[it\_]{}\_0 - i \_0\^t ds e\^[i(t-s)\_]{} (s) \[int-pGP-sys-1\]\ (t) & = & e\^[it\_]{}\_0 - i \_0\^t ds e\^[i(t-s)\_]{} (s) , \[int-pGP-sys-2\] where we note that the second line is formally a self-consistent fixed point equation for $\opB\Gamma$. Let $I:=[0,T]$. We introduce the product space \[def-Wspace-1\] := L\^\_[tI]{}\_\^ L\^1\_[tI]{}\_\^ , which we endow with the norm (, ) \_ := \_[L\^\_[tI]{}\_\^]{} + \_[L\^1\_[tI]{}\_\^]{} . Clearly, $( \Wspacexi , \\| \, \cdot \, \\|_{\Wspacexi} )$ is a Banach space. Then, we introduce the system (t) & = & e\^[it\_]{}\_0 - i \_0\^t ds e\^[i(t-s)\_]{} (s) \[int-pGP-sys-Wspace-1\]\ (t) & = & e\^[it\_]{}\_0 - i \_0\^t ds e\^[i(t-s)\_]{} (s) , \[int-pGP-sys-Wspace-2\] which is formally equivalent to the system (\[int-pGP-sys-1\]), (\[int-pGP-sys-2\]). Our main result in this section is the following existence and uniqueness theorem, which corresponds to part (i) in Theorem \[thm-main-0\]. \[thm-locwp-main-1\] Assume that $\alpha\in\alphaset(d,p)$ where $d\geq1$ and $p\in\{2,4\}$. Then, the following holds. For every $\Gamma_0\in\cH_{{\xi_1}}^\alpha$, there exist constants $T>0$ and $0<\xi_2\leq\xi_1$ such that for $t\in I=[0,T]$, there exists a unique solution $(\Gamma(t),\Xi(t))$ of the system (\[int-pGP-sys-Wspace-1\]), (\[int-pGP-sys-Wspace-2\]), in the space $\Wspace$. Moreover, the relation (t)=(t)L\^1\_[t I]{}\_[\_2]{}\^, holds for the solution, and the component $\Gamma(t)$ satisfies the $p$-GP hierarchy (\[seclwp-pGP\]), \[eq-GPhier-thm-1\] i\_t + \_ - = 0, with initial condition $\Gamma(0)=\Gamma_0\in\cH_{\xi_1}^\alpha$. Moreover, there exists a constant $C(T,d,p,\xi_1,\xi_2)$ such that \_[L\^1\_[tI]{}\_[\_2]{}\^]{} C(T,d,p,\_1,\_2) \_0 \_[\_[\_1]{}\^]{} holds. We note that our local existence and uniqueness result differs than the one proven in [@esy1; @esy2] in that performing the contraction principle on the space $\Wspace$ presumes finiteness of $\\| \, \opB \Gamma \, \\|_{L^1_{t\in I}\cH_{\xi_2}^\alpha}$. However, [^1] it is a priori conceivable that there exist solutions of the $p$-GP hierarchy $\Gamma(t)\in \cH_{\xi_2}^\alpha$ for which $\\| \, \opB \Gamma \, \\|_{L^1_{t\in I}\cH_{\xi_2}^\alpha}$ is not finite. We note that finiteness of $\\| \, \opB \Gamma \, \\|_{L^1_{t\in I}\cH_{\xi_2}^\alpha}$ is an essential element of this analysis, and has been assumed without proof in [@klma] as a key ingredient. It was previously unknown whether this condition is generally satisfied for solutions of the GP hierarchy. Using Theorem \[thm-locwp-main-1\], we can prove that $\\| \, \opB \Gamma \, \\|_{L^1_{t\in I}\cH_{\xi_2}^\alpha}<\infty$ is a sufficient and necessary condition for a solution $\Gamma(t)\in\cH_{\xi_2}^\alpha$ of (\[seclwp-pGP\]), with initial condition $\Gamma(0)\in\cH_{\xi_1}^\alpha$, to be unique. Accordingly, we arrive at the following result which corresponds to part (ii) in Theorem \[thm-main-0\]. \[thm-locwp-main-1-L1bd\] Assume that $\alpha\in\alphaset(d,p)$ where $d\geq1$ and $p\in\{2,4\}$. Assume moreover that for $\Gamma_0\in\cH_{{\xi_1}}^\alpha$ there exist constants $T>0$ and $0<\xi_2\leq\xi_1$ such that for $t\in I=[0,T]$, there exists a solution $\Gamma(t)$ of the $p$-GP hierarchy (\[seclwp-pGP\]) in the space $L^\infty_{t\in I}\cH_{\xi_2}^\alpha$. Then, the solution $\Gamma(t)\in L^\infty_{t\in I}\cH_{\xi_2}^\alpha$ is [unique]{} if and only if $\\| \, \opB \Gamma \, \\|_{L^1_{t\in I}\cH_{\xi}^\alpha}<\infty$ holds for some $\xi>0$. If the latter holds, then in fact, there exists a constant $C(T,d,p,\xi_1,\xi_2)$ such that \[eq-Strichartz-Wspace-1-1\] \_[L\^1\_[tI]{}\_[\_2]{}\^]{} C(T,d,p,\_1,\_2) \_0 \_[\_[\_1]{}\^]{} is satisfied. We first prove Theorem \[thm-locwp-main-1\]. We make the key observation that the fixed point equation (\[int-pGP-sys-Wspace-2\]) determining the component $\Xi(t)$ of the desired solution $(\Gamma(t),\Xi(t))$ is self-contained, and independent of the component $\Gamma(t)$. Therefore, we can invoke the Picard fixed point principle on the space $L^1_{t\in I}\cH_{\xi_2}^\alpha$ (the second factor of the product space $\Wspace$) to first find a unique $\Xi\in L^1_{t\in I}\cH_{\xi_2}^\alpha$ solving (\[int-pGP-sys-Wspace-2\]). We recall that the fixed point equation for $\Xi(t)$ is given by \[eq-opBGamma-fixedpt-1\] (t) = e\^[it\_]{} \_0 - i \_0\^t ds e\^[i(t-s)\_]{}(s), in the space $L^1\cH_{\xi_2}^\alpha$. We define, for an admissible sequence of density matrices $\widetilde\Gamma=(\widetilde\gamma^{(k)})_{k\in\N}$, \[eq-Duh-j-def-1\]\ & := & (-i)\^j\_0\^t dt\_1 \_0\^[t\_[j-1]{}]{}dt\_j e\^[i(t-t\_1)\_\^[(k+1)]{}]{}B\_[k+2]{}e\^[i(t\_1-t\_2)\_\^[(k+2)]{}]{}\ && B\_[k+3]{} B\_[k+j+1]{} e\^[i t\_j \_\^[(k+j+1)]{}]{} \^[(k+j+1)]{}(t\_j) . Then, any solution of (\[eq-opBGamma-fixedpt-1\]) satisfies the fixed point equation (obtained from iterating the Duhamel formula $k$ times for the $k$-th component of $\opB\Gamma$) ()\^[(k)]{}(t) = \_[j=1]{}\^[k-1]{}B\_[k+1]{}\_j(\_0)\^[(k+1)]{}(t) + B\_[k+1]{}\_[k]{}()\^[(k+1)]{}(t) . To formulate a Picard-type fixed point argument, we define ()=( ()\^[(k)]{})\_[k]{} where the $k$-th component is given by \[eq-PhiGamma-fixed-pt\] ()\^[(k)]{}(t) = \_[j=1]{}\^[k-1]{}B\_[k+1]{}\_j(\_0)\^[(k+1)]{}(t) + B\_[k+1]{}\_[k]{}()\^[(k+1)]{}(t) . Similarly as in [@chpa], we use different approaches when $d\geq2$ and when $d=1$. In dimension $d=1$, and for both the cubic and quintic GP hierarchy, we use a spatial a priori bound as in [@chpa] where we refer for details. In dimensions $d\geq2$, we apply the Klainerman-Machedon spacetime bounds similarly to [@klma] and [@kiscst]. This is explained in detail in the Appendix. We first consider the case $d\geq2$. In the sequel, we will often abbreviate H\^\_k := H\^(\^[dk]{}\^[dk]{}) . We invoke Propositions \[prp-spacetime-bd-1\] and \[prp-spacetime-bd-2\] in the Appendix. They generalize the spacetime bounds and “board game" arguments developed in [@klma]. Proposition \[prp-spacetime-bd-2\] implies that for $\Gamma_0=(\gamma_0^{(k)})_{k\in\N}$,\ & < & k C\^k \_[j=1]{}\^[k-1]{} (c T)\^[2]{} \^[(k+j+1)]{}\_0\_[H\^\_[k+j+1]{}]{}\ & < & k (C \_1\^[-1]{})\^[k]{} \_[j=1]{}\^[k-1]{} (c T \_1\^[-2]{})\^[2]{} \_1\^[k+j+1]{} \^[(k+j+1)]{}\_0\_[H\^\_[k+j+1]{}]{}\ & < & (c T \_1\^[-2]{}) k (C \_1\^[-1]{})\^[k]{} \_[j=1]{}\^[k-1]{} \_1\^[k+j+1]{} \^[(k+j+1)]{}\_0\_[H\^\_[k+j+1]{}]{} . \[eq-Duhexp-aux-1\] Therefore,\ & < & (c T \_1\^[-2]{}) \_[k=1]{}\^k (C )\^k \_[j=1]{}\^[k-1]{} \_1\^[k+j+1]{} \^[(k+j+1)]{}\_0\_[H\^\_[k+j+1]{}]{}\ & < & (c T \_1\^[-2]{}) \_[k=1]{}\^k (C )\^k \_[=1]{}\^[2k]{} \_1\^ \^[()]{}\_0\_[H\^\_]{}\ & < & (c T \_1\^[-2]{}) \_[k=1]{}\^k (C )\^k \_0\_[\_[\_1]{}\^]{}\ & < & (c T \_1\^[-2]{}) \_0\_[\_[\_1]{}\^]{} \[eq-Duhexp-aux-2\] for $\xi_2=\eta\xi_1$, with $0<\eta\leq1$ sufficiently small. This implies that, for $I=[0,T]$, and any $T>0$, ( \_[j=1]{}\^[k-1]{}B\_[k]{}\_j(\_0)\^[(k+1)]{}(t) )\_[k]{} L\^1\_[tI]{}\^\_[\_2]{} , provided that $\Gamma_0\in\cH_{\xi_1}^\alpha$, and $\xi_2=\eta\xi_1$ with $\eta>0$ sufficiently small. Our next step is to prove that $\Phi$ is a contraction on $L^1_{t\in I}\cH^\alpha_{\xi_2}$. To this end, we use the bound (\_1)\^[(k)]{} -(\_2)\^[(k)]{} \_[L\_[tI]{}\^1H\^\_k]{} k (C T)\^ \_1\^[(2k)]{} - \_2\^[(2k)]{} \_[L\^1\_[tI]{}H\^\_[2k]{}]{} which is proven in Proposition \[prp-spacetime-bd-2\] in the Appendix. We obtain \[eq-PhiGamma-contr-1\]\ & = & \_[k=1]{}\^\_2\^k (\_1)\^[(k)]{} -(\_2)\^[(k)]{} \_[L\_[tI]{}\^1H\^\_k]{}\ && \_[k=1]{}\^k (C T \_2\^[-2]{})\^ \_2\^[2k]{} \_1\^[(2k)]{} - \_2\^[(2k)]{} \_[L\^1\_[tI]{}H\^\_[2k]{}]{}\ && \_[k1]{}{ k (C T \_2\^[-2]{})\^ } \_[k=1]{}\^ \_2\^[2k]{} \_1\^[(2k)]{} - \_2\^[(2k)]{} \_[L\^1\_[tI]{}H\^\_[2k]{}]{}\ && ( C T \_2\^[-2]{})\^[12]{} \_1 - \_2 \_[L\^1\_[tI]{}\_[\_2]{}\^]{}, for $T>0$ sufficiently small. This implies that $\Phi$ is a contraction on $L^1_{t\in I}\cH^\alpha_{\xi_2}$ provided that $T$ is sufficiently small. Consequently, for a every initial condition $\Gamma_0\in\cH_{\xi_1}^\alpha$, there exist $T>0$ and $\xi_2=\eta\xi_1$ with $0<\eta\leq1$ such that there exists a unique solution $\Xi(t)$ of (\[eq-opBGamma-fixedpt-1\]) in $L^1_{t \in I}\cH^\alpha_{\xi_2}$, Repeating the above arguments, we find that this solution satisfies \[eq-Strichartz-cH-1\] \_[L\^1\_[tI]{}\_[\_2]{}\^]{} &&(c T \_1\^[-2]{}) \_0\_[\_[\_1]{}\^]{} + ( C T \_2\^[-2]{} )\^[12]{} \_[L\^1\_[tI]{}\_[\_2]{}\^]{} . Hence, in particular, \[eqn-BGamma-spacetime-bd-1\] \_[L\^1\_[tI]{}\_[\_2]{}]{} && \_0\_[\_[\_1]{}\^]{} for sufficiently small $T>0$. Given the solution $\Xi$ in $L^1_{t \in I}\cH^\alpha_{\xi_2}$ obtained from the contraction argument (\[eq-PhiGamma-contr-1\]), the right hand side of the integral equation (\[int-pGP-sys-Wspace-1\]) is completely determined by $\Gamma_0$ and $\Xi(s)$ where $s\in I$. Accordingly, $\Gamma(t)$ is given by \[int-pGP-sys-Wspace-1-2\] (t) = e\^[it\_]{}\_0 - i \_0\^t ds e\^[i(t-s)\_]{} (s) where the solution $\Xi(s)$ found in step [(1)]{} above has been substituted into the integral. Evidently, \[eq-Gamma-Duh-opBGamma-1\] (t) \_[ \_[\_2]{}\^]{} & & \_0 \_[\_[\_2]{}\^]{} + \_0\^t ds e\^[i(t-s)\_]{}(s)\_[\_[\_2]{}\^]{}\ & & \_0 \_[\_[\_2]{}\^]{} + \_0\^t ds (s)\_[\_[\_2]{}\^]{}\ & & \_0 \_[\_[\_2]{}\^]{} + (s) \_[L\^1\_[sI]{}\_[\_2]{}\^]{} , using the unitarity of $e^{i(t-s)\opDelta_\pm}$ with respect to $\cH_{\xi_2}^\alpha$. By (\[eqn-BGamma-spacetime-bd-1\]), we conclude that the last line is bounded, hence $\Gamma(t)$ determined by (\[int-pGP-sys-Wspace-1-2\]) indeed lies in $L^\infty_{t\in I}\cH_{\xi_2}^\alpha$. To prove uniqueness, we assume that $(\Gamma_1(t),\Xi_1(t))$ and $(\Gamma_2(t),\Xi_2(t))\in\Wspace$ correspond to solutions of the system (\[int-pGP-sys-Wspace-1\]), (\[int-pGP-sys-Wspace-2\]) to the same initial condition $\Gamma_1(0)=\Gamma_2(0)\in\cH_{\xi_1}^\alpha$, for $t\in[0,T]$. Then, linearity implies that $(\Gamma_1(t)-\Gamma_2(t),\Xi_1(t))-\Xi_2(t))\in\Wspace$ solves the system (\[int-pGP-sys-Wspace-1\]), (\[int-pGP-sys-Wspace-2\]) with initial condition $\Gamma_1(0)-\Gamma_2(0)=(0)_{k\in\N}$. Hence, (\[eq-Gamma-Duh-opBGamma-1\]) implies that \_1(t)-\_2(t) \_[ \_[\_2]{}\^]{} & & ( 1 + ) \_1(0)-\_2(0)\_[\_[\_1]{}\^]{}\ &=&0. This implies that $\Gamma_1(t)=\Gamma_2(t)$, for all $t<T$. This proves local existence and uniqueness of solutions for the system (\[int-pGP-sys-Wspace-1\]), (\[int-pGP-sys-Wspace-2\]). Next, we observe that applying $\opB$ to $\Gamma(t)$ produces the rhs of the fixed point equation (\[eq-opBGamma-fixedpt-1\]) for $\Xi(t)$. Therefore, we immediately find that \[eq-XiopBGammadiff-1\] = L\^1\_[tI]{}\_[\_2]{}\^. Finally, it is easy to verify for the unique solution $(\Gamma(t),\Xi(t))\in\Wspace$ of (\[int-pGP-sys-Wspace-1\]), (\[int-pGP-sys-Wspace-2\]) that i\_t (t) + \_(t) = (t) . Therefore, using (\[eq-XiopBGammadiff-1\]), i\_t (t) + \_(t) - (t) \_[L\^1\_[tI]{}\_[\_2]{}\^]{} = 0 , which implies (\[eq-GPhier-thm-1\]). For the quintic GP hierarchy, all steps of the above proof can be adopted with minor modifications. A key difference is the fact that (\[eq-Gamma-Duh-opBGamma-1\]) is replaced by \[eq-Gamma-Duh-opBGamma-quintic-1\] (t) \_[ \_[\_2]{}\^]{} & & \_0 \_[\_[\_2]{}\^]{} + \_[L\^1\_[tI]{}\_[\_2]{}\^]{}\ & & ( 1 + ) \_0\_[\_[\_1]{}\^]{} . This concluded the proof of Theorem \[thm-locwp-main-1\] for $d\geq2$. In the case $d=1$, we can straightforwardly adapt the proof given in [@chpa] of the uniqueness of solutions of the quintic GP hierarchy in $d=1$. The spacetime bounds of Proposition \[prp-spacetime-bd-1\] is not available in $d=1$ since it would produce divergent bounds. However, the spatial bounds in $d=1$ proven in [@chpa] apply for both the cubic and the quintic GP hierarchy, under the assumption that $\alpha>\frac12$. The result is that we get a factor $t$ instead of $t^{\frac12}$, in all of the bounds found above for the cubic GP hierarchy that produced a factor $t^{\frac12}$. Accordingly, we find \[eq-Gamma-Duh-opBGamma-1D-1\] (t) \_[ \_[\_2]{}\^]{} & & ( 1 + ) \_0\_[\_[\_1]{}\^]{} for the cubic GP hierarchy instead of (\[eq-Gamma-Duh-opBGamma-1\]), and \[eq-Gamma-Duh-opBGamma-quintic-1D-1\] (t) \_[ \_[\_2]{}\^]{} & & ( 1 + ) \_0\_[\_[\_1]{}\^]{} for the quintic GP hierarchy instead of (\[eq-Gamma-Duh-opBGamma-quintic-1\]), respectively. Hence, we obtain local wellposedness for sufficiently small $T>0$. In conclusion, we have proved the existence and uniqueness of solutions of the GP hierarchy (\[seclwp-pGP\]) in the space $\Wspace$ defined in (\[def-Wspace-1\]). It is now straightforward to prove Theorem \[thm-locwp-main-1-L1bd\]. $"\Rightarrow"$. We first prove that uniqueness of $\Gamma\in L^\infty_{t\in I}\cH_{\xi_2}^\alpha$ implies $\\| \opB\Gamma \\|_{L^1_{t \in I} \cH^\alpha_{\xi}}<\infty$ for some $\xi>0$. Let $t'\in I_1=[0,T_1]$. Assume $\Gamma(t+t')$ is the unique solution in $\cH_{\xi_2}^1$ of the $p$-GP hierarchy for the initial condition $\Gamma(t)\in \cH_{\xi_1}^\alpha$, for $t'\in I_1$, and $0<\xi_1<1$ chosen appropriately. We may also use $\Gamma(t)$ as an initial condition for the system (\[int-pGP-sys-Wspace-1\]), (\[int-pGP-sys-Wspace-2\]) in $\mathfrak{W}_{\xi_2}^\alpha(I_2)$. Thereby, we obtain a unique solution $( \Gamma'(t+t') , \Xi(t+t') $ in $\mathfrak{W}_{\xi_2}^\alpha(I_2)$, for $t' \in I_2 = [0,T_2]$, for $T_2>0$ and $0<\xi_2\leq\xi_1$ chosen small enough. Let $I:=[0,\min\{T_1,T_2\}]$. From Theorem \[thm-locwp-main-1\], we infer that $\Gamma'(t+t')$ is also an element of $\cH_{\xi_2}^\alpha$, for $t\in I$, and solves the GP hierarchy equation with initial condition $\Gamma'(t)=\Gamma(t)$ for $t'=0$. Since by assumption, $\Gamma(t+t')$ is the unique solution of the GP hierarchy in $\cH_{\xi_2}^\alpha$, solving the same GP hierarchy equation, and with the same initial condition $\Gamma(t)$ at $t'=0$. Thus, $\Gamma(t+t')=\Gamma'(t+t')$. But since the solution $( \Gamma'(t+t') , \Xi(t+t') )$ in $\Wspace$ for the system (\[int-pGP-sys-Wspace-1\]), (\[int-pGP-sys-Wspace-2\]) has the property that $\\| \opB\Gamma'(t + . ) \\|_{L^1_{t'\in I} \cH^\alpha_{\xi_2}} \leq C(T,d,p,\xi_1,\xi_2) \, \\| \Gamma(t) \\|_{\cH_{\xi_1}^\alpha}$ is bounded, we conclude that also, $\\| \opB\Gamma(t + . ) \\|_{L^1_{t'\in I} \cH^\alpha_{\xi_2}} \leq C(T,d,p,\xi_1,\xi_2) \, \\| \Gamma(t) \\|_{\cH_{\xi_1}^\alpha}$. $"\Leftarrow"$. Next, we prove that $\\| \opB\Gamma \\|_{L^1_{t \in I} \cH^\alpha_{\xi}}<\infty$ for some $\xi>0$ implies uniqueness of $\Gamma\in L^\infty_{t\in I}\cH_{\xi_2}^\alpha$, for some $\xi_2$. We may set $\xi=\xi_2$ by choosing $\min\{\xi,\xi_2\}$. We note that since $\Gamma\in L^\infty_{t\in I}\cH_{\xi_2}^\alpha$ and $\\| \opB\Gamma \\|_{L^1_{t \in I} \cH^\alpha_{\xi}}<\infty$, it is clear that $(\Gamma,\opB\Gamma)\in\Wspace$. Accordingly, Theorem \[thm-locwp-main-1\] implies that $\Gamma$ is the unique solution in $\Wspace$ for the initial condition $\Gamma(0)=\Gamma_0\in\cH_{\xi_1}^\alpha$ for which $\\| \opB\Gamma \\|_{L^1_{t \in I} \cH^\alpha_{\xi}}<\infty$ is satisfied. Finally, Theorem \[thm-locwp-main-1\] implies that whenever $\\| \opB\Gamma \\|_{L^1_{t \in I} \cH^\alpha_{\xi}}<\infty$ holds for some $\xi>0$, the stronger estimate (\[eq-Strichartz-Wspace-1-1\]) is satisfied. $\;$\ \ On the global existence and uniqueness of solutions for defocusing GP hierarchies {#sec-gwp-GP-1} ================================================================================= It is proved for the $d=2,3$ defocusing cubic case in [@esy1; @esy2; @kiscst], and for the $d=1,2$ defocusing quintic case in [@chpa], that whenever $\Gamma(t)=(\gamma^{(k)}(t))_{k\in\N}$ is obtained from the $N\rightarrow\infty$ limit of a bosonic $N$-particle BBGKY hierarchy with repulsive interactions, \[eq-GPsol-defoc-apriorien-1\] ( S\^[(k,1)]{} \^[(k)]{}\_0 ) < C\^k for some constant $C$, then \[eq-GPsol-defoc-apriorien-2\] ( S\^[(k,1)]{} \^[(k)]{}(t) ) < C\_0\^k with $C_0$ independent of $t\geq0$. This follows from energy conservation in the $N$-particle system of which the GP hierarchy is the $N\rightarrow\infty$ limit. We first claim that this implies that $\Gamma(t)\in\cH_{\xi_1}^1$ for some $\xi_1>0$, and all $t\in\R$. To see this, we consider a fixed $k$. Let $\gamma^{(k)}$ be non-negative, normalized trace class, $\tr(\gamma^{(k)})=1$, and hermitean. Then, we have that \^[(k)]{}(\_k;\_k’) = \_j \_j \|\_j(\_k’)\_j(\_k)\| for an orthogonal basis $\psi_j$ of $L^2(\R^{dk})$ with $\lambda_j\geq0$ and $\sum\lambda_j=1$. Then, \^[(k)]{} \_[H\^1]{} & = & \_[j,j’]{} \_j \_[j’]{} \| \_[\_k]{}\_j \| \_[\_k]{}\_[j’]{} \|\^2\ && (\_j \_j \_[\_k]{}\_j \^2 )\^2\ &=& ( ( S\^[(k,1)]{} \^[(k)]{} ) )\^2 . Thus, for a solution $\gamma^{(k)}(t)$ of the (cubic or quintic) GP hierarchy with initial condition satisfying (\[eq-GPsol-defoc-apriorien-1\]), we have that \^[(k)]{}(t) \_[H\^1]{} < C\_0\^k with $C_0$ independent of $t$. Thus, for $\xi_1$ sufficiently small, we obtain the a priori bound (t)\_[\_[\_1]{}\^1]{} ( \_[k=1]{}\^(C\_0)\^[ 2k]{} )\^[12]{} < , for all $t\in\R$. This implies the existence of a solution $\Gamma(t)\in\cH_{\xi_1}^1$ globally in time. The key question is whether $\Gamma(t)\in \cH_{\xi_1}^1$ is the unique solution of the $p$-GP hierarchy with initial condition $\Gamma_0\in\cH_{\xi_1}^1$. To this end, we prove the following uniqueness result. \[thm-defocGP-globalwp-1\] Assume that $1\leq d\leq3$ for $p=2$, and in $1\leq d\leq 2$ for $p=4$ such that $1 \in\alphaset(d,p)$. Assume that $\Gamma(t)\in\cH_{\xi_1}^1$, $t\in\R$, is a solution of the defocusing $p$-GP hierarchy for the initial condition $\Gamma(0)=\Gamma_0$, and satisfies the a priori bound $\\|\Gamma(t)\\|_{\cH_{\xi_1}^1}<c_0<\infty$ for $t\in\R$. Moreover, assume that for every sufficiently short open interval $I\in\R$, there exists a constant $\xi=\xi(I)$ with $0<\xi\leq \xi_1$ such that $\\|\opB\Gamma\\|_{L^1_{t\in I}\cH_{\xi}^1}<\infty$. Then, there exists a constant $\xi_2=\xi_2(\xi_1)$ depending only on $\xi_1$, and a constant $T_0=T_0(d,p,\xi_1,\xi_2)$ such that the pair $(\Gamma,\opB\Gamma)$ is the unique solution of the system (\[int-pGP-sys-Wspace-1\]), (\[int-pGP-sys-Wspace-2\]) associated to the defocusing ($\mu=+1$) $p$-GP hierarchy with initial condition $\Gamma(0)=\Gamma_0\in\cH_{\xi_1}^1$, in the space := \_[I,\|I\|<T\_0]{}ace , for $0<\xi_2\leq\xi_1$ sufficiently small. Hence in particular, \_[L\^1\_[tI]{}\_[\_2]{}\^1]{} C(T\_0,d,p,\_1,\_2) c\_0 for all $I\in\R$ with $\|I\|<T_0$. Assume that $I'\subset\R$ is such that $\\|\opB\Gamma\\|_{L^1_{t\in I'}\cH_{\xi}^1}<\infty$. Let $I:=[\tau,T]\subset I'$ Then, Theorem \[thm-locwp-main-1-L1bd\] implies that in fact, there exists a constant $\xi_2\leq\xi_1$ such that \_[L\^1\_[tI ]{}\_[\_2]{}\^1]{} < C\_0 () \_[\_[\_1]{}\^1]{} < C\_0 c\_0 , for a constant $C_0=C_0(\|I\|,d,p,\xi_1,\xi_2)$. This estimate holds independently of the value of $\tau\in\R$. Therefore, we can cover $\R$ with intervals $\R=\cup_j I_j$ where $I_j=[\tau_j,\tau_{j+1}]$, $j\in\Z$, and $\|I_j\|=\|I\|=: T_0$. Assume that the initial time $t=0\in I_0$. Then, Theorem \[thm-locwp-main-1-L1bd\] implies that there exists a unique (forward and backward in time) solution $(\Gamma,\opB\Gamma)$ of the system (\[int-pGP-sys-Wspace-1\]), (\[int-pGP-sys-Wspace-2\]) in $\mathfrak{W}_{\xi_2}^1(I_0)$ for the initial condition initial condition $\Gamma_0$. By iteration, we use $\Gamma(\tau_{1})$ as the initial condition for $t\in I_1$, and $\Gamma(\tau_0)$ as the terminal condition for $t\in I_{-1}$, and continue recursively, thereby covering $\R=\cup I_j$. In conclusion, we obtain that there exists a unique global in time solution in $\Wspaceglob$ with initial condition $\Gamma_0\in\cH_{\xi_1}^1$, under the assumptions of the theorem. As noted in Remark \[rem-BBGKYlim-1\], every solution $\Gamma(t)\in\cH_{\xi_1}^1$ extracted by a diagonal argument from the $N\rightarrow\infty$ limit of the $N$-particle BBGKY hierarchies (with repulsive interactions) studied by Erdös-Schlein-Yau in [@esy1; @esy2], Kirkpatrick-Schlein-Staffilani in [@kiscst], and Chen-Pavlović in [@chpa], satisfies $\\| \, \opB \Gamma \, \\|_{L^1_{t\in I}\cH_{\xi}^\alpha}<\infty$ for some $\xi>0$, and all sufficiently short intervals $I\subset\R$. This follows a posteriori from the uniqueness of those solutions, which has been established in these works with independent methods, combined with Theorem \[thm-locwp-main-1-L1bd\] in the work at hand. Accordingly, $(\Gamma,\opB\Gamma)$ is automatically an element of $\Wspaceglob$ whenever $\Gamma(t)$ is obtained from an $N\rightarrow\infty$ limit of the above noted type. $\;$\ \ Lower bound on the blowup rates {#sec-Proof-blowuprate-1} =============================== In this section, we establish Theorem \[thm-blowuprate-L2crit-1\]. We adapt a standard proof given for $L^2$-critical focusing NLS to the GP hierarchy; see for instance [@ra]. Let $p \in \{2,4\}$. Similarly as in (\[eq-pseudoconf-gamma-1\]), one finds that the $p$-GP hierarchy is invariant under the rescaling \[eq-pseudoconf-gamma-2\]\ && := \^[(k)]{}( t +((t))\^[-2]{} , ((t))\^[-1]{}\_k ; ((t))\^[-1]{}\_k’ ) If $\Gamma(t)=(\gamma^{(k)}(t))_{k\in\N}$ solves the $p$-GP hierarchy, then $\cS_{\lambda,t}\Gamma=(\cS_{\lambda,t}\gamma^{(k)} )_{k\in\N}$ is also a solution of the $p$-GP hierarchy. The proof can be straightforwardly adapted from the one given in Section \[sec-confinv-1\]. \ \ Blowup in finite time means that there exists $T^<\infty$ such that $\Av_{H^\alpha}(\Gamma(t))\rightarrow\infty$ as $t\rightarrow T^$. To prove a lower bound on the blowup rate, we may assume that $1<\Av_{H^\alpha}(\Gamma(t))<\infty$ at a fixed time $t$, and choose (t) = (\_[H\^]{}((t)))\^ > 1 . We note that $\frac{4}{d}<p<\frac{4}{d-2\alpha}$ implies that $2\alpha-d+\frac4p>0$. Let S\^[(k,)]{}\_[(t)]{} := \_[j=1]{}\^k ((t))\^[-1]{} \_[x\_j]{}\^ ((t))\^[-1]{} \_[x\_j’]{} \^ where $\langle b\nabla_{x}\rangle=\sqrt{1-b^2\Delta_{x}}$ for any $b\in\R$. Clearly, \[eq-Sdot-quasiblowup-L2-aux-1\]\ & = & ((t))\^[k(d-)]{} ( S\^[(k,)]{}\_[(t)]{} \^[(k)]{} ) (t+((t))\^[-2]{}) \_[L\^2\_[\_k,\_k’]{}]{} , and \[eq-scaleSkalpha-bd-1\] ((t))\^[-2k]{} S\^[(k,)]{} S\^[(k,)]{}\_[(t)]{} S\^[(k,)]{} since we are assuming that $\lambda(t)>1$. We define \_<(,t,) := ((t))\^[-4p+d-2]{} = (\_[H\^]{}((t)))\^[-1]{} and \[eq-xibigxi-id-1\] \_>(,t,) := ((t))\^[-4p+d ]{} = (\_[H\^]{}((t)))\^[ ]{} . Clearly, (\[eq-Sdot-quasiblowup-L2-aux-1\]) and (\[eq-scaleSkalpha-bd-1\]) imply that \[eq-sandwich-bd-1\] (t+((t))\^[-2]{}) \_[\_[\_<(,t,)]{}\^]{} & & \_[,t]{}() \_[ \^\_]{}\ && (t+((t))\^[-2]{}) \_[\_[\_>(,t,)]{}\^]{} . As a consequence of the definition of $\Av_{H^\alpha}(\Gamma(t))$, it follows that for $\tau=0$, 0 < (t ) \_[\_[\_<(,t,)]{}\^]{} < c for any $0<\xi<1$. To ensure that $\\| \Gamma(t+(\lambda(t))^{-2}\tau) \\|_{\cH_{\xi_>(\xi,t,\lambda)}^\alpha}<c$, we use the fact that according to (\[eq-xibigxi-id-1\]), $\xi_>(\xi,t,\lambda) $ can be made arbitrarily small by choosing $\xi$ small. We note that our assumption $\frac{4}{d}<p<\frac{4}{d-2\alpha}$ implies that $2\alpha-d+\frac4p>0$ and $d-\frac4p>0$, so that the exponent on the rhs of (\[eq-xibigxi-id-1\]) is positive. If blowup occurs, such that $\Av_{H^\alpha}(\Gamma(t))\rightarrow\infty$ as $t\nearrow T^$, the above considerations necessitate the choice of values of $\xi$ (whose reciprocal determines the energy scale) tending to zero as $t\nearrow T^$. Thus, for $\xi_1>0$ sufficiently small, \_[,t]{}(0) \_[ \^\_[\_1]{}]{} (t ) \_[\_[\_>(\_1,t,)]{}\^]{} Due to Theorem \[thm-locwp-main-1\], we may pick $0<\xi_2=\eta\xi_1\ll\xi_1<1$, such that there exists a solution $\cS_{\lambda,t}\gamma^{(k)} (\tau)\in\cH_{\xi_2}^\alpha$ if $\tau\in[0,\tau_{\max}]$, for $\tau_{\max}>0$ sufficiently small. But this implies that (t+((t))\^[-2]{}) \_[\_[\_<(\_2,t,)]{}\^]{} \_[,t]{}() \_[ \^\_[\_2]{}]{} < for $\tau\in[0,\tau_{\max}]$ so that there is no blowup if $\tau$ lies in that interval. Therefore, the blowup time $T^$ is bounded from below by T\^\* > t+((t))\^[-2]{}\_ , and hence, ( \_[H\^]{}((t)) )\^[12]{} = (t)\^[(-+)]{} > . This proves ($a$). \ \ It is easy to see that \[eq-Lp-rescale\] \_[,t]{}\^[(k)]{} () \_[L\^r\_[\_k,\_k’]{}]{} = ((t))\^[-2k(2p-)]{} \^[(k)]{} (t+((t))\^[-2]{}) \_[L\^r\_[\_k,\_k’]{}]{} , which, in turn, implies that \_[,t]{}\^[(k)]{} (0) \_[\^r\_]{} &=& \_[k1]{}\^k \_[,t]{}\^[(k)]{} (0) \_[L\^r\_[\_k,\_k’]{}]{}\ & = & \_[k1]{} \^k ((t))\^[-2k(2p-)]{} \^[(k)]{} (t) \_[L\^r\_[\_k,\_k’]{}]{}\ & = & \_[k1]{} ( )\^[k]{} \^[(k)]{} (t) \_[L\^r\_[\_k,\_k’]{}]{} . \[bigLp-lambda\] However is bounded for every $\xi<1$, if we choose (t) = ( \_[L\^r]{}((t)) )\^ . Now we argue as in the proof of the part ($a$) by using the local well-posedness Theorem \[thm-locwp-main-1\] to conclude that the $H^\alpha$ blowup time $T^$ is bounded from below by T\^\ > t+((t))\^[-2]{}\_ . Therefore ( \_[L\^r]{}((t)) )\^[12]{} = (t)\^[(-)]{} > . Hence ($b$) is proved. Proof of pseudoconformal invariance {#sec-confinv-1} =================================== In this section, we prove Theorem \[thm-pseudoconf-gamma-1\]. We recall the pseudoconformal transformations \[eq-pseudoconf-gamma-1\]\ && := e\^[-i]{} \^[(k)]{}( , ; ) , for any $b\in\R\setminus\{0\}$. Similarly as in the case of NLS, one can verify that\ &=& e\^[-i]{}\ && ((i\_t+ \_[\_k]{}-\_[\_k’]{})\^[(k)]{}) ( , ; ) . Now we shall prove the pseudoconformal invariance of the quintic GP hieararchy when $d=1$. In particular, we find that\ &=& e\^[-i]{} dx\_[k+1]{} dx\_[k+2]{} dx\_[k+1]{}’ dx\_[k+2]{}’\ &&(x\_j-x\_[k+1]{})(x\_j-x\_[k+1]{}’) (x\_j-x\_[k+2]{})(x\_j-x\_[k+2]{}’)\ && \^[(k)]{}( , ; )\ &=& e\^[-i]{}\ && \^[(k)]{}( , ; ) ,\ &=& e\^[-i]{}\ && B\_[j;k+1,k+2]{}\^1\^[(k+2)]{} ( , ; ) with $B_{j;k+1,k+2}=B_{j;k+1,k+2}^1-B_{j;k+1,k+2}^2$; in $B_{j;k+1,k+2}^2$, the variable $x_j$ in $B_{j;k+1,k+2}^1$ is replaced by $x_j'$. Notably, we have used that e\^[-i]{}\|\_[x\_[k+1]{}=x\_[k+2]{}=x\_[k+1]{}’=x\_[k+2]{}’=x\_j]{} = 1 . Thus, when $d=1$, we obtain\ &=& e\^[-i]{}\ && ( (i\_t+ \_[\_k]{}-\_[\_k’]{})\^[(k)]{} - \_[j=1]{}\^k B\_[j;k+1,k+2]{}\^[(k+2)]{} ) ( , ; )\ &=&0 . This proves pseudoconformal invariance of the quintic GP hierarchy in dimension $d=1$. For the cubic GP hierarchy, the operators $B_{j;k+1,k+2}$ are replaced by operators $B_{j;k+1}$ which contract $x_j$, $x_j'$ only with $x_{k+1}$ and $x_{k+1}'$, (i\_t+ \_[\_k]{}-\_[\_k’]{})\^[(k)]{} - \_[j=1]{}\^k B\_[j;k+1]{}\^[(k+1)]{} = 0 . The same considerations as above then produce\ &=& e\^[-i]{}\ && ( (i\_t+ \_[\_k]{}-\_[\_k’]{})\^[(k)]{} - \_[j=1]{}\^k B\_[j;k+1]{}\^[(k+1)]{} ) ( , ; )\ &=&0 if $d=2$. This proves Theorem \[thm-pseudoconf-gamma-1\]. The Klainerman-Machedon spacetime bounds {#sect-boundsfreeGP-1} ======================================== We present the Klainerman-Machedon spacetime bounds in dimensions $d\geq2$ in the form required for this paper, with $\alpha\in\alphaset(d,p)$; see (\[eq-alphaset-def-1\]). In the regime $\alpha>\frac d2-\frac{1}{2(p-1)}$, we present a simple argument to prove the result. In the endpoint case $(d,p)=(3,2)$ and $\alpha=1$, we invoke a result of [@klma]. \[prp-spacetime-bd-1\] Let $p=2,4$ account for the cubic and quintic GP hierarchy, respectively, and assume that $\alpha\in\alphaset(d,p)$. Let $\gamma^{(k+\frac p2)}$ be the solution of i\_t\^[(k+p2)]{}(t,\_[k+p2]{};\_[k+p2]{}’) + (\_[\_[k+p2]{}]{}-\_[\_[k+p2]{}’]{}) \^[(k+p2)]{}(t,\_[k+p2]{};\_[k+p2]{}’) = 0 with initial condition \^[(k+p2)]{}(0,) = \_0\^[(k+p2)]{}\^. Then, there exists a constant $C$ such that\ && C S\^[(k+p2,)]{} \_0\^[(k+p2)]{} \_[L\^2 (\^[d(k+p2)]{}\^[d(k+p2)]{})]{} holds. For notational convenience, we discuss the proof for the quintic GP hierarchy where $p=4$. Let $(\tau,\uu_k,\uu_k')$, $\uq:=(q_1,q_2)$, and $\uq':=(q_1',q_2')$ denote the Fourier conjugate variables corresponding to $(t,\ux_k,\ux_k')$, $(x_{k+1},x_{k+2})$, and $(x_{k+1}',x_{k+2}')$, respectively. Without any loss of generality, we may assume that $j=1$ in $B_{j;k+1,k+2}$. Then, abbreviating () := ( + (u\_1+q\_1+q\_2-q\_1’-q\_2’)\^2 + \_[j=2]{}\^k u\_j\^2 + \|\|\^2 - \|\_k’\|\^2 - \|’\|\^2 ) we find\ & = & \_dd\_k d\_k’ \_[j=1]{}\^ku\_j\^[2]{} u\_j’\^[2]{}\ && ( dd’ () \^[(k+2)]{}(,u\_1+q\_1+q\_2-q\_1’-q\_2’,u\_2,…,u\_k,;\_k’,’) )\^2 . Using the Schwarz estimate, this is bounded by &&\_dd\_k d\_k’ I\_(,\_k,\_k’) dd’ ()\ && u\_1+q\_1+q\_2-q\_1’-q\_2’ \^[2]{} q\_1\^[2]{} q\_2\^[2]{} q\_1’\^[2]{} q\_2’ \^[2]{} \_[j=2]{}\^ku\_j\^[2]{} \_[j’=1]{}\^ku\_[j’]{}’\^[2]{}\ && \| \^[(k+2)]{}(,u\_1+q\_1+q\_2-q\_1’-q\_2’,u\_2,…,u\_k,;\_k’,’) \|\^2 where \[eq-Ialpha-def-1\]\ && := d d’ . Similarly as in [@klma; @kiscst], we observe that $$\label{eq-u1-aux-bd-1} \bra u_1\ket^{2\alpha} \, \leq \, C\Big[ \, \bra u_1+q_1+q_2-q_1'-q_2'\ket^{2\alpha} + \bra q_1\ket^{2\alpha} + \bra q_2\ket^{2\alpha} + \bra q_1'\ket^{2\alpha} + \bra q_2'\ket^{2\alpha} \, \Big] \,, $$ so that I\_(,\_k,\_k’) \_[=1]{}\^5 J\_where $J_\ell$ is obtained from bounding the numerator of (\[eq-Ialpha-def-1\]) using (\[eq-u1-aux-bd-1\]), and from canceling the $\ell$-th term on the rhs of (\[eq-u1-aux-bd-1\]) with the corresponding term in the denominator of (\[eq-Ialpha-def-1\]). Thus, for instance, J\_1 < d d’ , and each of the terms $J_\ell$ with $\ell=2,\dots,5$ can be brought into a similar form by appropriately translating one of the momenta $q_i$, $q_j'$. Further following [@klma; @kiscst], we observe that the argument of the delta distribution equals + (u\_1+q\_1+q\_2-q\_1’)\^2 + \_[j=2]{}\^k u\_j\^2 + \|\|\^2 - \|\_k’\|\^2 - (q\_1’)\^2 - 2(u\_1+q\_1+q\_2-q\_1’)q\_2’ , and we integrate out the delta distribution using the component of $q_2'$ parallel to $(u_1+q_1+q_2-q_1')$. This leads to the bound \[eq-J1-bound-aux-1\] J\_1 & < & C\_C dd q\_1’ where C\_ := \_ . Clearly, $C_\alpha$ is finite for any $\alpha>\frac12$. Moreover, it is clear that $J_1$ is monotonically decreasing in $\alpha$. For the cubic GP hierarchy, the above arguments lead to the condition that instead of (\[eq-J1-bound-aux-1\]), the integral \[eq-J1-bound-aux-2\] d q\_1 must be bounded. \ \ We first consider the case $p=4$ corresponding to the quintic GP hierarchy, and argue as follows. To bound (\[eq-J1-bound-aux-1\]), we pick a spherically symmetric function $h\geq0$ with rapid decay away from the unit ball in $\R^d$, such that $h^\vee(x)\geq0$ decays rapidly outside of the unit ball in $\R^d$, and \[eq-convuppbd-1\] < h\( \_[B\_1]{} + )(q) . (for example, $h(u)= c_1 e^{-c_2 u^2}$, for suitable constants $c_1,c_2$), where $\chi_{B_1} +\chi_{B_1}^c=1$ is a smooth partition of unity with $\chi_{B_1}$ supported on the unit ball, with $\chi_{B_1}(u)=1$ for $\|u\|\leq\frac12$, and $\chi_{B_1}(u)=0$ for $\|u\|>\frac12$. Clearly, $h \chi_{B_1}$ and $h\frac{\chi_{B_1}^c}{\| \, \cdot \, \|^{2\alpha}}$ are both in $L^\infty$, for any $\alpha>0$. Then, assuming that $\alpha<\frac d2$, inserting this into (\[eq-J1-bound-aux-1\]), the most singular part is given by \[eq-J1-convFTprod-bd-1\]\ &=& C\_C dx ( )\^(x) ( (h\)\^(x) )\^3\ &=& C\_C’ dx (h\^(x))\^3 ( ( (\_[B\_1]{}\^c)\^ ) (x) )\^3\ &<& C\_C’ dx (h\^(x))\^3 ( )\^3 . For sufficiently large $C'$, this is an upper bound on all of the remaining terms that are obtained from substituting the bound (\[eq-convuppbd-1\]) into (\[eq-J1-bound-aux-1\]). We have here used that $(\chi_{B_1}^c)^\vee=1^\vee-\chi_{B_1}^\vee=\delta-\chi_{B_1}^\vee$, so that $\|((\chi_{B_1}^c)^\vee\frac{1}{\| \, \cdot \, \|^{d-2\alpha}})(x)\| \leq C \, \frac{1}{\|x\|^{d-2\alpha}}$ holds for $\alpha<\frac d2$. We conclude that (\[eq-J1-convFTprod-bd-1\]) is finite provided that the singularity at $x=0$ is integrable, since $h^\vee(x)$ falls off rapidly as $\|x\|\rightarrow\infty$. In dimension $d$, this is the case if the exponents in the denominator satisfy d-1 + 3d - 6 < d , such that \[eq-alpha-lowbd-1\] > d2-16 . This proves the claim for the quintic GP hierarchy, i.e., for $p=4$. In order to prove the lower bound (\[eq-alpha-lowbd-1\]) on $\alpha$, we have assumed that $\alpha<\frac d2$, which is consistent with it. Now, since $J_1$ is monotonically decreasing in $\alpha$, we arrive at the asserted result. For the cubic GP hierarchy, the same considerations lead to the condition that $(\ref{eq-J1-bound-aux-1}) \, < \, \infty$ if $\alpha \, > \, \frac d2-\frac12$. For a general $p$-GP hierarchy, one obtains the condition $\alpha>\frac d2-\frac{1}{2(p-1)}$. \ \ In the situation $d=3$ and $p=2$ of the cubic GP hierarchy in 3 dimensions, we have the endpoint case $\frac d2-\frac1{2(p-1)}=1$. Klainerman and Machedon have proven in [@klma] that (\[eq-Ialpha-def-1\]) is bounded in this case. Next, we prove the iterated spacetime estimates for the cubic GP hierarchy used in Section \[sec-locwp-1\], which involve the “boardgame estimates” of [@klma], which are motivated by the Feynman graph techniques in [@esy1; @esy2; @ey]. The corresponding results for the quintic GP hierarchy are obtained in an analogous manner, and we refer to [@chpa] for details. \[prp-spacetime-bd-2\] Assume $\alpha$ as in Proposition \[prp-spacetime-bd-1\], for the cubic GP hierarchy ($p=2$). For brevity, let H\^\_k := H\^(\^[dk]{}\^[dk]{}) . Then, for $k\geq1$ and $j\leq k$, and $t\in I=[0,T]$, B\_[k+1]{}\_j(\_0)\^[(k+1)]{}(t) \_[L\^1\_[tI]{}H\^\_k]{} < k C\^k (c T)\^[2]{} \^[(k+j+1)]{}\_0\_[H\^\_[k+j+1]{}]{} . Moreover, \[eq-Boardgame-rem-est-1\] B\_[k+1]{}\_[k]{}()\^[(k+1)]{} \_[L\^1\_[tI]{}H\^\_k]{} < k C\^k (c T)\^[k2]{} ()\^[(2k)]{} \_[L\^1\_[t I]{}H\^\_[2k]{}]{} , where $\duh_j( \, \cdot \,)$ is defined in (\[eq-Duh-j-def-1\]). Let $I=[0,T]$. Using an argument presented as a “board game", it is proven in [@klma] that the following holds. Let ${\mathcal E}_{j,k+1}$ denote the space of sequences $\umu=(\mus(1),\dots,\mus(j))$ where $\mu(i)\in\{1,\dots,k+i\}$, where for every $i\in\{1,\dots,j\}$, one has $\mu(i)\geq\mu(i')$ for all $i'>i$. The elements of ${\mathcal E}_{j,k+1}$ parametrize $(k+j)\times j$ matrices in so-called “special upper echelon form" (see [@klma] for definitions). The cardinality of this set satisfies $\|{\mathcal E}_{j,k+1}\|\leq C^{j+k}$. For every $\umu\in{\mathcal E}_{j,k+1}$, one associates the term\ &:=&\_[D]{} dt\_1 dt\_[j]{} e\^[i(t-t\_1)\_\^[(k+1)]{}]{}B\_[(1),k+2]{}e\^[i(t\_1-t\_2)\_\^[(k+2)]{}]{}\ && e\^[i(t\_[j-1]{}-t\_[j]{})\_\^[(k+j)]{}]{} B\_[(j),k+j+1]{}e\^[i t\_j \_\^[(k+j+1)]{}]{} \^[(k+j+1)]{}\_0 for a measurable subset $D\subset[0,t]^j$. Then, it is proven in [@klma] that \[eq-klma-boardgame-dec-1\]\ && \_[\_[j,k+1]{}]{} B\_[k+1]{} ( \_j(\_0)\^[(k+1)]{}(t) )\_ \_[L\^2\_[tI]{}H\^]{} . For the proof of (\[eq-klma-boardgame-dec-1\]) in the case of the cubic GP hierarchy, we refer to [@klma]. For the case of the quintic GP hierarchy, we refer to [@chpa]. We have, under the given assumptions on $\alpha$, that for $I=[0,T]$ and $D\subset I^j$,\ && \_[=1]{}\^k \_[D]{} dt\_1 dt\_[j]{} B\_[,k+1]{}e\^[i(t-t\_1) \_\^[(k+1)]{}]{} B\_[(1),k+2]{}e\^[i(t\_1-t\_2)\_\^[(k+2)]{}]{}\ && e\^[i(t\_[j-1]{}-t\_[j]{})\_\^[(k+j)]{}]{} B\_[(j),k+j+1]{} e\^[i t\_j \_\^[(k+j+1)]{}]{} \^[(k+j+1)]{}\_0 \_[L\^1\_[tI]{}H\^\_k]{}\ && k \_[I]{} dt \_D dt\_1 dt\_[j]{} B\_[,k+1]{}e\^[i(t - t\_1)\_\^[(k+1)]{}]{}B\_[(1),k+2]{}e\^[i(t\_1-t\_2)\_\^[(k+2)]{}]{}\ && e\^[i(t\_[j-1]{}-t\_[j]{})\_\^[(k+j)]{}]{} B\_[(j),k+j+1]{} e\^[i t\_j \_\^[(k+j+1)]{}]{} \^[(k+j+1)]{}\_0 \_[ H\^\_k]{}\ && k \_[I\^[j+1]{}]{} dt dt\_1 dt\_[j]{} B\_[,k+1]{}e\^[i(t - t\_1)\_\^[(k+1)]{}]{}B\_[(1),k+2]{}e\^[i(t\_1-t\_2)\_\^[(k+2)]{}]{}\ && e\^[i(t\_[j-1]{}-t\_[j]{})\_\^[(k+j)]{}]{} B\_[(j),k+j+1]{} e\^[i t\_j \_\^[(k+j+1)]{}]{}\^[(k+j+1)]{}\_0 \_[ H\^\_k]{} . \[eq-BGamma-Duhj-aux-1\] By Cauchy-Schwarz with respect to the integral in $t$, this is bounded by && k T\^[12]{} \_[I\^[j]{}]{} dt\_1 dt\_[j]{} B\_[,k+1]{}e\^[i(t - t\_1)\_\^[(k+1)]{}]{}B\_[(1),k+2]{}e\^[i(t\_1-t\_2)\_\^[(k+2)]{}]{}\ && e\^[i(t\_[j-1]{}-t\_[j]{})\_\^[(k+j)]{}]{} B\_[(j),k+j+1]{} e\^[i t\_j\_\^[(k+j+1)]{}]{} \^[(k+j+1)]{}\_0 \_[ L\^2\_t(I)H\^\_k]{}\ && k T\^[12]{} \_[I\^[j]{}]{} dt\_1 dt\_[j]{} B\_[,k+1]{}e\^[i(t - t\_1)\_\^[(k+1)]{}]{}B\_[(1),k+2]{}e\^[i(t\_1-t\_2)\_\^[(k+2)]{}]{}\ && e\^[i(t\_[j-1]{}-t\_[j]{})\_\^[(k+j)]{}]{} B\_[(j),k+j+1]{} e\^[i t\_j \_\^[(k+j+1)]{}]{} \^[(k+j+1)]{}\_0 \_[ L\^2\_t()H\^\_k]{} . Using Proposition \[prp-spacetime-bd-1\] and unitarity of $e^{-i t_1\Delta_\pm^{(k+1)}}$, this is bounded by && k (cT)\^[12]{} \_[I\^[j]{}]{} dt\_1 dt\_[j]{} B\_[(1),k+2]{}e\^[i(t\_1-t\_2)\_\^[(k+2)]{}]{}\ && e\^[i(t\_[j-1]{}-t\_[j]{})\_\^[(k+j)]{}]{} B\_[(j),k+j+1]{} e\^[i t\_j \_\^[(k+j+1)]{}]{} \^[(k+j+1)]{}\_0 \_[ H\^\_[k+1]{}]{} \[ap-usingKM1\]\ &=& k (c T)\^[12]{} \_[I\^[j-1]{}]{} dt\_2 dt\_[j]{} B\_[(1),k+2]{} e\^[i(t\_1-t\_2)\_\^[(k+2)]{}]{}\ && e\^[i(t\_[j-1]{}-t\_[j]{})\_\^[(k+j)]{}]{} B\_[(j),k+j+1]{} e\^[i t\_j \_\^[(k+j+1)]{}]{} \^[(k+j+1)]{}\_0 \_[L\^1\_[t\_1]{}(I)H\^\_[k+1]{}]{} . \[ap-Hol\] Iterating the same steps as above, we find, after $j$ steps, & &\ && k (c T)\^ \_[I]{} dt\_[j]{} B\_[(j-1),k+j]{} e\^[i(t\_[j-1]{}-t\_[j]{})\_\^[(k+j)]{}]{} \[ap-KMiterated\]\ && B\_[(j),k+j+1]{} e\^[i t\_j \_\^[(k+j+1)]{}]{} \^[(k+j+1)]{}\_0 \_[L\^2\_[t\_[j-1]{}I]{}H\^\_[k+j-1]{}]{}\ && k (c T)\^ \_[I]{} dt\_[j]{} B\_[(j),k+j+1]{} e\^[i t\_[j]{} \_\^[(k+j+1)]{}]{} \^[(k+j+1)]{}\_0 \_[H\^\_[k+j]{}]{}\ & & k (c T)\^ \^[(k+j+1)]{}\_0 \_[H\^\_[k+j+1]{}]{} . \[A2final\] In the last step, we used Cauchy-Schwarz in $t_j$, and Proposition \[prp-spacetime-bd-1\]. Then, estimating by $C^{j+k}$ the number of terms in the sum over $\umu\in {\mathcal E}_{j,k+1}$, \[eq-BGamma-Duhj-combin-bd-1\] B\_[k+1]{}\_j(\_0)\^[(k+1)]{}(t) \_[L\^1\_[tI]{}H\^]{} k C\^k (c T)\^[2]{} \^[(k+j+1)]{}\_0\_[H\^]{} , as claimed. In the same manner, we prove (\[eq-Boardgame-rem-est-1\]). In this case, we have\ && \_[=1]{}\^k \_[D]{} dt\_1 dt\_[k]{} B\_[,k+1]{}e\^[i(t-t\_1) \_\^[(k+1)]{}]{} B\_[(1),k+2]{}e\^[i(t\_1-t\_2)\_\^[(k+2)]{}]{}\ && e\^[i t\_[k]{} \_\^[(2k)]{}]{} B\_[ 2k+ 1]{} \^[(2k+ 1)]{}(t\_k) \_[L\^1\_[tI]{}H\^\_k]{}\ && k \_[I]{}dt \_[D]{} dt\_1 dt\_[k]{} B\_[,k+1]{}e\^[i(t-t\_1) \_\^[(k+1)]{}]{} B\_[(1),k+2]{}e\^[i(t\_1-t\_2)\_\^[(k+2)]{}]{}\ && e\^[i t\_[k]{} \_\^[(2k)]{}]{} B\_[ 2k+ 1]{} \^[(2k+ 1)]{}(t\_k) \_[ H\^\_k]{} Applying the same arguments as above between (\[eq-BGamma-Duhj-aux-1\]) and (\[ap-KMiterated\]), with $j=k$, one finds the upper bound & &\ && k (c T)\^ \_[I]{} dt\_[k]{} B\_[(k-1),2k ]{} e\^[i t\_[k]{} \_\^[(2k )]{}]{} \[ap-KMiterated-BGamma\]\ && B\_[ 2k+ 1]{} \^[(2k+ 1)]{}(t\_k) \_[L\^2\_[t\_[k-1]{}I]{}H\^\_[2k -1]{}]{}\ && k (c T)\^ \_[I]{} dt\_[k]{} B\_[ 2k+ 1]{} \^[(2k+ 1)]{}(t\_k) \_[H\^\_[2k]{}]{}\ & = & k (c T)\^ B\_[ 2k+ 1]{} \^[(2k+ 1)]{}(t\_k) \_[L\_[tI]{}\^1H\^\_[2k ]{}]{}\ & = & k (c T)\^ ()\^[(2k)]{}(t\_k) \_[L\_[tI]{}\^1H\^\_[2k ]{}]{} . \[A2final-BGamma\] Invoking the argument used to establish (\[eq-BGamma-Duhj-combin-bd-1\]), we arrive at (\[eq-Boardgame-rem-est-1\]). For more details, we refer to [@chpa]. Acknowledgements {#acknowledgements .unnumbered} ---------------- We thank S. Klainerman and N. Tzirakis for inspiring discussions. We are deeply indebted to B. Schlein for extremely useful comments that helped us to significantly improve a previous version. The work of T.C. was supported by NSF grant DMS-0704031. The work of N.P. was supported NSF grant number DMS 0758247 and an Alfred P. Sloan Research Fellowship. [99]{} R. Adami, G. Golse, A. Teta, [Rigorous derivation of the cubic NLS in dimension one]{}, J. Stat. Phys. [127]{}, no. 6, 1194–1220 (2007). M. Aizenman, E.H. Lieb, R. Seiringer, J.P. Solovej, J. Yngvason [Bose-Einstein Quantum Phase Transition in an Optical Lattice Model]{}, Phys. Rev. A [70]{}, 023612 (2004). V. Bach, T. Chen, J. Fröhlich and I. M. Sigal, [Smooth Feshbach map and operator-theoretic renormalization group methods]{}, J. Funct. Anal., [203]{} (1), 44-92 (2003). T. 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Schwarz, [Mean-field- and classical limit of many-body Schrödinger dynamics for bosons]{}, Comm. Math. Phys. [271]{}, no. 3, 681–697 (2007). J. Fröhlich, A. Knowles, A. Pizzo, [Atomism and quantization]{}, J. Phys. A [40]{}, no. 12, 3033–3045 (2007). J. Fröhlich, A. Knowles, S. Schwarz [On the Mean-Field Limit of Bosons with Coulomb Two-Body Interaction]{}, Preprint arXiv:0805.4299. K. Hepp, [The classical limit for quantum mechanical correlation functions]{}, Comm. Math. Phys. [35]{}, 265–277 (1974). S. Klainerman, M. Machedon, [On the uniqueness of solutions to the Gross-Pitaevskii hierarchy]{}, Commun. Math. Phys. [279]{}, no. 1, 169–185 (2008). K. Kirkpatrick, B. Schlein, G. Staffilani, [Derivation of the two dimensional nonlinear Schrödinger equation from many body quantum dynamics]{}, Preprint arXiv:0808.0505. E.H. Lieb, R. Seiringer, [Proof of Bose-Einstein condensation for dilute trapped gases]{}, Phys. Rev. Lett. [88]{}, 170409 (2002). E.H. Lieb, R. Seiringer, J.P. Solovej, J. Yngvason, [The mathematics of the Bose gas and its condensation]{}, Birkhäuser (2005). E.H. Lieb, R. Seiringer, J. Yngvason, [A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas]{}, Commun. Math. Phys. [224]{} (2001). G. Perelman, [On the formation of singularities in solutions of the critial nonlinear Schrödinger equation]{}, Ann. H. Poinc., [2]{} (4), 605–673 (2001). P. Raphael, [Stability and blow up for the non linear Schrödinger Equation]{}, Lecture notes for CMI workshop, Zürich (2008). I. Rodnianski, B. Schlein, [Quantum fluctuations and rate of convergence towards mean field dynamics]{}, preprint arXiv:math-ph/0711.3087. B. Schlein, [Derivation of Effective Evolution Equations from Microscopic Quantum Dynamics]{}, Lecture notes for the minicourse held at the 2008 CMI Summer School in Zurich. H. Spohn, [Kinetic Equations from Hamiltonian Dynamics]{}, Rev. Mod. Phys. [52]{}, no. 3, 569–615 (1980). T. Tao, [Nonlinear dispersive equations. Local and global analysis*]{}, CBMS 106, eds: AMS, 2006. [^1]: We thank B. Schlein for calling our attention to this fact.	ArXiv
--- author: - 'Jan-Peter Calliess$^{1}$ [^1] [^2]' title: '*Lipschitz Optimisation for Lipschitz Interpolation$^$** ' --- [^1]: \*This paper is an extended version of a conference paper that will appear in the Proceedings of the American Control Conference (ACC 2017). [^2]: $^{1}$Jan-Peter Calliess is with the Engineering Department, University of Cambridge, UK. [[email protected]]{}	ArXiv
--- abstract: 'Let $R$ be a ring spectrum and $ {\mathcal{E}}\to X$ an $R$-module bundle of rank $n$. Our main result is to identify the homotopy type of the group-like monoid of homotopy automorphisms of this bundle, $hAut^R({\mathcal{E}})$. This will generalize the result regarding $R$-line bundles proven by the authors in [@cjgauge]. The main application is the calculation of the homotopy type of $BGL_n(End (({\mathcal{L}}))$ where ${\mathcal{L}}\to X$ is any $R$-line bundle, and $End ({\mathcal{L}})$ is the ring spectrum of endomorphisms. In the case when such a bundle is the fiberwise suspension spectrum of a principal bundle over a manifold, $G \to P \to M$, this leads to a description of the $K$-theory of the string topology spectrum in terms of the mapping space from $M$ to $BGL (\Sigma^\infty (G_+))$.' author: - \| Ralph L. Cohen [^1]\ Department of Mathematics\ Stanford University\ Bldg. 380\ Stanford, CA 94305, USA - \| John D.S Jones\ Mathematics Institute\ Zeeman Building\ Warwick University\ Coventry, CV4 7AL, UK title: 'Homotopy automorphisms of $R$-module bundles, and the $K$-theory of string topology' --- Introduction {#introduction .unnumbered} ============ Let $R$ be a ring spectrum. In several places in the recent literature, the notion of an $R$-module bundle ${\mathcal{E}}\to X$ of rank $n$ has been defined and described [@units], [@5author], [@lind]. This is a parameterized $R$-module spectrum ${\mathcal{E}}$ over $X$, where each fiber $E_x$ admits an $R$-module equivalence $E_x {\xrightarrow}{\simeq} \vee_{n} R$. In analogy to vector bundles, it was proved in [@lind] that equivalence classes of rank $n$ $R$-module bundles over $X$ are in bijective correspondence with the set of homotopy classes, $[X, BGL_n(R)] = \pi_0(Map(X, BGL_n(R))$. The main theorem in this paper is the identification of the homotopy type of the group-like monoid of homotopy automorphisms, ${hAut^R}({\mathcal{E}})$. This is the space of self equivalences of ${\mathcal{E}}$ living over the identity of $X$ that preserve the $R$-module structure. A precise definition will be given in the text of the paper. \[main\] Let $R$ be a ring spectrum and $X$ a connected space of the homotopy type of a $CW$-complex. Let ${\mathcal{E}}\to X$ be an $R$-module bundle of rank $n$. Then there is an equivalence of group-like monoids, $$hAut^R({\mathcal{E}}) \simeq \Omega Map_{\mathcal{E}}(X, BGL_n(R))$$ where the subscript in this mapping space refers to the path component of maps that classify ${\mathcal{E}}$. The special case of this theorem when $n=1$ was proved by the authors in [@cjgauge]. As discussed there, this result is important in string topology. Namely, given a principal bundle over a manifold $G \to P \to M$, if we let $${\mathcal{L}}= \Sigma^\infty_M (P_+)$$ be the fiberwise suspension spectrum of $P$ with a fiberwise disjoint basepoint, then the string topology spectrum of $P$, ${\mathcal{S}}(P) = P^{-TM}$ is equivalent, as ring spectra, to the endomorphism ring $End^{\Sigma^\infty (G_+)}(\Sigma^\infty_M (P_+))$. Thus if $R = \Sigma^\infty (G_+)$, and ${\mathcal{E}}= \Sigma^\infty_M (P_+)$, then the above theorem, in the case $n= 1$, describes the homotopy type of the group-like monoid of units, $GL_1({\mathcal{S}}(P))$. Theorem \[main\] in its general setting will have following implication to string topology. Let ${\mathcal{L}}\to X$ be an $R$-line bundle, and let $\oplus_n {\mathcal{L}}\to X$ be the Whitney-sum of $n$-copies of ${\mathcal{L}}$. This is an $R$-module bundle of rank $n$. \[bgln\] There is a homotopy equivalence $$BGL_n({End^R_X {\mathcal{L}}}) \simeq Map_{\oplus_n{\mathcal{L}}}(X, BGL_nR).$$ As a special case we obtain the following result about the general linear groups of the string topology spectrum. \[string\] If $G \to P \to M$ is a principal bundle over a manifold and ${\mathcal{L}}= \Sigma^\infty_M (P_+)$, there is a homotopy equivalence $$BGL_n ({\mathcal{S}}(P)) \simeq Map_{\oplus_n{\mathcal{L}}}(M, BGL_n(\Sigma^\infty(G_+)).$$ In particular there is an equivalence, $$BGL_n(LM^{-TM}) \simeq Map_{\iota_n}(M, BGL_n(\Sigma^\infty(\Omega M_+)).$$ Here $\iota_n$ classifies $\oplus_n {\mathcal{L}}$, where ${\mathcal{L}}= \Sigma^\infty_M ({\mathcal{P}}_+)$, and ${\mathcal{P}}\to M$ is a universal bundle in the sense that ${\mathcal{P}}$ is contractible. Our next main result describes how these results have $K$-theoretic consequences for these ring spectra. First note that the string topology spectrum ${\mathcal{S}}(P)$ is a nonconnective ring spectrum. Indeed its homology groups are nontrivial through dimension $-n$, where $n$ is the dimension of the manifold $M$. If ${\mathcal{S}}$ is a ring spectrum, let $K_{conn} (S)$ denote the algebraic $K$-theory spectrum of its connective cover $K({\mathcal{S}}_0)$. This $K$-theory spectrum has zero-space $\Omega^\infty K_{conn} ({\mathcal{S}}) = K_0(\pi_0({\mathcal{S}})) \times BGL({\mathcal{S}})^+$, where the superscript $+$ denotes a group completion that will be described in the text of the paper. We will show that Corollary \[string\] implies the following result about $K$-theory. \[ktheory\] Given an $R$-line bundle ${\mathcal{L}}\to M$, there is a homology isomorphism $$\alpha : Map_{{\mathcal{L}}} (M, BGL(\Sigma^\infty (G_+))) \to \Omega_0^\infty K_{conn}({\mathcal{S}}(P)).$$ The subscript $0$ denotes the path component of the basepoint in $\Omega^\infty K_{conn}({\mathcal{S}}(P))$. $Map_{{\mathcal{L}}} (M, BGL(\Sigma^\infty (G_+))$ is the homotopy colimit of the mapping spaces $Map_{\oplus_n {\mathcal{L}}} (M, BGL_n(\Sigma^\infty (G_+))$. This theorem can be viewed as a statement about the group completion of the above mapping spaces. However we point out that this is not the same as the mapping space to the group completion, which would be the zero space of the mapping spectrum $Map_0 (M, K(\Sigma^\infty (G_+))$. This spectrum calculates the ${K(\Sigma^\infty (G_+))}$-cohomology of $M$. However, as we will show below, we can define a homomorphism of $K$-theory groups, $$\label{kgcoho} \gamma: K_{conn}^{-q}({\mathcal{S}}(P)) \to {K(\Sigma^\infty (G_+))}^{-q}(M)$$ which gives a partial geometric understanding of the ${K(\Sigma^\infty (G_+))}$-cohomology theory in terms of the algebraic $K$-theory of the string topology spectrum. The situation when $q=0$ was studied in detail by Lind in [@lind]. We conclude by observing two important applications of Theorem \[ktheory\]. \[application\] Let $M$ be a closed manifold. There are homology equivalences $$\begin{aligned} Map_{\Sigma^\infty_M ({\mathcal{P}}_+)} (M, BGL(\Sigma^\infty (\Omega M_+)) &\to \Omega^\infty_0 K_{conn}(LM^{-TM}) \notag \\ Map_{{\mathbb{S}}} (M, BGL({\mathbb{S}})) &\to \Omega_0^\infty K_{conn}(DM) \notag\end{aligned}$$ where ${\mathbb{S}}$ is the sphere spectrum, and ${LM^{-TM}}$ is the Thom spectrum of the virtual bundle $-TM$ over $M$, pulled back over $LM$ via the map $e : LM \to M$ that evaluates a loop at the basepoint of the circle. $DM$ denotes the Spanier-Whitehead dual of the manifold $M$, which is an $E_\infty$-ring spectrum. We point out that in these cases, the map $\gamma$ defined above (\[kgcoho\]) gives homomorphisms $$\gamma : K_{conn}^{-q}(LM^{-TM}) \to A(M)^{-q}(M) \quad \text{and} \quad \gamma : K_{conn}^{-q}(D(M)) \to A(point)^{-q}(M).$$ The algebraic $K$-theory of nonconnective spectra was defined in terms of Waldhausen categories of modules by Blumberg and Mandell in [@blumbergmandell]. When $X$ is simply connected, they related the Waldhausen category defining $K(D(X))$ to the Waldhausen category defining $A(X) = K(\Sigma^\infty (\Omega X_+))$. It would be interesting to relate Corollary \[application\] regarding the $K_{conn}$-theory to their results. This paper is organized as follows. In Section 1 we will prove Theorem \[main\] and derive Corollaries \[bgln\] and \[string\]. In Section 2 we describe the $K$-theoretic implications of Thoerem \[main\], and in particular we prove Theorem \[ktheory\]. Automorphisms of $R$-module bundles =================================== Let $R$ be a ring spectrum. Let ${\mathcal{E}}\to X$ be an $R$-module bundle of rank $n$, in the sense of Lind [@lind]. This is a parameterized spectrum over $X$ in the sense of May and Sigurdsson [@maysigurd], where for each $x \in X$, the fiber ${\mathcal{E}}_x$ is an $R$-module spectrum of rank $n$. We denote the category of such bundles by $R-mod_n (X)$. Again, this category was defined in [@lind]. It was shown there that equivalence classes of such bundles are classified by homotopy classes of maps $X \to BGL_n(R)$. Fix a particular map $\gamma_{\mathcal{E}}: X \to BGL_n(R)$ classifying ${\mathcal{E}}$. This choice defines a basepoint in the mapping space $\gamma_{\mathcal{E}}\in Map_{\mathcal{E}}(X, BGL_n(R))$. The endomorphisms of ${\mathcal{E}}$ in $R-mod_n(X)$ is a parameterized spectrum which we denote by $End_M^R({\mathcal{E}}) \to X$. For every $x \in X$ this defines a fiber spectrum $End_M^R({\mathcal{E}})_x$ which is equivalent to the ring of endomorphisms $End^R(\vee_n R)$. $End_M^R({\mathcal{E}}) $ is a parameterized ring spectrum under composition. By taking a fibrant replacement if necessary, we can take sections to produce an ordinary spectrum $$End^R({\mathcal{E}}) = \Gamma_M(End^R_M ({\mathcal{E}})).$$ The parameterized ring structure on $End_M^R({\mathcal{E}})$ defines a ring spectrum structure on $End^R({\mathcal{E}})$. \[haut\] We define the group-like monoid ${hAut^R}({\mathcal{E}})$ to be the units of the ring spectrum of endomorphisms, $${hAut^R}({\mathcal{E}}) = GL_1(End^R({\mathcal{E}})).$$ We are now ready to prove Theorem \[main\]. Consider the fiber bundle of infinite loop spaces given by taking the zero spaces of the fibrant model of $End^R_M({\mathcal{E}})$: $$\Omega^\infty (End^R(\vee_n R)) \to \Omega^\infty_M End^R_M({\mathcal{E}}) \to M.$$ By restricting to path components of those $R$-module endomorphisms that consist of equivalences, we get a subbundle, which we will call $ {\mathcal{G}}L_n({\mathcal{E}})$: $$GL_n(R) \to {\mathcal{G}}L_n({\mathcal{E}}) \to M.$$ Notice that ${hAut^R}({\mathcal{E}})$ can be described as the space of sections of this bundle, $ {hAut^R}({\mathcal{E}}) = \Gamma_M({\mathcal{G}}L_n({\mathcal{E}}))$. We now observe that the fiber homotopy type of the bundle ${\mathcal{G}}L_n ({\mathcal{E}}) \to M$ has another description. The homotopy class of map $\gamma_{\mathcal{E}}: M \to BGL_n(R)$ that classifies the $R$-module bundle ${\mathcal{E}}$, also classifies a principal $GL_n(R)$-bundle over $M$: $$GL_n(R) \to P_{\mathcal{E}}\to M.$$ Here we are replacing the group-like topological monoid $GL_n(R)$ by a topological group, which by abuse of notation, we continue to refer to as $GL_n(R)$. As was shown in [@lind], the relationship between $P_{\mathcal{E}}$ and the parameterized spectrum ${\mathcal{E}}$ is that there is an equivalence, $$\Omega^\infty_M {\mathcal{E}}\simeq P_{\mathcal{E}}\times_{GL_n(R)} \Omega^\infty (\vee_n R),$$ where we are continuing the abuse of notation to allow $\Omega^\infty (\vee_n R)$ to refer to a weakly homotopy equivalent infinite loop space that carries an action of the group $GL_nR$. (See [@maysigurd] and [@lind] for details.) Now consider the corresponding adjoint bundle, $GL_n(R) \to P_{\mathcal{E}}^{Ad} \to X$. In this notation $P_{\mathcal{E}}^{Ad} $ is the homotopy orbit space $P_{\mathcal{E}}^{Ad} = P_{\mathcal{E}}\times_{GL_n(R)} GL_n(R)$ where $GL_n(R)$ is acting on itself by the adjoint action (conjugation). A standard observation about adjoint bundles implies that $P_{\mathcal{E}}^{Ad} $ is the bundle whose fiber over $x \in M$ is the space of $GL_n(R)$-equivariant automorphisms of the fiber of $P_{\mathcal{E}}$ at $x$. The section space $\Gamma_M(P_{\mathcal{E}}^{Ad})$ is therefore the group of equivariant automorphisms of $P_{\mathcal{E}}$ living over the identity on $M$. This is known as the gauge group, ${\mathcal{G}}(P_{\mathcal{E}})$. From this viewpoint it becomes clear that there is a map of fibrations $ P_{\mathcal{E}}^{Ad} \to {\mathcal{G}}L_n({\mathcal{E}})$ over $M$ (after taking appropriate fibrant and cofibrant replacements), which is an equivalence on the fibers. Therefore there is an equivalence of their spaces of sections, as group-like $A_\infty$-spaces. $$\label{autgl} \phi: {\mathcal{G}}(P_{\mathcal{E}}) = \Gamma_M( P_{\mathcal{E}}^{Ad}) \simeq \Gamma_M({\mathcal{G}}L_1({\mathcal{E}})) = hAut^R_X({\mathcal{E}}).$$ Now a well known theorem of Atiyah and Bott [@atiyahbott] says that the classifying space of the gauge group ${\mathcal{G}}(P)$ of a principal bundle $G \to P \to M$ is equivalent to the mapping space, $B{\mathcal{G}}(P) \simeq Map_P(X, BG)$. Theorem \[main\] now follows by applying this Atiyah-Bott equivalence to the principal bundle $GL_n(R) \to P_{\mathcal{E}}\to X.$ We now consider an application of Theorem \[main\] to an important special case. Let ${\mathcal{L}}\to X$ be an $R$-line bundle, and let $\oplus_n {\mathcal{L}}\to X$ be the Whitney-sum of $n$-copies of ${\mathcal{L}}$. This is an $R$-module bundle of rank $n$, which is defined to be the pullback under the diagonal map $\Delta^n : X \to X^n$ of the exterior $n$-fold product ${\mathcal{L}}^n \to X^n$. Consider the endomorphism spectrum $End^R({\mathcal{L}})$. As noted above, this is a ring spectrum. We first need the following observation: \[GLN\] There is an equivalence of group-like monoids $$\lambda : {hAut^R}(\oplus_n {\mathcal{L}}) \simeq GL_n(End^R({\mathcal{L}})).$$ Given a ring spectum $S$, recall that $GL_n(S)$ is defined to be the group-like monoid of units in the endomorphism ring, $$GL_n(S) = GL_1(End^S(\vee_n S)).$$ Notice that there is a natural equivalence $End^S(\vee_n S) = \prod_n (\vee_n S)$. Also notice that $\vee_n End^R({\mathcal{L}}) \simeq Mor^R({\mathcal{L}}, \oplus_n {\mathcal{L}})$, where $Mor^R$ refers to the $R$-module morphisms in the category of parameterized spectra over $X$. Thus when $S = End^R({\mathcal{L}})$, we have a natural equivalence $$End^S(\vee_n S) \simeq \prod_n(Mor^R({\mathcal{L}}, \oplus_n {\mathcal{L}}) = End^R (\oplus_n {\mathcal{L}}).$$ Furthermore this equivalence clearly preserves the ring structure. It therefore induces an equivalence of their group-like monoids of units, $$GL_1(End^S(\vee_n S)) \simeq GL_1(End^R(\oplus_n {\mathcal{L}}).$$ The left side is by definition $GL_n (End^R({\mathcal{L}}))$, and the right side is by definition ${hAut^R}(\oplus_n {\mathcal{L}})$. We now notice that Theorem \[main\] and Lemma \[GLN\] together imply the following. \[bgln\] There is a homotopy equivalence $$BGL_n({End^R_X {\mathcal{L}}}) \simeq Map_{\oplus_n{\mathcal{L}}}(X, BGL_nR).$$ When $G \to P \to M$ is a principal bundle over a manifold and ${\mathcal{L}}= \Sigma^\infty_M (P_+)$, we obtain the following result about the general linear groups of the string topology spectrum, \[string\] There is a homotopy equivalence $$\beta : BGL_n ({\mathcal{S}}(P)) {\xrightarrow}{\simeq} Map_{\oplus_n {\mathcal{L}}}(M, BGL_n(\Sigma^\infty(G_+))$$ where ${\mathcal{L}}$ is the line bundle $\Sigma^\infty (G_+) \to \Sigma^\infty_M(P_+) \to M$. In particular there is an equivalence $$BGL_n(LM^{-TM}) \simeq Map_{\iota_n}(M, BGL_n(\Sigma^\infty(\Omega M_+)).$$ $K$-theoretic implications ========================== The goal of this section is to use Corollary \[string\] to prove Theorem \[ktheory\] as stated in the introduction. We will then use it to derive descriptions of the $K$-theory of the connective covers of the Spanier-Whitehead dual $D(M) = Map (\Sigma^\infty (M_+), {\mathbb{S}})$ and of the string topology ring spectrum ${LM^{-TM}}$. To do this we need to understand the equivalence given in Corollary \[bgln\] more carefully, so that we can deduce $K$-theoretic consequences. As in that corollary, let ${\mathcal{L}}\to M$ be an $R$-line bundle and let $P_{\mathcal{L}}\to M$ denote the principal $GL_1R$-bundle defined by the classifying map of ${\mathcal{L}}$, $\gamma_{\mathcal{L}}: M \to BGL_1(R)$. Let $\oplus_n P_{\mathcal{L}}$ denote the principal bundle classified by $$M {\xrightarrow}{\Delta} \prod_n M {\xrightarrow}{\gamma_{\mathcal{L}}^n } \prod_n BGL_1(R) {\xrightarrow}{\mu} BGL_n(R).$$ Here $\mu$ is the usual block addition. This is the principal $GL_n(R)$-bundle associated to the $n$-fold Whitney sum $\oplus_n {\mathcal{L}}$. Combining equivalence (\[autgl\]) with Lemma \[GLN\] defines an equivalence of group-like $A_\infty$-spaces, $$\begin{CD} \Psi_n : {\mathcal{G}}(\oplus_n P_{\mathcal{L}}) @>\phi > \simeq > hAut^R(\oplus_n {\mathcal{L}}) @>\lambda >\simeq > GL_n(End^R({\mathcal{L}})). \end{CD}$$ Notice that these equivalences respect the standard inclusions of wreath products, that makes the following diagrams commute: $$\label{wreath} \begin{CD} \Sigma_k \int {\mathcal{G}}(\oplus_n P_{\mathcal{L}}) @>{\hookrightarrow}>> {\mathcal{G}}((\oplus_{nk} P_{\mathcal{L}}) \\ @V 1 \times \Psi_n^k VV @VV\Psi_{nk} V \\ \Sigma \int (GL_n(End^R({\mathcal{L}}))) @>>{\hookrightarrow}> GL_{nk}(End^R({\mathcal{L}})). \end{CD}$$ Using the Barratt-Eccles $E_\infty$-operad, we can conclude the following: The disjoint unions $\coprod_{n \geq 0} B{\mathcal{G}}(\oplus_n P_{\mathcal{L}})$ and $\coprod_{n\geq 0} BGL_n(End^R({\mathcal{L}}))$ have $E_\infty$-algebra structures, and the equivalence $$\sqcup B\Psi_n : \coprod_{n \geq 0} B{\mathcal{G}}(\oplus_n P_{\mathcal{L}}) \to \coprod_{n\geq 0} BGL_n(End^R({\mathcal{L}}))$$ is an equivalence of $E_\infty$-spaces. We are now ready to deduce $K$-theoretic information from the above analysis. If $X$ is an $E_\infty$ space, let $X^+$ denote the corresponding infinite loop space given by its group completion. In other words, $$X^+ = \Omega B(X).$$ Given a ring spectrum $S$, let $S_0$ denote its connective cover. We denote by $K_{conn}(S)$ the algebraic $K$-theory spectrum $K(S_0)$. A connected component of the zero space of this spectrum is given by $\Omega_0^\infty K_{conn}(S) = \left(\coprod_n BGL_n(S)\right)_0^+$, the component of the basepoint in the group completion. We note that the usual algebraic $K$-theory spectrum $K(S)$ is defined as the $K$-theory of the Waldhausen-category of perfect modules over $S$ [@blumbergmandell]. When $S$ is not connective, these spectra are not necessarily equivalent, where they are for connective ring spectra. We can now conclude the following. \[firstK\] If $R$ is a ring spectrum and $R \to {\mathcal{L}}\to M$ is an $R$-line bundle over a closed manifold $M$, then each path component of the zero space of the $K$-theory spectrum of the endomorphism ring has the following homotopy type: $$\Omega_0^\infty (K_{conn}(End^R({\mathcal{L}}))) \simeq \left(\coprod_{n\geq 0} BGL_n(End^R({\mathcal{L}}))\right)_0^+ \simeq \left(\coprod_{n \geq 0} B{\mathcal{G}}(\oplus_n P_{\mathcal{L}})\right)_0^+.$$ We now proceed with the proof of Theorem \[ktheory\] as stated in the introduction. As is usual, let $BGL (End ({\mathcal{L}}))$ denote the homotopy colimit of the inclusion maps $BGL_n (End ({\mathcal{L}})) {\hookrightarrow}BGL_{n+1} (End ({\mathcal{L}}))$ induced by the usual inclusions of the groups $GL_n (End ({\mathcal{L}})) {\hookrightarrow}GL_{n+1}(End({\mathcal{L}}))$. Then by the group-completion theorem of McDuff-Segal [@mcduffsegal], there is a homology equivalence, $$\alpha : {\mathbb{Z}}\times BGL (End({\mathcal{L}})) \to \left(\coprod_{n\geq 0} BGL_n(End^R({\mathcal{L}}))\right)^+ \simeq {\mathbb{Z}}\times \Omega_0^\infty (K_{conn}(End^R({\mathcal{L}}))).$$ Furthermore, by Proposition \[firstK\] there is an equivalence $${\mathbb{Z}}\times {\operatorname{hocolim}}_n B{\mathcal{G}}(\oplus_n P_{\mathcal{L}}) \simeq {\mathbb{Z}}\times BGL (End({\mathcal{L}})).$$ The maps in the colimit are induced by the inclusions of gauge groups, $j_n: {\mathcal{G}}(\oplus_n P_{\mathcal{L}}) {\hookrightarrow}{\mathcal{G}}(\oplus_{n+1}P_{\mathcal{L}})$ given by thinking of an automorphism of $\oplus_n P$ as an automorphism of $\oplus_{n+1} P$ by taking the direct sum with the identity map on the last factor. Now for any principal bundle $G \to P \to M$, consider the Atiyah-Bott equivalence [@atiyahbott] mentioned in the proof of Theorem \[main\]: $$\beta : B{\mathcal{G}}(P) \simeq Map_P(M, BG).$$ This equivalence is given by the observation that the equivariant mapping space $Map^G(P, EG)$ is a contractible space with a free action of the gauge group ${\mathcal{G}}(P) = Aut^G_M(P)$ given by precomposition. The orbit space of this action is then $Map_P (M, BG)$. The examples relevant here are the principal bundles $GL_n(R) \to \oplus_n P_{\mathcal{L}}\to M$. In this case we have an equivalence $$\beta_n: B{\mathcal{G}}(\oplus_n P_{\mathcal{L}}) {\xrightarrow}{\simeq} Map_{\oplus_n{\mathcal{L}}}(M, BGL_nR).$$ With respect to these equivalences, the inclusions $j_n : B{\mathcal{G}}(\oplus_n P_{\mathcal{L}}) \to B{\mathcal{G}}(\oplus_{n+1} P_{\mathcal{L}})$ are given up to homotopy by maps $$q_n : Map_{\oplus_n{\mathcal{L}}}(M, BGL_nR) \to Map_{\oplus_{n+1}{\mathcal{L}}}(M, BGL_{n+1}R)$$ defined by sending $\phi : M \to BGL_n(R)$ to the composition $$q_n(\phi) : M {\xrightarrow}{\Delta} M \times M {\xrightarrow}{\phi \times \gamma_{\mathcal{L}}} BGL_n(R) \times BGL_1(R) {\xrightarrow}{\mu} BGL_{n+1}(R)$$ where $\gamma_{\mathcal{L}}$ is the fixed basepoint in $Map_{\mathcal{L}}(M, BGL_1(R))$, and $\mu$ is the usual block pairing. If we denote the homotopy colimit of these maps $hocolim_n Map_{\oplus_n{\mathcal{L}}}(M, BGL_nR)$ by $Map_{\mathcal{L}}(M, BGL(R))$, we then have a homology equivalence $${\mathbb{Z}}\times Map_{\mathcal{L}}(M, BGL(R)) {\xrightarrow}{\simeq} {\mathbb{Z}}\times {\operatorname{hocolim}}_n B{\mathcal{G}}(\oplus_n P_{\mathcal{L}}) \simeq {\mathbb{Z}}\times BGL (End({\mathcal{L}})) \to {\mathbb{Z}}\times \Omega_0^\infty K_{conn}(End ({\mathcal{L}})),$$ which is the statement of Theorem \[ktheory\]. We now proceed to prove Corollary \[application\], describing $K_{conn}$ of both the string topology spectrum, and of the Spanier-Whitehead dual of a manifold. In the case of the string topology spectrum ${\mathcal{S}}(M) = {LM^{-TM}}$, this is the string topology of the universal principal bundle. That is, a principal bundle over $M$ fiber homotopy equivalent to the path-loop fibration $\Omega M \to {\mathcal{P}}_M \to M$. In the case of the Spanier-Whitehead dual, $DM$, one sees that if one considers the trivial principal bundle, when $G = \{id\}$, then the string topology spectrum of this bundle is simply the dual, $DM = Map(M, {\mathbb{S}})$. Both of these examples are analyzed in more detail in [@cjgauge]. Corollary \[application\] follows by applying Theorem \[ktheory\] to these special cases. [99]{} M. Ando, A. J. Blumberg, D. J. Gepner, M. J. Hopkins, and C. Rezk. Units of ring spectra and Thom spectra preprint, arXiv:0810.4535 M. Ando, A. J. Blumberg, D. Gepner, M. J. Hopkins, and C. Rezk, An $\infty$-categorical approach to $R$-line bundles, $R$-module Thom spectra, and twisted $R$-homology, preprint (2012). M.F. Atiyah, Thom complexes , Proc. London Math. Soc. (3) , no. 11 (1961), 291–310. M. Atiyah, and R. Bott The Yang-Mills equations over Riemann surfaces, Phil. Trans. R. Soc. Lond. A 308, 523-615 (1982) A. Blumberg and M. 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Mathematical Surveys and Monographs, vol. 132, Amer. Math. Soc., 2006 Mathematical Surveys and Monographs, [vol. 132]{}, Amer. Math. Soc., 2006 D. McDuff, G.B. Segal, Homology fibrations and the “group completion" theorem, Invent. Math., 31, (1976), 279 - 284. [^1]: The first author was partially supported by a grant from the NSF.	ArXiv
--- abstract: \| With heavy quark limit and hierarchy approximation $\lambda_{QCD}\ll m_D\ll m_B$, we analyze the $B\to D^0\overline D^0$ and $B_s\to D^0\overline D^0$ decays, which occur purely via annihilation type diagrams. As a roughly estimation, we calculate their branching ratios and CP asymmetries in Perturbative QCD approach. The branching ratio of $B\to D^0\overline D^0$ is about $3.8\times10^{-5}$ that is just below the latest experimental upper limit. The branching ratio of $B_s\to D^0\overline D^0$ is about $6.8\times10^{-4}$, which could be measured in LHC-b. From the calculation, it could be found that this branching ratio is not sensitive to the weak phase angle $\gamma$. In these two decay modes, there exist CP asymmetries because of interference between weak and strong interaction. However, these asymmetries are too small to be measured easily. author: - \| Ying Li[^1] and Juan Hua\ [Physics Department, Yantai University, Yantai 264005, China]{} title: 'Study of Pure Annihilation Decays $B_{d,s} \to D^{0} \overline D^{0}$ ' --- Introduction {#sc:intro} ============ In the Standard Model (SM), CP-violation (CPV) arises from a complex phase in the Cabibbo-Kobayashi- Maskawa (CKM) quark mixing matrix, and the angles of unitary triangle are defined as [@Yao:2006px]: $$\begin{aligned} \beta=\arg \Bigl[-\frac{V_{cb}^V_{cd}}{V_{tb}^V_{td}}\Bigl],~~~~ \alpha=\arg \Bigl[-\frac{V_{tb}^V_{td}}{V_{ub}^V_{ud}}\Bigl],~~~~ \gamma=\arg \Bigl[-\frac{V_{ub}^V_{ud}}{V_{cb}^V_{cd}}\Bigl].\label{ckmangle}\end{aligned}$$ In order to test SM and search for new physics, many measurements of CP-violation observables can be used to constrain these above angles. It is well known that we measure $\beta$ precisely using the golden decay mode $B\to J/\psi K_s$; the angle $\alpha$ can be determined with decay $B\to\pi\pi$ and $\gamma$ could be measured precisely in Large Hadron Collider (LHC) with decay mode $B_s\to D_sK$. ![The quark level Feynman diagrams for $B_d \to D^{0} \bar D^{0}$ process []{data-label="fig0"}](fig1.eps) Besides the above channels mentioned, many other channels are used to cross check the measurements. Among these decays, $B\to DD$ decay is considered to test the $\beta$ measurement. For $B\to DD$ decay, the analysis based on $SU(3)$ symmetry [@Savage:1989ub], iso-spin symmetry [@Xing:1999yx] and factorization approach [@Xing:1998ca] have been done in last several years. However, the the calculation of decay $B^0 \to D^0\overline D^0$ has difficulties. It is a pure-annihilation diagram decay, also named W-exchange diagram decay, which is power suppressed in factorization language. The quark diagrams of this decay are shown in Figure \[fig0\]. Theoretically, QCD factorization approach (QCDF) [@Beneke:1999br] and soft collinear effective theory (SCET)[@Bauer:2001yt] can not deal decays with two heavy charmed mesons effectively. In Ref.[@Keum:2003js; @Lu:2003xc], perturbative QCD (PQCD) has been exploited to $B$ meson decays with one charm meson in the final states and the results agree with experimental data well. Specially, the pure annihilation type B decays with charmed meson were studied in Ref.[@Lu:2003xc]. In the standard model picture, the $W$ boson exchange causes $\bar{b}d \to \bar{c}c $, and the $\bar{u}u$ quarks are produced from a gluon. This gluon attaches to any one of the quarks participating in the $W$ boson exchange. In decay $B \to D^{0} \overline D^{0}$, the momentum of the final state $D$ meson is $\frac{1}{2} m_B (1-2 r^2)$, with $r=m_D/m_B$. If we consider heavy quark limit and hierarchy approximation $\lambda_{QCD}\ll m_D\ll m_B$, the $D$ meson momentum is nearly $m_B/2$. According to the distribution amplitude used in Ref.[@Keum:2003js], the light quark in $D$ meson carrying nearly $40\%$ of the $D$ meson momentum. So, this light quark is still a collinear quark with 1 $\mathrm{GeV}$ energy, like that in $B\to DM$ [@Keum:2003js; @Lu:2003xc], $B\to K(\pi)\pi$ [@kls; @luy] decays. The gluon could be viewed as a hard gluon approximatively, so we can treat the process perturbatively where the four-quark operator exchanges a hard gluon with $u \bar u$ quark pair. Of course, we are able to calculate the diagrams if charm quark and up quark exchange. As a roughly estimation, we give the branching ratio and CP-violation of $B_{d,s} \to D^{0} \overline D^{0}$. In this article, the analytic formulas for the decay amplitudes will be shown in the next section. In section \[sc:result\], we give the numerical results and summarize this article in section \[sc:summary\]. Analytic formulas {#sc:analy} ================= For simplicity, we set $B$ meson at rest in our calculation. In light-cone coordinates, the momentum of $B$, $D^0$ and$\overline D^0$ are: $$\begin{aligned} P_B=\frac{M_B}{\sqrt{2}}(1,1,\vec{0}); P_2=\frac{M_B}{\sqrt{2}}(1-r^2,r^2,\vec{0}); P_3=\frac{M_B}{\sqrt{2}}(r^2,1-r^2,\vec{0}).\end{aligned}$$ we define the light (anti-)quark momenta in $B$, $D^0$ and $\overline D^0$ mesons as $k_1$, $k_2$, and $k_3$ as: $$k_1 = (x_1P_1^+,0,{\bf k}_{1T}),\ \ k_2 = (x_2 P_2^+,0,{\bf k}_{2T}),\ \ k_3 = (0, x_3 P_3^-,{\bf k}_{3T}). \label{eq:momentun2}$$ In PQCD, we factorize the decay amplitude into soft($\Phi$), hard($H$), and harder ($C$) dynamics characterized by different scales, [@kls; @luy] $$\begin{gathered} \mathcal{A}\sim \int\!\! d x_1 d x_2 d x_3 b_1 d b_1 b_2 d b_2 b_3 d b_3 \mathrm{Tr} \Bigl[ C(t) \Phi_B(x_1,b_1) \Phi_{D}(x_2,b_2) \Phi_D(x_3, b_3) H(x_i, b_i,t) S_t(x_i)\, e^{-S(t)} \Bigr]. \label{eq:convolution2}\end{gathered}$$ In above equation, $b_i$ is the conjugate space coordinate of the transverse momentum ${\bf k}_{iT}$, and $t$ is the largest energy scale. $C$ is Wilson coefficient, and $\Phi$ is the wave function. The last term, $e^{-S(t)}$, contains two kinds of contributions. One is due to the resummation of the large double logarithms from renormalization of ultra-violet divergence $\ln tb$, the other is from resummation of double logarithm $\ln^2 b$ from the overlap of collinear and soft gluon corrections, which is called Sudakov form factor. The hard part $H$ can be calculated perturbatively, and it is channel dependent. More explanation of above formula and review about PQCD can be found in many reference, such as [@kls; @luy; @Ali:2007ff]. As a heavy meson, the $B$ meson wave function is not well defined, neither is $D$ meson. In heavy quark limit, we take them as: $$\Phi_{B}(x,b) = \frac{i}{\sqrt{6}} \left[ \not \! P + M_B \right] \gamma_5\phi_B(x,b),$$ $$\Phi_{D}(x,b) = \frac{i}{\sqrt{6}}\gamma_5 \left[ \not \! P + M_D \right] \phi_D(x,b).$$ The Lorentz structure of two mesons are different because the $B$ meson is initials state and $D$ meson is final state. The effective Hamiltonian $\bar b\to \bar q(q=d,s)$ is given by [@Buchalla:1996vs]: $$\begin{gathered} \mathcal{H}_\mathrm{eff} = \frac{G_F}{\sqrt{2}}\Bigl\{ V_{cq}V_{cb}^* \Bigl[ C_1(\mu) O_1^c(\mu) + C_2(\mu) O_2^c(\mu) \Bigl]+ V_{uq}V_{ub}^* \Bigl[ C_1(\mu) O_1^u(\mu) + C_2(\mu) O_2^u(\mu) \Bigl]\\ -V_{tb}^V_{tq}\sum_{i=3}^{10}C_i(\mu) O_i(\mu)\Bigl\},\label{hami}\end{gathered}$$ where $C_{i}(\mu)(i=1,\cdots,10)$ are Wilson coefficients at the renormalization scale $\mu$ and the four quark operators $O_{i}(i=1,\cdots,10)$ are $$\begin{array}{ll} O_1^c = (\bar{b}_ic_j)_{V-A}(\bar{c}_jq_i)_{V-A}, & O_2^c = (\bar{b}_ic_i)_{V-A} (\bar{c}_jq_j)_{V-A}, \\ O_1^u = (\bar{b}_iu_j)_{V-A}(\bar{u}_jq_i)_{V-A}, & O_2^u = (\bar{b}_iu_i)_{V-A} (\bar{u}_jq_j)_{V-A}, \\ O_3 = (\bar{b}_iq_i)_{V-A}\sum_{q} (\bar{q}_jq_j)_{V-A}, & O_4 = (\bar{b}_iq_j)_{V-A}\sum_{q} (\bar{q}_jq_i)_{V-A}, \\ O_5 = (\bar{b}_iq_i)_{V-A}\sum_{q} (\bar{q}_jq_j)_{V+A}, & O_6 = (\bar{b}_iq_j)_{V-A} \sum_{q} (\bar{q}_jq_i)_{V+A}, \\ O_7 = \frac{3}{2}(\bar{b}_iq_i)_{V-A} \sum_{q} e_q(\bar{q}_jq_j)_{V+A}, & O_8 = \frac{3}{2}(\bar{b}_iq_j)_{V-A}\sum_{q} e_q (\bar{q}_jq_i)_{V+A}, \\ O_9 = \frac{3}{2}(\bar{b}_iq_i)_{V-A}\sum_{q} e_q(\bar{q}_jq_j)_{V-A}, & O_{10} = \frac{3}{2}(\bar{b}_iq_j)_{V-A}\sum_{q} e_q(\bar{q}_jq_i)_{V-A}. \label{eq:effectiv} \end{array}$$ Here $i$ and $j$ are $SU(3)$ color indices; in $O_{3,...,10}$ the sum over $q$ runs over the quark fields that are active at the scale $\mu=O(m_{b})$, i.e., $q\in \{u,d,s,c,b\}$. For Wilson coefficients, we will also use the leading logarithm summation for QCD corrections, although the next-to -leading order calculation already exists [@Buchalla:1996vs]. This is the consistent way to cancel the explicit $\mu$ dependence in the theoretical formulae. ![The leading order Feynman diagrams for $B_d \to D^{0} \overline D^{0}$ process in PQCD approach[]{data-label="fig1"}](fig2.eps) According to the effective Hamiltonian in eq.(\[hami\], \[eq:effectiv\]), the lowest order diagrams of $B \to D^0\overline D^0$ are drawn in Fig. \[fig1\]. We first calculate the usual factorizable diagrams (a), (b), (c) and (d). For the $(V-A)(V-A)$ operators, their contributions of (a) and (c) are always canceled by diagrams (b) and (d) respectively because of current conservation. For the $(V-A)(V+A)$ operators, these diagrams can not give contribution, either. That’s to say, factorizable diagrams have no contribution. For non-factorizable diagrams (e), (f), (g) and (h), we find the hard part for $(V-A)(V-A)$ operators are same to $(V-A)(V+A)$ operators. We group the contribution of diagrams (e) and (f), denoted by $M_a$, as follows: $$\begin{gathered} M_{a}[C_i] = \frac{64 \pi C_FM_B^2}{\sqrt{2N_C}} \int_0^1 \!\! dx_1 dx_2 dx_3 \int_0^\infty \!\! b_1 db_1\, b_2 db_2\ \phi_B(x_1,b_1) \phi_{D}(x_2,b_2)\phi_{D}(x_3,b_2) \\ \times \Bigl\{ \Bigl[x_1+x_2+(2x_3-x_2)r^2\Bigl] C_i(t_{a}^1)E(t_{a}^1) h_a^{(1)}(x_1, x_2,x_3,b_1,b_2) \\ +\Bigl[-x_3+(2x_1-2x_2+x_3)r^2\Bigl] C_i(t_{a}^2)E(t_{a}^2) h_a^{(2)}(x_1, x_2,x_3,b_1,b_2) \Bigr\}, \label{eq:Ma}\end{gathered}$$ where $C_F = 4/3$ is the group factor of $\mathrm{SU}(3)_c$ gauge group, and $C_i$ is Wilson coefficient. The function $E_m$ is defined as $$E(t) = \alpha_s(t)\, e^{-S_B(t)-S_D(t)-S_D(t)},$$ and $S_B$, $S_D$ result from Sudakov factor and single logarithms due to the renormalization of ultra-violet divergence. The functions $h_a$ is the Fourier transformation of virtual quark and gluon propagators. It is defined by $$\begin{aligned} &h^{(j)}_a(x_1,x_2,x_3,b_1,b_2) = \nonumber \\ & \biggl\{ \frac{\pi i}{2} \mathrm{H}_0^{(1)}(M_B\sqrt{x_2x_3(1-2r^2)}\, b_1) \mathrm{J}_0(M_B\sqrt{x_2x_3(1-2r^2)}\, b_2) \theta(b_1-b_2) \nonumber \\ & \qquad\qquad\qquad\qquad + (b_1 \leftrightarrow b_2) \biggr\} \times\left( \begin{matrix} \mathrm{K}_0(M_B F_{a(j)} b_1), & \text{for}\quad F^2_{a(j)}>0 \\ \frac{\pi i}{2} \mathrm{H}_0^{(1)}(M_B\sqrt{\|F^2_{a(j)}\|}\ b_1), & \text{for}\quad F^2_{a(j)}<0 \end{matrix}\right), \label{eq:propagator1} \end{aligned}$$ with: $$\begin{aligned} F^2_{a(1)} &=&-x_1-x_2-x_3+x_1x_3+x_2x_3+(x_2+x_3-x_1x_3-2x_2x_3)r^2;\\ F^2_{a(2)}&=&x_2x_3-x_1x_3+(x_1x_3-2x_2x_3)r^2.\end{aligned}$$ In above equation, $\mathrm{H}_0^{(1)}(z) = \mathrm{J}_0(z) + i\, \mathrm{Y}_0(z)$. In order to reduce the large logarithmic radiative corrections, the hard scale $t$ in the amplitudes is selected as the largest energy scale in the hard part: $$\begin{aligned} t_{a}^j &=& \mathrm{max}(M_B \sqrt{\|F^2_{a(j)}\|}, M_B \sqrt{(1-2r^2)x_2x_3 }, 1/b_1,1/b_2). \end{aligned}$$ Analogically, we can get the $M_{b}$, which comes from the contribution of diagrams (g) and (h): $$\begin{gathered} M_{b}[C_i]= \frac{64 \pi C_FM_B^2}{\sqrt{2N_C}} \int_0^1 \!\! dx_1 dx_2 dx_3 \int_0^\infty \!\! b_1 db_1\, b_2 db_2\ \phi_B(x_1,b_1) \phi_{D}(x_2,b_2)\phi_{D}(x_3,b_2) \\ \times \Bigl\{ \Bigl[1-x_3+(2+2x_1-2x_2+x_3)r^2\Bigl]C_i(t_{b}^1) E(t_{b}^1) h_b^{(1)}(x_1, x_2,x_3,b_1,b_2) \\ +\Bigl[x_1+x_2-1+(-2-x_2+2x_3)r^2\Bigl]C_i(t_{b}^2) E(t_{b}^2) h_b^{(2)}(x_1, x_2,x_3,b_1,b_2) \Bigr\}, \label{eq:Mb}\end{gathered}$$ and the functions are defined as: $$\begin{aligned} &h^{(j)}_b(x_1,x_2,x_3,b_1,b_2) = \nonumber \\ & \biggl\{ \frac{\pi i}{2} \mathrm{H}_0^{(1)}(M_B\sqrt{1-x_2-x_3+x_2x_3+(x_2+x_3-2x_2x_3)r^2}\, b_1) \nonumber\\ &~~~~~~~~ \times\mathrm{J}_0(M_B\sqrt{1-x_2-x_3+x_2x_3+(x_2+x_3-2x_2x_3)r^2}\, b_2) \theta(b_1-b_2) \nonumber \\ & \qquad\qquad\qquad\qquad + (b_1 \leftrightarrow b_2) \biggr\} \times\left( \begin{matrix} \mathrm{K}_0(M_B F_{b(j)} b_1), & \text{for}\quad F^2_{b(j)}>0 \\ \frac{\pi i}{2} \mathrm{H}_0^{(1)}(M_B\sqrt{\|F^2_{b(j)}\|}\ b_1), & \text{for}\quad F^2_{b(j)}<0 \end{matrix}\right); \label{eq:propagator2} \end{aligned}$$ $$\begin{aligned} F^2_{b(1)}&=&-1-x_1x_3+x_2x_3+(x_1x_3-2x_2x_3)r^2,\\ F^2_{b(2)}&=&1-x_1-x_2-x_3+x_1x_3+x_2x_3+(x_2+x_3-x_1x_3-2x_2x_3)r^2, \end{aligned}$$ $$\begin{aligned} t_{b}^j &=& \mathrm{max}(M_B \sqrt{\|F^2_{b(j)}\|}, M_B \sqrt{1-x_2-x_3+x_2x_3+(x_2+x_3-2x_2x_3)r^2 }, 1/b_1,1/b_2). \end{aligned}$$ So, the decay amplitude of decay $B_d \to D^{0} \overline D^{0}$ can be read as: $$\begin{aligned} \mathcal{A}_1&=&V_{cb}^V_{cd}M_{a}[C_2]-V_{tb}^V_{td}M_{a}[C_5+C_7]+V_{ub}^V_{ud}M_{b}[C_2] -V_{tb}^V_{td}M_{b}[C_5+C_7]\nonumber\\ &=&V_{cb}^V_{cd}T_1 -V_{tb}^V_{td}P_1\nonumber\\ &=&V_{tb}^V_{td}P_1(1+z_1e^{i(\beta+\delta_1)}),\label{bdd1}\end{aligned}$$ where $\beta$ is weak phase angle defined in Eq.(\[ckmangle\]), and $\delta_1$ is the strong phase, which plays an important role in studying CP-violation. In above calculation, we denote that $$\begin{aligned} T_1&=&M_{a}[C_2]-M_{b}[C_2], \nonumber \\ P_1&=&M_{a}[C_5+C_7]+M_{b}[C_5+C_7]+M_{b}[C_2],\end{aligned}$$ and $$\begin{aligned} z_1=\left\|\frac{V_{cb}^V_{cd}}{V_{tb}^V_{td}} \right\|\left\|\frac{T_1}{P_1}\right\|,\end{aligned}$$ which describes the ratio between tree diagram and penguin diagram. The corresponding charge conjugate decay is $$\begin{aligned} \label{bdd2} \overline{\mathcal{A}_1}=V_{tb}V_{td}^P_1(1+z_1e^{i(-\beta+\delta_1)}).\end{aligned}$$ Therefore, the averaged decay width $\Gamma$ for $B^0 \to D^0 \overline D^0$ decay is then given by $$\Gamma(B^0 \to D^0 \overline D^0) = \frac{G_F^2 M_B^3}{128\pi} (1-2r^2)\|V_{tb}^V_{td}P_1\|^2 \bigl\|1+z_1^2+2z_1\cos \beta \cos\delta_1\|. \label{eq:neut_width1}$$ >From this equation, we know that the averaged branching ratio is a function of CKM angle $\beta$, if $z_1\neq 0$. Derived from Eq.(\[bdd1\]) and Eq.(\[bdd2\]), the direct CP-violation can be formulated as: $$\begin{aligned} \label{cp1} A_{CP}^{dir}(B\to D^{0} \overline D^{0})=\frac{\|A_{B_d\to D^0\overline D^0}\|^2-\|A_{\overline B_d\to \overline D^0D^0}\|^2}{\|A_{B_d\to D^0\overline D^0}\|^2+\|A_{\overline B_d\to \overline D^0D^0}\|^2}=\frac{-2z_1\sin\beta\sin\delta_1}{1+z_1^2+2z_1\cos \beta \cos\delta_1}. \end{aligned}$$ For $ B^0_s \to D^0 \overline D^0$ and its conjugate decay, we write the decay amplitudes and rearrange them as: $$\begin{aligned} \mathcal{A}_2&=&V_{cb}^V_{cs}M_{a}[C_2]-V_{tb}^V_{ts}M_{a}[C_5+C_7]+V_{ub}^V_{us}M_{b}[C_2] -V_{tb}^V_{ts}M_{b}[C_5+C_7]\nonumber\\ &=&V_{ub}^V_{us}M_{b}[C_2]-V_{tb}^V_{ts}\Bigl\{M_{a}[C_5+C_7] +M_{b}[C_5+C_7]-\frac{V_{cb}^V_{cs}}{V_{tb}^V_{ts}}M_{a}[C_2]\Bigl\}\nonumber\\ &=&V_{ub}^V_{us}T_2-V_{tb}^V_{ts}P_2\nonumber\\ &=&V_{ub}^V_{us}T_2\Bigl[1+z_2e^{i(-\gamma+\delta_2)}\Bigl],\\ \overline{\mathcal{A}_2}&=&V_{ub}V_{us}^T_2\Bigl[1+z_2e^{i(\gamma+\delta_2)}\Bigl],\label{bdd3}\end{aligned}$$ where $T_2$, $P_2$ and $z_2$ are defined as: $$\begin{aligned} T_2&=&M_{b}[C_2],\nonumber\\ P_2&=&M_{a}[C_5+C_7] +M_{b}[C_5+C_7]-\frac{V_{cb}^V_{cd}}{V_{tb}^V_{ts}}M_{a}[C_2],\nonumber\\ z_2&=&\left\|\frac{V_{tb}^V_{ts}}{V_{ub}^V_{us}} \right\|\left\|\frac{T_2}{P_2}\right\|.\end{aligned}$$ So, the averaged decay width and direct CP violation can be formulated as: $$\begin{aligned} \label{eq:neut_width2} \Gamma(B_s \to D^0\overline D^0) &=& \frac{G_F^2 M_B^3}{128\pi} (1-2r^2) \bigl\|V_{ub}V_{us}^T_2\bigr\|^2 (1+z_2^2+2z_2\cos\delta_2\cos\gamma),\\ A_{CP}^{dir}(B_s \to D^0\overline D^0)&=&\frac{\|A_{B_s\to D^0\overline D^0}\|^2-\|A_{\overline B_s\to \overline D^0D^0}\|^2}{\|A_{B_s\to D^0\overline D^0}\|^2+\|A_{\overline B_s\to \overline D^0D^0}\|^2}=\frac{2z_2\sin\gamma\sin\delta_2}{1+z_2^2+2z_2\cos \gamma \cos\delta_2}.\end{aligned}$$ In our calculation, we set $m_c\approx m_D$, just because $m_D -m_c \approx \Lambda_{QCD}$ and $\frac{\Lambda_{QCD}}{m_B} \rightarrow 0$ in the heavy quark limit. Numerical Results {#sc:result} ================= For $B$ meson, the distribution amplitude is well determined by charmless $B$ decays [@kls; @luy], which is chosen as $$\phi_B(x,b) = N_B x^2(1-x)^2 \exp \left[ -\frac{M_B^2\ x^2}{2 \omega_b^2} -\frac{1}{2} (\omega_b b)^2 \right],\label{waveb}$$ with parameters $\omega_b=0.4\mbox{ GeV}$, and $N_B=91.745\mbox{ GeV}$ which is the normalization constant using $f_B=190 \mbox{ MeV}$. For $B_s$ meson, we use the same wave function according to SU(3) symmetry, where $\omega_b=0.4\mbox{ GeV}$, $N_{B_s}=119.4 \mbox{ GeV}$ and $f_{B_s}=230 \mbox{ MeV}$. Since the $c$ quark is much heavier than the $u$ quark, the $c$ quark shares more momentum, and this function should be asymmetric with respect to $x = 1/2$. The asymmetry is parameterized by $a_D$. Similar to the $b$-dependence on the wave function of $B$ meson, for controlling the size of charmed mesons, we also introduce the intrinsic $b$-dependence on those of charmed mesons. Hence, we use the wave function of $D$ meson as [@Chen:2003px] $$\phi_{D}(x,b) = \frac{3}{\sqrt{2 N_c}} f_{D} x(1-x)\Bigl[ 1 + a_{D} (1 -2x)\Bigl]\exp \left[-\frac{1}{2} (\omega_D b)^2 \right].\label{waved}$$ We use $a_{D}=0.7$ and $\omega_{D}=0.4$ in above function. Other parameters, such as meson mass, decay constants, the CKM matrix elements and the lifetime of $B$ meson are list [@Yao:2006px; @Follana:2007uv]: $$\begin{gathered} M_B = 5.28 \mbox{ GeV},\ M_{B_S} = 5.36 \mbox{ GeV},\ M_{D} = 1.87 \mbox{ GeV},\ f_{D} = 210 \mbox{ MeV}, \nonumber \\ \|V_{ud}\|=0.974, \ \|V_{ub}\|= 4.3 \times 10^{-3},\ \|V_{cd}\|=0.23, \ \|V_{cb}\|= 41.6\times 10^{-3} \nonumber\\ \|V_{td}\|= 7.4\times 10^{-3}, \ \|V_{tb}\|=1.0,\ \|V_{us}\|=0.226, \ \|V_{cs}\|=0.957,\nonumber \\ \|V_{ts}\|=41.6\times 10^{-3}, \quad\tau_{B_d^0}=1.54\times 10^{-12}\mbox{ s,}\quad \tau_{B_s^0}=1.46\times 10^{-12}\mbox{ s}.\label{parameterZ}\end{gathered}$$ $B_d\to D^0\overline D^0$ $B_s\to D^0\overline D^0$ ------------- --------------------------- --------------------------- $T(e)+T(f)$ $68+17 \, i$ $66 +27 \, i$ $P(e)+P(f)$ $ 0.80+ 0.23\, i$ $0.77 +3.68\, i$ $T(g)+T(h)$ $ 9.81- 2.99\, i$ $14.0-0.6\, i$ $P(g)+P(h)$ $ 0.08- 0.02\, i$ $-0.01 +0.01\, i$ : Amplitudes ($10^{-3}$ GeV) of $B_d\to D^0\overline D^0$ and $B_s\to D^0\overline D^0$.[]{data-label="tb:amplitudes"} ![The branching ratio of $B_s\to D^0\overline D^0$ changes with CKM angle $\gamma$.[]{data-label="fig3"}](fig3.eps) With these parameters fixed, we calculate the decay amplitudes of the $B^0\to D^0\overline D^0$ and $B_s\to D^0\overline D^0$ decays in Table \[tb:amplitudes\]. From the table, we notice that the main contribution comes from the tree diagram (e) and (f). And our predictions for the branching ratio of each mode corresponding to $\beta=23^\circ$ and $\gamma=63^\circ$ are listed, $$\begin{aligned} BR(B_d\to D^0\overline D^0) &=& 2.3\times 10^{-5}; \nonumber\\ BR(B_s\to D^0\overline D^0) &=& 6.8\times 10^{-4}.\end{aligned}$$ In Fig. \[fig3\], we plot the branching ratio of $B_s\to D^0\overline D^0$ with different $\gamma$. In this figure, we find the branching ratio is not sensitive to CKM angle $\gamma$. For the experimental side, there are only upper limits given at $90\%$ confidence level for decay $B_d\to D^0\overline D^0$, $$\begin{aligned} BR(B_d\to D^0\overline D^0) &< &6.0\times 10^{-5}; ~~~~~~BarBar\cite{Aubert:2006ia}\nonumber\\ BR(B_d\to D^0\overline D^0) &< &4.2\times 10^{-5}. ~~~~~~Belle\cite{:2007sk}\end{aligned}$$ Obviously, our result is consistent with the data. For $B_d\to D^0\overline D^0$ decay mode, $z_1$ is about 6.5, and the strong phase $\delta_1$ is $34^\circ$, so $A_{CP}^{dir}$ is about $-6\%$ with the definition in Eq.(\[cp1\]). As decay mode $B_s\to D^0\overline D^0$ is concerned, $z_2$ is about $205$ and $\delta_2=155^\circ$, and the relation between direct CP violation and $\gamma$ is shown in Fig.\[fig4\]. From the figure, we read the CP asymmetry is about $0.4\%$, which is rather tiny. It is necessary to state that the $z_1$ and $z_2$ are not the true ratio between tree contribution and penguin, because mathematical technics are used in Eq. (\[bdd1\]) and (\[bdd3\]). ![The direct CP-violation of $B_s\to D^0\overline D^0$ changes with CKM angle $\gamma$.[]{data-label="fig4"}](fig4.eps) In addition to the perturbative annihilation contributions, there is also a hadronic picture for the $B_d\to D^0\overline D^0$, named soft final states interaction[@Cheng:2004ru]. The $B$ meson decays into $D^+$ and $D^-$, the secondary particles then exchanging a $\rho$ meson, then scatter into $D^0\overline D^0$ through final state interaction afterwards. For $B_s$ decay, the $B_s$ meson decays into $D_s^+$ and $D^+$ then scatters into $D^0\overline D^0$ by exchanging a Kaon. But this picture cannot be calculated accurately because of lack of many effective vertexes, and we will ignore this contribution here, though it may be important [@Cheng:2004ru]. ---------------------------- ------------------------------- ------------------------------- ----------------------------------------- ----------------------------------------- $BR(B_d\to D^0\overline D^0)$ $BR(B_s\to D^0\overline D^0)$ $A_{CP}^{dir}(B_d\to D^0\overline D^0)$ $A_{CP}^{dir}(B_s\to D^0\overline D^0)$ $(\times 10^{-5})$ $(\times 10^{-4})$ $(\%)$ $(\%)$ $\omega_b(B\setminus B_s)$ $0.35\setminus0.45$ 4.3 7.8 -7.2 0.4 $0.40\setminus0.50$ 3.8 6.8 -5.3 0.4 $0.45\setminus0.55$ 3.2 5.9 -5.8 0.4 $\omega_D$ $0.35$ 5.0 9.7 -4.2 0.3 $0.40$ 3.8 6.8 -5.3 0.4 $0.45$ 2.2 4.2 -7.8 0.5 $a_D$ $0.6$ 3.2 5.9 -6.9 0.4 $0.7$ 3.8 6.8 -5.3 0.4 $0.8$ 4.3 7.8 -6.1 0.4 ---------------------------- ------------------------------- ------------------------------- ----------------------------------------- ----------------------------------------- : The sensitivity of the decay branching ratios and CP asymmetries to change of ¥ø$\omega_b$, $\omega_D$ and $a_D$[]{data-label="table"} There are many uncertainties in our calculation such as higher order corrections, the parameters listed in Eq.(\[parameterZ\]) and the distribution amplitudes of heavy mesons. We will not discuss uncertainty taken by high order correction as we only roughly estimate the branching ratios and CP asymmetries, though high order corrections have been done for some special channels [@Li:2005kt; @Li:2006jv] and showed $15-20\%$ uncertainty. The parameters in Eq.(\[parameterZ\]), fixed by experiments, are proportional to the amplitudes, so we will not analyze this kind uncertainty either. In our calculation, we find that the results are sensitive to the distribution amplitudes, especially to that of $D$ meson. Since the heavy $D$ wave function is less constrained, we set $a_{D}\in (0.6-0.8)\mbox{ GeV}$ and $\omega_{D}\in(0.35-0.45)\mbox{ GeV}$ to exploit the uncertainty. Table \[table\] shows the sensitivity of the branching ratios to change of $\omega_b$, $\omega_D$ and $a_{D}$. It is found that uncertainty of the predictions on PQCD is mainly due to $\omega_D$, which describes the behavior in end-point region of $D$ meson, however it is very hard to be determined. Considering the experimental upper limit, our results favor large $\omega_b$, large $\omega_D$ and small $a_D$. At last, we give the prediction of branching ratios with err bar as follows: $$\begin{aligned} BR(B_d\to D^0\overline D^0) &=&(3.8^{+0.5+1.2+0.5}_{-0.6-1.6-0.6})\times 10^{-5} \left(\frac{f_B\cdot f_D\cdot f_D}{190\mathrm{ GeV}\cdot 210\mathrm{ GeV}\cdot 210\mathrm{ GeV}}\right)^2; \nonumber\\ BR(B_s\to D^0\overline D^0) &=&(6.8^{+1.0+2.9+1.0}_{-0.9-2.6-0.9})\times 10^{-4} \left(\frac{f_{B_s}\cdot f_D\cdot f_D}{230\mathrm{ GeV}\cdot 210\mathrm{ GeV}\cdot 210\mathrm{ GeV}}\right)^2.\end{aligned}$$ We believe that the $B_d\to D^0\overline D^0$ will be measured soon because this ratio is just below the upper limit, and $B_d\to D^0\overline D^0$ will be measured in LHC-b in next year as a channel to cross check the $\gamma$ measurements. Summary {#sc:summary} ======= With heavy quark limit and hierarchy approximation $\lambda_{QCD}\ll m_D\ll m_B$, we analyze the $B\to D^0\overline D^0$ and $B_s\to D^0\overline D^0$ decays, which occur purely via annihilation type diagrams. As a roughly estimation, we calculate the branching ratios and CP asymmetries in PQCD approach. The branching ratios are still sizable. The branching ratio of $B\to D^0\overline D^0$ is about $3.8\times10^{-5}$, which is just below the experimental upper limited result[@Aubert:2006ia; @:2007sk], and we think that it will be measured in near future. For $B_s\to D^0\overline D^0$, the branching ratio is about $6.8\times10^{-4}$, which could be measured in LHC-b. From the calculation, it is found that this branching ratio is not sensitive to angle $\gamma$. In these two decays, there exist CP asymmetries because of interference between weak and strong interaction, though they are very small. Acknowledgements {#acknowledgements .unnumbered} ================ Y.Li thanks Institute of High Energy Physics for their hospitality during visit where part of this work was done. This work is partly supported by Foundation of Yantai University under Grant No.WL07B19. 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--- abstract: 'Causal Dynamical Triangulations (CDT) is a non-perturbative quantisation of general relativity. Hořava-Lifshitz gravity on the other hand modifies general relativity to allow for perturbative quantisation. Past work has given rise to the speculation that Hořava-Lifshitz gravity might correspond to the continuum limit of CDT. In this paper we add another piece to this puzzle by applying the CDT quantisation prescription directly to Hořava-Lifshitz gravity in 2 dimensions. We derive the continuum Hamiltonian, and we show that it matches exactly the Hamiltonian derived from canonically quantising the Hořava-Lifshitz action. Unlike the standard CDT case, here the introduction of a foliated lattice does not impose further restriction on the configuration space and, as a result, lattice quantisation does not leave any imprint on continuum physics as expected.' author: - Lisa Glaser - 'Thomas P. Sotiriou' - Silke Weinfurtner bibliography: - 'ref.bib' title: 'Extrinsic curvature in $2$-dimensional Causal Dynamical Triangulation' --- Introduction ============ Causal Dynamical Triangulations (CDT) is a non-perturbative approach to quantum gravity that discretises spacetime into a foliated simplicial manifold. It is an attempt to extend to gravity the lattice methods that have proven very powerful for quantum chromodynamics. CDT has made it possible to numerically explore the path integral over geometries in both $3$ and $4$ dimensions [@ambjorn_nonperturbative_2001; @ambjorn_spectral_2005; @benedetti_spectral_2009; @ambjorn_second-order_2011; @ambjorn_effective_2014; @ambjorn_recent_2015; @benedetti_spacetime_2015]. In $2$ dimensions the model can be solved analytically [@ambjorn_non-perturbative_1998] and gives rise to a continuum Hamiltonian. Extending lattice methods to gravity is not straightforward. Instead of calculating field configurations on a fixed lattice, the lattice itself becomes the object of the dynamics. The presence of a time foliation is crucial. The precursor to CDT is the theory of Dynamical Triangulations (DT), where the discretisation is implemented by approximating spacetime through simplicial complexes [@ambjorn_dynamical_1995], with each $d$ dimensional simplicial complex consisting of $d$-simplices of flat space glued together along their $d-1$ dimensional faces. In these configurations curvature is concentrated at the $d-2$ dimensional faces of the simplices. The action on the space of simplicial complexes is the Regge action for discretised spacetimes [@regge_general_1961]. Simulations of this theory uncovered the existence of two phases, neither of which resembles a continuum spacetime in a suitable limit. The first is known as the crumpled phase. Simplices are all glued together as closely as possible and in the limit of infinite size the Hausdorff dimension is infinite as well. The other phase is the branched polymer phase, where the simplices form long chains and the Hausdorff dimension of the resulting space is $2$ [@ambjorn_dynamical_1995]. The solution Ambjørn and Loll proposed for this problem was to force the simplicial complex to have a foliated structure [@ambjorn_non-perturbative_1998]. This gives rise to a unique timelike direction. The length of a timelike edge over the length of a spacelike edge is a free parameter, $a_t$. The path integral over these foliated simplicial complexes shows that the resulting geometries are much better behaved. The $2$-dimensional model can be solved analytically in different ways [@durhuus_spectral_2009; @ambjorn_non-perturbative_1998], which lead to the same result. These approaches have been extended to include matter [@ambjorn_new_2012] or local topology changes [@ambjorn_topology_2008]. In $3$ and $4$ dimensions analytic methods are no longer fruitful and CDT has been explored through computer simulations. These have shown that there exists a region in CDT parameter space in which the average Hausdorff dimension of geometries agrees with the dimension of the building blocks, and in which the evolution of spacelike slices follows a mini-superspace action [@ambjorn_semiclassical_2005; @benedetti_spacetime_2015]. This phase has also given rise to the first predictions of a varying spectral dimension [@ambjorn_spectral_2005], which has been found independently in many other approaches [@lauscher_fractal_2005; @horava_spectral_2009; @sotiriou_dispersion_2011](see also Ref. [@carlip_dimensional_2015] for a review and comparison). Using foliated simplicial complexes might have led to a phase with desirable properties, but introducing a foliation is a thorny issue. Even though the path integral in CDT sums over different foliations it only sums over geometries that actually admit a global foliation. It is thus unclear if one should expect to recover general relativity in the continuum limit or a theory in which all geometries admit a global foliation. Hořava-Lifshitz (HL) gravity [@horava_quantum_2009] is a typical example of a theory with this characteristic. It is a continuum theory with a preferred foliation whose defining symmetry are foliation-preserving diffeomorphisms. Due to the existence of this foliation, one can add higher-order spatial derivatives without increasing the number of time derivatives. This leads to a modification of the propagators at high momenta that renders the theory power-counting renormalizable. In fact, a certain version of HL gravity called [projectable]{} [@horava_quantum_2009; @sotiriou_phenomenologically_2009] has recently been shown to be renormalizable beyond power counting in 4-dimensions [@Barvinsky:2015kil]. In this version the lapse function of the preferred foliation is assumed to be space-independent, which drastically reduces the number of terms in the action and makes the theory tractable. On the other hand, there are serious infrared viability issues concerning projectable 4-dimensional HL gravity [@horava_quantum_2009; @Sotiriou:2009bx; @Koyama:2009hc; @Weinfurtner:2010hz; @Mukohyama:2010xz; @Sotiriou:2010wn] and this suggest that the full non-projectable version [@Blas:2009qj] might be phenomenologically preferable.[^1] It has been shown in Ref. [@horava_spectral_2009] that the spectral dimension in HL gravity exhibits qualitatively the same behaviour as in CDT in 4 dimensions, i.e. it changes from 4 to 2 in the ultraviolet. Ref. [@Sotiriou:2011mu] has focused on the simpler case of 3 dimensions, but it has shown that the complete flow of the spectral dimension of (non-projectable) HL gravity from 3 to 2 can reproduce precisely the flow of of the spectral dimension in 3-dimensional CDT. Interestingly, a certain resemblance can also be found when comparing the Lifshitz phase diagram to the phase diagram of CDT. The measured volume profile of spacelike slices in CDT can be fit with a mini-superspace action derived from either HL gravity or general relativity [@ambjorn_cdt_2010; @benedetti_spacetime_2015]. These are indications for a connection between HL gravity and CDT in the continuum limit. A strong piece of evidence that CDT and HL gravity might be related comes from comparing the Hamiltonians of the $2$d theories. This comparison has been done with projectable HL gravity. In CDT a continuum Hamiltonian can be derived from the analytic solution of the $2$d theory, while in projectable HL gravity a Hamiltonian can be derived through canonical quantisation. These two Hamiltonians have been compared and found to agree, up to a specific rescaling [@ambjorn_2d_2013]. The CDT action in 2 dimensions is the discretized version of the Einstein–Hilbert action $$\begin{aligned} \label{saction} S_{2d CDT}= \frac{1}{2 \kappa} \int {\mathrm{d}}x^2 \sqrt{-g} (R- 2 \Lambda) \to \lambda \,{\mathcal{N}}\;,\end{aligned}$$ where $\kappa$ is a dimensionless parameter, $g$ is the determinant of the two dimensional metric $g_{\mu\nu}$, $R$ is the corresponding Ricci scalar and $\Lambda$ the cosmological constant. ${\mathcal{N}}$ is the total number of simplices and $\lambda$ is the discrete analogue of the cosmological constant. The action of projectable HL gravity in 2 dimensions is [@Sotiriou:2011dr] $$\begin{aligned} \label{hlaction} S_{2d HL} = \frac{1}{2 \kappa} \int {\mathrm{d}}x {\mathrm{d}}t N \sqrt{h} [(1-\lambda_{HL}) K^2 - 2 \Lambda] \;,\end{aligned}$$ where $K$ is the mean curvature of the slices of the preferred foliation, $h$ is the induced metric, and $\lambda_{HL}$ is an extra coupling with respect to GR. For $\lambda_{HL}=1$ the only term that survives is the cosmological constant. This is also the case for the Einstein–Hilbert action (modulo topological consideration) considering that the Ricci scalar is a total divergence in 2 dimensions. Though one can in principle absorb the coefficient of $K^2$ in the HL gravity action by suitably redefining the cosmological constant and multiplying the action by a suitable coefficient, this can only be done if no coupling to matter is present and strictly when $\lambda_{HL}\neq 1$. The fact that the discretised version of the Einstein-Hilbert action and the canonical quantisation of action (\[hlaction\]) lead to the same Hamiltonian, up to a rescaling, that can be interpreted as fixing $(1-\lambda_{HL})$, is quite intriguing. It implies that lattice regularisation of general relativity via CDT does not lead back to general relativity in the continuum limit, but instead to a theory with a preferred foliation. Since CDT restrict the configuration space to that of foliated triangulations, a possible interpretation would be that this restriction leaves its imprint in the continuum limit. In this perspective, there seems to be a mismatch between the configuration space and the symmetries of the action in CDT. It is thus very tempting to promote the configuration space restriction into an actual symmetry of the (continuum) action, [i.e.]{} start from a discretisation of an action that is invariant under only foliation-preserving diffeomorphisms, as is the case for HL gravity. To this end, instead of applying the CDT prescription to a discretised version of action as in Ref. [@ambjorn_2d_2013], we apply it to a discretised version of action (\[hlaction\]). We derive the corresponding continuum Hamiltonian and we compare it with both the standard CDT continuum Hamiltonian and the Hamiltonian one obtains after canonically quantising HL gravity. We show that, for all boundary conditions, we can recover the Hamiltonian for HL gravity, including a free parameter corresponding to $\lambda_{HL}$. That is, the initial action, and the continuum action one would infer by assigning an action to the continuum Hamiltonian match exactly and share the same continuum symmetries, unlike the case of standard CDT, studied in Ref. [@ambjorn_2d_2013]. The rest of the paper is organised as follows. In Section \[sec:ext\] we find a discrete realisation of the extrinsic curvature squared term for $2$d CDT, which we include in the action in Section \[sec:sum\] where we also solve the resulting model analytically. In Section \[sec:H\] we use this analytic solution to derive the Hamiltonian for $2$d CDT with extrinsic curvature terms included, and compare this to the Hamiltonian of projectable $2$d HL gravity. \[sec:ext\]A discrete extrinsic curvature ========================================= Our first task is to find an appropriate discretisation for the extrinsic curvature of constant time slices. To this end we will follow the lines of Ref. [@dittrich_counting_2006], where the extrinsic curvature was used to define trapped surfaces in a triangulation. It is convenient to actually consider the extrinsic curvature of half-integer time slices $t + \frac{1}{2}$. This avoids the curvature singularities at the d-2 simplices in the integer $t$ slices. The extrinsic curvature is concentrated at the joints, or d-1 simplices. The extrinsic curvature of a spacelike surface $\Sigma$ in a manifold $M$ is given by $$\begin{aligned} K_{ab}=- h^{c}_{a} \nabla_c n_b\end{aligned}$$ with $n^b$ a unit vector normal to the surface $\Sigma$ and $h_{ac}$ the induced metric on $\Sigma$. To calculate the extrinsic curvature of the half integer $t$-slices we need the unit vectors normal to the two pieces of the constant time surface $n^{a}_{(i)}$ and the spacelike unit tangent vectors along the constant time surface $s^{a}_{(i)}$, $$\begin{aligned} n^{a}_{(i)}&= \text{cosh}\left( \rho_{(i)} \right) \mathbf{e^{a}_0} + \text{sinh}\left( \rho_{(i)} \right) \mathbf{e^{a}_1} \,,\\ s^{a}_{(i)}&= \text{sinh}\left( \rho_{(i)} \right) \mathbf{e^{a}_0} + \text{cosh}\left( \rho_{(i)} \right) \mathbf{e^{a}_1} \;.\end{aligned}$$ $\rho_{(i)}$ is the angle between the normal vector of the $t+\frac{1}{2}$ surface and the d-1 simplex at which the curvature is located. For the 2d case this is sketched in Figure \[fig:upupdowndown\]. The triangles used in CDT are isosceles with a spacelike edge of length $\ell$ at the base and two timelike edges of length $a_t \ell$. Hence, the angle $\rho$ depends on the base angle $\alpha$, $$\begin{aligned} \alpha= \text{arccos} \left( \frac{1}{2 a_t} \right) \;.\end{aligned}$$ The relative length parameter $a_t$ lies in the interval $\frac{1}{2}<a_t < \infty$, with the limiting cases clearly being excluded as degenerate, since for $a_t=\frac{1}{2}$ the triangle becomes a spacelike line, and for $a_t= \infty$ it turns into two parallel timelike lines. This gives us a range for the angle $0 < \alpha < \frac{\pi}{2}$, as we would expect for the base angle of a triangle. Using this and Fig. \[fig:upupdowndown\] we can determine that for the down-down transition $\rho$ is given as $$\begin{aligned} \label{eq:rho} \rho(x_1) &= \alpha - \frac{\pi}{2} & \rho(x_2) &= -\alpha + \frac{\pi}{2}\,, \end{aligned}$$ whereas for the up-up transition $\rho$ has the opposite sign. One can embed any two triangles into a local Minkowski system such that the kink between them is flat. The covariant derivative then simplifies to the normal coordinate derivative. It is straightforward to see from Fig. \[fig:upupdowndown\] that the derivative of the normal vector will diverge as one moves over the kink. In Ref. [@dittrich_counting_2006] this is resolved by introducing a class of smoothing functions $\delta_\epsilon$ that converge to the delta function as $\epsilon \to 0$. The angle can then be written as $$\begin{aligned} \rho(x)= \rho_{(1)} + \Delta \rho \int_{-\epsilon}^{x} \delta_\epsilon(x') {\mathrm{d}}x'\;,\end{aligned}$$ with $\Delta \rho=\rho_{(2)}-\rho_{(1)}$. The induced metric can be written as $h_{a}^c=\eta^{c}_{a} +n_{a}n^{c}$, and one can then calculate the extrinsic curvature as $$\begin{aligned} \label{Kdef} K^{ab}(x)= - \delta_\epsilon(x) \Delta \rho \;\text{cosh}(\rho(x)) s^a(x) s^b(x)\;.\end{aligned}$$ From this one can calculate the integrated extrinsic curvature scalar as $$\begin{aligned} K&=\int \lim_{\epsilon \to 0} K(x){\mathrm{d}}x= \int \lim_{\epsilon \to 0} K^{ab}(x)\eta_{ab} {\mathrm{d}}x\\ &=\int \delta_\epsilon(x) \Delta \rho \;\text{cosh}(\rho(x)){\mathrm{d}}x \;.\label{eq:Kint}\end{aligned}$$ Plugging the $\rho$ values from equation into equation above, we find the integrated extrinsic curvature. The integrated curvature when passing over down-down, up-up, and up-down transitions are, respectively, $$\begin{aligned} K_{{\downarrow \downarrow}}&= (2 \alpha-\pi) \text{cosh}(\alpha - \pi/2) \; ;\\ K_{{\uparrow \uparrow}}&= -(2 \alpha-\pi) \text{cosh}(-\alpha + \pi/2) \; ;\\ K_{\uparrow \downarrow} &=0 \;.\end{aligned}$$ Here we are not actually interested in the integrated extrinsic curvature itself, but instead in the integral over the extrinsic curvature squared. Defining $K^2$ by taking the square of is problematic due to the presence of the smoothing function $\delta_\epsilon$. This issue can be easily avoided. The smoothing function has been introduced in eq. (\[Kdef\]) in order to regularise the curvature on the kink. One can do the same for $K^2$ by defining $$\begin{aligned} \label{K^2def} K^2(x)= - \delta_\epsilon(x) (\Delta \rho)^2 \;\left[\text{cosh}(\rho(x))\right]^2\;.\end{aligned}$$ That can be understood as “pilling off” the smoothing function from the definition of $K^{ab}(x)$ in eq. (\[Kdef\]) before taking the square and then regularising the result. One can then simply define the integrated squared extrinsic curvature as $$\begin{aligned} K^2=\int \lim_{\epsilon \to 0} K^2 (x){\mathrm{d}}x \;.\end{aligned}$$ Using this prescription the contribution to the extrinsic curvature squared at each d-1 simplex is $$\begin{aligned} K_{{\downarrow \downarrow}}^2 &= (2 \alpha-\pi)^2 \text{cosh}^2(\alpha - \pi/2)\\ K_{{\uparrow \uparrow}}^2 &= (2 \alpha-\pi)^2 \text{cosh}^2(-\alpha + \pi/2) \;.\end{aligned}$$ Due to the symmetry properties of the hyperbolic cosine these are the same, hence we shall call this term $K ^2$. Since $0 < \alpha < \pi/2$ one has that $\pi^2 \text{cosh}\left(\pi/2\right)^2 >K^2>0$. We can tune the contribution from each edge by changing the relative edge-length between space and time, but we can not make the contribution vanish or exceed a certain value. \[sec:sum\]Summing over the simplicial configurations ===================================================== We can now include the extrinsic curvature squared term in the simplicial action for a triangulation $T$ $$\begin{aligned} S(T)= \lambda {\mathcal{N}}+ \mu \sum_{\text{transitions}} K^2 \;,\end{aligned}$$ where $\mu$ is the discrete coupling equivalent to $(1-\lambda)/(2 \kappa)$ and transitions refers to all ${{\uparrow \uparrow}}, {{\downarrow \downarrow}}$ transitions, since for $\uparrow \downarrow$ transitions the extrinsic curvature vanishes. Using this discrete action we can calculate the sum over configurations following the method set out in [@ambjorn_non-perturbative_1998]. The first step is to calculate the transition function $T^{(s)}_{i j}(g,a,1)$ for a transition from $i$ initial edges to $j$ final edges in one time-step. In ordinary CDT each configuration from $i$ to $j$ edges has the same weight, since it has the same overall number of triangles. However, in our case the curvature square term adds different weights to different configurations. After calculating $T^{(s)}_{i j}(g,a,1)$ the next step is to calculate the generating function $\theta^{(s)}(x,y \| g,a,1)$. Switching from the transition function to the generating function is similar to switching from a micro canonical ensemble to a grand canonical ensemble in thermodynamics. Using the generating function makes many calculations easier, especially taking the continuum limit, in which necessarily $i,j \to \infty$. It is possible to calculate the generating function for $t$ time-steps by gluing together several generating functions, but for us this step is unnecessary. Instead we will take the continuum limit and expand the generating function to obtain the Hamiltonian of the theory. Ref. [@di_francesco_integrable_2000] has modified the CDT action by adding a term that contributes at ${{\uparrow \uparrow}}, {{\downarrow \downarrow}}$ transitions. The key motivation for adding this terms was to capture the influence of higher curvature corrections. In simplicial triangulations the curvature at a given vertex is proportional to $v-6$, where $v$ is the number of triangles adjacent to the vertex. Hence, in order to construct a term that influences the local curvature they propose to add to the action the terms $\|v_1 -3\|$ and $\|v_2-3\|$, where $v_1$ is the number of triangles adjacent to a vertex in the slice above it and $v_2$ is the number of triangles adjacent in the slice below it. Attaching a weight of $a^{\|v_1-3\|/2+\|v_2-3\|/2}$ to each vertex is equivalent to attaching a weight of $a$ to each ${{\uparrow \uparrow}}$ or ${{\downarrow \downarrow}}$ transition. The generating function for such a modification has been calculated in Ref. [@di_francesco_integrable_2000]. The extra term in our action leads to the same contribution to the discrete path integral as that considered in Ref. [@di_francesco_integrable_2000]. Hence, even though the physical motivation we used to justify this modification of the action is distinct from that used in Ref. [@di_francesco_integrable_2000], we can nonetheless use the results obtained there. As we will discuss in more detail latter, the Hamiltonian depends on the boundary conditions and there are more than one options. Ref. [@ambjorn_non-perturbative_1998] applied the closed loop conditions, whereas Ref. [@di_francesco_integrable_2000] solve their model using so called staircase boundary conditions. The latter require that the strip of spacetime has a triangle pointing up on its leftmost edge and a triangle pointing down on its rightmost edge. This is called a staircase because it resembles one in the dual graph description. Each of these up/down pointing final triangles has a weight of $\sqrt{g}$ attached. This is necessary to match to the original result for periodic boundary conditions, as will be explained later. ![\[fig:Tij\]A triangulation going from $i$ initial to $j$ final edges is split into $k$ bunches of $n_r,m_r$ upwards / downwards pointing triangles.](sliceTij.pdf){width="\columnwidth"} In order to write the sum over all triangulations we define $g=Exp(-\lambda)$ and $a=Exp(- \mu K^2)$. For $\mu = 0$ the extrinsic curvature contribution vanishes and we recover the standard CDT results. The one time-step transfer matrix connecting $i$ initial to $j$ final edges is given by $$\begin{aligned} T^{(s)}_{i j}(g,a,1)&= \sum^{\text{min}(i,j)}_{ k=1} {\hspace{-15pt}}\sum_{\substack{n_r,m_r \\ r=1,2,\dots,k \\ \sum n_r =i \sum m_r=j}} {\hspace{-15pt}}g^{i+j-1} a^{\sum (n_r-1) + \sum (m_r -1)} \\ &= \sum^{\text{min}(i,j)}_{ k=1} {\hspace{-15pt}}\sum_{\substack{n_r,m_r \\ r=1,2,\dots,k \\ \sum n_r =i \sum m_r=j}} {\hspace{-15pt}}g^{i+j-1} a^{i-k + j-k} \;.\end{aligned}$$ The $i$ intitial and $j$ final edges can be divided into $k$ bunches of adjacent upwards-pointing triangles and $k$ bunches of adjacent downwards-pointing triangles. We denote the number of triangles in the $r$-th bunch of upwards-pointing triangles as $n_r$ and of the $r$-th bunch of downwards-pointing triangles as $m_r$. This is illustrated in Fig. \[fig:Tij\]. ![image](staircase+anti.pdf){width="90.00000%"} Each composition of $i$ into $k$ terms and $j$ into $k$ terms gives the same weight for a fixed $k$. The sum over the compositions $n_r$ and $m_r$ is then just the number of different compositions, leading to $$\begin{aligned} T^{(s)}_{ij}(g,a,1)&= g^{i+j-1} a^{i+j} \sum^{\text{min}(i,j)}_{ k=1} a^{-2k} \binom{i-1}{k-1} \binom{j-1}{k-1}\;.\end{aligned}$$ While each composition into $k$ terms has the same weight, the factor $a$ changes the weight for different $k$, hence leading to a different weighting of the individual geometries than that found in standard CDT. The next step is to introduce the generating function, for a single time step $$\begin{aligned} & \theta^{(s)}(x,y \| g,a,1) = \sum_{i,j} x^i y^j T^{(s)}_{i,j}(g,a) \label{eq:xysum}\\ &= \frac{1}{g} \sum_{ k\geq 1} a^{-2k} \sum_{i\geq k} \binom{i-1}{k-1} ( a g x)^i \sum_{j\geq k} \binom{j-1}{k-1} ( a g y)^j \label{eq:sums}\\ &= \frac{1}{g} \sum_{ k\geq 1} a^{-2k} \frac{ (agx)^k}{(1-agx)^k} \frac{(agy)^k}{(1-agy)^k} \label{eq:eries}\\ &= \frac{ g x y}{1- ag(x+y) - g^2(1-a^2)xy} \;. \label{eq:onestep}\end{aligned}$$ Diagonalising this single-step generating function and taking it $t$-th power yields a generating function for multiple time steps, $t$. Finally, Di Francesco [et al.]{} take the continuum limit of this function. We will not repeat this calculation here and instead directly derive a continuum Hamiltonian using equation and the composition rule for $t$-step generating functions. For this we need to understand the radius of convergence of the sums in eq. . In order to take a continuum limit the coupling constants $g,x,y$ need to be tuned towards their critical values $x_c,y_c,g_c$ which are reached at the radius of convergence. At these critical values all terms in the sum in eq. make contributions of the same order of magnitude. This becomes intuitive when looking at eq. to determine the values of $x_c,y_c$ at the critical point. The series converges for $x,y<1$, but only in the limit $x,y \to 1$ do loops of all lengths contribute equally. Since the continuum limit consists of taking the length of the edges to zero, while taking the number of edges to infinity, we see that only the limit $x,y \to 1$ will lead to loops of non zero macroscopic length. With $x_c,y_c =1$ fixed we can then determine the radius of convergence of . We find two possible solutions $g_c=1/(\pm 1 +a)$. Since our solution should smoothly connect to the standard solution for which $a=1, g_c= 1/2$, we conclude that $$\begin{aligned} \label{eq:xcycgc} x_c&=1 &y_c&=1& g_c&=\frac{1}{1+a} \;.\end{aligned}$$ In addition to the different couplings, the number of geometries included in the sum is also dependent on the boundary conditions imposed. As already mentioned before, Di Francesco [et al.]{} impose staircase boundary conditions, as these allow one to easily count the possible compositions. Ordinarily CDT is solved with periodic boundary conditions with or without a marked point. For our discussion it will be useful to calculate everything for all three of these possible boundary conditions, since we will find that they all find an interpretation in the continuum. In order to compare the result for staircase boundary conditions with the known results for periodic boundary conditions with one marked point on the in-going boundary Di Francesco [et al.]{} [@di_francesco_integrable_2000] glue the staircase together with an anti-staircase (Fig. \[fig:stairs\]). An anti-staircase is defined such that the outermost triangles can be glued onto those of the staircase in a way that reproduces the periodic results. See Fig. \[fig:stairs\]. This gluing leads to a two loop correlator with periodic boundary conditions and marked points on both the in-going and outgoing loop. Attempting to glue the staircase into a single loop would have resulted in a seam with an enforced pattern with down-up down-up (or up-down up-down) pointing triangles. The number of configurations with the anti-staircase boundary condition is the same as that of staircase configurations, hence the one step generating functions are identical. Gluing the configurations together corresponds to simply multiplying the generating functions, and dividing by $xy$ to remove doubled boundary links. One then has that $$\begin{aligned} \theta^{(2)}(x,y \| g,a,1)&= \frac{\theta^{(s)}(x,y \| g,a)^2}{xy} \nonumber \\ &= \frac{g^2 x y }{(1- a g(x+y) - g^2(1-a^2)xy)^2} \;.\end{aligned}$$ This is the one step generating function for a propagator with a point marked on both the ingoing and outgoing loops.[^2] To convert it to the generating function for the propagator with a marked point only on the incoming loop we unmark the outgoing loop by dividing the amplitude $T^{(2)}_{ij}(g,a,1)$ by a factor of $i$. In the generating function this corresponds to calculating $$\begin{aligned} &\theta^{(1)}(x,y\|g,a,1)= \int_{0}^{y} \frac{{\mathrm{d}}\tilde y}{\tilde y} \theta^{(2)}(x, \tilde y\| g,a,1) \;.\end{aligned}$$ We then find $$\begin{aligned} \label{eq:onestepmark} &\theta^{(1)}(x,y\|g,a,1)= \nonumber \\ &\frac{g^2 x y }{(1-a g x)(1- a g (x+y) - g^2(1-a^2)xy)} \end{aligned}$$ which in the limit $a \to 1$ agrees with the result in [@ambjorn_non-perturbative_1998]. To complete the possible cases we can also calculate the unmarked propagator with periodic boundary conditions, by removing the mark from the incoming loop through $\int_{0}^{x} {\mathrm{d}}\tilde x/ \tilde x$, and find $$\begin{aligned} \label{eq:onestepnomark} \theta^{(0)}(x,y\|g,a,1)&= \log\left( \frac{(1-a g y)(1-a g x)}{1- a g(x+y) -g^2 (1-a^2) x y } \right) \;.\end{aligned}$$ \[sec:H\]Deriving a Hamiltonian =============================== We can derive a Hamiltonian for the development of the loop-loop correlator by combining eq. with the composition rule for the generating functions. For the generating functions with one marked point, or staircase boundary conditions one has $$\begin{aligned} &\theta(x,y\|g,a,t_1+t_2) =\nonumber \\ & \oint \frac{{\mathrm{d}}z}{2 \pi i z} \theta(x,z^{-1}\|g,a,t_1) \theta(z,y\|g,a,t_2) \;,\end{aligned}$$ where the contour is chosen such that the singularities of $\theta(x,z^{-1}\|g,a,t_1)$ are included but those of $\theta(z,y\|g,a,t_2)$ are not. This gluing rule is the same for $\theta^{(s)}(x,y\|g,a,t)$ and $\theta^{(1)}(x,y\|g,a,t)$, since in both cases there is only one consistent way to glue two geometries together along the final / initial boundary. For $\theta^{(0)}(x,y\|g,a,t)$ the composition rule is slightly more complicated [@zohren_causal_2009] $$\begin{aligned} &\theta^{(0)}(x,y\|g,a,t_1+t_2) \nonumber \\ &=\oint \frac{{\mathrm{d}}z'}{2 \pi i z'^2} \partial_z \theta^{(0)}(x,z\|g,a,t_1)\bigg\|_{z=\frac{1}{z'}} {\hspace{-15pt}}\theta^{(0)}(z',y\|g,a,t_2)\;,\end{aligned}$$ taking into account that the final/ initial loops of length $l$ can be consistently glued together in $l$ different ways. Inserting or , or to the suitable one of the two expressions above corresponds to calculating them for $t_1=1$, with $t_2=t-1$. This yields $$\begin{aligned} \theta^{(s)}(x,y\|g,a,t) =& \frac{ g x \; \theta^{(s)}\left(\frac{g a + g^2 x(1-a^2)}{1- a g x},y\|g,a,t-1\right) }{ g a + g^2 x (1-a^2)} \\ \theta^{(1)}(x,y\|g,a,t) =& \frac{ g x \;\theta^{(1)}\left(\frac{g a + g^2 x(1-a^2)}{1- a g x},y\|g,a,t-1\right) }{ (1-a g x)( a + g x (1-a^2))}\\ \theta^{(0)}(x,y\|g,a,t)=& \theta^{(0)}\left(\frac{g a + g^2 x(1-a^2)}{1- a g x},y\|g,a,t-1\right) \nonumber\\ & -\theta^{(0)}\left(g a,y\|g,a,t-1\right)\;.\end{aligned}$$ One can calculate the continuum Hamiltonian via an expansion in the lattice spacing. In the continuum limit the lattice length $\ell$ is taken to zero in such a way that the coupling constants $x,y,g$ are tuned towards their critical points $x_c,y_c,g_c$ which we determined in eq. . We assume the following scaling around these values $$\begin{aligned} \label{eq:lattice1} x&= e^{- \ell X}= 1- \ell X +\frac{1}{2} \ell^2 X^2 +O(\ell^3) \\ y&= e^{- \ell Y} = 1- \ell Y +\frac{1}{2} \ell^2 Y^2+O(\ell^3)\\ g&= \frac{1}{a+1} e^{- \ell^2 \Lambda} = \frac{1}{a+1} (1- \ell^2 \Lambda) +O(\ell^4)\end{aligned}$$ with $a$, and hence $a_t$, kept constant. Since the length of each time step also scales to zero we introduce $t = \tau/\ell$. The scaling we chose is consistent with that in Ref. [@ambjorn_non-perturbative_1998], albeit with a slight modification to match the condition $g \to \frac{1}{a+1}$ in the $\ell \to 0$ limit. It also matches the scaling chosen in Ref. [@di_francesco_integrable_2000] up to a redefinition of the cosmological constant, $\Lambda \to a \Lambda/2$. $\Lambda$ is a numerical constant and such a redefinition is legitimate. However it will become clear that the scaling we chose is preferable when we compare our Hamiltonian to the literature. We denote the continuum propagators as $$\begin{aligned} \Theta(X,Y\|\Lambda ,a, \tau)&= \lim_{\ell \to 0} \;\ell\; \theta(x,y\|g,a,t) \end{aligned}$$ where $x,y,g,t$ are understood as the functions of $\ell$ defined in and $\theta$ without a superscript denotes any of the 3 generating functions $\theta^{(s)}$, $\theta^{(1)}$, and $\theta^{(0)}$. We can then expand $\theta(x,y\|g,a,t)$ to first order in $\ell$. This leads to a heat kernel equation $$\begin{aligned} \label{eq:HPDE} \partial_\tau \Theta(X,Y\| \tau , \sqrt{\Lambda},a) &=- H_X \Theta(X,Y\| \tau , \sqrt{\Lambda},a) \;, \end{aligned}$$ with the Hamiltonians $$\begin{aligned} H^{(s)}_X&= a X + (a X^2 -2 \Lambda )\partial_X \\ H^{(1)}_X&= 2 a X + (a X^2 -2 \Lambda )\partial_X \\ H^{(0)}_X&= (a X^2 -2 \Lambda )\partial_X \;.\end{aligned}$$ From these we can calculate the Hamiltonian acting on $G(L_1,L_2\|\tau, \sqrt{\Lambda},a)$ with an inverse Laplace transform, $$\begin{aligned} H^{(s)}_L&= -a L \partial_L^2 -a \partial_L + 2 \Lambda L \\ H^{(1)}_L&= -a L \partial_L^2 + 2 \Lambda L \\ H^{(0)}_L&= -a L \partial_L^2 -2 a \partial_L + 2 \Lambda L \;.\end{aligned}$$ It is worth pointing out that the Hamiltonian for the staircase boundary condition is the same as one could derive for an amplitude with two marked points, assuming again that the correct gluing rule is used. We can now compare these Hamiltonians with the Hamiltonian derived for HL gravity in Ref. [@ambjorn_2d_2013], where we have reinstated a constant $\zeta=1/(4(1-\lambda_{HL}))$ that is absorbed into the loop length in that paper. The Hamiltonian from HL gravity actually has three possible forms, depending on the ordering of the operators. The ordering choice corresponds to the different possible boundary conditions that can be imposed in CDT. The three possible Hamiltonians are[^3] $$\begin{aligned} H_{-1}&= - \zeta L \partial_L^2 + 2 \Lambda L \,,\\ H_{0}&=- \zeta L \partial_L^2 - \zeta \partial_L +2 \Lambda L \,,\\ H_{1}&=- \zeta L \partial_L^2 -2\zeta \partial_L +2 \Lambda L \,.\end{aligned}$$ Identifying $\zeta$ with $a$ there is a complete matching, with $H_{-1}$ matching the Hamiltonian for the single marked loop $H^{(1)}_L$, $H_{0}$ matching the one for the staircase boundary conditions $H^{(s)}_L$, and $H_{1}$ is matching the Hamiltonian for the an unmarked loop $H^{(0)}_L$. Conclusions =========== In this paper we have applied the CDT prescription for quantisation to a discretisation of the action of projectable HL gravity instead of the Einstein-Hilbert action. We have calculated the corresponding continuum Hamiltonians for different boundary conditions and we have shown that they match exactly the Hamiltonians one obtains from the canonical quantisation of HL gravity for different orderings of the operators. This result is far from surprising and it seems to support the idea that the introduction of a lattice in the quantisation scheme leaves continuum physics unaffected even when the lattice is dynamical. However, this issue is more subtle and this can be better appreciated when our results are interpreted in conjunction with the result of Ref. [@ambjorn_2d_2013]. It was shown there that the continuum Hamiltonian for standard 2d CDT agrees with the Hamiltonian for projectable 2d HL gravity up to a rescaling of the loop length $L$ and the cosmological constant $\Lambda$ in HL gravity by a factor $\zeta=1/[4(1-\lambda_{HL})]$. In other words, the starting action did not have a preferred foliation, the final Hamilton did, presumably due to the fact that the configuration space is CDT is restricted to foliated triangulations. Hence, in that case lattice quantisation does seem to leave an imprint on continuum physics. Combining these two results suggest strongly that if the lattice quantisation scheme is compatible with the symmetries of the original action then it does not affect continuum physics, whereas if the introduction of the lattice introduces further restrictions to the configuration space, then it actually modifies the continuum theory. In standard CDT the requirement that the triangulation be foliated is incompatible between the symmetries of the Einstein-Hilbert action (full diffeomorphisms) and this seems to lead to the generation of the extrinsic curvature terms in the continuum Hamiltonian. Considering the process of taking the continuum limit as a form of renormalisation, one can compare this situation with work on the renormalisation group flow in HL gravity. The large number of couplings of HL gravity in more than 2 dimensions make a complete study challenging, but first studies of part of the parameter space have been done [@contillo_renormalization_2013; @rechenberger_functional_2013; @dodorico_asymptotic_2014]. Of particular interest is that they show that the isotropic plane $\lambda_{HL}=1$, which contains GR, is not a fixed plane of the flow [@contillo_renormalization_2013]. Hence, one expects to leave this plane through the generation of symmetry breaking terms. As already mentioned, the continuum Hamiltonian(s) we derived here are in full agreement with the Hamiltonian(s) of HL gravity, whereas they only agree with the continuum Hamiltonian(s) of standard CDT derived in Ref. [@ambjorn_2d_2013] up to a rescaling of parameters. In the continuum theory this rescaling would correspond to a redefinition of the coupling constant and the cosmological constant, and it could also be seen as a reparametrization of time or the spatial coordinate. Hence, as already discussed in the introduction, it is only allowed without loss of generality if there is no coupling to matter. More generically, it would correspond to a fixing of the HL coupling $\lambda_{HL}$ \[to a value different than that corresponding to general relativity\]. This is a salient point that certainly deserves further investigation. Some notes of caution are in order. Firstly, gravity in $2$ dimensions is significantly different that in higher dimensions, and hence special care needs to be taken in trying to generalize results in $2$d to higher dimension. For example, 2-dimensional general relativity and HL gravity are topological theories and hence quantisation is trivial. In fact, HL gravity is renormalizable in 2d without any anisotropy between space and time. Secondly, the discretisation of the extrinsic curvature squared term is not unique. Our choice was guided by a balance between physical motivation and solvability. An appropriate discretisation should lead to a good continuum limit and the one we chose manifestly does. However, alternative discretisation scheme do exist [@hamber_higher_1984; @ambjorn_quantum_1993]. Clearly, it would be very interesting to generalise our results to higher dimensions. While this might not be possible analytically, it can be done numerically. Some results for simulations of CDT plus higher curvature terms in $2+1$d already exist [@anderson_quantizing_2012]. It would also be particularly interesting to reexamine the discrete RG flow for CDT in $3$ or $4$ dimensions [@ambjorn_renormalization_2014], taking into account extrinsic curvature and higher derivative terms. The authors would like to thank Jan Ambjørn and Renate Loll for helpful discussions. The work leading to this invention has received funding from the European Research Council under the European Union Seventh Framework Programme (FP7/2007-2013) / ERC Grant Agreement n. 306425 “Challenging General Relativity”. S.W. acknowledges financial support provided under the Royal Society University Research Fellow (UF120112), the Nottingham Advanced Research Fellow (A2RHS2) and the Royal Society Project (RG130377) grants. [^1]: Other restricted version of HL gravity exist as well [@horava_quantum_2009; @Horava:2010zj; @Sotiriou:2010wn; @Vernieri:2011aa], but we will not discuss them here. [^2]: The superscript $^{(2)}$ indicates the two marked points, similarly $^{(s)}$ indicates the staircase boundaries,$^{(1)}$ indicates a single marked point and $^{(0)}$ indicates no marked points. [^3]: The subscripts here are identical to those in Ref. [@ambjorn_2d_2013], which were chosen to reflect the measure on which the Hamiltonian is hermitian.	ArXiv
--- abstract: 'We prove that almost every finite collection of matrices in $GL_d( \mathbb{R} )$ and $SL_d({{\mathbb R}})$ with positive entries is Diophantine. Next we restrict ourselves to the case $d=2$. A finite set of $SL_2({{\mathbb R}})$ matrices induces a (generalized) iterated function system on the projective line $\RP^1$. Assuming uniform hyperbolicity and the Diophantine property, we show that the dimension of the attractor equals the minimum of 1 and the critical exponent.' address: - 'Yuki Takahashi, Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel' - 'Boris Solomyak, Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel' author: - BORIS SOLOMYAK - YUKI TAKAHASHI bibliography: - 'bib.bib' title: 'Diophantine property of matrices and attractors of projective iterated function systems in $\RP^1$' --- Introduction and main results ============================= Diophantine property of matrices -------------------------------- Recently there has been interest in Diophantine properties in non-Abelian groups. The following is a variant of [@GJS1999 Definition 4.2]. Let $\Ak = \{A_i\}_{i\in \Lam}$ be a finite subset of a topological group $G$ equipped with a metric $\varrho$. Write $A_\bi = A_{i_1}\cdots A_{i_n}$ for $\bi = i_1\ldots i_n$. We say that the set $\Ak$ is [Diophantine]{} if there exists a constant $c>0$ such that for every $n\in \N$, we have $$\label{Dioph1} \bi,\bj\in \Lam^n,\ A_\bi\ne A_\bj \implies \varrho(A_\bi,A_\bj) > c^n.$$ The set $\Ak$ is [strongly Diophantine]{} if there exists $c>0$ such that for all $n\in \N$, $$\label{Dioph2} \bi,\bj\in \Lam^n,\ \bi\ne \bj \implies \varrho(A_\bi,A_\bj) > c^n.$$ Clearly, $\Ak$ is strongly Diophantine if and only if it is Diophantine and generates a free semigroup. Gamburd, Jacobson, and Sarnak [@GJS1999 Definition 4.2] gave a definition of a Diophantine set, which is equivalent to ours, except that they always consider symmetric sets (that is, $g\in \Ak\ \Rightarrow\ g^{-1}\in \Ak$). Diophantine-type questions in groups arise in connection with spectral gap estimates, see [@GJS1999; @Bourgain2014]. See [@ABRS2015; @ABRS2018] for a recent discussion of Diophantine properties in groups and related problems. In [@ABRS2018] a Lie group $G$ is called Diophantine, if almost every $k$ elements of $G$, chosen independently at random according to the Haar measure, together with their inverses, form a Diophantine set in $G$. Gamburd et al. [@GJS1999] conjectured that $SU_2({{\mathbb R}})$ is Diophantine. More generally, it is conjectured that semi-simple Lie groups are Diophantine. Kaloshin and Rodnianski [@KR2001] proved a weaker Diophantine-type property: for a.e. $(A,B) \in SO_3({{\mathbb R}})\times SO_3({{\mathbb R}})$, there exists $c>0$ such that for any $n{\geqslant}1$ and any two distinct words $W_1, W_2$ over the set $\Ak=\{A,B,A^{-1},B^{-1}\}$ of length $n$, $$\\|W_1-W_2\\|{\geqslant}c^{n^2}.$$ It is mentioned in [@KR2001] that their method is general, and applies to $SU_2({{\mathbb R}})$ as well, and also to $m$-tuples of matrices for any $m{\geqslant}2$. Next we state our first result. For any collection of linearly independent vectors $v_1,\ldots,v_{d}$ in ${{\mathbb R}}^{d}$ consider the simplicial cone $$\label{cone} \Sig=\Sig_{v_1,\ldots,v_{d}} = \{x_1 v_1 + \cdots + x_{d} v_{d}:\ x_1,\ldots,x_{d}{\geqslant}0\}.$$ If a matrix $A\in GL_{d}({{\mathbb R}})$ satisfies $$A({\Sig}{\smallsetminus}\{0\}) \subset \Sig^\circ,$$ we say that $\Sig$ is [strictly invariant]{} for $A$. Given a cone $\Sig=\Sig_{v_1,\ldots,v_{d}}$, denote by $\Xk_{\Sig,m}$ (respectively, $\Yk_{\Sig,m}$) the set of all $GL_{d}({{\mathbb R}})$ (respectively, $SL_{d}({{\mathbb R}})$) $m$-tuples of matrices for which $\Sig$ is strictly invariant. We consider $\Xk_{\Sig,m}$ as an open subset of ${{\mathbb R}}^{d^2m}$ and $\Yk_{\Sig,m}$ as a $(d^2-1)m$-dimensional manifold. \[main\_thm\] Let $\Sig=\Sig_{v_1,\ldots,v_{d}}$ be a simplicial cone in ${{\mathbb R}}^{d}$ and $m{\geqslant}2$. [(i)]{} For a.e. $\mathcal{A} \in \mathcal{X}_{\Sig, m}$, the $m$-tuple $\mathcal{A}$ is strongly Diophantine. In particular, a.e. $m$-tuple of positive $GL_{d}({{\mathbb R}})$ matrices is strongly Diophantine. [(ii)]{} For a.e. $\mathcal{A} \in \mathcal{Y}_{\Sig, m}$, the $m$-tuple $\mathcal{A}$ is strongly Diophantine. In particular, a.e. $m$-tuple of positive $SL_{d}({{\mathbb R}})$ matrices is strongly Diophantine. 1. Unfortunately, our results do no cover any example of a symmetric set, since the strict invariance property cannot hold for a matrix $A$ and $A^{-1}$ simultaneously. 2\. Every $m$-tuple of matrices with algebraic entries is Diophantine (but not necessarily strongly Diophantine), see, e.g., [@GJS1999 Prop.4.3]. 3\. It is well-known that Diophantine numbers in ${{\mathbb R}}$ form a set of full measure, which is, however, meagre in Baire category sense (its complement contains a dense $G_\delta$ set). Baire category genericity of non-Diophantine $m$-tuples in $SU_2({{\mathbb R}})$ has been pointed out in [@GJS1999]. In $G=SL_d({{\mathbb R}})$ the situation is different, since there are, for example, open sets of $m$-tuples in $G\times G$ which satisfy (\[Dioph2\]). For instance, if ${{\mathbb R}}^d_+$ is mapped by $A,B$ into closed cones that are disjoint, except at the origin, then (\[Dioph2\]) holds for $\{A,B\}$. On the other hand, there are open sets in $(SL_d({{\mathbb R}}))^m$ in which non-Diophantine pairs are dense. For instance, the set of elliptic matrices in $SL_2({{\mathbb R}})$ is open, and a standard argument shows that a generic $m$-tuple that contains an elliptic matrix is not Diophantine. The scheme of the proof of Theorem \[main\_thm\] is as follows. We consider the induced action of the matrices on the projective space, and show that, given a non-degenerate family of $m$-tuples strictly preserving an open set, depending on a parameter real-analytically, for all parameters outside an exceptional set of zero Hausdorff dimension, the induced iterated function system (IFS) satisfies a version of the “exponential separation condition”. This property implies the strong Diophantine condition for the matrices. We then locally foliate the space of $m$-tuples of matrices and apply Fubini’s Theorem. The result on the zero-Hausdorff dimensional set of exceptions uses the notion of [order-$k$ transversality]{}, which is a modified version of that which appeared in the work of Hochman [@Hochman2014; @Hochman2015]. The strict open set preservation property is needed to ensure that the induced IFS is contracting (uniformly hyperbolic). Projective IFS and linear cocycles ---------------------------------- Let $\Ak = \{A_i\}_{i\in \Lam}$ be a finite collection of $SL_d({{\mathbb R}})$ matrices. The linear action of $SL_d({{\mathbb R}})$ on ${{\mathbb R}}^{d}$ induces an action on the projective space $\RP^{d-1}$, and thus $\Ak$ defines an IFS $\Phi_\Ak=\{\varphi_A\}_{A\in \Ak}$ on $\RP^{d-1}$, called a [(real) projective IFS]{}. Such IFS were studied by Barnsley and Vince [@BV2012], and by De Leo [@Leo2015a; @Leo2015b]. Following [@BV2012], we say that the IFS $\Phi_\Ak$ has an attractor $K$ if for every nonempty compact set $B$ in a neighborhood of $K$, we have $\lim_{k\to \infty}\Phi_\Ak^k(B) = K$ in the Hausdorff metric, where $\Phi_\Ak(B) = \bigcup_{A\in \Ak} \varphi_A(B)$. Let $\Ak = \{A_i\}_{i\in \Lam}$ be a finite collection of $GL_d({{\mathbb R}})$ matrices and $\Phi_\Ak$ the associated IFS on $\RP^{d-1}$. An alternative, but closely related viewpoint, is to consider the linear cocycle $A: \Lam^{{\mathbb Z}}\to SL_d({{\mathbb R}})$ over the shift on $\Lam^{{\mathbb Z}}$, defined by $A(\bi) = A_{i_1}$. Strict contractivity of the projective IFS turns out to be equivalent to [uniform hyperbolicity]{} of the cocycle [@BG2009]. Here we restrict ourselves to the case of $d=2$, which was investigated in great detail by Avila, Bochi, and Yoccoz [@ABY2010]. There is a natural identification between $[0, \pi)$ and the projective space $\RP^1$. Below we use this identification freely, and whenever necessary we view $[0, \pi)$ as $\mathbb{R} / \pi \mathbb{Z}$. For $A\in GL_2({{\mathbb R}})$ denote the action of $A$ on $[0, \pi)\cong \RP^1$ by the symbol $\varphi_A$. Denote by $d_{\Prob}$ the metric on $\RP^1$ induced from the identification with ${{\mathbb R}}/\pi{{\mathbb Z}}$. Below we work with $m$-tuples of $SL_2({{\mathbb R}})$-matrices, since the action of $GL_2({{\mathbb R}})$ factors through the $SL_2({{\mathbb R}})$ action in the obvious way, via $A\mapsto (\det A)^{-1}A$. A [multicone]{} is a proper nonempty open subset $U$ of $\RP^1$, having finitely many connected components with disjoint closures. In the following theorem we extracted the results relevant for us from [@ABY2010; @BV2012] (note that [@BV2012] considers real projective IFS of any dimension). \[th-unihyp\] Let $\Ak = \{A_i\}_{i\in \Lam}$ be a family of $SL_2({{\mathbb R}})$ matrices and let $\Phi_\Ak$ be the associated IFS on $\RP^1$. The following are equivalent: [(i)]{} the IFS $\Phi_\Ak$ has an attractor $K \ne \RP^1$; [(ii)]{} the associated linear cocycle over $\Lam^{{\mathbb Z}}$ is uniformly hyperbolic; [(iii)]{} there is a multicone $U$, such that $\Phi_\Ak(\ov{U}) \subset U$; [(iv)]{} there is nonempty open set $V\subset \RP^1$ such that $\Phi_\Ak$ is contractive on $\ov{V}$, with respect to a metric equivalent to $d_{\Prob}$. Following [@ABY2010], we will call a multicone satisfying $\Phi_\Ak(\ov{U})\subset U$, a [strictly invariant multicone]{} for the family of matrices and for the IFS. There are examples, see [@ABY2010], which show that one may need a multicone having $k$ components, for any given $k{\geqslant}2$, even for a pair of $SL_2({{\mathbb R}})$ matrices $\{A_1,A_2\}$. Our next result concerns the dimension of the attractor. Following De Leo [@Leo2015a], consider the $\zeta$-function $$\zeta_\Ak(t) = \sum_{n{\geqslant}1} \sum_{\bi \in \Lam^n} \\|A_\bi\\|^{-t},$$ and define the [critical exponent]{} of $\Ak$ by \[critic\] s\_= \_[t0]{} {t: \_(t) = }. \[thm-attr\] Let $\Ak= \{A_i\}_{i\in \Lam}$ be a finite set of $SL_2({{\mathbb R}})$ matrices which has a strictly invariant multicone (or satisfies any of the equivalent conditions from Theorem \[th-unihyp\]), and let $K$ be the attractor of the associated IFS $\Phi_\Ak$ on $\RP^1$. Assume that at least two of the maps $\varphi_{A_i}$ have distinct attracting fixed points. If $\Ak$ is strongly Diophantine, then $\dim_H(K) = \min\{1,\half s_\Ak\}$, where $s_\Ak$ is the critical exponent (\[critic\]). In the special case when the IFS $\Phi_\Ak$ satisfies the Open Set Condition, this result is due to De Leo [@Leo2015a Th.4]. Recall that the strong Diophantine condition holds, in particular, when $\Ak$ generates a free semigroup and all the entries of $A_i$ are algebraic. \[rem-DeLeo\] [It is further shown in [@Leo2015a] that for $\Ak$ hyperbolic (and in some parabolic cases), $$s_\Ak = \lim_{r\to \infty} \frac{N_{\Ak}(r)}{\log r},$$ where $N_{\Ak}(r)$ is the number of elements of norm ${\leqslant}r$ of the semigroup generated by $\Ak$. An analogy is pointed out with the classical results on Kleinian and Fuchsian groups, see, e.g., [@Sullivan1984]. ]{} Let $\Phi=\Phi_\Ak$. An alternative way to express the dimension, and one we actually use in the proof, is in terms of [Bowen’s pressure formula]{} \[Bowen1\] P\_(s) = 0,where $P_\Phi(\cdot)$ is the pressure function associated with the IFS $\Phi$. Throughout the paper we use the notation $$\varphi_\bi = \varphi_{i_1}\ldots \varphi_{i_n}.$$ The pressure is defined by \[Bowen0\] P\_(t) = \_[n]{} \_[\^n]{} \_’\^t, where $\\|\cdot\\|$ is the supremum norm on $\ov{U}$. As will be clear from the Bounded Distortion Property, the definition of $P_\Phi(t)$ does not depend on the choice of strictly invariant multicone $U$, and moreover, \[critic2\] 2s = s\_. It is a classical result, going back to Bowen [@Bowen1979] and Ruelle [@Ruelle1982], see also [@Falconer_Tech], that if $\{\varphi_i\}_{i\in \Lam}$ is a hyperbolic IFS on ${{\mathbb R}}$ of smoothness $C^{1+{{\varepsilon}}}$, satisfying the Open Set Condition, then the dimension of the attractor $K$ is given by the Bowen’s equation. In the case that the maps $\varphi_i$ are affine, $s>0$ is the unique solution of $$\sum_{i\in \Lam} r_i^s = 1,$$ where $r_i \in (0,1)$ is the contraction ratio of $\varphi_i$. For an IFS with overlaps this is not necessarily true. In [@SSU1], Simon, Solomyak, and Urbański showed that for a one-parameter family of nonlinear IFS with overlaps (hyperbolic and some parabolic) satisfying the [order-1 transversality condition]{}, for Lebesgue-a.e. parameter the dimension of the attractor is given by $$\label{dimul} \dim_H (K) = \min\{1,s\},$$ where $s$ is from (\[Bowen1\]) and the pressure is given by (\[Bowen0\]). \[def-sep\] Let ${{\mathcal F}}= \{f_i\}_{i\in \Lam}$ be an IFS on a metric space $(X,\varrho)$, that is, $f_i:X\to X$. We say that ${{\mathcal F}}$ satisfies the exponential separation condition on a set $J\subset \Xk$ if there exists $c > 0$ such that for all $n\in \N$ we have $$\label{exp_sep} \sup_{x \in J} \varrho(f_\bi(x), f_\bj(x) ) > c^{n},\ \ \mbox{for all}\ \bi,\bj\in \Lam^{n}\ \ \mbox{with}\ \ i_1\ne j_1\ \ \mbox{and}\ \ f_\bi\not \equiv f_\bj.$$ If, in addition, the semigroup generated by ${{\mathcal F}}$ is free, that is, $f_\bi \equiv f_\bj \ \Longleftrightarrow\ \bi=\bj$, we say that ${{\mathcal F}}$ satisfies the [strong exponential separation condition]{}. If these properties hold for infinitely many $n$, then we say that ${{\mathcal F}}$ satisfies the [(strong) exponential separation condition on $J$ along a subsequence]{}. It is rather straightforward to show that the (strong) Diophantine condition for an $m$-tuple in $SL_2({{\mathbb R}})$ matrices is equivalent to the (strong) exponential separation condition for the associated projective IFS (see Lemma \[lem-Dioph\] below). In [@Hochman2014 Cor. 1.2], Hochman proved (\[dimul\]) for an affine IFS $\mathcal{F} = \{ f_i \}_{i \in \Lambda}$ satisfying the exponential separation condition on $J=\{0\}$ along a subsequence. Thus our Theorem \[thm-attr\] is, in a sense, a generalization of Hochman’s result to the case of contractive projective IFS. [In fact, Hochman [@Hochman2014] used the condition (\[exp\_sep\]) without the requirement $i_1\ne j_1$. However, for an IFS $\{f_i\}_{i\in \Lam}$ on an interval $J\subset {{\mathbb R}}$, such that $$\inf_{x\in J, i\in \Lam}\|f_i'(x)\|{\geqslant}r_{\min}>0,$$ requiring $i_1\ne j_1$ in (\[exp\_sep\]) does not weaken the exponential separation condition — it only affects the constant $c$. This follows from the estimate $$\|f_\bi(x) -f_\bj(x)\| =\|f_{(\bi\wedge \bj)\bu}(x) - f_{(\bi\wedge \bj)\bv}(x) \|{\geqslant}r_{\min}^n \|f_\bu(x) - f_\bv(x)\|, \ \ \bi,\bj\in \Lam^n,$$ where $\bi\wedge \bj$ is the common initial segment of $\bi$ and $\bj$, so that $u_1 \ne v_1$. ]{} IFS of linear fractional transformations ---------------------------------------- It is well-known that the action of $GL_2({{\mathbb R}})$ on $\RP^1$ can be expressed in terms of linear fractional transformations. For $$A = \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} \in GL_2(\mathbb{R}),$$ let $f_A(x) = (ax + b)/(cx + d)$, and define $\psi : [0, \pi) \to \mathbb{R^}$ by $\psi (\theta) = \cos\theta/\sin\theta$, where ${{\mathbb R}}^ = {{\mathbb R}}\cup \{\infty\}$. It is easy to see that the following diagram commutes: \(m) \[matrix of math nodes, row sep=3em, column sep=4em, minimum width=2em\] [ \[0, ) & [\[0, )]{}\ &\ ]{}; (m-1-1) edge node \[left\] [$\psi$]{} (m-2-1) edge node \[above\] [$\varphi_A$]{} (m-1-2) (m-2-1.east\|-m-2-2) edge node \[above\] [$f_A$]{} (m-2-2) (m-1-2) edge node \[left\] [$\psi$]{} (m-2-2); Observe that $\psi$ is smooth, and on any compact subset of $(0,\pi)$ the derivatives of $\psi$ and $\psi^{-1}$ are bounded. The following is then an immediate corollary of Theorem \[thm-attr\]. \[cor-ifs\] Let $\mathcal{F} = \{ f_i \}_{ i \in \Lambda }$ be a finite collection of linear fractional transformations with real coefficients. Assume that there exists $U\subset {{\mathbb R}}$, a finite union of bounded open intervals with disjoint closures, such that $f_i( \overline{U} ) \subset U$ for all $i \in \Lambda$. If $\mathcal{F}$ satisfies the strong exponential separation condition on $\ov{U}$, then we have $ \dim_H (K) = \min\{ 1, s \}, $ where $s > 0$ is the unique zero of the pressure function $P_{{\mathcal F}}$. Furstenberg measure ------------------- Let $\mathcal{A} = \{ A_i \}_{i \in \Lambda}$ be a finite collection of $SL_2(\mathbb{R})$ matrices, and let $p = ( p_i )_{i \in \Lambda}$ be a probability vector. Assume that $p_i > 0$ for all $i \in \Lambda$ (we always assume this for any probability vector). We consider the finitely supported probability measure $\mu$ on $SL_2({{\mathbb R}})$: \[def-mu\] = \_[i]{} p\_i \_[A\_i]{}. Our standing assumption is that $\mathcal{A}$ generates an unbounded and totally irreducible subgroup (i.e., does not preserve any finite set in $\RP^1$). Then there exists a unique probability measure $\nu$ on $\RP^1$ satisfying $\mu\cdot\nu = \nu$, that is, $$\label{Furst1} \nu = \sum_{i \in \Lambda} p_i A_i \nu,$$ where $A_i \nu$ is the push-forward of $\nu$ under the action of $A_i$, see [@Furstenberg1963]. The measure $\nu$ is the [stationary measure]{}, or the [Furstenberg measure]{}, for the random matrix product $A_{i_n}\cdots A_{i_1}$ where the matrices are chosen i.i.d. from $\Ak$ according to the probability vector $p$. The properties of the Furstenberg measure for $SL_2({{\mathbb R}})$ random matrix products, such as absolute continuity, singularity, Hausdorff dimension, etc., were studied by many authors, including [@Ledrappier1983; @BL1985]. In [@Pincus1994; @Lyons2000; @SSU2] this investigation was linked with the study of IFS consisting of linear fractional transformations. The reader is referred to [@HS2017] for a discussion of more recent applications. We will recall the main result of [@HS2017], since it will be the main tool in proving Theorem \[thm-attr\]. Let $\chi_{\mathcal{A}, p}$ be the Lyapunov exponent, which is the almost sure value of the limit $$\label{Lyap1} \lim_{n \to \infty} \frac{1}{n} \log \\| A_{i_1 \cdots i_n } \\|,$$ where $i_1, i_2, \cdots \in \Lambda$ is a sequence chosen randomly according to the probability vector $p = ( p_i )_{ i \in \Lambda}$. The Lyapunov exponent is usually defined as the almost sure value of the limit $$\label{Lyap2} \lim_{n \to \infty} \frac{1}{n} \log \\| A_{i_n \cdots i_1 } \\|,$$ but it is easy to see that (\[Lyap1\]) and (\[Lyap2\]) define the same value (e.g., by Egorov’s Theorem). Under the standing assumptions, the limit exists almost surely and is positive [@Furstenberg1963]. The Hausdorff dimension of a measure $\nu$ is defined by $$\dim_H(\nu) = \inf\{\dim_H (E):\ \nu(E^c)=0\}.$$ For a probability vector $p = ( p_i )_{i \in \Lambda}$, we denote the entropy $H(p)$ by $$H(p) = - \sum_{i \in \Lambda} p_i \log p_i.$$ \[thmHS\] Let $\mathcal{A} = \{ A_i \}_{i \in \Lambda}$ be a finite collection of $SL_2(\mathbb{R})$ matrices. Assume that $\mathcal{A}$ is strongly Diophantine and generates an unbounded and totally irreducible subgroup. Let $p = ( p_i )_{i \in \Lambda}$ be a probability vector, and let $\nu$ be the associated Furstenberg measure. Then we have $$\label{eq-Fursten} \dim_H (\nu) = \min \Big\{ 1, \frac{H( p )}{ 2 \chi_{ \mathcal{A}, p } } \Big\}.$$ Theorem \[main\_thm\] implies, in particular, that the dimension formula (\[eq-Fursten\]) holds for the Furstenberg measure associated with a.e. finite family of positive matrices (independent of the probability vector). Next we address the question: what is the Hausdorff dimension of the support of the Furstenberg measure? Sometimes, the support is all of $\RP^1$, in which case the answer is trivially one. The definition (\[Furst1\]) implies that the support is invariant under the IFS $\Phi$ induced by $\Ak$. Thus, Theorem \[thm-attr\] has the following immediate corollary: Let $\Ak= \{A_i\}_{i\in \Lam}$ be a Diophantine set of $SL_2({{\mathbb R}})$ matrices which has a strictly invariant multicone, $\mu$ a finitely supported measure defined by (\[def-mu\]), and $\nu$ the associated Furstenberg measure. Then $\dim_H({\mathrm{supp} \,}\nu) = \min\{1,\half s_\Ak\}$, where $s_\Ak$ is the critical exponent of $\Ak$. Denote by $\Hk_m$ the set of $m$-tuples in $SL_2({{\mathbb R}})$ which have a strictly invariant multicone. Avila (see [@Yoccoz2004 Prop.6]) proved that the interior of the complement of $\Hk_m$ in $(SL_2({{\mathbb R}}))^m$ is ${{\mathcal E}}_m$, where ${{\mathcal E}}_m$ is the set of $m$-tuples which generate a semigroup containing an elliptic matrix. Observe that if an elliptic matrix is conjugate to an irrational rotation, then certainly the invariant set (support of the Furstenberg measure) is all of $\RP^1$. On the other hand, if it is conjugate to a rational rotation, then the semigroup generated by $\Ak$ contains the identity and the strong Diophantine property fails. We expect that our methods can be extended to cover strongly Diophantine families on the boundary of $\Hk_m$, which include parabolic systems. Structure of the paper ---------------------- The rest of the paper is organized as follows. In the next section we prove Theorem \[main\_thm\]. In Section 3 we consider projective IFS and prove Theorem \[thm-attr\]. Finally, in Section 4 we include proofs of some standard technical results for the reader’s convenience. Diophantine property of $GL_{d+1}(\mathbb{R})$ and $SL_{d+1}(\mathbb{R})$ matrices ================================================================================== For notational reasons it is convenient to consider $GL_{d+1}({{\mathbb R}})$ instead of $GL_d({{\mathbb R}})$. $GL_{d+1}({{\mathbb R}})$ actions --------------------------------- Let $A\in GL_{d+1}({{\mathbb R}})$ be a matrix that strictly preserves a cone $\Sig=\Sig_{v_1,\ldots,v_{d+1}}\subset {{\mathbb R}}^{d+1}$. Without loss of generality, we can assume that $\Sig{\smallsetminus}\{0\}$ is contained in the halfspace $\{x\in {{\mathbb R}}^{d+1}: x_{d+1}>0\}$. It is convenient to represent the induced action of $A$ on $\RP^{d}$ on the affine hyperplane $\{x\in {{\mathbb R}}^{d+1}: x_{d+1}=1\}$, and consider the corresponding action on ${{\mathbb R}}^{d}$. To be precise, for $x= (x_1,\ldots,x_{d})\in {{\mathbb R}}^{d}$, we consider $(x,1) = (x_1,\ldots,x_{d},1)\in {{\mathbb R}}^{d+1}$ and let $$f_A(x) = {\rm P}_{d}\Bigl(\frac{A(x,1)}{A(x,1)_{d+1}}\Bigr),\ \ \mbox{when}\ \ A(x,1)_{d+1}\ne 0,$$ where ${\rm P}_{d}$ is the projection onto the first $d$ coordinates. The components of $f_A$ are rational functions, which are, of course, real-analytic on their domain. Consider $$\ov{V} := {\rm P}_{d}(\Sig \cap \{x\in {{\mathbb R}}^{d+1}: x_{d+1}=1\}).$$ By assumption, $f_A$ is well-defined on $\ov{V}$, and we have $f_A(\ov{V})\subset V$. We will also consider the action of $A$ on the unit sphere, given by $$\varphi_A(x):=A\cdot x =\frac{Ax}{\\|x\\|},$$ for a unit vector $x\in \Sp^{d}$. Consider also $\ov{U}$, the intersection of $\Sig$ with the upper hemisphere. We have $\varphi_A(\ov{U}) \subset U$. Lines through the origin provide a 1-to-1 correspondence between $\ov{U}$ and $\ov{V}$, which is bi-Lipschitz in view of the assumption $\Sig {\smallsetminus}\{0\} \subset \{x\in {{\mathbb R}}^{d+1}: x_{d+1}>0\}$. It is well-known [@Birk1957] (see also [@BV2012 Section 9]) that strictly preserving a cone implies that $\varphi_A$ is a strict contraction in the Hilbert metric on $\ov{U}$, which is by-Lipschitz with the round metric. We thus obtain the following: \[lem-Hilbert\] Suppose that the finite family $\Ak = \{A_i\}_{i\in \Lam} \subset GL_d({{\mathbb R}})$ strictly preserves a simplicial cone $\Sig =\Sig_{v_1,\ldots,v_{d+1}}\subset \{x\in {{\mathbb R}}^{d+1}: x_{d+1}>0\}\cup \{0\}$. Then the associated IFS ${{\mathcal F}}_\Ak = \{f_A\}_{A\in \Ak}$ is real-analytic and uniformly hyperbolic on $\ov{V}\subset {{\mathbb R}}^{d}$, in the sense that there exist $C>0$ and $\gam\in (0,1)$ such that $$\max_{x\in \ov{V}} \\|f'_\bi(x)\\| {\leqslant}C\gam^n,\ \ \mbox{for all}\ \bi\in \Lam^n,$$ where $f_\bi = f_{A_{i_1}}\circ \cdots f_{A_{i_n}}$ and $\\|f_\bi'(x)\\|$ is the operator norm of the differential at the point $x$. From exponential separation to the Diophantine property ------------------------------------------------------- Recall the strong exponential separation condition (Definition \[def-sep\]). \[prop-Dioph2\] Let $\Ak$ be a finite family of $GL_{d+1}({{\mathbb R}})$ matrices, and let $\Phi_\Ak$ be the induced IFS on $\Sp^d$. If $\Phi_\Ak$ satisfies the strong exponential separation on a nonempty set, then $\Ak$ is strongly Diophantine. Let $C_1 = \max_{i\in \Lam}\{1,\\|A_i\\|\}$ and $C_2 = \max_{i\in \Lam} \{1,\\|A_i^{-1}\\|\}$. Suppose that $\bi\ne \bj$ in $\Lam^n$. Let us write $$\bi = (\bi\wedge \bj) \bu,\ \ \bj = (\bi\wedge \bj) \bv,$$ where $\bi\wedge \bj$ is the common initial segment of $\bi$ and $\bj$, so that $\bu = u_1\ldots u_k,\ \bv = v_1\ldots v_k$ for some $k{\leqslant}n$, with $u_1 \ne v_1$. We have $$\label{tiu1} \\|A_\bi- A_\bj\\| {\geqslant}\\|A_{\bi\wedge\bj}^{-1}\\|^{-1} \\|A_\bu - A_\bv\\| {\geqslant}C_2^{-n} \\|A_\bu - A_\bv\\|.$$ \[lem-claim\] For any $A, B \in GL_{d+1}(\mathbb{R})$ and any unit vector $x\in {{\mathbb R}}^{d+1}$, we have $$\\| A \cdot x - B \cdot x \\| {\leqslant}\\| A^{-1} \\| \bigl( 1 + \\| B \\| \\|B^{-1}\\|\bigr)\cdot \\| A - B \\|.$$ We have $$\begin{aligned} \\| A \cdot x - B \cdot x \\| &= \left\\| \frac{Ax}{\\| Ax \\|} - \frac{Bx}{\\| Bx \\|} \right\\| \\ &{\leqslant}\left\\| \frac{Ax}{ \\| Ax \\|} - \frac{ Bx }{ \\| Ax \\| } \right\\| + \left\\| \frac{Bx}{ \\| Ax \\| } - \frac{ Bx }{ \\| Bx \\| } \right\\| =: R_1 + R_2. \end{aligned}$$ Since $$1 = \\| A^{-1} (Ax) \\| {\leqslant}\\| A^{-1} \\| \\| Ax \\|,$$ we have $\\| Ax \\|^{-1} {\leqslant}\\| A ^{-1}\\|$. Therefore, $$R_1 {\leqslant}\\| A - B \\| \cdot \\| A^{-1} \\|.$$ Similarly, $$\begin{aligned} R_2 &{\leqslant}\\| B \\| \cdot \big\| \\| Ax \\| - \\| Bx \\| \big\| \cdot \\| Ax \\|^{-1} \\| Bx \\|^{-1} \\ &{\leqslant}\\| B \\| \cdot \\| A - B \\| \cdot \\| A^{-1} \\| \\| B^{-1} \\|, \\ \end{aligned}$$ and the desired estimate follows. Applying the lemma to $A_\bu$ and $A_\bv$ yields, in view of $\\|A_\bw\\|{\leqslant}C_1^n$, $\\|A_\bw^{-1}\\|{\leqslant}C_2^n$ for any $\bw \in \Lam^k$, $k{\leqslant}n$: $$\label{eq-tiu2} \\|A_\bu - A_\bv\\| {\geqslant}2^{-n} C_1^{-n} C_2^{-2n} \\|A_\bu\cdot x- A_\bv \cdot x\\|.$$ Now we continue with the proof of the lemma. By assumption, $\Phi_\Ak$ satisfies the exponential separation condition on a nonempty set. Let $c\in (0,1)$ be the constant from the definition (\[exp\_sep\]). It follows that there exists $x\in \Sp^d$ such that $$\\|A_\bu\cdot x - A_\bv\cdot x\\|{\geqslant}c^k{\geqslant}c^n.$$ Combining this inequality with (\[eq-tiu2\]) and (\[tiu1\]) yields $$\\|A_\bi - A_\bj\\|{\geqslant}2^{-n} C_1^{-n} C_2^{-3n}c^n,$$ confirming the strong Diophantine property. Dimension of exceptions for one-parameter real-analytic families ---------------------------------------------------------------- We consider a one-parameter family of real-analytic IFS on a compact subset of ${{\mathbb R}}^d$, and show that under some mild assumptions it satisfies the exponential separation condition outside of a Hausdorff dimension zero set. This section is based on [@Hochman2014 Section 5.4] and [@Hochman2015 Section 6.6], but we had to make a substantial number of modifications in the definitions and proofs. Let $\mathcal{J}$ be a compact interval in ${{\mathbb R}}$ and $V$ a bounded open set in ${{\mathbb R}}^d$. Suppose that for each $i \in \Lambda$ we are given a real-analytic function $$f_i:\,\ov{V}\times \Jk\to V.$$ This means that it is real-analytic on some neighborhood of $\ov{V}\times \Jk$. We will sometimes write this function as $$f_{i,t}(x) = f_i(x,t),\ \ x\in \ov{V},\ \ t\in \Jk.$$ Denote $\mathcal{F}_t = \{ f_{i, t} \}_{ i \in \Lambda }$. This is a real-analytic IFS on $\ov{V}$, depending on the parameter $t\in \Jk$ real-analytically. For $\bi = i_1\ldots i_n$ we write $f_{\bi,t} = f_{i_1,t} \circ \cdots \circ f_{i_n,t}$. Further, assume that this family of IFS is uniformly hyperbolic in the following sense: there exist $C>0$ and $0 < \gam<1$, such that \[gluk1\] f’\_[,t]{}(x) C\^n, \^n, x, t. Here in the left-hand side is the norm of differential with respect to $x\in {{\mathbb R}}^d$. Fix $x_0 \in V$. For any finite sequence $\bi \in\Lambda^n$ we define $$F_\bi(t) = f_{\bi,t}(x_0).$$ Of course, this depends on $x_0$, but we suppress it from notation. For $\bi\in \Lam^\N$ we have $$\label{conver} \Pi_t(\bi) = F_\bi(t) := \lim_{n\to \infty} F_{\bi\|_n}(t),$$ where $\Pi_t:\Lam^\N\to {{\mathbb R}}^d$ is the natural projection corresponding to the IFS ${{\mathcal F}}_t$ and $\bi\|_n = i_1\ldots i_n$. Notice that this limit is well-defined, independent of $x_0$, and is uniform in $t\in \Jk$, by uniform hyperbolicity (\[gluk1\]). \[lem-analytic\] The function $F_\bi(\cdot)$ is real-analytic on $\Jk$, for any $\bi\in \Lam^\N$. Moreover, $F_{\bi\|_n}(\cdot) \to F_\bi(\cdot)$ uniformly on $\Jk$ for all $\bi\in \Lam^\N$, together with derivatives of all orders. By assumption, for every $\bi \in \Lam^n$, the function $F_\bi$ extends to a holomorphic function in a complex neighborhood of $\Jk$, and we are going to prove that for all $\bi \in \Lam^\N$ the sequence $F_{\bi\|_n}$ converges to $F_\bi$ on a sufficiently small neighborhood uniformly. In order to achieve this, note that since $f_i(x,t):\,\ov{V}\times \Jk \to V$ is real-analytic, it can be extended to a holomorphic (complex-analytic) function $\wtil{f}_i(z,\tau)$, defined on a neighborhood of $\ov{V}\times \Jk$ in ${{\mathbb C}}^d\times {{\mathbb C}}$. Denote by $[\ov{V}]^\delta$ the $\delta$-neighborhood of $\ov{V}$ in ${{\mathbb C}}^d$ and let $\wtil{f}_{i,\tau} = \wtil{f}_i(\cdot,\tau)$. Choose $\ell \in\N$ so that $C\gam^\ell < 1/2$. Then $\\|f'_{\bi,t}(x)\\|<1/2$ for $\bi\in \Lam^\ell$, and $x\in \ov{V}$. By continuity, there exists $\delta>0$ such that $\wtil{f}_{\bi,t}$, with $\bi\in \Lam^\ell$, is holomorphic on $[\ov{V}]^\delta$ and $$\label{niki1} \\|{\wtil{f}_{\bi,t}}'(z)\\|< 1/2,\ \mbox{for all}\ \bi\in \Lam^\ell,\ z\in [\ov{V}]^\delta,\ t\in \Jk.$$ Here in the left-hand side is the norm of the differential with respect to $z\in {{\mathbb C}}^d$. Thus, each $\wtil{f}_{\bi,t}$, with $\bi\in \Lam^\ell$, is a strict contraction on $[\ov{V}]^\delta$, and since $\wtil{f}_{\bi,t}({\ov{V}}) = f_{\bi,t}(\ov{V})\subset V$, we obtain that $[\ov{V}]^\delta$ is mapped into its interior by $\wtil{f}_{\bi,t}$, for $t\in \Jk$. Then the same must be true for all $\tau$ in a sufficiently small complex neighborhood of $\Jk$, which we denote by $\Ok$. We can find a constant $L>0$ such that $$\label{niki2} \\|\wtil{f}_{\bj,\tau}'(z)\\|{\leqslant}L,\ \mbox{for all}\ \bj \ \mbox{such that}\ \ \|\bj\| {\leqslant}\ell-1,\ z\in [\ov{V}]^\delta,\ \tau\in \Ok,$$ since there are finitely many holomorphic functions involved. Now, it follows from (\[niki1\]) and (\[niki2\]) that the function $\wtil{f}_{\bj,t}$, for [all]{} $\bj\in \bigcup_{n=1}^\infty \Lam^n$ and $t\in \Jk$, is well-defined and holomorphic in ${{\mathcal W}}:= [\ov{V}]^{\delta/L}$, and moreover, it maps ${{\mathcal W}}$ into $[V]^\delta$. In addition, $\wtil{{{\mathcal F}}}^\ell_\tau = \{\wtil{f}_{\bi,\tau}\}_{\bi \in \Lam^\ell}$ is a strictly contracting IFS on ${{\mathcal W}}$, depending on $\tau\in \Ok$ holomorphically. It follows that the finite iterates $\wtil{f}_{\bi\|_{n},\tau}$ converge to $\Pi_\tau(\bi)$, the natural projection for $\wtil{{{\mathcal F}}}_\tau$, as $n\to \infty$, uniformly for $\tau\in \Ok$. The uniform limit of holomorphic functions in an open set in ${{\mathbb C}}$ is holomorphic, and since $F_\bi(t) = \Pi_t(\bi)$ is the restriction of a holomorphic map to an interval on the real line, it is real-analytic. The uniform convergence of holomorphic functions implies uniform convergence of their derivatives as well. Next, for $\bi,\bj \in \bigcup_{n=1}^\infty \Lam^n \cup \Lam^{\N}$, let $$\Delta_{\bi, \bj}(t) = F_\bi(t) - F_\bj(t) \in {{\mathbb R}}^d.$$ For any ${{\varepsilon}}> 0$, let $$E_{{{\varepsilon}}} = \bigcap_{N=1}^{\infty} \bigcup_{n > N} \Big( \bigcup_{\bi, \bj \in \Lambda^n, i_1 \neq j_1} \, ( \Delta_{\bi, \bj} )^{-1} B_{{{\varepsilon}}^n} \Big)$$ and $$\label{def-E} E = \bigcap_{{{\varepsilon}}> 0} E_{{{\varepsilon}}},$$ where $B_{{{\varepsilon}}^n} = \{x\in {{\mathbb R}}^d:\ \\|x\\|{\leqslant}{{\varepsilon}}^n\}$. It is easy to see that if $t \notin E$ then $\mathcal{F}_t$ satisfies the strong exponential separation condition. [In [@Hochman2014; @Hochman2015] Hochman considered the case where $\mathcal{F}_t$ is an affine IFS. He defined the sets $E'_{{{\varepsilon}}}$ and $E'$ as follows: $$E'_{{{\varepsilon}}} = \bigcup_{N=1}^{\infty} \bigcap_{n > N} \Big( \bigcup_{\bi, \bj \in \Lambda^n, \bi \neq \bj} ( \Delta_{\bi, \bj} )^{-1} B_{{{\varepsilon}}^n} \Big)$$ and $$\label{def-Etag} E' = \bigcap_{{{\varepsilon}}> 0} E'_{{{\varepsilon}}}.$$ If $t \notin E'$ then $\mathcal{F}_t$ satisfies the strong exponential separation condition [along a subsequence]{}. ]{} For a family of IFS ${{\mathcal F}}_t,\ t\in \Jk$, as above, and for $\bi,\bj\in \Lam^\N$ let $\Delta_{\bi,\bj}(t) = F_\bi(t) - F_\bj(t)$. We say that the family is [non-degenerate]{} if $$\label{nondegen} \Delta_{\bi, \bj}(\cdot) \equiv 0 \iff \bi = \bj \ \ \mbox{for all $\bi, \bj \in \Lambda^{\mathbb{N}}$.}$$ We next prove the following: \[thm\_important\] Suppose that the family of IFS ${{\mathcal F}}_t,\ t\in \Jk$, is non-degenerate. Then the set $E$ from (\[def-E\]) has Hausdorff dimension zero, and therefore, ${{\mathcal F}}_t$ satisfies the strong exponential separation condition on $\Jk$, outside of a set of zero Hausdorff dimension. \[cor\_sep\] For a family of IFS ${{\mathcal F}}_t,\ t\in \Jk$, as above, assume that there exists $t_0 \in \mathcal{J}$ such that the sets $\{ f_{i, t_0}( \overline{V} ) \}_{i \in \Lambda}$ are pairwise disjoint. Then (\[nondegen\]) holds, and hence the set $E$ from (\[def-E\]) has Hausdorff dimension zero. Hochman [@Hochman2014; @Hochman2015] proved, for a non-degenerate family of affine IFS, with a real-analytic dependence on parameter, that the set $E'$ from (\[def-Etag\]) has packing dimension zero. For any smooth function $F : \mathcal{J} \to {{\mathbb R}}^d$, denote $F^{(p)}(t) = \frac{d^p}{dt^p} F(t)$. The family $\{ \mathcal{F}_t \}_{t \in \mathcal{J} }$ is said to be transverse of order $k$ if there exists $c > 0$ such that for all $n \in \mathbb{N}$ and $\bi, \bj \in \Lambda^{n}$, with $i_1 \neq j_1$, we have $$\forall t \in \mathcal{J} \ \ \exists p \in \{ 0, \cdots, k \} \text{\, s.t. } \\| \Delta^{(p)}_{\bi, \bj}(t) \\| > c.$$ Here the norm $\\|\cdot\\|$ is simply the Euclidean norm in ${{\mathbb R}}^d$. [The above definition is different from [@Hochman2014] and it simplifies the proof of Theorem \[thm\_important\]. ]{} \[delta\_prop\] Suppose that the non-degeneracy condition (\[nondegen\]) holds. Then $\{ \mathcal{F}_t \}_{t \in \mathcal{J}}$ is transverse of order $k$ for some $k \in \mathbb{N}$. Suppose that for all $k\in \N$ the family $\{ \mathcal{F}_t \}_{t \in \mathcal{J}}$ is not transverse of order $k$. Then by assumption, for $\{c_k\}$ with $c_k < 1/k$, we can choose $n(k)$, $\bi^{(k)}, \bj^{(k)} \in \Lambda^{n(k)}$ with $i^{(k)}_1 \neq j^{(k)}_1$ and a point $t_k \in \mathcal{J}$ such that $$\\| \Delta^{(p)}_{ \bi^{(k)}, \bj^{(k)} } (t_k) \\| < c_k$$ for $0 {\leqslant}p {\leqslant}k$. Passing to a subsequence $\{ k_{l} \}$, we can assume that $t_{k_{l}} \to t_0\in \Jk$, $\bi^{( k_{l} )} \to \bi \in \Lambda^{\mathbb{N}}$ and $\bj^{( k_{l} )} \to \bj \in \Lambda^{\mathbb{N}}$, with $i_1\ne j_1$. Arguing as in the proof of Lemma \[lem-analytic\], the complex extension of $\Delta_{\bi^{( k_{l} )}, \bj^{( k_{l} )}}$ converges to the complex extension of $\Delta_{\bi, \bj}$ uniformly on a complex neighborhood of $\Jk$, and hence the same holds for $p$-th derivatives. Thus for all $p {\geqslant}0$, we have $$\\| \Delta^{(p)}_{\bi, \bj} (t_0) \\| = \lim_{l \to \infty} \\| \Delta^{(p)}_{ \bi^{(k_{l}),} \bj^{(k_{l})} } (t_{k_{l}}) \\| = 0.$$ Since $\Delta_{\bi, \bj}$ is real-analytic, the vanishing of its derivatives implies $\Delta_{\bi, \bj} \equiv 0$ on $\Jk$, contradicting (\[nondegen\]), since $\bi\ne \bj$ by construction. For a $C^k$-smooth function $F : \mathcal{V} \to \mathbb{C}$, write $$\\| F \\|_{ \mathcal{V}, k } = \max_{ p \in \{0, \cdots, k\} } \sup_{t \in \mathcal{V}} \| F^{(p)}(t) \|,\ \ \\|F\\|_{\Vk} = \\|F\\|_{\Vk,0},$$ and similarly for vector-functions. \[key\_lem\] Let $k \in \mathbb{N}$ and let $F : \mathcal{J} \to \mathbb{R}$ be a $k$ times continuously differentiable function. Let $M = \\| F \\|_{\mathcal{J}, k}$, and let $0 < c < 1$ be such that for every $t \in \mathcal{J}$ there is $p \in \{ 0, \cdots, k\}$ with $\| F^{(p)}(t) \| > c$. Then there exists $C=C_{c,M,\|J\|}{\geqslant}1$ such that for every $0 < \rho < (c/2)^{2^k}$, the set $F^{-1} (-\rho, \rho) \cap \mathcal{J}$ can be covered by $C^k$ intervals of length ${\leqslant}2( \rho/c )^{1/2^k} $ each. \[lem-conc\] If $\{ \mathcal{F}_t \}_{t \in \mathcal{J} }$ is transverse of order $k {\geqslant}1$ on the compact interval $\mathcal{J}$, then the set $E$ from (\[def-E\]) has Hausdorff dimension zero. Extending the real-analytic functions to the complex plane, as in Lemma \[lem-analytic\], since $$\sup_n \sup_{\bi, \bj \in \Lambda^n, i_1 \neq j_1} \\| \Delta_{\bi, \bj} \\|_{\Ok} < \infty$$ on a neighborhood $\Ok$ of $\Jk$, and $\Delta_{\bi, \bj}(\cdot)$ is holomorphic on $\Ok$ for all $\bi, \bj \in \Lambda^n$, we have $$\label{normal} M:=\sup_{n} \sup_{\bi, \bj \in \Lambda^n, i_1 \neq j_1} \\| \Delta_{\bi, \bj} \\|_{ \mathcal{J} , k} < \infty.$$ Let $$\label{En} E_{{{\varepsilon}}, n} = \bigcup_{\bi, \bj \in \Lambda^n, i_1 \neq j_1} ( \Delta_{\bi, \bj} )^{-1}( B_{{{\varepsilon}}^n} ).$$ Then $$\label{E} E_{{{\varepsilon}}} = \bigcap_{N = 1}^{\infty} \bigcup_{n > N} E_{{{\varepsilon}}, n}.$$ Let $\bi, \bj \in \Lambda^n$, with $i_1\ne j_1$, and assume that $\\|\Delta_{\bi,\bj}(t)\\|< {{\varepsilon}}^n$. By Lemma \[key\_lem\] applied to a component of $\Delta_{\bi,\bj}$, for ${{\varepsilon}}$ sufficiently small, the set $$( \Delta_{\bi, \bj} )^{-1} ( B_{{{\varepsilon}}^n} )$$ may be covered by $C^k$ intervals of length ${\leqslant}2( {{\varepsilon}}^n \cdot c^{-1} )^{1/2^k}$. It follows that the set $E_{{{\varepsilon}},n}$ from (\[En\]) may be covered by $O( \| \Lambda \|^{2n} \cdot C^k )$ intervals of length ${\leqslant}( {{\varepsilon}}^n \cdot c^{-1} )^{1/2^k}$. Fix $s>0$ and write $\Hk^s$ for the $s$-dimensional Hausdorff measure. We obtain from (\[E\]) that $$\Hk^s(E_{{\varepsilon}}) {\leqslant}O(1)\cdot \sum_{n{\geqslant}1} \|\Lam\|^{2n} C^k {\bigl({{\varepsilon}}^n \cdot c^{-1}\bigr)}^{s/2^k} < \infty$$ for ${{\varepsilon}}$ sufficiently small. It follows that $\Hk^s(E)=0$. This is now immediate from Lemmas \[delta\_prop\] and \[lem-conc\]. Proof of Theorem \[main\_thm\] ------------------------------ The next lemma follows by an application of Fubini’s Theorem. \[Fubini\] Let $F \subset \mathbb{R}^n$ and let $v \in \mathbb{R}^n$ be a nonzero vector. Assume that for every $x_0 \in \mathbb{R}^n$, the set $\{ x_0 + t v : t \in \mathbb{R} \} \cap F$ has 1-dimensional Lebesgue measure $0$. Then the set $F$ has $n$-dimensional Lebesgue measure $0$. \(i) Let $\Sig=\Sig_{v_1,\ldots,v_{d+1}}$ be a simplicial cone in ${{\mathbb R}}^{d+1}$. Let $\Uk \subset \Xk_{\Sig,m}$ be a small open set in $(GL_{d+1}({{\mathbb R}}))^m$ of $m$-tuples of matrices for which $\Sig$ is strictly invariant. Choose vectors $w_i \in {{\mathbb R}}^{d+1} \ (i \in \Lambda)$, with distinct directions, in such a way that \[w-cond\] w\_i \_[ A\_i v\_1, …,A\_i v\_[d+1]{} ]{} (A\_i)\_[i ]{} . This is possible when $\Uk$ is sufficiently small. Let $(A_i)_{i \in \Lambda} \in \Uk$, and for each $t {\geqslant}0$ and $i\in \Lam$ let $A_{i,t}$ be such that $$A_{i, t} v_j = A_i v_j + t w_i,\ j=1,\ldots,d+1.$$ Condition (\[w-cond\]) guarantees that $\{A_{i,t}v_j\}_{i=1}^{d+1}$ is linearly independent, and hence $A_{i,t}\in GL_{d+1}({{\mathbb R}})$ for all $t>0$. This is a consequence of the following elementary claim. [Claim.]{} [Let $y_1,\ldots,y_{d+1}\in {{\mathbb R}}^{d+1}$ be linearly independent, and suppose that $w = \sum_{k=1}^{d+1} a_k y_k$ for some $a_k{\geqslant}0$. Then the family $\{y_1 + w,\ldots, y_{d+1} + w\}$ is linearly independent as well.]{} We have $$\sum_{j=1}^{d+1} c_j \Bigl(y_j + \sum_{k=1}^{d+1} a_k y_k\Bigr) = 0\ \ \Longrightarrow \ \ \sum_{j=1}^{d+1} \Bigl(c_j + a_j \sum_{k=1}^{d+1} c_k \Bigr) y_j = 0,$$ hence $c_j + a_j \sum_{k=1}^{d+1} c_k = 0$ for all $j$. If $\sum_{k=1}^{d+1} c_k\ne 0$, we obtain a contradiction, in view of $a_j{\geqslant}0,\ j=1,\ldots, d+1$; thus $c_j=0,\ j=1,\ldots, d+1$, as claimed. Let $\Ak_t = \{A_{i,t}\}_{i\in \Lam}$ be the family of matrices defined above, for $t{\geqslant}0$, and let ${{\mathcal F}}_t={{\mathcal F}}_{\Ak_t}$ be the corresponding one-parameter family of IFS on the set $\ov{V} \subset {{\mathbb R}}^d$ obtained by projection of $\Sig \cap \{x\in {{\mathbb R}}^{d+1}: x_d=1\}$ onto ${{\mathbb R}}^d$. Notice that the cone $\Sig$ is strictly preserved by all $\Ak_t,\ t{\geqslant}0$, by construction, hence by Lemma \[lem-Hilbert\], these IFS are all uniformly hyperbolic. It is easy to see that the IFS and their dependence on $t$ is real-analytic, since the IFS are given by rational functions. Condition (\[gluk1\]) holds for $t\in [0,M]$, for any $M<\infty$, by uniform hyperbolicity and compactness. Finally, observe that, given ${{\varepsilon}}>0$, for $t$ sufficiently large, we have $$f_{i,t}(\ov{V}) \subset {\rm P}_d\bigl(\Sig_{{\varepsilon}}(w_i)\cap \{x_{d+1}=1\}\bigr),$$ where $\Sig_{{\varepsilon}}(w_i)$ is the cone of vectors ${{\varepsilon}}$-close to $w_i$ in direction. By construction, $w_i$ are all distinct, hence Corollary \[cor\_sep\] applies. We obtain that for all $t\in [0,\infty)$ outside a set of Hausdorff dimension zero, the IFS ${{\mathcal F}}_t$ satisfies the exponential separation condition, and then Proposition \[prop-Dioph2\] implies that the $m$-tuple of matrices $(A_{i,t})_{i\in \Lam}$ is Diophantine for all $t$ outside of a zero-dimensional set, so certainly for Lebesgue-a.e. $t$. Now Lemma \[Fubini\] yields the desired claim. \(ii) We consider $(SL_{d+1}({{\mathbb R}}))^m$ as a codimension-$m$ submanifold of $(GL_{d+1}({{\mathbb R}}))^m \subset {{\mathbb R}}^{(d+1)^2m}$. In the proof of part (i) we showed that for a.e. $(A_i)_{i\in \Lam} \in \mathcal{X}_{\Sig, m}$, the induced IFS on a subset of ${{\mathbb R}}^d$ satisfies the strong exponential separation condition. Suppose that there is a positive measure subset ${{\mathcal E}}\subset \Yk_{\Sig,m}$ for which the strong Diophantine condition is violated. Then for every $(A_i)_{i\in \Lam}\in {{\mathcal E}}$, the induced IFS $\Phi$ does not have strong exponential separation, by another application of Proposition \[prop-Dioph2\]. However, $(A_i)_{i\in \Lam}\in \Yk_{\Sig,m}$ and $(c_i A_i)_{i\in \Lam}\in \Xk_{\Sig,m}$, for any $c_i>0$, induce the same IFS on the projective space, and we get a set of positive measure in $\Xk_{\Sig,m}$ for which the strong Diophantine condition does not hold. This is a contradiction, and the theorem is proved completely. Dimension of the attractor ========================== Let $A \in SL_2(\mathbb{R})$. It is easy to see that $A^{*} A$ has eigenvalues $\\| A \\|^2$, $\\| A \\|^{-2}$. Let $(\cos t_A, \sin t_A)^{\mathrm{t}}$ be the unit eigenvector corresponding to the eigenvalue $\\| A \\|^{-2}$, where $t_A \in [0, \pi)$. We recall some basic properties of the map $\varphi_A$. For more details see sections 2.2, 2.3 and 2.4 in [@HS2017]. The following simple lemma is [@HS2017 Section 2.4]. \[trivial0\] Let $A \in SL_2(\mathbb{R})$. Then the induced map $\varphi_{A}$ expands by at most $\\| A \\|^2$ and contracts by at most $\\| A \\|^{-2}$. Furthermore, for any ${{\varepsilon}}> 0$ there exists $C_{{\varepsilon}}> 1$ such that $\\| A \\|^{-2} {\leqslant}\| \varphi_{A}'(x) \| < C_{{\varepsilon}}\\| A \\|^{-2}$ for all $x \in [0, \pi) {\smallsetminus}(t_A - {{\varepsilon}}, t_A + {{\varepsilon}})$. The following lemma is now immediate. \[trivial\] Let $U \subsetneq (0, \pi)$ be an open set. Then, for every ${{\varepsilon}}> 0$ there exists $C_{{\varepsilon}}> 1$ such that for any $A \in SL_2( \mathbb{R} )$ with $(t_A-{{\varepsilon}},t_A+{{\varepsilon}}) \subset U$, we have $$\pi - C_{{\varepsilon}}\\| A \\|^{-2} < \| \varphi_A( U ) \| < \pi. $$ Lemma \[trivial\] implies the following: \[trivial3\] Let $U \subsetneq (0, \pi)$ be an open set. Then, for every ${{\varepsilon}}> 0$ there exists $M =M({{\varepsilon}})> 0$ such that the following holds: for any $A \in SL_2( \mathbb{R} )$ that satisfies $\varphi_A( U ) \subset U$ and $\\| A \\| > M$, we have $(t_A-{{\varepsilon}},t_A+{{\varepsilon}}) \not\subset U$. Let $\mathcal{A} = \{A_i \}_{i \in \Lambda}$ be a finite collection of $SL_2(\mathbb{R})$ matrices and let $\Phi = \{\varphi_A\}_{A\in \Ak}$ be the corresponding IFS on $[0,\pi)\cong\RP^1$. Recall the notation: $$\Phi(E) = \bigcup_{A\in \Ak} \varphi_A(E).$$ Assume that there is a strictly invariant multicone $U\subset [0,\pi)$, that is, a nonempty open set having finitely many connected components with disjoint closures, such that $\ov{U}\ne \RP^1$ and $\Phi(\ov{U})\subset U$. By Theorem \[th-unihyp\], the associated cocycle is uniformly hyperbolic, which implies that there exist $c>0$ and $\lam>1$ such that $$\label{eq-hyp} \\|A_\bi\\| {\geqslant}c\lam^n\ \ \mbox{for all}\ \bi\in \Lam^n,\ n\in \N,$$ see [@Yoccoz2004] and [@ABY2010 Theorem 2.2]. Fix ${{\varepsilon}}>0$ such that the $(2{{\varepsilon}})$-neighborhood of $\Phi(\ov{U})$ is contained in $U$, and let $M=M({{\varepsilon}})$ from Lemma \[trivial3\]. By (\[eq-hyp\]), there exists $n_0\in \N$ such that $\\|A_\bi\\|>M$ for $\bi\in \Lam^n$, $n{\geqslant}n_0$. Lemma \[trivial3\] implies that $(t_{A_\bi}-{{\varepsilon}}, t_{A_\bi}+{{\varepsilon}}) \not \subset U$, hence $$(t_{A_\bi}-{{\varepsilon}}, t_{A_\bi}+{{\varepsilon}}) \cap \Phi(\ov{U}) = \emptyset,\ \ \mbox{for all}\ \bi\in \Lam^n,\ n{\geqslant}n_0.$$ Hence, by Lemmas \[trivial0\] and \[trivial\] we obtain $$\label{eq-BDP} \\|A_\bi\\|^{-2} {\leqslant}\|\varphi'_{\bi}(x)\| {\leqslant}C_{{\varepsilon}}\\|A_\bi\\|^{-2},\ \ \mbox{for all}\ \ x\in \ov{U},\ \bi\in \Lam^n,\ n{\geqslant}n_0.$$ Thus we obtain \[lem-contract\] [(i)]{} The Bounded Distortion Property holds for $\Phi$ on $U$: there exists $C'>1$ such that $$\label{eq-bdp} \frac{1}{C'} {\leqslant}\frac{\|\varphi_{\bi}'(x)\|}{\|\varphi_{\bi}'(y)\|} {\leqslant}C'\ \ \mbox{for all}\ \ x,y\in \ov{U}, \ \bi\in \Lam^n,\ n\in \N.$$ [(ii)]{} The IFS $\Phi^k$ is contractive on $U$ in the metric $d_{\Prob}$ for sufficiently large $k$. More precisely, there exists $C''>0$ such that $$\label{eq-HYP} \\|\varphi'_\bi\\|_{\ov{U}} {\leqslant}C'' \lam^{-2n},$$ where $\lam>1$ is from (\[eq-hyp\]). [(iii)]{} We have $$s = s_\Ak/2,$$ where $s$ is the solution of the Bowen’s equation $P_\Phi(s)=0$, with the pressure given by (\[Bowen0\]) and $s_\Ak$ is the critical exponent, given by (\[critic\]). Let $p = ( p_i )_{i \in \Lambda}$ be a probability vector, and let $x_0 \in U$. Let $\chi_{\Phi, p}$ be the almost sure value of the limit $$\label{lim} \lim_{n \to \infty} - \frac{1}{n} \log \| ( \varphi_{ i_1 \cdots i_n } )' (x_0) \|,$$ where $i_1, i_2, \cdots \in \Lambda$ is a sequence chosen randomly according to the probability vector $p = ( p_i )_{ i \in \Lambda}$. The equations (\[eq-BDP\]) and (\[Lyap1\]) imply \[lem-Lyap\] We have $\chi_{ \Phi, p } = 2 \chi_{ \mathcal{A}, p }$. By the Birkhoff Ergodic Theorem, applied to the shift transformation on $\Lam^\N$ with the measure $\mu=p^\N$, in view of the bounded distortion (\[eq-bdp\]), we have $$\label{eq-Lyap3} \chi_{\Phi,p} = \lim_{n \to \infty} - \frac{1}{n} \log \left\| ( \varphi_{ i_1 \cdots i_n } )^{'} ( \Pi (\bi)) \right\|=-\int_{\Lam^\N} \log\bigl\|\varphi_{i_1}'(\Pi(\bi))\bigr\|\,d\mu(\bi),$$ where $\Pi:\Lam^\N\to \RP^1$ is the natural projection corresponding to $\Phi$. \[lem-Dioph\] [(i)]{} Let $\Ak$ be a finite set of matrices in $GL_2({{\mathbb R}})$, and let $\Phi$ be the IFS induced by $\Ak$ on the projective line $\RP^1$. If $\Phi$ satisfies the strong exponential separation condition on a nonempty set, then $\mathcal{A}$ is strongly Diophantine. [(ii)]{} Let $\Ak$ be a finite set of matrices in $SL_2({{\mathbb R}})$, and let $\Phi$ be the IFS induced by $\Ak$ on the projective line $\RP^1$. Then $\Phi$ satisfies the strong exponential separation condition on a set containing at least three points if and only if $\mathcal{A}$ is strongly Diophantine. \(i) This is a special case of Proposition \[prop-Dioph2\], since exponential separation for the induced action on a subset of $\RP^1$ is equivalent to that for the induced action on a subset of the circle. \(ii) One direction, that the strong exponential separation for $\Phi$ implies the strong Diophantine property for $\Ak$, follows from (i). For the converse, we refer to [@HS2017 Lemma 2.5], which says that $SL_2({{\mathbb R}})$ is quantitatively separated by the action on three points of $\RP^1$. Recall that $\A$ is a finite set of $SL_2({{\mathbb R}})$ matrices satisfying the strong Diophantine condition and having a strictly invariant multicone $U$, and $\Phi=\Phi_\Ak$ is the associated IFS on $U$. Then $\Phi$ has a compact attractor $K$, and our goal is to show that $\dim_H(K) =s$, where $P_\Phi(s)=0$ and $P_\Phi$ is given by (\[Bowen0\]). It is known that $$\label{dim_ineq} \dim_H (K) {\leqslant}s,$$ see the appendix for a short proof. Let us show the opposite inequality. Let $d_n > 0$ be the solution of the equation $$\sum_{\bi\in \Lam^n} \| U_\bi \|^{d_n} = 1,$$ where $U_\bi =\varphi_\bi(U)$ and $\|\cdot\|$ denotes the Lebesgue measure on $[0,\pi)\cong\RP^1$. It is not hard to see that $$\label{dim_lim} \lim_{n \to \infty} d_n = s.$$ For convenience of the reader, we include the proof in the appendix, following [@SiSo1999]. Let $p^{(n)} = ( p^{(n)}_\bi )_{\bi \in \Lam^n}$ be the probability vector such that $p^{(n)}_\bi = \| U_\bi \|^{d_n}$. Let $\eta^{(n)}$ be the invariant probability measure for the IFS $\Phi^n$ on $U$, corresponding to $p^{(n)}$. Since $\eta^{(n)}$ is supported on $K$, we have $\dim \eta^{(n)} {\leqslant}\dim_{H} (K)$. We claim that $\Ak$ satisfies the assumptions of Theorem \[thmHS\]. Indeed, the existence of a strictly invariant multicone is known to imply that all the matrices in $\Ak$ are hyperbolic, hence the group generated by $\Ak$ is unbounded. Further, we assumed that not all attracting fixed points of $\Ak$ are the same, hence this group is totally irreducible. Thus the Furstenberg measure for $(\Ak^n,p^{(n)})$ is unique, and it coincides with $\eta^{(n)}$. Since $\Ak$ is Diophantine, we have that $\Ak^n$ is Diophantine as well. Now, by Theorem \[thmHS\] and Lemma \[lem-Lyap\] we have $$\frac{ H( p^{(n)} ) }{ \chi_{ \Phi^n, p^{(n)} } } {\leqslant}\dim_{H} (K).$$ We claim that there exists $C > 0$ such that $$\label{claim2} \chi_{\Phi^n, p^{(n)} } {\leqslant}-\sum_{ \bi \in \Lambda^n } \| U_\bi \|^{d_n} \log \| U_\bi \| + C\ \ \mbox{for all}\ \ n\in \N.$$ Indeed, by (\[eq-Lyap3\]), we have $$\chi_{\Phi^n, p^{(n)} } {\leqslant}\sum_{\bi \in\Lam^n} \mu([\bi])\cdot \log\bigl(\min_{x\in \ov{U}}\|\varphi_\bi'(x)\|\bigr)^{-1},$$ where $[\bi]$ is the cylinder set of sequences starting with $\bi$. Now $\mu([\bi])=\|U\|^{d_n}$, and $$\min_{x\in \ov{U}}\|\varphi_\bi'(x)\|{\geqslant}\frac{\|U_\bi\|}{C'\|U\|},$$ by the Bounded Distortion Property (\[eq-bdp\]). Therefore, $$\begin{aligned} \chi_{ \Phi^n, p^{(n)} } & {\leqslant}\sum_{ \bi \in \Lambda^n } \| U_\bi \|^{d_n} \log \frac{ C' \| U \| }{ \| U_\bi \| } \\ &= -\sum_{\bi \in \Lambda^n} \| U_\bi \|^{d_n} \log \| U_\bi \| + \log C' \| U \|. \end{aligned}$$ By (\[claim2\]), we have $$\begin{aligned} \frac{ H( p^{(n)} ) }{ \chi_{ \Phi^n, p^{(n)} } } &> \frac{ -\sum_{ \bi \in \Lambda^n } \| U_\bi \|^{d_n} \log \| U_\bi \|^{d_n} } { -\sum_{ \bi \in \Lambda^n } \| U_\bi \|^{d_n} \log \| U_\bi \| + C } \\ &= d_n \left( 1 + \frac{C}{ -\sum_{ \bi \in \Lambda^n } \| U_\bi \|^{d_n} \log \| U_\bi \| } \right)^{-1}. \end{aligned}$$ Since $\lim_{n \to \infty} d_n = s$ and $\lim_{n \to \infty} -\sum_{ \bi \in \Lambda^n } \| U_\bi \|^{d_n} \log \| U_\bi \| = \infty$, we obtain $s {\leqslant}\dim_{H} (K)$, as desired. Finally, $s=s_\Ak$ by Lemma \[lem-contract\](iii). Appendix: the proof of (\[dim\_ineq\]) and (\[dim\_lim\]) {#appendix} ========================================================= Proof of (\[dim\_lim\]) [@SiSo1999] ----------------------------------- We have a projective IFS $\Phi = \{\varphi_i\}_{i\in \Lam}$ on a strictly invariant multicone $U$. Observe that $$P_\Phi(t) = \lim_{n\to \infty} \frac{1}{n} \log \sum_{\bi \in \Lam^n} \\|\varphi_\bi'\\|^t = \lim_{n \to \infty} \frac{1}{n} \log \sum_{\bi \in \Lambda^n} \left\| U_{\bi} \right\|^t,$$ by the Bounded Distortion Property (\[eq-bdp\]). Let $$Q_n = \frac{1}{n} \log \sum_{\bi \in \Lambda^n} \left\| U_{\bi} \right\|^s.$$ Since $P_\Phi(s ) = 0$, we have $\lim_{n \to \infty} Q_n = 0$. Let $r_1>0$ be such that $r_1 {\leqslant}\|\varphi_i'(x)\|$ for all $i \in \Lambda$ and $x \in \ov{U}$. Recall (\[eq-HYP\]), which says that $${\\|\varphi'_\bi\\|}_{\ov{U}} {\leqslant}C'' \lam^{-2n},$$ for $C''>0$ and $\lam>1$. Then $r^n_1 \|U\| {\leqslant}\|U_\bi\| < C''\lam^{-2n} \|U\|$ for $\bi\in\Lam^n$, and hence we have $$( r^n_1 \|U\| )^{s-d_n} \cdot \|U_\bi\|^{d_n} < \|U_\bi\|^s < C'' ( \lam^{-2n} \|U\| )^{s-d_n} \cdot \|U_\bi\|^{d_n}.$$ In view of $\sum_{\bi\in \Lambda^n} \|U_\bi\|^{d_n} = 1$, we have $$\frac{1}{n} \log ( r^n_1 \|U\| )^{s-d_n} < Q_n < \frac{1}{n} \bigl[\log C'' + (s-d_n)(\log\|U\| - 2n\log\lam)\bigr],$$ and it follows that $$\frac{Q_n -(\log C'')/n}{ -2\log \lam+ ( \log \|U\| ) / n } < s - d_n < \frac{Q_n}{ \log r_1 + ( \log \|U\| ) / n },$$ which implies $d_n\to s$, as desired. Proof of (\[dim\_ineq\]) ------------------------ Fix ${{\varepsilon}}> 0$. Then for sufficiently large $n$ we have $d_n < s + {{\varepsilon}}/2$. Thus $$\sum_{\bi \in \Lambda^n} \| U_\bi \|^{s + {{\varepsilon}}} < \sum_{\bi \in \Lambda^n} \| U_\bi \|^{d_n + {{\varepsilon}}/2} < ( r_2^n \|U\| )^{{{\varepsilon}}/2} \to 0, \text{\ as } n \to \infty.$$ Therefore, the $(s + {{\varepsilon}})$-dimensional Hausdorff measure of $K$ is zero. By the definition of the Hausdorff dimension, this proves (\[dim\_ineq\]). Acknowledgements {#acknowledgements .unnumbered} ================ The authors would like to acknowledge Balazs Barańy for helpful comments and for telling us about the papers [@ABY2010; @BG2009].	ArXiv
--- abstract: 'Aspect Term Extraction (ATE), a key sub-task in Aspect-Based Sentiment Analysis, aims to extract explicit aspect expressions from online user reviews. We present a new framework for tackling ATE. It can exploit two useful clues, namely opinion summary and aspect detection history. Opinion summary is distilled from the whole input sentence, conditioned on each current token for aspect prediction, and thus the tailor-made summary can help aspect prediction on this token. Another clue is the information of aspect detection history, and it is distilled from the previous aspect predictions so as to leverage the coordinate structure and tagging schema constraints to upgrade the aspect prediction. Experimental results over four benchmark datasets clearly demonstrate that our framework can outperform all state-of-the-art methods.[^1]' author: - \| Xin Li$^1$, Lidong Bing$^2$, Piji Li$^1$, Wai Lam$^1$, Zhimou Yang$^3$\ $^1$Key Laboratory of High Confidence Software Technologies, Ministry of Education (CUHK Sub-Lab),\ Department of Systems Engineering and Engineering Management,\ The Chinese University of Hong Kong, Hong Kong\ $^2$Tencent AI Lab, Shenzhen, China\ $^3$College of Information Science and Engineering, Northeastern University, China\ {lixin, wlam, pjli}@se.cuhk.edu.hk, [email protected], [email protected]\ bibliography: - 'ijcai18.bib' title: 'Aspect Term Extraction with History Attention and Selective Transformation[^2]' --- Introduction ============ Aspect-Based Sentiment Analysis (ABSA) involves detecting opinion targets and locating opinion indicators in sentences in product review texts [@liu2012sentiment]. The first sub-task, called Aspect Term Extraction (ATE), is to identify the phrases targeted by opinion indicators in review sentences. For example, in the sentence “I love the operating system and preloaded software”, the words “operating system” and “preloaded software” should be extracted as aspect terms, and the sentiment on them is conveyed by the opinion word “love”. According to the task definition, for a term/phrase being regarded as an aspect, it should co-occur with some “opinion words” that indicate a sentiment polarity on it [@pontiki-EtAl:2014:SemEval]. Many researchers formulated ATE as a sequence labeling problem or a token-level classification problem. Traditional sequence models such as Conditional Random Fields (CRFs) [@chernyshevich:2014:SemEval; @toh-wang:2014:SemEval; @toh-su:2016:SemEval; @yin2016unsupervised], Long Short-Term Memory Networks (LSTMs) [@liu-joty-meng:2015:EMNLP] and classification models such as Support Vector Machine (SVM) [@manek2016aspect] have been applied to tackle the ATE task, and achieved reasonable performance. One drawback of these existing works is that they do not exploit the fact that, according to the task definition, aspect terms should co-occur with opinion-indicating words. Thus, the above methods tend to output false positives on those frequently used aspect terms in non-opinionated sentences, e.g., the word “restaurant” in “the restaurant was packed at first, so we waited for 20 minutes”, which should not be extracted because the sentence does not convey any opinion on it. There are a few works that consider opinion terms when tackling the ATE task. [@wang-EtAl:2016:EMNLP20164] proposed Recursive Neural Conditional Random Fields (RNCRF) to explicitly extract aspects and opinions in a single framework. Aspect-opinion relation is modeled via joint extraction and dependency-based representation learning. One assumption of RNCRF is that dependency parsing will capture the relation between aspect terms and opinion words in the same sentence so that the joint extraction can benefit. Such assumption is usually valid for simple sentences, but rather fragile for some complicated structures, such as clauses and parenthesis. Moreover, RNCRF suffers from errors of dependency parsing because its network construction hinges on the dependency tree of inputs. CMLA [@wang2017coupled] models aspect-opinion relation without using syntactic information. Instead, it enables the two tasks to share information via attention mechanism. For example, it exploits the global opinion information by directly computing the association score between the aspect prototype and individual opinion hidden representations and then performing weighted aggregation. However, such aggregation may introduce noise. To some extent, this drawback is inherited from the attention mechanism, as also observed in machine translation [@luong-pham-manning:2015:EMNLP] and image captioning [@xu2015show]. To make better use of opinion information to assist aspect term extraction, we distill the opinion information of the whole input sentence into opinion summary[^3], and such distillation is conditioned on a particular current token for aspect prediction. Then, the opinion summary is employed as part of features for the current aspect prediction. Taking the sentence “the restaurant is cute but not upscale” as an example, when our model performs the prediction for the word “restaurant”, it first generates an opinion summary of the entire sentence conditioned on “restaurant”. Due to the strong correlation between “restaurant’ and “upscale” (an opinion word), the opinion summary will convey more information of “upscale” so that it will help predict “restaurant” as an aspect with high probability. Note that the opinion summary is built on the initial opinion features coming from an auxiliary opinion detection task, and such initial features already distinguish opinion words to some extent. Moreover, we propose a novel transformation network that helps strengthen the favorable correlations, e.g. between “restaurant’ and “upscale”, so that the produced opinion summary involves less noise. Besides the opinion summary, another useful clue we explore is the aspect prediction history due to the inspiration of two observations: (1) In sequential labeling, the predictions at the previous time steps are useful clues for reducing the error space of the current prediction. For example, in the B-I-O tagging (refer to Section \[sec:task\]), if the previous prediction is “O”, then the current prediction cannot be “I”; (2) It is observed that some sentences contain multiple aspect terms. For example, “Apple is unmatched in product quality, aesthetics, craftmanship, and customer service” has a coordinate structure of aspects. Under this structure, the previously predicted commonly-used aspect terms (e.g., “product quality”) can guide the model to find the infrequent aspect terms (e.g., “craftmanship”). To capture the above clues, our model distills the information of the previous aspect detection for making a better prediction on the current state. Concretely, we propose a framework for more accurate aspect term extraction by exploiting the opinion summary and the aspect detection history. Firstly, we employ two standard Long-Short Term Memory Networks (LSTMs) for building the initial aspect and opinion representations recording the sequential information. To encode the historical information into the initial aspect representations at each time step, we propose truncated history attention to distill useful features from the most recent aspect predictions and generate the history-aware aspect representations. We also design a selective transformation network to obtain the opinion summary at each time step. Specifically, we apply the aspect information to transform the initial opinion representations and apply attention over the transformed representations to generate the opinion summary. Experimental results show that our framework can outperform state-of-the-art methods. ![image](fig/model_times.pdf){width="80.00000%"} The Proposed Model ================== The ATE Task {#sec:task} ------------ Given a sequence $X = \{x_1,...,x_{T}\}$ of $T$ words, the ATE task can be formulated as a token/word level sequence labeling problem to predict an aspect label sequence $Y = \{y_1,...,y_{T}\}$, where each $y_i$ comes from a finite label set $\mathcal{Y}= \{B, I, O\}$ which describes the possible aspect labels. As shown in the example below:\ $B$, $I$, and $O$ denote beginning of, inside and outside of the aspect span respectively. Note that in commonly-used datasets such as [@pontiki-EtAl:2016:SemEval], the gold standard opinions are usually not annotated. Model Description ----------------- As shown in Figure \[fig:architecture\], our model contains two key components, namely Truncated History-Attention (THA) and Selective Transformation Network (STN), for capturing aspect detection history and opinion summary respectively. THA and STN are built on two LSTMs that generate the initial word representations for the primary ATE task and the auxiliary opinion detection task respectively. THA is designed to integrate the information of aspect detection history into the current aspect feature to generate a new history-aware aspect representation. STN first calculates a new opinion representation conditioned on the current aspect candidate. Then, we employ a bi-linear attention network to calculate the opinion summary as the weighted sum of the new opinion representations, according to their associations with the current aspect representation. Finally, the history-aware aspect representation and the opinion summary are concatenated as features for aspect prediction of the current time step. ### Building Memory As Recurrent Neural Networks can record the sequential information [@graves2012supervised], we employ two vanilla LSTMs to build the initial token-level contextualized representations for sequence labeling of the ATE task and the auxiliary opinion word detection task respectively. [**]{} For simplicity, let $\text{LSTM}^{\mathcal{T}}(x_t)$ denote an LSTM unit where $\mathcal{T} \in \{A, O\}$ is the task indicator. In the following sections, without specification, the symbols with superscript $A$ and $O$ are the notations used in the ATE task and the opinion detection task respectively. We use Bi-Directional LSTM to generate the initial token-level representations $h^{\mathcal{T}}_t \in \mathbb{R}^{2\mathrm{dim}^{\mathcal{T}}_h}$ ($\mathrm{dim}^{\mathcal{T}}_h$ is the dimension of hidden states): $$h^{\mathcal{T}}_t = [\overrightarrow{\text{LSTM}}^{\mathcal{T}}(x_t); \overleftarrow{\text{LSTM}}^{\mathcal{T}}(x_t)], t \in [1, T]. $$ ### Capturing Aspect History In principle, RNN can memorize the entire history of the predictions [@graves2012supervised], but there is no mechanism to exploit the relation between previous predictions and the current prediction. As discussed above, such relation could be useful because of two reasons: (1) reducing the model’s error space in predicting the current label by considering the definition of B-I-O schema, (2) improving the prediction accuracy for multiple aspects in one coordinate structure. We propose a Truncated History-Attention (THA) component (the THA** block in Figure \[fig:architecture\]) to explicitly model the aspect-aspect relation. Specifically, THA caches the most recent $N^{A}$ hidden states. At the current prediction time step $t$, THA calculates the normalized importance score $s^t_i$ of each cached state $h^A_i$ ($i \in [t-N^A, t-1]$) as follows: $$a^t_i = \mathrm{\bf v}^\top \mathrm{tanh}(\mathrm{\bf W}_1 h^A_i + \mathrm{\bf W}_2 h^A_t + \mathrm{\bf W}_3 \tilde{h}^A_i),$$ $$s^t_i = \mathrm{Softmax}(a^t_i).$$ $\tilde{h}^A_i$ denotes the previous history-aware aspect representation (refer to Eq. \[eq:haar\]). $\mathrm{\bf v} \in \mathbb{R}^{2\mathrm{dim}^A_h}$ can be learned during training. $\mathrm{\bf W}_{1,2,3} \in \mathbb{R}^{2\mathrm{dim}^A_h \times 2\mathrm{dim}^A_h}$ are parameters associated with previous aspect representations, current aspect representation and previous history-aware aspect representations respectively. Then, the aspect history $\hat{h}^A_t$ is obtained as follows: $$\hat{h}^A_t =\sum^{t-1}_{i=t-N^A} s^t_i \times \tilde{h}^A_i.$$ To benefit from the previous aspect detection, we consolidate the hidden aspect representation with the distilled aspect history to generate features for the current prediction. Specifically, we adopt a way similar to the residual block [@he2016deep], which is shown to be useful in refining word-level features in Machine Translation [@wu2016google] and Part-Of-Speech tagging [@bjerva2016semantic], to calculate the history-aware aspect representations $\tilde{h}^A_t$ at the time step $t$: $$\label{eq:haar} \tilde{h}^A_t = h^A_t + \mathrm{ReLU}(\hat{h}^A_t),$$ where ReLU is the relu activation function. ### Capturing Opinion Summary Previous works show that modeling aspect-opinion association is helpful to improve the accuracy of ATE, as exemplified in employing attention mechanism for calculating the opinion information [@wang2017coupled; @li-lam:2017:EMNLP2017]. MIN [@li-lam:2017:EMNLP2017] focuses on a few surrounding opinion representations and computes their importance scores according to the proximity and the opinion salience derived from a given opinion lexicon. However, it is unable to capture the long-range association between aspects and opinions. Besides, the association is not strong because only the distance information is modeled. Although CMLA [@wang2017coupled] can exploit global opinion information for aspect extraction, it may suffer from the noise brought in by attention-based feature aggregation. Taking the aspect term “fish” in “Furthermore, while the fish* is unquestionably fresh, rolls tend to be inexplicably bland.”* as an example, it might be enough to tell “fish” is an aspect given the appearance of the strongly related opinion “fresh”. However, CMLA employs conventional attention and does not have a mechanism to suppress the noise caused by other terms such as “rolls”. Dependency parsing seems to be a good solution for finding the most related opinion and indeed it was utilized in [@wang-EtAl:2016:EMNLP20164], but the parser is prone to generating mistakes when processing the informal online reviews, as discussed in [@li-lam:2017:EMNLP2017]. To make use of opinion information and suppress the possible noise, we propose a novel Selective Transformation Network (STN) (the STN block in Figure \[fig:architecture\]), and insert it before attending to global opinion features so that more important features with respect to a given aspect candidate will be highlighted. Specifically, STN first calculates a new opinion representation $\Hat{h}^O_{i,t}$ given the current aspect feature $\tilde{h}^A_t$ as follows: $$\label{eq:new_op_re} \Hat{h}^O_{i,t} = h^O_i + \mathrm{ReLU}(\mathrm{\bf W}_4 \tilde{h}^A_t + \mathrm{\bf W}_5 h^O_i),$$ where $\mathrm{\bf W}_{4}$ and $\mathrm{\bf W}_{5} \in \mathbb{R}^{2\mathrm{dim}^O_h \times 2\mathrm{dim}^O_h}$ are parameters for history-aware aspect representations and opinion representations respectively. They map $\tilde{h}^A_t$ and $h^O_i$ to the same subspace. Here the aspect feature $\tilde{h}^A_t$ acts as a “filter” to keep more important opinion features. Equation \[eq:new\_op\_re\] also introduces a residual block to obtain a better opinion representation $\Hat{h}^O_{i,t}$, which is conditioned on the current aspect feature $\tilde{h}^A_t$. For distilling the global opinion summary, we introduce a bi-linear term to calculate the association score between $\tilde{h}^A_t$ and each $\Hat{h}^O_{i,t}$: $$\label{eq:w_i_t} w_{i,t} = \mathrm{Softmax}(\mathrm{tanh}(\tilde{h}^A_t \mathrm{\bf W}_{bi} \Hat{h}^O_{i,t} + \mathrm{\bf b}_{bi})),$$ where $\mathrm{\bf W}_{bi}$ and $\mathrm{\bf b}_{bi}$ are parameters of the Bi-Linear Attention layer. The improved opinion summary $\Hat{h}^O_t$ at the time $t$ is obtained via the weighted sum of the opinion representations: $$\Hat{h}^O_t = \sum^{T}_{i=1} w_{i,t} \times \Hat{h}^O_{i, t}.$$ Finally, we concatenate the opinion summary $\Hat{h}^O_t$ and the history-aware aspect representation $\tilde{h}^A_t$ and feed it into the top-most fully-connected (FC) layer for aspect prediction: $$f^A_t = [\tilde{h}^A_t : \Hat{h}^O_t],$$ $$P(y^A_{t}\|x_t) = \mathrm{Softmax}(\mathrm{\bf W}^A_{f} f^A_t + \mathrm{\bf b}^A_f).$$ Note that our framework actually performs a multi-task learning, i.e. predicting both aspects and opinions. We regard the initial token-level representations $h^O_i$ as the features for opinion prediction: $$P(y^O_i\|x_i) = \mathrm{Softmax}(\mathrm{\bf W}^O_{f} h^O_i + \mathrm{\bf b}^O_f).$$ $\mathrm{\bf W}^{\mathcal{T}}_{f}$ and $\mathrm{\bf b}^{\mathcal{T}}_f$ are parameters of the FC layers. Joint Training -------------- All the components in the proposed framework are differentiable. Thus, our framework can be efficiently trained with gradient methods. We use the token-level cross-entropy error between the predicted distribution $P(y^{\mathcal{T}}_t\|x_t)$ ($\mathcal{T} \in \{A, O\}$) and the gold distribution $P(y^{\mathcal{T}, g}_t\|x_t)$ as the loss function: $$\ \begin{split} \mathcal{L}_{\mathcal{T}} &= -\frac{1}{~T~}\sum^{T}_{t=1} P(y^{\mathcal{T}, g}_{t}\|x_{t}) \odot \log [P(y^{\mathcal{T}}_{t}\|x_{t})]. \end{split}$$ Then, the losses from both tasks are combined to form the training objective of the entire model: $$\mathcal{J}(\theta)=\mathcal{L}_A+\mathcal{L}_O,$$ where $\mathcal{L}_A$ and $\mathcal{L}_O$ represent the loss functions for aspect and opinion extractions respectively. Experiment ========== Datasets -------- To evaluate the effectiveness of the proposed framework for the ATE task, we conduct experiments over four benchmark datasets from the SemEval ABSA challenge [@pontiki-EtAl:2014:SemEval; @pontiki-EtAl:2015:SemEval; @pontiki-EtAl:2016:SemEval]. Table \[dataset\_statistic\] shows their statistics. $D_1$ (SemEval 2014) contains reviews of the laptop domain and those of $D_2$ (SemEval 2014), $D_3$ (SemEval 2015) and $D_4$ (SemEval 2016) are for the restaurant domain. In these datasets, aspect terms have been labeled by the task organizer. Gold standard annotations for opinion words are not provided. Thus, we choose words with strong subjectivity from MPQA[^4] to provide the distant supervision [@Mintz_distantIE]. To compare with the best SemEval systems and the current state-of-the-art methods, we use the standard train-test split in SemEval challenge as shown in Table \[dataset\_statistic\]. Comparisons ----------- We compare our framework with the following methods: - CRF-1: Conditional Random Fields with basic feature templates[^5]. - CRF-2: Conditional Random Fields with basic feature templates and word embeddings. - Semi-CRF: First-order Semi-Markov Conditional Random Fields [@sarawagi2004semi] and the feature templates in @cuong2014conditional are adopted. - LSTM: Vanilla bi-directional LSTM with pre-trained word embeddings. - IHS\_RD [@chernyshevich:2014:SemEval], DLIREC [@toh-wang:2014:SemEval], EliXa [@sanvicente-saralegi-agerri:2015:SemEval], NLANGP [@toh-su:2016:SemEval]: The winning systems in the ATE subtask in SemEval ABSA challenge [@pontiki-EtAl:2014:SemEval; @pontiki-EtAl:2015:SemEval; @pontiki-EtAl:2016:SemEval]. - WDEmb [@yin2016unsupervised]: Enhanced CRF with word embeddings, dependency path embeddings and linear context embeddings. - MIN [@li-lam:2017:EMNLP2017]: MIN consists of three LSTMs. Two LSTMs are employed to model the memory interactions between ATE and opinion detection. The last one is a vanilla LSTM used to predict the subjectivity of the sentence as additional guidance. - RNCRF [@wang-EtAl:2016:EMNLP20164]: CRF with high-level representations learned from Dependency Tree based Recursive Neural Network. - CMLA [@wang2017coupled]: CMLA is a multi-layer architecture where each layer consists of two coupled GRUs to model the relation between aspect terms and opinion words. To clarify, our framework aims at extracting aspect terms where the opinion information is employed as auxiliary, while RNCRF and CMLA perform joint extraction of aspects and opinions. Nevertheless, the comparison between our framework and RNCRF/CMLA is still fair, because we do not use manually annotated opinions as used by RNCRF and CMLA, instead, we employ an existing opinion lexicon to provide weak opinion supervision. Settings -------- We pre-processed each dataset by lowercasing all words and replace all punctuations with `PUNCT`. We use pre-trained GloVe 840B vectors[^6] [@pennington2014glove] to initialize the word embeddings and the dimension (i.e., $\mathrm{dim}_w$) is 300. For out-of-vocabulary words, we randomly sample their embeddings from the uniform distribution $\mathcal{U}(-0.25, 0.25)$ as done in [@kim:2014:EMNLP2014]. All of the weight matrices except those in LSTMs are initialized from the uniform distribution $\mathcal{U}(-0.2, 0.2)$. For the initialization of the matrices in LSTMs, we adopt Glorot Uniform strategy [@glorot2010understanding]. Besides, all biases are initialized as 0’s. The model is trained with SGD. We apply dropout over the ultimate aspect/opinion features and the input word embeddings of LSTMs. The dropout rates are empirically set as 0.5. With 5-fold cross-validation on the training data of $D_2$, other hyper-parameters are set as follows: $dim^A_h=100$, $dim^O_h=30$; the number of cached historical aspect representations $N^A$ is 5; the learning rate of SGD is 0.07. Main Results ------------ As shown in Table \[tab:main\_results\], the proposed framework consistently obtains the best scores on all of the four datasets. Compared with the winning systems of SemEval ABSA, our framework achieves 5.0%, 1.6%, 1.4%, 1.3% absolute gains on $D_1$, $D_2$, $D_3$ and $D_4$ respectively. Our framework can outperform RNCRF, a state-of-the-art model based on dependency parsing, on all datasets. We also notice that RNCRF does not perform well on $D_3$ and $D_4$ (3.7% and 3.9% inferior than ours). We find that $D_3$ and $D_4$ contain many informal reviews, thus RNCRF’s performance degradation is probably due to the errors from the dependency parser when processing such informal texts. CMLA and MIN do not rely on dependency parsing, instead, they employ attention mechanism to distill opinion information to help aspect extraction. Our framework consistently performs better than them. The gains presumably come from two perspectives: (1) In our model, the opinion summary is exploited after performing the selective transformation conditioned on the current aspect features, thus the summary can to some extent avoid the noise due to directly applying conventional attention. (2) Our model can discover some uncommon aspects under the guidance of some commonly-used aspects in coordinate structures by the history attention. CRF with basic feature template is not strong, therefore, we add CRF-2 as another baseline. As shown in Table \[tab:main\_results\], CRF-2 with word embeddings achieves much better results than CRF-1 on all datasets. WDEmb, which is also an enhanced CRF-based method using additional dependency context embeddings, obtains superior performances than . Therefore, the above comparison shows that word embeddings are useful and the embeddings incorporating structure information can further improve the performance. ![image](fig/3-1_times.pdf){height="27mm" width="100.00000%"} \[fig:1a\] ![image](fig/3-2_times.pdf){height="27mm" width="100.00000%"} \[fig:1b\] ![image](fig/2-1_times.pdf){height="25mm" width="100.00000%"} \[fig:1c\] ![image](fig/2-2_times.pdf){height="25mm" width="100.00000%"} \[fig:1d\] Ablation Study -------------- To further investigate the efficacy of the key components in our framework, namely, THA and STN, we perform ablation study as shown in the second block of Table \[tab:main\_results\]. The results show that each of THA and STN is helpful for improving the performance, and the contribution of STN is slightly larger than THA. “OURS w/o THA & STN” only keeps the basic bi-linear attention. Although it performs not bad, it is still less competitive compared with the strongest baseline (i.e., CMLA), suggesting that only using attention mechanism to distill opinion summary is not enough. After inserting the STN component before the bi-linear attention, i.e. “OURS w/o THA”, we get about 1% absolute gains on each dataset, and then the performance is comparable to CMLA. By adding THA, i.e. “OURS”, the performance is further improved, and all state-of-the-art methods are surpassed. Attention Visualization and Case Study -------------------------------------- In Figure \[fig:my\_label\], we visualize the opinion attention scores of the words in two example sentences with the candidate aspects “maitre-D” and “bathroom”. The scores in Figures \[fig:1a\] and \[fig:1c\] show that our full model captures the related opinion words very accurately with significantly larger scores, i.e. “incredibly”, “unwelcoming” and “arrogant” for “maitre-D”, and “unfriendly” and “filthy” for “bathroom”. “OURS w/o STN” directly applies attention over the opinion hidden states $h_i^O$’s, similar to what CMLA does. As shown in Figure \[fig:1b\], it captures some unrelated opinion words (e.g. “fine”) and even some non-opinionated words. As a result, it brings in some noise into the global opinion summary, and consequently the final prediction accuracy will be affected. This example demonstrates that the proposed STN works pretty well to help attend to more related opinion words given a particular aspect. Some predictions of our model and those of LSTM and OURS w/o THA & STN are given in Table \[tab:predictions\]. The models incorporating attention-based opinion summary (i.e., OURS and OURS w/o THA & STN) can better determine if the commonly-used nouns are aspect terms or not (e.g. “device” in the first input), since they make decisions based on the global opinion information. Besides, they are able to extract some infrequent or even misspelled aspect terms (e.g. “survice” in the second input) based on the indicative clues provided by opinion words. For the last three cases, having aspects in coordinate structures (i.e. the third and the fourth) or long aspects (i.e. the fifth), our model can give precise predictions owing to the previous detection clues captured by THA. Without using these clues, the baseline models fail. Related Work ============ Some initial works [@hu2004miningA] developed a bootstrapping framework for tackling Aspect Term Extraction (ATE) based on the observation that opinion words are usually located around the aspects. [@popescu-etzioni:2005:HLTEMNLP] and [@qiu2011opinion] performed co-extraction of aspect terms and opinion words based on sophisticated syntactic patterns. However, relying on syntactic patterns suffers from parsing errors when processing informal online reviews. To avoid this drawback, [@liu-xu-zhao:2012:EMNLP-CoNLL; @liu2013opinion] employed word-based translation models. Specifically, these models formulated the ATE task as a monolingual word alignment process and aspect-opinion relation is captured by alignment links rather than word dependencies. The ATE task can also be formulated as a token-level sequence labeling problem. The winning systems [@chernyshevich:2014:SemEval; @sanvicente-saralegi-agerri:2015:SemEval; @toh-su:2016:SemEval] of SemEval ABSA challenges employed traditional sequence models, such as Conditional Random Fields (CRFs) and Maximum Entropy (ME), to detect aspects. Besides heavy feature engineering, they also ignored the consideration of opinions. Recently, neural network based models, such as LSTM-based [@liu-joty-meng:2015:EMNLP] and CNN-based [@poria2016aspect] methods, become the mainstream approach. Later on, some neural models jointly extracting aspect and opinion were proposed. [@wang-EtAl:2016:EMNLP20164] performs the two task in a single Tree-Based Recursive Neural Network. Their network structure depends on dependency parsing, which is prone to error on informal reviews. CMLA [@wang2017coupled] consists of multiple attention layers on top of standard GRUs to extract the aspects and opinion words. Similarly, MIN [@li-lam:2017:EMNLP2017] employs multiple LSTMs to interactively perform aspect term extraction and opinion word extraction in a multi-task learning framework. Our framework is different from them in two perspectives: (1) It filters the opinion summary by incorporating the aspect features at each time step into the original opinion representations; (2) It exploits history information of aspect detection to capture the coordinate structures and previous aspect features. Concluding Discussions ====================== For more accurate aspect term extraction, we explored two important types of information, namely aspect detection history, and opinion summary. We design two components, i.e. truncated history attention, and selective transformation network. Experimental results show that our model dominates those joint extraction works such as RNCRF and CMLA on the performance of ATE. It suggests that the joint extraction sacrifices the accuracy of aspect prediction, although the ground-truth opinion words were annotated by these authors. Moreover, one should notice that those joint extraction methods do not care about the correspondence between the extracted aspect terms and opinion words. Therefore, the necessity of such joint extraction should be obelized, given the experimental findings in this paper. [^1]: Codes and datasets are available at <https://github.com/lixin4ever/HAST>. [^2]: The work was done when Xin Li was an intern at Tencent AI Lab. The project is substantially supported by a grant from the Research Grant Council of the Hong Kong Special Administrative Region, China (Project Code: 14203414). [^3]: Technically, opinion summary is the linear combination of the opinion representations generated from LSTM. [^4]: http://mpqa.cs.pitt.edu/ [^5]: http://sklearn-crfsuite.readthedocs.io/en/latest/ [^6]: https://nlp.stanford.edu/projects/glove/	ArXiv
--- abstract: 'NO$\nu$A is an accelerator-based neutrino oscillation experiment which has a great potential to measure the last unknown mixing angle $\theta_{13}$, the neutrino mass hierarchy, and the CP-violation phase in lepton sector with 1) 700 kW beam, 2) 14 mrad off the beam axis, 3) 810 km long baseline. The Near Detector on the Surface is fully functioning and taking both NuMI and Booster beam data. The far detector building achieved beneficial occupancy on April 13. This proceeding will focus on the DAQ software system.' author: - 'X. C. Tian, on behalf of the NO$\nu$A Collaboration' title: 'NO$\nu$A Data Acquisition Software System' --- Introduction ============ The next generation long-baseline neutrino experiments [@NOvA; @T2K; @LBNE] aim to measure the third mixing angle $\theta_{13}$, determine whether CP is violated in the lepton sector, and resolve the neutrino mass hierarchy. The NuMI Off-axis electron-neutrino ($\nu_e$) Appearance (NO$\nu$A) experiment is the flagship experiment of the US domestic particle physics program which has the potential to address most of the fundamental questions in neutrino physics raised by the Particle Physics Project Prioritization Panel (P5). NO$\nu$A has two functionally identical detectors (Fig. \[detector\]), a 222 ton near detector located underground at Fermilab and a 14 kiloton far detector located in Ash River, Minnesota with a baseline of 810 km. The detectors are composed of extruded PVC cells loaded with titanium dioxide to enhance reflectivity. There are 16,416 and 356,352 cells for the near and far detector, respectively. Each cell has a size of 3.93 cm transverse to the beam direction and 6.12 cm along the direction filled with liquid scintillator (mineral oil plus 5% pseudocumene). The corresponding radiation length is 0.15 $X_0$ and the Moliere radius is 10 cm, ideal for the identification of electron-type neutrino events. The “Neutrinos at the Main Injector” (NuMI) will provide a 14 mrad off-axis neutrino beam to reduce neutral current backgrounds and which peaks at 2 GeV, corresponding to the first oscillation maximum for this detector distance. The accelerator and NuMI upgrades will double the protons per year delivered to the detector which is $6\times 10^{20}$ protons per year. For further details on the current status of NO$\nu$A experiment, please see Ref. [@Gavin]. ![The NO$\nu$A detectors. The far (near) detector has 29 (6) blocks and each block is made of 32 scintillator PVC planes, and 928 (192) planes in total. The near detector also has a muon catcher which is composed of 13 scintillator PVC planes and 10 steel planes.[]{data-label="detector"}](detector){width="100.00000%"} Charged particles from neutrino interactions or cosmic ray muons will emit scintillation light in the scintillator. The scintillation light is collected by a loop of wavelength shifting fibers (WLS) and a 32-pixel Avalanche Photo-diode (APD) attached to the fibers converts the light pulse into electrical signals. The Data Acquisition (DAQ) System as shown in Fig. \[daq-all\] will concentrate the data from those APDs into a single stream that can be analyzed and archived. The DAQ can buffer the data and wait for a trigger decision that the data should be recorded or rejected. Online trigger processors will be used to analyze the data stream to correlate data with similar time stamps and to look for clusters of hits indicating an interesting event. Additional functionality for dealing with flow control, monitoring, system operations and alarms is also included [@NOvA-TDR]. The event types that NO$\nu$A DAQ will record include beam neutrino events, cosmic ray muons, and other physics events (supernova neutrinos, high energy neutrinos, [etc.]{}) Every 2.2 s (will be reduced to 1.3 s with the accelerator and NuMI upgrades), a 10 $\mu$s beam spill will be generated and time stamped by a GPS based timing system at Fermilab. All hits that occur in a 30 $\mu$s window centered on the 10 $\mu$s spill are recorded for further processing. The event rates are 30 neutrino events per spill for the near detector and 1,400 $\nu_e$ beam events per year for the far detector. The randomly selected cosmic ray muons used for calibration and monitoring are taken to give 100 times the number of beam neutrino events. The cosmic ray muon rate is 50 Hz (200 kHz) for the near (far) detector. Other interesting physics process such as a supernova explosion at 10 kpc will result in thousands of neutrinos within 10 seconds in the far detector. For the near detector, the data rates are 75 TB per year through DAQ system and 1 TB per year written to disk. For the far detector, the data rates are 12,000 TB per year through DAQ sytem and 25 TB per year written to disk. ![An schematic overview of the NO$\nu$A DAQ system. The data stream is from left to right.[]{data-label="daq-all"}](daq.png){width="\textwidth"} NO$\nu$A Data Acquisition System ================================ The primary task for the DAQ is to record the data from APDs for further processing. The data flows through Front End Boards (FEBs), Data Concentrate Modules (DCMs), Buffer Nodes (BNs), DataLogger (DL) and then archived on disk or tape as shown in Fig. \[daq-all\]. Each APD is digitized by a FEB continuously without dead time. The data from a group of FEBs up to 64 are consolidated by the DCM into 5 ms time slices which are routed to downstream Buffer Nodes. The data is buffered in the Buffer Nodes for a minimum of 20 s waiting for the spill trigger. A spill signal is required to arrive within the buffering time so that the spill time can be correlated with the time-stamped data to determine if the hits occurred in or out spill. The triggered data from Buffer Nodes will be merged to form an event in the DataLogger and the event will be written to file for storage or shared memory for monitoring. The power distribution system (PDS) provides power to FEBs, APDs, ThermoElectric Coolers (TECs) [^1], DCMs and Timing Distribution Units (TDUs). The Run Control provides the overall control of the DAQ system. The following sections will describe the key subsystems of the NO$\nu$A DAQ system. Front End Boards (FEBs) ----------------------- The front end electronics (Fig. \[feb\]) is responsible for amplifying and integrating the signals from the APD arrays, determining the amplitude of the signals and their arrival time and presenting that information to the DAQ. The FEBs are operated in trigger-less, continuous readout mode with no dead time, and the data is zero suppressed based on Digital Signal Processing (DSP) algorithms. Data above a pre-programmed threshold is settable at the channel level to allow different thresholds to be set depending on the particular characteristics of a given channel. Data above that threshold will be time-stamped and compared to a NuMI timing signal in the DAQ system to determine if the event was in or out of spill. Major components of the FEB are the carrier board connector location at the left, which brings the APD signals to the NO$\nu$A ASIC, which performs integration, shaping, and multiplexing. The chip immediately to the right is the ADC to digitize the signals, and FPGA for control, signal processing, and communication. The ASIC is customized to maximize the sensitivity of the detector to small signals from long fibers in the far detector. The average photoelectrons (PEs) yield at the far end of an extrusion module is 30, and the noise is 4 PEs. The FPGA on the FEB uses a Digital Signal Processing algorithm to extract the time and amplitude of signals from the APD. Each FEB reads out 32 channels corresponding to 32 pixels of one APD. Higher detector activity during beam spill at the near detector requires higher time resolution, therefore FEBs sample APDs pixels at 8 MHz at the near detector, 2 MHz at the far detector. The FEBs are capable of limited waveform digitization and waveform readout. ![Schematic of the APD module and the front-end electronics board showing the major components. []{data-label="feb"}](feb-apd.png){width="100.00000%"} Data Concentrator Modules (DCMs) -------------------------------- The DCM (Fig. \[dcm\]) is a custom component of the DAQ. Each DCM is responsible for consolidating the data into 5 ms time slices received from up to 64 FEBs to internal data buffers and for transferring the data out to the Event Builder Buffer Farm nodes over Gigabit Ethernet. The DCMs also pass timing and control information from the timing system to the FEBs. The DCM mainly consists of a FPGA, an embedded Power PC processor, and connectors as shown in Fig. \[dcm\]. Data from FEB consists of a header and hit information for a given timeslice. The mid-sized FPGA on the DCM concatenates and combines the hit information for all 64 FEBs to time slices on the order of 50 $\mu$s (MicroSlices). The data is then read from the FPGA by the embedded Power PC processor running embedded Linux. The DCM application software running on the processor has the responsibility of concatenating and packaging the data into 5 ms time slices (MilliSlice) received from the DCM FPGA and distributing the resulting data packets to Event Builder Buffer Nodes over a Gigabit Ethernet. The application software also handles communication with Run Control and the DAQ Monitor System, provides the high-level interface for programming and configuring the DCM and FEB hardware components, and provides support for operating in simulated data input mode. Data is transferred from the DCMs to different Buffer Nodes in a round robin fashion with each DCM starting with a different Buffer Node. Timing information from the timing system is used to divide time into slices such that data packets are transmitted to a different Buffer Node in each slice. In this way, DCMs stay synchronized in their rotation, and no two DCMs ever transmit to the same Buffer Node at once. Buffer Nodes are receiving data from only one DCM at a time, and all Buffer Nodes are receiving data on all time slices. The DCM also connects to the timing system which passes a single common timing clock to all FEBs to maintain a sync to common time better than $\pm1$ 64MHz clock tick detector wide. ![The block diagram of the Data Concentrate Module. The DCM FPGA concatenates and combines the data received from up to 64 FEBs on dedicated serial links to time slices on the order of 50 $\mu$s (MicroSlices). The DCM CPU runs kernel device drivers responsible for the programming of the DCM and FEB FPGAs, readout of the MicroSlices and FEB status buffers prepared by the DCM FPGA. []{data-label="dcm"}](dcm.pdf){width="90.00000%"} Buffer Nodes (BNs) ------------------ The Buffer Nodes are used to buffer the raw hit data from the entire detector for a minimum of 20 s until a trigger is received from the Global Trigger system (Fig. \[bn\]). The trigger is a time window and the Buffer Nodes use the time window to search for all data within that frame of time and route it to DataLogger process. All data for one time frame will be buffered in the same Buffer Node. Each Buffer Node is independent and contains distinct detector data, such that when a given time window is requested, all the Buffer Nodes must perform a search on the available data and at least one should return the desired data The Buffer Node will need to aggregate the data from each of the DCMs for the time frame and insure data from all DCMs is seen. This is a form of event building where data for a large time frame is built as a single event. For any frame seen by a Buffer Node, data must be received from each DCM. This provides a system integrity check and also allows more performant searching when a trigger is received. While all the data from a given time frame will be contained in one farm node, an actual 30 $\mu$s trigger, which defines an event, may span these large time frame boundaries and thus be split across two farm nodes. Each farm node will need to determine if it has any data for the trigger window in its memory buffer and send it to the logging process. ![The diagram of the Buffer Node. []{data-label="bn"}](bn){width="100.00000%" height="4.5cm"} ![The internal structure of the Buffer Node. Data is distributed from the DCMs to the Buffer Nodes in a round-robin fashion with traffic shaping.[]{data-label="dcm-bn"}](dcm-bn){width="100.00000%" height="4.5cm"} DataLogger (DL) --------------- The DataLogger (DL) is responsible for merging the triggered data blocks from Buffer Nodes to form events and writing them to local files or online monitoring systems through a Data Dispatcher process (Fig. \[dl\]). It receives input from the Global Trigger and the Buffer Nodes. It writes completed events to run/subrun output disk files and continuously populates a shared memory segment with events for online monitoring. The incomplete events called DataAtoms consist of TriggerBlock coming from Global Trigger and/or DataBlocks coming from Buffer Nodes. Each DataAtom, whether it is a TriggerBlock or DataBlocks or a combination of both has an associated Trigger Number which is used to build events. The complete events are written to disk - data streams are formed corresponding to each trigger type for the run plus a stream for all events. Failure to build an event which was incomplete is an indication of a DAQ problem and generates a warning message sent to the Run Control, with the resulting incomplete event written to disk. These files consist of various headers, data blocks, tails, and checksum words. Events are also written to a shared memory segment which is formed at run setup on the DataLogger node. This memory segment is overwritten at a constant rate with new events. An Event Viewer is available to view the data structure of these events and an online Event Display can also be run on them. ![The diagram of DataLogger. []{data-label="dl"}](dl){width="100.00000%" height="4.5cm"} Trigger ------- The Global Trigger system is responsible for receiving a beam spill signal, or other triggering conditions, such as the periodic calibration “Pulser” trigger and instructs the NO$\nu$A DAQ’s data buffer system to save a set of data, rather than the default buffer action which is to forget the oldest data in order to make room for current data. Among the variety of triggers, the most important is the beam spill trigger originating in the NuMI beam-line, to allow the beam neutrino induced interactions to be recorded. The beam spill signal will be generated at Fermilab in response to the signal firing the kicker magnet. The actual time will be logged and transmitted to the far detector Global Trigger via the Internet. Upon receipt of the beam spill time the Global Trigger system will then generate the time windows based on the signal and send that signal to the buffering processes on each of the buffer farm nodes. In the absence of actual spill triggers the system will generate random triggers to allow for tracking of calibration and monitoring of the detector. The calibration source for NO$\nu$A is the copious rate of cosmic rays hitting this large detector on the surface. While the data rate of these cosmic rays is too great to consider saving all of them, a subset is needed to calibrate, measure the background, and monitor the detector. Thus, a Calibration Trigger will be issued at configurable intervals to save blocks of time as determined by operational requirements to contain a useful sample of cosmic ray muons. Additionally, Data Driven trigger processing information will be sent from the Buffer Nodes to the Global Trigger, which will use that information to decide if the appropriate Data Driven trigger will be issued. Timing System ------------- The timing system consists one Master Timing Distribution Unit (MTDU) and serval Slave Timing Distribution Units (STDUs) which is used to to synchronize both near and far detector timing systems to a known time standard. MTDU is cabled to their corresponding STDUs in a daisy-chained fashion, and similarly each STDU will be cabled to the 12 DCMs on its double block in a daisy-chain configuration. MTDU derives clock from Global Positioning System (GPS) and distributes to the first STDU in chain. All the FEBs/DCMs are synchronized to this common high precision 16 MHz clock reference which is distributed by STDU. MTDU also generates command and sync signals as directed by Run Control, and synchronizes all detector DAQ components on both near and far detectors. The Near Detector MTDU differs from the Far Detector MTDU in that it also connects to the Main Injector timing system, and synchronizes itself to the Main Injector. The timing system is self-compensating for cable/transmission propagation delays between units (timing units and DCMs). Message Systems --------------- The message systems include message passing and message facility. The message passing system which is capable of high message bandwidth will be used to transport control and monitoring messages between processes in the DAQ. It will support the sending of messages to individual processes as well as groups of processes, and it will provide support for reply messages that are generated in response to request messages. General message passing is handled using the FNAL “Responsive Message System (RMS)” which uses OpenSplice DDS for low level message transmission. The NO$\nu$A specific layers provide ease-of-use. The message facility system will provide the infrastructure for all of the distributed processes in the DAQ to report status messages of various severities in a consistent manner to a central location, and it will provide the tools for displaying and archiving the messages for later analysis. The message facility system should act as a gate keeper to the Run Control clients with respect to error messages. It is the responsibility of this part of the system to determine which messages need to be seen by Run Control. The archiving of status messages for later analysis will also be valuable for diagnosing problems after they have occurred. Run Control ----------- The Run Control system provides a graphical interface for operators to control data taking, control logic to carry out the operators’ requests, and monitoring functionality to automatically react to exceptional conditions. It is written in C++ using QT [@QT] technology and client/server model. All DAQ components implement a well defined state model and under the command of Run Control, make transitions between states. The Run Control application must communicate with other applications, including the run history and configuration databases, trigger, message systems, and the detector control system. For debugging and commissioning purposes of the large detector of essentially one type of detector sub-system, the Run Control will support partitioning of the resources. There will need to be one central resource manager that tracks assignments of DCMs and Buffer Nodes to partitions. The resource manager will provide the Run Control client the necessary information for using only the hardware and applications reserved for its partition. Run Control will interact with the configuration database to provide appropriate information to processes on how to configure themselves, download or receive calibration parameters, [etc.]{} In addition, it is responsible for saving all relevant information on the configuration of the run, including partition information, to a run history database for offline access. Any changes in the configuration would require a new run to be started. Monitoring Systems ------------------ The DAQ is monitored at different levels by five different applications. The real time monitors include: 1) DAQ Monitor monitors the DAQ health and performance using Ganglia [@Ganglia]; 2) Memory Viewer displays the bytes of raw data; 3) Online Event Display displays reconstructed events; 4) Online Monitor monitors the run metrics. The offline monitor is data check which checks the detector performance by looking at metrics of multi-run periods. Please refer to [@Susan] for details. DAQ Performance on Near Detector On the Surface (NDOS) ====================================================== The Near Detector On the Surface (NDOS) DAQ system has been up and running since fall of 2010. The first neutrino candidate was observed on Dec.15, 2010. The NDOS has recorded 8.4$\times10^{18}$ Protons On Target (POTs) from NuMI neutrino beam, and 5.6$\times10^{19}$ POTs from NuMI anti-neutrino beam. NDOS also accumulated 3.0$\times10^{19}$ POTs from Booster neutrino beam. Please refer to [@Minerba] for details. Many DAQ performance gains and fixes of bugs have resulted from NDOS commissioning and running, for example, the DCM data through-put has gained a factor of two as a result of optimizing software and network (Fig. \[dcmgain\] left). The number of active channels (Fig. \[dcmgain\] right), live time and quality of data continue to improve over time. ![The Data Concentrate Module gain as a function of time (left) and the number of active channels as a function of time (right).[]{data-label="dcmgain"}](daqgain "fig:"){width="48.00000%"} ![The Data Concentrate Module gain as a function of time (left) and the number of active channels as a function of time (right).[]{data-label="dcmgain"}](channel "fig:"){width="48.00000%"} Conclusion ========== Much of the DAQ system has been designed, implemented and deployed to the NO$\nu$A prototype detector, Near Detector on the Surface (NDOS) which has been up and running from the Fall of 2010. The system is robust enough that can run the readout for the NDOS at near 100% live readout to disk. Many performance gains and fixes of bugs have been made as a result of commissioning the Prototype Detector. The far detector will start construction starting from Jan. 2012 and expect to readout the first di-block early next year. The far detector DAQ system is under development and the system is ready for the rate conditions that will see at the far detector early next year. The author would like to thank the NO$\nu$A Collaboration for the help on preparing the presentation and proceeding. [9]{} D. S. Ayres [et al.]{} (NO$\nu$A Collaboration), arXiv:hep-ex/0503053. Y. Itow [et al.]{} (T2K Collaboration), arXiv:hep-ex/0106019. M. C. Sanchez [et al.]{} (LBNE DUSEL Collaboration), [AIP Conf. Proc.]{} [1222]{}, 479 (2010). G. S. Davies (NO$\nu$A Collaboration), DPF proceeding. “Data Acquisition System”, Chapter 15, NOvA Technical Design Report. http://qt.nokia.com/ http://ganglia.sourceforge.net/ S. Lein (NO$\nu$A Collaboration), DPF proceeding. M. Betancourt (NO$\nu$A Collaboration), DPF proceeding. [^1]: TEC controller cools down the APDs to -15$^\circ$C to keep the noise contribution from photoconversion regions small.	ArXiv
--- abstract: 'A recent scanning tunneling microscopy (STM) experiment reports the observation of charge density wave (CDW) with period of approximately 8a in the halo region surrounding the vortex core, in striking contrast to the approximately period 4a CDW that are commonly observed in the cuprates. Inspired by this work, we study a model where a bi-directional pair density wave (PDW) with period 8 is at play. This further divides into two classes, (1) where the PDW is a competing state of the d wave superconductor and can exist only near the vortex core where the d wave order is suppressed, and (2) where the PDW is the primary order, the so called “mother state” that persists with strong phase fluctuations to high temperature and high magnetic field and lies behind the pseudogap phenomenology. We study the charge density wave structures near the vortex core in these models. We emphasize the importance of the phase winding of the d-wave order parameter. The PDW can be pinned by the vortex core due to this winding and become static. Furthermore, the period 8 CDW inherits the properties of this winding, which gives rise to a special feature of the Fourier transform peak, namely, it is split in certain directions. There are also a line of zeros in the inverse Fourier transform of filtered data. We propose that these are key experimental signatures that can distinguish between the PDW-driven scenario from the more mundane option that the period 8 CDW is primary. We discuss the pro’s and con’s of the options considered above. Finally we attempt to place the STM experiment in the broader context of pseudogap physics of underdoped cuprates and relate this observation to the unusual properties of X-ray scattering data on CDW carried out to very high magnetic field.' author: - Zhehao Dai - 'Ya-Hui Zhang' - 'T. Senthil' - 'Patrick A. Lee' bibliography: - 'reference.bib' title: 'Pair density wave, charge density wave and vortex in high Tc cuprates' --- [^1] [^2] Introduction ============ The pseudogap phase has long been considered a central puzzle in the study of the cuprate high temperature superconductors[@keimer2015quantum]. After decades of studies, the phenomenology is well established. The pseudogap temperature is now demonstrated to signal a genuine phase transition: some form of broken crystalline symmetry has been shown to occur from ultrasound attenuation [@shekhter2013bounding], second harmonic generation[@HsieNaturePhysicshzhao2017global], and the anisotropy of the spin susceptibility[@MatsudaNaturePhysicssato2017thermodynamic; @Matsuda2unpublished]. Just below this temperature, neutron scattering has detected the onset of intra-cell magnetic moments[@bourges2011novel] which have been interpreted in terms of orbital loop currents[@varma2006theory], even though this experimental finding has recently been challenged, at least in the case of YBCO[@HaydenPRBcroft2017noevidence; @bourges2017comment]. At lower temperatures, short range order charge density wave (CDW) order emerges, often, but not always, suppressed by the onset of superconductivity[@blackburn2013x; @ghiringhelli2012long; @BlancoPhysRevB.90.054513; @GrevenPRB96tabis2017synchrotron]. In high magnetic field the CDW order in YBCO dramatically increases its range, as seen in NMR[@JulienNature477191wu2011magnetic; @Julien2arXivzhou2017spin; @wu2013emergence]. X ray scattering reveals that it is unidirectional and becomes stacked in phase between layers[@changNatureComm72016magnetic; @ZX1science350949gerber2015three; @ZX2PNAS11314647jang2016ideal]. There seems to be two distinct forms of CDW co-existing, one long ranged ordered and uni-directional, while the other is short ranged and exists in both directions. It is quite mysterious why they have the same incommensurate period. At very low temperature quantum oscillations have been observed which have been interpreted as the emergence of small electron-like pockets (for a review, see Ref. ). Of course, the appearance of a pseudogap in the single particle spectrum near the anti-node which persists to very high temperature is what gave this phenomenon its name in the first place. The phenomenology is so rich and complicated that it seems to defy any unifying theme, leading to notions such as “competing orders” or “intertwined order”. Adding to this complexity, a recent STM experiment detected CDW with period 8a co-existing with the previously observed period 4a CDW in the “halo” surrounding the vortex core[@EdkinsSeamusRecentSTM]. In this broader context, a key question we would like to address is this. Does this observation simply increase the complexity of the problem or is it the breakthrough that provides the key to crack open the pseudo-gap problem? It should be noted that the double period CDW is expected in a scenario based on the existence of pair density wave (PDW) when it co-exists with the d-wave superconducting order. In this paper we review different scenarios that can lead to the double period CDW and discuss the pro’s and con’s of each of them. Most importantly, we propose a refinement of the STM experiment which we believe can unambiguously distinguish between different scenarios, including different versions of PDWs, like Canted PDW and Uni-directional PDW. A PDW is a superconductor with a pairing order parameter which is periodic in space. It was first introduced by Larkin and Ovchinnikov[@larkin1965inhomogeneous] and by Fulde and Ferrell[@fulde1964superconductivity] as a way to overcome the Pauli limiting effect of a magnetic field. The notion of PDW in the context of the cuprates has a long history. Himeda, Kato and Ogata [@himeda2002stripe] found in 2002 by projected Monte Carlo studies that the PDW is the preferred ground state in the presence of stripe order. Starting from the standard stripe picture [@tranquada1995jm]of a period 8 spin density wave (SDW) and a period 4 CDW, they found that the d wave superconductor is more stable if the sign of the order parameter is reversed at the hole poor region of the CDW, leading to a period 8 PDW. We shall refer to this state as the stripe-PDW. They proposed that if the stripe-PDW is stacked perpendicular to each other from one layer to the next, the resulting state has drastically reduced Josephson coupling and may explained the disappearance of the Josephson plasma edge observed in Nd doped LaSr2CuO4 (LSCO)[@tajimaPRL862001c]. Strong anisotropy in the transport properties was discovered in the LBCO $\text{La}_{2-x}\text{Ba}_x\text{CuO}_4$ system[@PhysRevLett.99.067001] and since that time the theory of layer de-coupled PDW and related phases has been greatly advanced.[@PhysRevLett.99.127003; @PhysRevB.79.064515] For a review, see Ref. . The next development is the introduction of a Landau theory description. [@PhysRevLett.99.127003; @PhysRevB.79.064515; @agterberg2008dislocations; @berg1NatPhys2009charge] Agterberg and Tsunetsugu[@agterberg2008dislocations] described the coupling of PDW with various subsidiary orders such as CDW and magnetization waves. By examining the interplay between the PDW vortex and the dislocation in the CDW, they showed that it is possible to suppress the PDW order by phase fluctuations, while the subsidiary CDW order remains long ranged. Berg, Fradkin and Kivelson[@berg1NatPhys2009charge] constructed a phase diagram using renormalization group arguments which include regions in parameter space where the primary PDW order is destroyed while CDW and a novel charge 4e superconductor survive. Berg et al [@berg22009NTPhysstriped]suggested that the stripe PDW may have a more general applicability than the low temperature behaviors in the LBCO family, ie, it may be behind the pseudo-gap phase. Part of their argument is based on the spectral property of such a uni-directional PDW. We comment that while this state produces what looks like a Fermi arc, the gap is actually small near the antinode in the direction perpendicular to the stripe orientation[@baruch2008PRB77spectral; @berg22009NTPhysstriped]. This kind of two gap structure has difficulties with STM and ARPES data. Stimulated by a detailed angle resolved photo-emission (ARPES) study of the single layer cuprate Bi2201[@heSci3312011single], one of us [@lee2014amperean] proposed that the unusual features of the spectra can be explained by postulating a bi-directional PDW state as the underlying state of the pseudogap. The pairing is produced by singlet pairing of electrons with momenta $K_i+p$ and $K_i-p$ where the $K_i$’s are located at or near the Fermi surface at the anti-nodal points. (see fig 1a) This gives rise to a bi-directional PDW. The pair carries momenta $P_1$ and ${-P_1}$ which equal twice the momentum K near the $(\pi, 0)$ antinode and are along the x-axis. There is a similar pair $P_2$ and $-P_2$ which are along the y-axis. There are 4 order parameters: $\D_{P_1}$, $\D_{-P_1}$, $\D_{P_2}$ and $\D_{-P_2}$. While Lee proposed using the idea of Amperean pairing[@PhysRevLett.98.067006] as the microscopic origin of the PDW, most of the paper was phenomenological, and explored the consequences of an assumed PDW. As such many of the conclusions are quite general. Nevertheless we would like to emphasize that the motivation for introducing the bi-directional PDW is fundamentally different from that for the uni-directional PDW[@berg22009NTPhysstriped; @fradkin2015colloquium], which is rooted in the phenomena observed in the LSCO/LBCO family at relatively low temperatures. Our view is that the recently discovered CDW which survives up to 150K are distinct from the stripe physics associated with LSCO/LBCO. The wave-vector decreases with increasing doping, whereas the stripe wave vector increases linearly up to about 0.125 doping and saturate, following the Yamada plot[@yamada1998PRB57doping]. For YBCO the period is incommensurate and close to 3, very different from the period 4 CDW associated with 1/8 doping in LSCO. Finally there is no sign of the SDW that is “intertwined” with the stripes. As phenomenology the bi-directional PDW produces the pseudogap at the antinodes and the Fermi arcs near the nodes. (strictly speaking these are the electron-like segments of closed orbits made up of Bogoliubov quasi-particles.) It explains why the gap closes at the end of the Fermi arcs with states moving up from lower energy, while a CDW-generated gap will necessarily close by a state coming down in energy. As opposed to conventional pairing, the spectrum is not particle-hole symmetric at each k point, which explains why the momentum of the minimum gap is shifted away from the original Fermi surface. In addition, CDW at wave-vectors $Q=2P_1$ and $2P_2$ naturally emerges as subsidiary orders. The states at the Fermi arc’s play two important roles. First they greatly suppress the superfluid density and therefore the phase stiffness, so that the PDW is subject to strong phase fluctuations over most of the phase diagram in the H-T plane. Secondly the normal state gives rise to a linear term in the entropy, which lowers the free energy and stabilizes it at finite temperatures, even if it is not the true ground state. In addition, in the superconducting state, a CDW with period $P_1$ and $P_2 (=Q/2)$ naturally appears if the PDW phase is pinned to that of the d wave pairing and reference was made to an STM experiment on YBCO where CDW at $Q$ and $Q/2$ have been reported[@Yeh1; @Yeh2], where $Q=0.28 (2\pi/a)$ matches what is now determined by X-ray scattering. We should point out that other workers have also associated PDW with the pseudogap phenomenon. Zelli , Kallin and Berlinsky[@ZelliQtmOscPhysRevB.86.104507] used the quasi-particle orbits produced by an uni-directional PDW order to produce quantum oscillations. A related proposal was recently made by M. Norman and J.C. Davis.[@Normanunpublished] We will comment on this below. Yu et al[@yuPNAS126672016magnetic] have interpreted their high magnetic field phase diagram in terms of a possible PDW. Two distinct pair fluctuation lifetimes have been reported in tunneling experiments, possibly indicative of the presence of two kinds of superconductors[@koren2016observation]. Other papers consider a PDW with the same wave-vector and on equal footing as the CDW and are less relevant to the present discussion[@PepinPhysRevB.90.195207; @WangPhysRevLett.114.197001]. Next, an interesting observation was made by Agterberg et al[@Agt2PhysRevB.91.054502] that by shifting the momenta K from the zone boundary line, a new state is formed where the PDW carries momenta $P_1$ and a $P'_1$ which is not equal to ${-P_1}$ and similarly for $P'_2$. (see fig 1a) We shall refer to this state as canted PDW, referring to the canting of the pairing momenta as seen in fig 1a. Agterberg et al [@Agt2PhysRevB.91.054502] showed that this state breaks time reversal and inversion symmetry, but preserves the product and that this is precisely the symmetry of the loop current model of Varma[@varma2006theory] which has been used to interpret the neutron scattering data. Advanced numerical methods applied to the t-J models have found evidence for stripe-PDW as a competing state.[@PhysRevLett.113.046402] Interestingly the energy is found to be quite insensitive to the hole filing per period, in contrast to the original stripe picture which strongly prefers half a hole per period. In the remaining of this paper, we address the adequacy of each of the following scenario’s as the explanation of the double period CDW, put in the broader context of the pseudogap phenomenology. (1) The Q/2 CDW is the primary order, while the Q CDW is subsidiary. (2) The Q/2 PDW is a competing order, or an example of “intertwined order” where several order parameters such as PDW, CDW, SDW and d wave pairing are intimately related to each other. In this picture, the PDW exists only in the vortex halo and vanishes outside. (3) The PDW is the primary order, the “mother state” that exists at a high energy scale and lurks behind a large segment of the phase diagram in the temperature/magnetic field plane. In order to explain the pseudogap at the anti-nodes the PDW is assumed to be bi-directional. While its order is destroyed by phase fluctuations, there are several subsidiary orders that emerge at lower temperatures which account for the observed complexity of the phase diagram. We shall also include a discussion of the canted PDW. Throughout this paper we assume the PDW to be bi-directional. A recent paper by Wang et al.[@wang2018] addresses issues related to PDW in the STM experiment and there are similarities and differences with the present work. They consider the d wave superconductivity and PDW as competing states inside the vortex halo and construct a sigma model description combining the two orders. They focus their calculations to uni-directional PDW. They argue against the persistence of PDW outside the vortex halo. As such their picture is closer in spirit to scenario (2) as outlined above. Recent STM results on period-8 density wave =========================================== First we give a short summary of the recent low temperature STM experiment in $\BSCCO$[@EdkinsSeamusRecentSTM]. The doping is about 0.17. At zero field patches of 4a CDW are observed. These appear locally uni-directional and have $d$ form factors. The correlation length is very short, about twice of the lattice spacing. At a finite field of 8.25T, by subtracting off the zero field data, period 4a and period 8a CDWs are revealed in the “halo” region around the vortex core. These appear to be bi-directional and have s-wave form factors. The signals are symmetric when the voltages are reversed. We distinguish bi-directional from “checkerboard” order, which consists of local patches of uni-directional stripes. From the widths of the Fourier transform peaks, the correlation length of the 8a and 4a CDW is about 8a and 4a respectively, comparable to their wavelengths. By examining the signals that are odd upon reversing the voltage, another 4a CDW is found which has $d$ form factors. Its correlation length is about 5a and it is uni-directional, running in the same direction from vortex to vortex. Purely on symmetry grounds, the observation of period 8a bidirectional charge order in the presence of a background superconductor implies that there are also period-8 modulations in the pair order parameter. Specifically if the Fourier component $\rho_{Q/2}$ of the density at a wave vector $Q/2$ is non-zero, then it implies a non-zero Fourier component $\Delta_{Q/2} \sim \Delta_d \rho_{Q/2}$ in the pairing order parameter (where $\Delta_d$ is the order parameter for the standard $d$-wave superconductor). An important question then is whether the observed period-8 modulations are driven primarily by the pinning of soft fluctuations of $\rho_{Q/2}$ (and $\Delta_{Q/2}$ is a subsidiary) or whether the driver is pinning of soft fluctuations of $\Delta_{Q/2}$ (and the observed $\rho_{Q/2}$ is a subsidiary). We will call the former CDW-driven and the latter PDW-driven. Clearly this is not a symmetry-based distinction and it is natural to wonder if the question is meaningful at all. However we will argue in this paper that there are, in fact, two distinct possibilities for the observed period-8 charge order which have distinct experimental signatures. It is natural to associate these two distinct possibilities with the (looser) distinction between CDW-driven and PDW-driven mechanisms. Basic features of bi-directional PDW ==================================== In this section, we explore the implications of the PDW-Driven scenario, and contrast it with the CDW-driven scenario. We will particularly emphasize the two distinct structures of the period-8 charge order and their experimental distinctions. PDW with long range order {#Sec: PDW with long range order} ------------------------- The new CDW recently found in $\BSCCO$ has a momentum close to $2\pi/8$, half of the momentum of the well-known short-range CDW at zero field. In the PDW-driven scenario, we consider a bi-directional PDW order with the same momentum, that is roughly the momentum between tips of the bare Fermi surface in the anti-nodal direction[@lee2014amperean]. Bi-directional PDW state with such a momentum is previously proposed by one of the authors [@lee2014amperean]. Following this proposal, we write down a mean field Hamiltonian H &=& \_[k,]{} \_k c\^\_[k,]{}c\_[k,]{}\ &+& \_[k]{} \^\\_[P\_1]{}(k) c\_[k,]{}c\_[-k + P\_1,]{} + \^\\_[P’\_1]{}(k) c\_[k,]{}c\_[-k + P’\_1,]{}\ &+& \_[k]{} \^\\_[P\_2]{}(k) c\_[k,]{}c\_[-k + P\_2,]{} + \^\\_[P’\_2]{}(k) c\_[k,]{}c\_[-k + P’\_2,]{}\ &+& h.c. \[Eq: long range PDW mean field\] We used the notation: $P_1 = 2K_1,\ P'_1 = 2K'_1$ — as shown in Fig. \[Fig: PDW band stucture, pocket\](a) $K_1$ and $K'_1$ are located at or near the Fermi surface at anti-nodal points, generically incommensurate with the B.Z.; Similarly, $P_2 = 2K_2,\ P'_2 = 2K'_2$. The 4 PDW order parameters generate CDW order $\rho_{Q_x}$ and $\rho_{Q_y}$ in second order perturbation even though we do not include them explicitly in the Hamiltonian. \_[Q\_x]{}\~\_[P\_1]{}\^\\_[P’\_1]{}, \_[Q\_y]{}\~\_[P\_2]{}\^\\_[P’\_2]{}. \[Eq: subsidiary doubled CDW\] We associate this subsidiary CDW as the well-known short-range CDW at zero field; it has momenta $Q_x = P_1 - P'_1$, $Q_y = P_2 - P'_2$, with magnitude $Q\simeq 2\pi/4$ in the recent STM experiment. In principle, we can also add CDW in (1,1) direction, e.g. $\rho \sim \D_{P_1}\D^_{P'_2} + \dots$. However, this CDW is absent in the recent STM experiment; we explain the reason in detail in the next subsection. Naively one may expect that if the PDW has local $d$ form factor, the CDW generated by Eq.\[Eq: subsidiary doubled CDW\] has $s$ form factor. This argument is not generally correct, because $s$ and $d$ form factor for a finite-momentum order parameter has no sharp symmetry distinction [^3] It is a local property, which is not captured by the long wavelength description of a Landau order parameter. In fact, when we solve our mean field Hamiltonian with only $d$ wave PDW as input, the CDW that emerges at $Q$ is predominantly $d$ wave. In view of the experimental observation of $s$ symmetry CDW near the vortex core, this may simply indicate that the mean field theory is not adequate to give a microscopic description. Nevertheless, we want to convey the message that this result shows that it is entirely possible that a $d$ wave CDW can emerge as a subsidiary order. We define the common phases $\theta_{\text{P},x},\ \theta_{\text{P},y}$ and relative phases $\phi_x,\ \phi_y$ of the PDW order parameters, and the phases of Q CDW order parameters as \_[P\_1]{} = \|\_[P\_1]{}\|e\^[i(\_[,x]{} + \_x)]{}&,& \_[P’\_1]{} = \|\_[P’\_1]{}\|e\^[i(\_[,x]{} - \_x)]{}\ \_[P\_2]{} = \|\_[P\_2]{}\|e\^[i(\_[,y]{} + \_y)]{}&,& \_[P’\_2]{} = \|\_[P’\_2]{}\|e\^[i(\_[,y]{} - \_y)]{}\ \_[Q\_x]{} = \|\_[Q\_x]{}\|e\^[i\_x]{}&,& \_[Q\_y]{} = \|\_[Q\_y]{}\|e\^[i\_y]{}, As shown in Eq.\[Eq: subsidiary doubled CDW\], $\gamma_x = 2\phi_x$ and $\gamma_y = 2\phi_y$ are the phase difference between PDW order parameters, hence the phases of the subsidiary CDW order parameter [^4]; they are proportional to the shift of density wave pattern in real space. On the other hand, $\theta_{\text{P},x}$ and $\theta_{\text{P},y}$ carry charge 2 under external electromagnetic field; when coexist with uniform d wave superconductivity $\|\D_d\|e^{i\theta_d}$, the relative phases $\theta_{\text{P},x}- \theta_d$ and $\theta_{\text{P},y} - \theta_d$, together with $\phi_x$ and $\phi_y$ determines the spatial pattern of new CDW orders with momenta $P_1, P'_1, P_2$, and $P'_2$, which are close to or equal to $Q/2$. We consider two scenarios: (1) $K_i$ and $K'_i$, $i = 1,2$ are located at the boundary of B.Z., shown as solid red dots in Fig. \[Fig: PDW band stucture, pocket\](a): $2K_1 = -2K'_1 = P_1 = -P'_1 = Q_x/2,\ 2K_2 = -2K'_2 = P_2 = -P'_2 = Q_y/2$ (2) $K_i$ and $K'_i$ are slightly shifted, shown as dashed red dots. The shifts in momenta can be either positive or negative, giving a $Z_2$ order parameter in each direction. We refer to this scenario as canted PDW. This possibility was discussed in Ref. in relation with loop current. It has a potential ability to account for T-reversal breaking and nematicity. Regarding the recent STM experiment, these two scenarios give similar predictions. We focus on the first scenario and comment on the second when necessary. Unlike the pairing in a conventional superconductor, where electrons forming a Cooper pair have opposite momenta and opposite velocity, this finite-momentum pairing groups electrons with momenta $K_i + \d k$ and $K_i - \d k$, (similarly, $K'_i + \d k$ and $K'_i - \d k$) and it has a strong effect only when these two momenta are both close to the Fermi surface. As a result, it opens a gap only in the anti-nodal direction (shown in Fig.\[Fig: PDW band stucture, pocket\](b)), and leaves a gapless surface of Bogoliubov quasi-particle in the nodal direction. Since PDW and CDW are generically incommensurate to the B.Z., we need to set a cutoff in momentum-space calculation. It was previously reported in Ref. [@lee2014amperean] by one of the author that a 5-band model describing the mixing of $c_{k}$, $c^{\dagger}_{-k+P_1}$, $c^{\dagger}_{-k+P'_1}$, $c_{k-Q_x}$ and $c_{k+Q_x}$ (similarly in y direction) produce Bogoliubov pockets with predominant electron weight on one side and predominent hole weight on the other side. In order to capture the effect of B.Z. folding caused by the subsidiary CDW, we increase the cutoff, and include the mixing among $c_{k+mQ_x+nQ_y}$ for $m,\ n$ up to $\pm 2$ (for details, see Appendix A). We used the Hamiltonian in Eq. \[Eq: long range PDW mean field\], the band structure in Appendix A, CDW momentum $Q=0.28(2\pi/a)$ measured in Ref. [@GrevenPRB96tabis2017synchrotron], PDW order parameter with d wave form factor \_[Q\_x/2]{}(k)= 2((k\_xQ\_x/4) - (k\_y))\ \_[Q\_y/2]{}(k)= 2((k\_x) - (k\_yQ\_y/4)), with $\D = 45$meV and no explicit CDW order parameter in the mean field Hamiltonian. We found that, the electron-like part of the 4 Bogoliubov pockets recombine into a predominantly electron-like pocket, similar to the Harrison-Sebastian pocket (shown in Fig. \[Fig: PDW band stucture, pocket\](b-c)). We believe that these pockets formed by mainly electron like segments will give rise to quantum oscillations. The reason is that while the Bogoliubov quasi-particles do not carry fixed charges, they carry a well defined current, because the holes are moving in the opposite direction. In these orbits, all the segments are electron like. In a semi-classical picture a wave-packet will travel in real space along a close contour that encloses the magentic flux. By the Onsager argument, we can expect quantum oscillations. In contrast, there are many closed orbits made up of segments that are part electron and part hole like.[@Normanunpublished] If we draw an arrow corresponding to the current, we find that at the corners where the electron-like and the hole-like segments meet, they both run into the corner and undergo Andreev scattering, ie the currents go into the condensate. In this case, even though the orbits look closed in momentum space, the wave-packets do not form closed orbit in real space, because part of the current goes into the condensate. Then Onsager’s argument no longer applies. For this reason we think it is unlikely that such orbits give rise to quantum oscillations, but only a detailed calculation can tell us the answer for sure. Analogous issues arise with the Fermi surface formed by Bogoliubov quasiparticles in a d-wave superconductor coexisting with loop current order. In that problem, detailed calculations[@allais2012loop; @wang2013quantum] indeed show that at $T = 0$ such a Fermi surface does not lead to quantum oscillations. We note that Zelli et al[@ZelliQtmOscPhysRevB.86.104507] claimed that oscillations corresponding to such orbits exist, but their conclusion is based on an approximate calculation. We believe this issue should be re-visited. Another point is that the PDW is a superconductor and in principle we should include vortices when we introduce the magnetic field. We provide the following argument. First it is known that quantum oscillations appear in the mixed state with a frequency which is the same as the same pockets in the normal state.[@PhysRevB.51.5927] This has been confirmed by numerical calculations with randomly pinned vortices in a d wave superconductor as long as the correlation length is not too short.[@PhysRevB.79.180510]. Of course to address quantum oscillations we need to think about a metallic state that emerges from fluctuations of a PDW ordered state. We will leave this problem aside in the present paper. We would like to mention that as we increase doping, the 4 electron pockets [^5] in Fig. \[Fig: PDW band stucture, pocket\](c) touches each other. In some parameter range, Fermi surface topology changes, and a hole pocket forms in the middle. This Lifshitz transition is predicted for Hg1201 at $10\%$ doping in Ref. [@GrevenPRB96tabis2017synchrotron], and for YBCO at a larger doping. However, distinguishing subtle changes of Fermi surface topology is beyond the scope of the current paper. ![Band structure of Bogoliubov quasi-particle and possible Fermi pockets in a PDW state. (a) An illustration of the bare Fermi surface, CDW momenta and PDW momenta. CDW momenta $Q_x$ and $Q_y$ are shown as yellow arrows. PDW momenta are $P_1 = 2K_1$, $P'_1 = 2K'_1$ in x direction, and $P_2 = 2K_2$, $P'_2 = 2K'_2$ in y direction. CDW is a subsidiary order of PDW, its momenta $Q_x = P_1 - P'_1$, $Q_y = P_2 - P'_2$. We consider two scenarios: (1) $K_i$ and $K'_i$ are located right at B.Z. boundary (solid red dots). $P_1 = -P'_1=Q_x/2$, $P_2 = -P'_2 = Q_y/2$. (2) $K_i$ and $K'_i$ are slightly shifted (dotted red circles); $P_1$ and $P'_1$ have a small y component, as shown in the inset figure (The small y component is exaggerated). (b)Electron weight on the Fermi-pocket of Bogoliubov quasi-particle. We used the band structure in Appendix A, CDW momentum $Q_x=Q_y=0.28(2\pi/a)$ measured in Ref. [@GrevenPRB96tabis2017synchrotron] PDW order parameter $\D_{Q/2}=45$meV, no explicit CDW order parameter in mean field Hamiltonian, and plotted the electron weight at Fermi energy and each momentum $k$ in the B.Z. (For details, see Appendix A). Electron weight is large on 4 “arcs” in the nodal direction. The anti-nodal direction is gapped out by PDW. (c) Details of the reconstructed electron-like pocket after B.Z. folding caused by CDW. We plotted the total electron weight at momenta up to $Q_x$ and $Q_y$. This pocket is formed by 4 segments with electron weight $>80\%$. It has the same shape as the Harrison-Sebastian pocket. Physically there is only one pocket, others are its copy shifted by $Q_x$ and $Q_y$. we only show the upper right quadrant of the B.Z.[]{data-label="Fig: PDW band stucture, pocket"}](image/PDW_pocket.pdf){width="7in"} Static short range PDW ---------------------- In this subsection, we discuss the situation where a short-range PDW coexists with d wave superconductivity. We focus on the setup of the recent STM experiment where a period-8 density wave was found in the vortex halo of d wave superconductor. To simplify the discussion, we consider the simplest scenario: $P_1 = -P'_1 = Q_x/2$, $P_2 = -P'_2 = Q_y/2$. We have 4 PDW order parameters: $\D_{\pm Q_x/2}$ and $\D_{\pm Q_y/2}$. We consider the following couplings between PDW, d wave, and CDW order parameters in a Landau theory in translation-invariant systems. We can write them in momentum space as F = &-&a\_[Q\_x]{}\^\\_[Q\_x/2]{}\_[-Q\_x/2]{} - b \_[Q\_x]{}\[\^2\_d\^[\2]{}\_[Q\_x/2]{}+\^[\2]{}\_d\^[2]{}\_[-Q\_x/2]{}\]\ &-& c\_[Q\_x/2]{} \[\^\\_d\_[-Q\_x/2]{} + \^\\_[Q\_x/2]{}\_d\] -…, \[Eq: Landau theory, momentum space\] where $a$, $b$, $c$ are real coupling constants. For simplicity, we write down only couplings in x direction. Couplings in y direction are similar. These momentum-space couplings are conceptually helpful, but the strong breaking of translation symmetry introduced by the vortex core brings in new physics that are better captured by a real-space analysis. Before we start, it is important to note that what the experimentalists found is not long-range PDW or CDW. Instead, STM experiment identified a static short-range charge order that lives only inside the vortex halo, with apparent correlation length comparable to its wavelength. Theoretically, a “short-range order” naturally fluctuate with time; the existence of static short-range order raises many questions — what pins the phases of the order parameters? — why does it appear only in vortex halo? One may tend to think of a phase competition between uniform d wave superconductivity and PDW, so that the latter may be greatly enhanced near the vortex core. However, a phase competition alone does not explain why the short-range order is static. The answer of these questions may lie in the following observation: just like the way spatial inhomogeneity pins short-range CDW, a spatial pattern of superconductivity close to the vortex core pins a short-range PDW. This static PDW then extends to a larger region with radius defined by its correlation length $\xi_P$. Outside $\xi_P$, there is still a PDW amplitude fluctuating with time, but the time average decays exponentially. For concreteness, we choose the origin to be the center of the vortex, $(r,\theta)$ to be the polar coordinate, $(x,y)$ to be the Cartesian coordinate, and write down the following ansatz for the amplitude of d wave and PDW: \[Eq: d wave amplitude in real space\] \_d() &=& \|\_d(r)\|e\^[i\_d]{}e\^[i]{}\ \_() &=& 2\|\_[Q\_x/2]{}\| e\^[-r/\_P]{} e\^[i\_[P,x]{}]{}(Qx/2 + \_[x]{})\ + &2&\|\_[Q\_y/2]{}\| e\^[-r/\_P]{} e\^[i\_[P,y]{}]{}(Qy/2 + \_[y]{}), \[Eq: PDW amplitude in real space\] where $\|\D_d(r)\|= r/\sqrt{r^2 + r^2_\text{core}}$. $e^{i\theta}$ encodes the $2\pi$ phase winding of d wave amplitude. We have three length scales. The radius of the vortex core: $r_\text{core}\simeq 3a$, the period of PDW: $4\pi/Q\simeq 8a$, and the radius of vortex halo, where field-enhanced CDW are found: we identify the halo size as $r_\text{halo}\sim\xi_P\sim 4\pi/Q$. A usual Landau theory with slowly-varying order parameters implicitly assumes that $r_\text{core}\gg 4\pi/Q$, $\xi_P\gg 4\pi/Q$. However, we are in the opposite limit: $4\pi/Q \sim \xi_P > r_\text{core}$. Since $\xi_P$ and $4\pi/Q$ are close to each other, and they are one order of magnitude larger than the lattice constant, we do not separate the exponential decay of order parameters $\D_{\pm Q_x/2}$ ($\D_{\pm Q_y/2}$) from the oscillatory part $\cos(Qx/2 + \phi_{x})$ ($\cos(Qy/2 + \phi_{y})$), as in a usual Landau theory. Instead, we take the ansatz in Eq. \[Eq: d wave amplitude in real space\] and Eq. \[Eq: PDW amplitude in real space\], and write down their couplings in real space together with charge density profile $\rho(r)$. \[Eq: Landau theory, real space\] F = &-&{a()\_() \^\\_()\ &+& b()\[\^2\_d() \^[\2]{}\_() + \^[\2]{}\_d() \^2\_()\]\ &+& c()\[\^\\_d()\_() + \_d()\^\\_()\]\ &+&s\[\^\\_d()\_() + \_d()\^\\_()\]}d\^2 We would like to remind the readers again that this free energy is not a Landau free energy in the usual sense, since we include the oscillatory part of PDW explicitly in $\D_\text{PDW}(\mathbf{r})$. The last term in Eq. \[Eq: Landau theory, real space\]: $-s\int \D^_d(\mathbf{r})\D_\text{PDW}(\mathbf{r}) d^2 \mathbf{r} + c.c.$ is the lowest-order symmetry-allowed term that describes the phase locking between PDW and d wave order parameter near a vortex core. In the case of spatially slowly-varying order parameters, this term usually vanishes because of momentum mismatch, eg. if the d wave superconductivity has uniform amplitude. However, close to the vortex core, the rapid changing of d wave amplitude strongly breaks translation symmetry. Furthermore the phase winds by $2\pi$ around the core, and near the core the winding is sufficiently rapid that it can phase match the finite wave-vector of the PDW. As a result the PDW is pinned to match the spatial pattern of vortex core so that free energy is minimized. Because of the phase winding, d wave amplitude changes sign across the origin and the overlap integral is optimized when PDW has the form $\sin(Qx/2)$ which also changes sign at the origin. Thus $\phi_{x}$ and $\phi_{y}$ are pinned to be $-\pi/2$. Then the overall phase, $\theta_{P,x} = \theta_d,\ \theta_{P,y} = \pi/2 + \theta_d$, are pinned so that the overlap is a positive real number. This pinning mechanism completely fixes the phases of PDW; a simple calculation of the overlap integral indicates the pinning is very effective in the vortex core. For details, see Fig. \[Fig: pinning by overlap\]. Of course, at the length scale of 10 lattice constants, everything except a microscopic model is merely an oversimplified illustration. Nonetheless, we believe this simple illustration captures the underlying physics of phase-locking between d wave and various PDW order parameters. This pinning mechanism is effective exactly because $4\pi/Q > r_\text{core}$ in the cuprates. In the opposite limit, d wave order parameter changes slowly. According to a usual Landau theory, this coupling cancels out. In the remaining part of this section, we discuss the consequences of this phase-locking on subsidiary charge order. We confirmed these consequences by an exact diagonalization study in the next section. ![(a) Overlap integral $\int \D^_d(\mathbf{r})\D_\text{PDW}(\mathbf{r}) d^2 \mathbf{r}$ as a function of $\phi_{x}$, for $\theta_x = 0$. We have set their maximum amplitude to 1 for both $\D_d(\mathbf{r})$ and $\D_\text{PDW}(\mathbf{r})$, and we normalize the integral by the overlap of PDW with itself inside the vortex core of radius $3a$. $\phi_x$ is pinned to $3\pi/2$. The large overlap implies the real-space pattern of PDW matches the pattern of d wave vortex core almost perfectly at $\phi_x = 3\pi/2$ — its amplitude is reduced only because d wave amplitude is reduced in the vortex core. (b) The integrand $\D^_d(\mathbf{r})\D_\text{PDW}(\mathbf{r})$ as a function of $\mathbf{r}$ near vortex core, for $\phi_{x} = 3\pi/2$, $\theta_x = 0$. Outside the vortex core, the integrand alternating between positive and negative because of momentum mismatch. However, with in the first period of PDW in the center, the integrand is always positive, giving a large overlap. This is because d wave and PDW both change sign across the origin. d wave change sign due to $2\pi$ phase winding, and PDW change sign because of the $\sin(Qx/2)$ factor.[]{data-label="Fig: pinning by overlap"}](image/overlap_integrand.pdf){width="3in"} Note that PDW does not have a vortex. Since PDW lives only in small patches, vortices are not required[@agterberg2015checkerboard], and it is energetically favorable to not have vortices in the PDW-driven scenario. This PDW order generates various CDWs in the vortex halo: \(1) bi-directional Q/2 CDW. According to Eq. (\[Eq: Landau theory, momentum space\] - \[Eq: Landau theory, real space\]), it has the following amplitude in real space \[Eq: CDW in real space\] \_() = F(r)(+ \_d - \_[P,]{})(Q\_ + \_) where $\a = x, y$, $Q_\a = Q_x, Q_y$ and $F(r) \sim 2c\|\D_d(r)\D_{Q_{\a}/2}\| e^{-r/\xi_P}$. The most interesting feature is that, apart from normal plane-wave factor, there is an additional factor $\cos(\theta + \theta_d - \theta_{P,\a})$ depending on the polar angle. A choice of the relative angle $\theta_d - \theta_{P,\a}$ selects a special angle along which $\rho_{\a}(\mathbf{r})$ vanishes. We point out that the pinning mechanism we discussed predicts that the amplitude $\rho_x$ vanishes in the vertical direction, when $\theta\sim\pm \pi/2$, while the amplitude $\rho_y$ vanishes in the horizontal direction, when $\theta\sim 0,\pi$. This choice restores C4 symmetry. Physically, this new feature originates from the $2\pi$ winding of d wave order parameter. We can identify two contributions to $\rho_{Q/2}$: $\D^_d\D_{Q/2}$ which carries -1 dislocation, and $\D_d\D^_{-Q/2}$ which carries +1 dislocation. The interference of these two terms give rise to a nodal direction in real space. This is an important prediction in PDW-driven scenario. On the contrary, in CDW-driven scenario, it is energetically favorable to put the dislocation in PDW amplitude, and the CDW amplitude is rather featureless. In the next section, we discuss the same feature in Fourier space, and propose follow-up experiments to distinguish PDW-driven and CDW-driven scenario. \(2) Q CDW. According to Eq. \[Eq: Landau theory, momentum space\] there are two contributions: \^A\_Q \~a\^\\_[-Q/2]{}\_[Q/2]{}, \[Eq: CDWA\] which we call $\text{CDW}_A$, and \^B\_Q \~b(\^[\2]{}\_d\^2\_[Q/2]{} + \^[2]{}\_d\^[\2]{}\_[-Q/2]{}), \[Eq: CDWB\] which we call $\text{CDW}_B$, which we can think of as a harmonic of $\rho_{Q/2}$. $\text{CDW}_A$ does not rely on the phase-locking between d wave and PDW; it is already pinned to be static short-range CDW by impurities at zero magnetic field, and it persists above $T_c$. On the other hand, a static $\text{CDW}_B$ rely on the phase-locking. Similar to Q/2 CDW, it is a superposition of +2 dislocation and -2 dislocation, and it exists only in vortex halo. In the case of spatially uniform PDW and CDW order, there is no distinction between the two. However, in a spatially inhomogeneous situation such as what we encounter near the vortex core, there is a physical distinction. For example, $\text{CDW}_A$ may be extended in space while $\text{CDW}_B$ may be localized near the vortex core. In this case the two CDW may have different local form factors, such as d or s wave. These form factors may in turn determine which one prefers to be bi-directional or uni-directional, because the coefficient of the quartic term that couples the amplitudes of the x and y oriented CDW may be different. In the STM data there already appears to be two kinds of CDW’s , one pinned to the vortex core and one which already exists at zero filed. We will make further use of this distinction in later discussions. Naively, one would expect a CDW with momentum $(Q/2,Q/2)$ appears in the second order — in real space this term may show up in the contribution $\rho(\mathbf{r})\sim a\D_\text{PDW}^(\mathbf{r})\D_\text{PDW}(\mathbf{r})$. However, the pinning in the vortex core requires \_() \~e\^[i\_d]{}((Qx/2) + i(Qy/2)),\ \_\^\()\_() \~\^2(Qx/2) + \^2(Qy/2), and the cross term $\sin(Qx/2)\sin(Qy/2)$ with momenta $(\pm Q/2, \pm Q/2)$ cancels out due to the $\pi/2$ relative phase. As a consequence, there is no $(2\pi/8, 2\pi/8)$ CDW in the leading order. In the fourth order, such a CDW is generated by the term $\D_{d}^{2}(r)\D^2_\text{PDW}(r)$, but the amplitude is weak and subject to broadening effect given by dislocations. The absence of $(2\pi/8, 2\pi/8)$ CDW is previously discussed in Ref. . It was pointed out that in the uniform case when PDW does not have a vortex, the relative phase between PDW in x and y direction determines whether $(2\pi/8, 2\pi/8)$ CDW is present or not. If the phase is zero it is present, while if it is $\pi/2$ bond currents are generated, producing a flux density wave at the same wave-vector instead. This flux density wave will be discussed in great detail in a later section. In the uniform case it is not known which phase is preferred. In our case we find that in the presence of a vortex, the phase choice $\pi/2$ is energetically favorable, therefore $(2\pi/8, 2\pi/8)$ CDW is absent in leading order. On the contrary, in CDW-driven scenario, naively the $(2\pi/8, 2\pi/8)$ CDW is comparable to the $(2\pi/4, 0)$ CDW. The absence of a $(2\pi/8, 2\pi/8)$ Fourier peak in STM data is an evidence favoring PDW-driven scenario. Next, we would like to comment on the correlation length of PDW in the recent STM experiment. In PDW-driven scenario, as discussed above, the Q/2 CDW has $2\pi$ phase winding around the vortex core. A simple calculation shows that this phase winding broadened the Fourier peak by roughly a factor of 2. Thus the intrinsic correlation length of Q/2 CDW and PDW should be close to 16 lattice constants, a little smaller than the half of the distance between neighboring vortex core. We end this section with some comments on the implications if a canted PDW is present. While the CDW generated by Eq(2) retains the wave-vector Q along the x and y axes, the double period CDW generated by the analog of the third term in Eq. \[Eq: Landau theory, momentum space\] now has wave-vector $P$ and $P'$. Similarly, its harmonic generated by the analog of the second term in Eq. \[Eq: Landau theory, momentum space\] have wave-vectors $2P$ and $2P'$. It is worth noting that we now have two distinct CDWs and the difference between A and B type CDW is now a sharp one that can be made even in a uniform system. A second point is that there is now an additional pinning mechanism. The term $(\Delta_d e^{i\theta(\mathbf{r})})^2 (\Delta_P \Delta_{P'})^$ is allowed if the local phase gradient matches the canting momentum $p = (P+P')/2$. This leads to a locking term at some distance from the vortex core where the phases are matched. The possible detection of the canting angle will be discussed in the next section. With the above understanding of PDW-driven scenario, we propose the following phenomenological picture explaining the recent STM experiment in $\BSCCO, \ 17\%$ doping, up to 8.5T: - short-range PDW is pinned by the vortex core and extends to its correlation length. - We estimate the intrinsic correlation length of PDW to be 16 lattice constants. The period-8 CDW appears to have a shorter correlation length $\sim$ 8 lattice constants as determined from the width of the Fourier transform peak by fitting it to a Gaussian. Part of this width is not intrinsic and is due to the $2\pi$ phase winding. - The period 8 CDW produces as a harmonic a period 4 CDW, which we have labeled as $\text{CDW}_B$. Its width is subject to the same blurring as the period 8 CDW. On the other hand, the static PDW near vortex core nucleates the period-4 $\text{CDW}_A$ by $\D^_{-Q/2}\D_{Q/2}$, which is not affected by the phase winding around the vortex. These two CDWs may have different form factors and different asymmetry factors between x direction and y direction. However it is hard to extract their correlation length separately based on the current data, since their Fourier peaks mix together. The width of $2\pi/4$ Fourier peak translates to a correlation length around 4a. This serves as a lower bound of the intrinsic correlation lengths of $\text{CDW}_A$ and $\text{CDW}_B$. - At zero field, $\D_{-Q/2}$ and $\D_{Q/2}$ fluctuate with time, we rely on their relative phase being pinned by spatial inhomogeneity to give a static $\text{CDW}_A$. This effect gives much weaker period-4 CDW puddles with a very short correlation length of order 2a. This CDW is unidirectional in each small puddle. We tentatively identify the unidirectional part of CDW both in zero field and in the vortex core as $\text{CDW}_A$. - The static-PDW-enhanced correlation length of $\text{CDW}_A$ is enough to give some overlap between neighboring vortices. It is energetically favorable for the unidirectional part to align its direction and stretch its phase between vortices smoothly to gain overlap energy. - PDW-driven model predicts the absence of $(2\pi/8,2\pi/8)$ peak. - Given the strong pinning effect and relatively small correlation length, these CDWs may not be able to overcome the local pinning effect and become phase coherent between halos. Experimental Proposal ===================== The disappearance of $(\frac{2\pi}{8},\frac{2\pi}{8})$ CDW order is surprising for a CDW-Driven model while it can be naturally explained in PDW-Driven model, as shown in last section. Despite this already existing evidence favoring PDW-Driven model, more experimental predictions need to be tested to fully settle down this issue. In this section we propose experiments to distinguish PDW-Driven and CDW-Driven scenario unambiguously. Besides, in PDW-Driven scenario our proposed experiment can extract the relative phase between PDW order parameter and $d$ wave order parameter, which is physical. The main prediction of PDW-Driven scenario is that CDW order parameter at $Q_x/2=(\frac{2\pi}{8},0)$ and $Q_y/2=(0,\frac{2\pi}{8})$ have the following profile as shown in Eq. \[Eq: CDW in real space\] $$\mathlarger \rho_{\mathbf {Q_{\a}/2}}(r,\theta)=e^{i\phi_a}F_P(r) \cos(\theta-\theta_a) \label{eq:pdw_profile}$$ where$(r,\theta)$ is the polar coordinate of real space around the vortex center and $a$ denotes $x$ or $y$ direction. $F_P(r)$ vanishes at $r=0$ and decays as $e^{-\frac{r}{\xi}}$ at large $r$. It has maximum at nonzero distance to center. $\theta_x=\theta_{P_x}-\theta_d$ and $\theta_y=\theta_{P_y}-\theta_d$ are the relative phases of PDW order parameters $\Delta_{\pm P_a}=\|\Delta_{P_a}\|e^{i\theta_{P_a}\pm i \phi_a}$ compared to d-wave order parameter $\Delta_D(r,\theta)=\|\Delta_D\|e^{i\theta_d}e^{i\theta}$ In contrast, CDW-Driven scenario shows quite distinct profile of period $8$ CDW order parameter: $$\mathlarger \rho_{\mathbf{Q_a/2}}(r,\theta)=e^{i\phi_a}F_c(r) \label{eq:cdw_profile}$$ $F_c(r)$ has maximum at $r=0$ and decays far away with $e^{-\frac{r}{\xi}}$. CDW order parameter doesn’t have angle dependence in this scenario. [0.45]{} ![Real Space Plot of on site LDoS $\nu^E(\mathbf{r})$ at $E=30$meV for PDW-Dirven and CDW-Driven model.[]{data-label="fig:real_space"}](image/pdw.png "fig:"){width="\textwidth"} [0.45]{} ![Real Space Plot of on site LDoS $\nu^E(\mathbf{r})$ at $E=30$meV for PDW-Dirven and CDW-Driven model.[]{data-label="fig:real_space"}](image/cdw.png "fig:"){width="\textwidth"} Clearly CDW order parameter profile from PDW-Driven and CDW-Driven models have both different radius dependence and angle dependence. A real space plot of LDoS can be found in Fig. \[fig:real\_space\]. The $\cos(\theta-\theta_a)$ factor in PDW-Driven model means a superposition of strength $\pm 1$ dislocation of CDW order parameter and in principle STM experiments can extract $\theta_a$. Here we will propose the following experimental predictions to distinguish the above two different CDW profiles. In the STM experiment, what is measured is the local density of states (LDoS) at a fixed energy $\nu(\mathbf{r},E)$. For a fixed energy, $\nu^E(\mathbf{r})=\nu(\mathbf{r},E)$ has the same symmetry as density and we expect it to follow Eq. \[eq:pdw\_profile\] and Eq. \[eq:cdw\_profile\]. Before going to specific predictions, it may be worthwhile to give one general suggestion to data analysis procedure of experimental data. For both PDW-Driven and CDW-Driven scenario, the phase of CDW order with momentum $Q_a$ is expected to be locked to position of vortex center. As a result, signals from different vortex halos are not coherent. Therefore, it’s better to shift the position of each vortex center to the origin when doing Fourier Transformation for each vortex halo. In this way we can make different vortex halos coherent and greatly enhance signals. The following are predictions for PDW-Driven scenario and how to detect it in experiment. As a benchmark, we show our numerical simulation data. The method of our simulation is summarized in Appendix.B. Profile of d wave order parameter is $\Delta_D(r,\theta)\sim \frac{r}{\sqrt{r^2+r_0^2}}$ with vortex core size $r_0=3.5$ lattice constants. We used a profile of PDW with $r$ dependence as $\Delta_P(r,\theta)\sim e^{1-\sqrt{r^2+\xi^2}/\xi}$ with correlation length $\xi=15$. In the following, local density of states $\nu^E(r)$ is obtained at fixed energy $E=30$ meV. Note we only show $d$ wave form of Bond LDoS because CDW generated by our model is dominated by $d$ wave. However, we expect our predictions in the following sections do not rely on form factor. Split Peaks for Period 8 CDW ---------------------------- The first prediction for PDW scenario is that the peak at $\mathbf{Q_a/2}$ is split to two peaks in the direction decided by $\theta_a$. Recall that the density modulation $\mathlarger \rho(\mathbf r)=\int^0_{-\infty} dE \nu^E(\mathbf r)$ is given by the integral of LDoS $\nu^E(\mathbf r)$ over the occupied states. We define the slowly varying complex amplitude $\nu^E_{\mathbf{Q_a/2}}(\mathbf r)$ by writing the real space local DoS as $\mathlarger \nu^E(\mathbf r)=\sum_{a}\mathlarger \nu^E_{\mathbf{Q_a/2}}(\mathbf r)e^{\frac{1}{2}i\mathbf{{Q}_a}\cdot \mathbf{r}}+h.c.$. This is the analog of $\mathlarger\rho_{\mathbf{Q_a/2}}$ discussed in the last section. We assume that $\nu^E_{\mathbf{Q_a/2}}(\mathbf r)$ has a similar real space profile as $\mathlarger\rho_{\mathbf{Q_a/2}}$ as given in Eq. \[eq:pdw\_profile\], i.e. it is confined to the vicinity of the vortex core and importantly, is proportional to $\cos(\theta-\theta_a)$. Recall that this factor encodes the phase winding of the d wave superconductor and is therefore an important signature for the PDW driven scenario. .This assumption is supported by our numerical simulations, and will be discussed and shown in greater detail later in Fig. \[fig:pdw\_angle\] and Fig. \[fig:local\]. We define $\tilde \nu^E(\mathbf q)$ to be the Fourier Transform of $\nu^E(\mathbf r)$. For $\mathbf q$ in the vicinity of $\mathbf{Q_a/2}$ we define $$\tilde A_a(\mathbf q)=\tilde \nu^E(\mathbf q-\mathbf{Q_a/2})=\sum_{\mathbf{r}}\mathlarger \nu^E_{\mathbf{Q_a/2}}(\mathbf r)e^{-i \mathbf{q}\cdot \mathbf{r}} \label{eq:conv}$$ Consider $a$ in the $x$ direction. When $\theta_a=0$, it’s easy to see that the absolute value of $\tilde A_a(\mathbf q)$ has two peaks in $x$ direction because of the $\cos \theta$ factor. This is because $\cos \theta=\frac{x}{\sqrt{x^2+y^2}}$ produces a line of zero in $\nu^E_{\mathbf{Q_a/2}}(\mathbf r)$ along the $y$ direction through the vortex core. $\nu^E_{\mathbf{Q_a/2}}(\mathbf r)$ is odd under $x \rightarrow -x$ and as a result $\tilde A_a(\mathbf{q_x}=0)=0$ and $\tilde A_a(\mathbf q)$ has a splitting along the $\mathbf{q_x}$ direction. The splitting is roughly $\delta q \sim \frac{1}{\xi}$. For general $a$ and general $\theta_a$, the line of zero in $\tilde A_a(\mathbf q)$ is rotated by an angle $\theta_a$. Therefore, the absolute value of $\mathlarger{\tilde \nu^E}(\mathbf q)$ should have two peaks at $\mathbf{q}\approx\mathbf{Q_a/2}$ with the splitting in the direction of $\theta_a$. This prediction is confirmed by numerical simulation results for both PDW-Driven model and CDW-Driven model in Fig. \[fig:fft\]. Here we show two different phase choices for PDW-Driven model. The splitting of period $8$ peak along the direction $\theta_a$ is very clear for PDW-Driven models while CDW-Driven model show one single peak. Therefore, we suggest to fit experimental data with a split-peak model. In our simulation, if we choose the vortex center as the origin, we found that $\tilde \nu^E(\mathbf q)$ is dominated by real part. Thus it is better to plot only real part of $\tilde \nu^E(\mathbf q)$. Besides, there should be a sign change at $\mathbf q=(\frac{1}{8}\frac{2\pi}{a},0)$ if we plot Re$\nu^E(q_x)$ along $q_y=0$ cut, as shown in Fig. \[fig:fftx\]. Again, this comes from the Fourier transformation of $\cos(\theta)$. [0.45]{} ![$\|\tilde \nu^E(q)\|$ with $E=30$ meV for PDW-Driven and CDW-Driven Models.[]{data-label="fig:fft"}](image/cdw_fft.png "fig:"){width="\textwidth"} [0.45]{} ![$\|\tilde \nu^E(q)\|$ with $E=30$ meV for PDW-Driven and CDW-Driven Models.[]{data-label="fig:fft"}](image/phase0.png "fig:"){width="\textwidth"} [0.45]{} ![$\|\tilde \nu^E(q)\|$ with $E=30$ meV for PDW-Driven and CDW-Driven Models.[]{data-label="fig:fft"}](image/phase1.png "fig:"){width="\textwidth"} Direct Visualization of “Dislocation” ------------------------------------- To have a direct visualization of profile shown in Eq. \[eq:pdw\_profile\] for a PDW-Driven model, we need to extract local CDW order parameter $\mathlarger \nu^E_{\mathbf{Q_a/2}}(x,y)$ from STM data $\nu^E(x,y)$. For each position $(x_0,y_0)$, we construct a new image by multiplying a gaussian mask: $$\bar \nu^E(\mathbf{r};\mathbf{r_0})=e^{-\frac{\|\mathbf{r}-\mathbf{r_0}\|^2}{2W^2}} \nu^E(\mathbf{r}) \label{eq:local_order}$$ We found that $W=8$ is a good choice in our simulation. Then we can extract local CDW order parameter $\mathlarger \nu^E_{\mathbf{Q_a/2}}(\mathbf{r_0})$ by a Fourier Transformation of $\tilde \nu^E(\mathbf{r};\mathbf{r_0})$: $$\mathlarger \nu^E_{\mathbf{Q_a/2}}(\mathbf{r_0})=\sum_{\mathbf{r}} \mathlarger{\bar \nu^E}(\mathbf{r};\mathbf{r_0})e^{-\frac{1}{2}i \mathbf{Q_a}\cdot \mathbf{r}}$$ After extracting $\mathlarger \nu^E_{\mathbf{Q_a/2}}(\mathbf{r_0})$ for each position, we can easily visualize it and decide whether there is a superposition of strength $\pm 1$ dislocations. The above algorithm can also be implemented by filter algorithm in momentum space directly as in Ref. : $$\mathlarger \nu^E_{\mathbf{Q_a/2}}(\mathbf{r_0})=\sum_{\mathbf q} \tilde \nu^E(\mathbf q)G(\mathbf{Q_a/2}-\mathbf q)e^{-i\mathbf{(Q_a/2-q)}\cdot \mathbf r_0}$$ where the filter is $G(\mathbf q)=\sum_re^{-\frac{\|\mathbf{r}\|^2}{2W^2}}e^{-i \mathbf q \cdot \mathbf r}=e^{-\frac{W^2}{2}\|\mathbf q\|^2}$. Here we show visualization for simulated data of $\|\mathlarger \nu^E_{\mathbf{Q_a/2}}\|^2$ from both CDW-Driven and PDW-Driven model in Fig. \[fig:local\]. The distinction is very obvious. For CDW-Driven model, $\nu^E_{\mathbf{Q_a/2}}$ has maximal intensity at vortex center. For PDW-Driven model, $\|\nu^E_{\mathbf{Q_a/2}}\|$ vanishes along a line across the vortex center in the direction of $\theta_a\pm\frac{\pi}{2}$, in agreement with a $\cos(\theta-\theta_a)$ angle dependence. Across the dark line, phase of local amplitude $\nu^E_{\mathbf{Q_a/2}}$ has a $\pi$ shift, as shown in Fig. \[fig:phase\_plot\]. We can see the phase of $\nu^E_{\mathbf{Q_a/2}}$ is $\phi_a$ or $\phi_a+\pi$. Therefore we can remove the overall phase by $\nu^E_{\mathbf{Q_a/2}}\rightarrow \nu^E_{\mathbf{Q_a/2}}e^{-i\phi_a}$ and make it real. Then angle dependence $\nu^E_{\mathbf{Q_a/2}}\sim \cos(\theta-\theta_a)$ can be visualized directly in Fig. \[fig:angle\]. For uni-directional PDW, Wang et al.[@wang2018] also noted the phase jump by $\pi$ by tracking the position of the DOS peaks in real space[@wang2018]. In Fig. \[fig:fixy\], we plot $Re \nu^E_{\mathbf{Q_x/2}}(x)$ at fixed y. For $y=0$, $\|\nu^E_{\mathbf{Q_x/2}}(x)\|$ gives the radius dependence $F(r)$. We can see that the maximum is at finite $r$. However, our simulation may overestimate the maximum because of boundary effects due to finite size. Finally, we comment on challenges to apply this algorithm to real experimental data. (1) The existence of multiple vortices and impurities modifies the $\cos(\theta-\theta_a)$ angle dependence. In general, there is no time reversal symmetry or any lattice symmetry left, and $\mathlarger \nu^E_{\mathbf{Q_a/2}}(\mathbf{r_0})$ is complex. Thus the line of zero we predicted in the simple model may not be exact. We still expect the real and imaginary parts of $\mathlarger \nu^E_{\mathbf{Q_a/2}}(\mathbf{r_0})$ to each have a line of zero but the lines will no longer coincide. As a result the line of zero’s shown in Fig. \[fig:local\](c-f) will partially fill in. (2) There is smooth background, which will add an offset to the $\cos(\theta-\theta_a)$ factor. If we assume background is smooth, it can be subtracted with sophisticated data analysis technique. [0.4]{} ![$\nu^E_{\mathbf{Q_a/2}}$ for PDW-Driven model with $\theta_x=0$ and $\theta_y=\frac{\pi}{2}$. []{data-label="fig:pdw_angle"}](image/angle_plot.png "fig:"){width="\textwidth"} [0.4]{} ![$\nu^E_{\mathbf{Q_a/2}}$ for PDW-Driven model with $\theta_x=0$ and $\theta_y=\frac{\pi}{2}$. []{data-label="fig:pdw_angle"}](image/fixy.png "fig:"){width="\textwidth"} [0.4]{} ![$\nu^E_{\mathbf{Q_a/2}}$ for PDW-Driven model with $\theta_x=0$ and $\theta_y=\frac{\pi}{2}$. []{data-label="fig:pdw_angle"}](image/Angle.png "fig:"){width="\textwidth"} [0.2]{} ![$\|\nu^E_{\mathbf{Q_a/2}}\|^2$ from CDW-Driven and PDW-Driven Models. $(a)$ and $(b)$ are from a CDW-Driven model; Others are from PDW-Driven models. $E=30$ meV.[]{data-label="fig:local"}](image/cdw_x_intensity.png "fig:"){width="\textwidth"} [0.2]{} ![$\|\nu^E_{\mathbf{Q_a/2}}\|^2$ from CDW-Driven and PDW-Driven Models. $(a)$ and $(b)$ are from a CDW-Driven model; Others are from PDW-Driven models. $E=30$ meV.[]{data-label="fig:local"}](image/cdw_y_intensity.png "fig:"){width="\textwidth"} [0.2]{} ![$\|\nu^E_{\mathbf{Q_a/2}}\|^2$ from CDW-Driven and PDW-Driven Models. $(a)$ and $(b)$ are from a CDW-Driven model; Others are from PDW-Driven models. $E=30$ meV.[]{data-label="fig:local"}](image/x_intensity.png "fig:"){width="\textwidth"} [0.2]{} ![$\|\nu^E_{\mathbf{Q_a/2}}\|^2$ from CDW-Driven and PDW-Driven Models. $(a)$ and $(b)$ are from a CDW-Driven model; Others are from PDW-Driven models. $E=30$ meV.[]{data-label="fig:local"}](image/y_intensity.png "fig:"){width="\textwidth"} [0.2]{} ![$\|\nu^E_{\mathbf{Q_a/2}}\|^2$ from CDW-Driven and PDW-Driven Models. $(a)$ and $(b)$ are from a CDW-Driven model; Others are from PDW-Driven models. $E=30$ meV.[]{data-label="fig:local"}](image/phase1_x_intensity.png "fig:"){width="\textwidth"} [0.2]{} ![$\|\nu^E_{\mathbf{Q_a/2}}\|^2$ from CDW-Driven and PDW-Driven Models. $(a)$ and $(b)$ are from a CDW-Driven model; Others are from PDW-Driven models. $E=30$ meV.[]{data-label="fig:local"}](image/phase1_y_intensity.png "fig:"){width="\textwidth"} Flux Density Wave ----------------- In PDW-Driven scenario, we will also get flux density wave. Orbital magnetic moment of each plaquette $M(\mathbf{r})$ can be estimated through the following equation: $$\begin{aligned} M(\mathbf{r}+\frac{\hat{x}}{2}+\frac{\hat{y}}{2})&=\frac{a^2}{4}\big(I(\mathbf{r},\mathbf{r+\hat{x}})+I(\mathbf{r+\hat{x}},\mathbf{r+\hat{x}+\hat{y}})\notag\\ &+I(\mathbf{r+\hat{x}+\hat{y}},\mathbf{r+\hat{y}})+I(\mathbf{r+\hat{y}},\mathbf{r})\big)\end{aligned}$$ where $a=3.5\textup{\AA}$ is lattice constant. $I(\mathbf{r},\mathbf{r+\hat{r}_a})$ is current going through bond from $\mathbf{r}$ to $\mathbf{r+\hat{r}_a}$ where $a$ denotes $x$ or $y$. $M(\mathbf{r})$ has density wave with momentum $\mathbf{Q_x/2}=(\frac{2\pi}{8},0)$, $\mathbf{Q_y/2}=(0,\frac{2\pi}{8})$. There is also density wave at diagonal direction $Q_{\pm,\pm}=(\pm \frac{2\pi}{8},\pm \frac{2\pi}{8})$. Real space and momentum space pattern of magnetic moment are shown in Fig. \[fig:fdw\]. Amplitude of density wave at momentum $(\frac{2\pi}{8},\frac{2\pi}{8})$ is around $0.005\mu_B$ and may be possible to be detected by neutron scattering experiment. The observation of flux density wave at this wave-vector offers the opportunity to definitively settle the question of uni-directional vs bi-directional PDW. [0.4]{} ![Flux Density Wave pattern from PDW-Driven model in vortex halo.[]{data-label="fig:fdw"}](image/fdw_real.png "fig:"){width="\textwidth"} [0.4]{} ![Flux Density Wave pattern from PDW-Driven model in vortex halo.[]{data-label="fig:fdw"}](image/fdw_fft.png "fig:"){width="\textwidth"} Other Types of PDW ------------------ This paper is mainly focused on bidirectional PDW model. However, other types of PDW state have been proposed before. In this section, we show signatures for Unidirectional PDW and Canted PDW model. Therefore STM experiments can rule out or support these kinds of PDW models. For Unidirectional PDW shown in Fig. \[fig:uni\_fft\] with only $x$ component, Fourier Transform data only show peak at $\mathbf{Q_x/2}$, not at $\mathbf {Q_y/2}$. There is still split of peak consistent with our previous discussions for bidirectional PDW. For canted PDW, we expect the peak in $\nu^E(q)$ deviates from $(1,0)$ and $(0,1)$ direction. For canted PDW model with shifted momentum $p=0.032\pi/a$: $\mathbf{P_1}=(\frac{2\pi}{8},p)$, $\mathbf{P'_1}=(-\frac{2\pi}{8},p)$ and $\mathbf{P_2}=(p,\frac{2\pi}{8})$,$\mathbf{P'_2}=(p,-\frac{2\pi}{8})$ this shift shows up in Fig. \[fig:canted\_fft\]. Because of condition $\tilde \nu^E(q)=\tilde \nu^E(-q)^$, we see double peak with shift $p$. In experiment it may be better to detect this feature with complex amplitude $\tilde \nu^E(q)$ instead of intensity $\|\tilde \nu^E(q)\|$. ![$\|\tilde \nu^E(q)\|$ for unidirectional PDW with phase $\theta_x=0$.[]{data-label="fig:uni_fft"}](image/uni_fft.png){width="40.00000%"} ![$\|\tilde \nu^E(\mathbf{q})\|$ for canted PDW with shifted momentum $p=0.032\pi/a$. Phase of PDW is $\theta_x=0$ and $\theta_y=\frac{\pi}{2}$.[]{data-label="fig:canted_fft"}](image/canted_fft.png){width="40.00000%"} If we can decide the value of shift momentum $\|p\|$ from Fourier Transformation data, then we can extract local order parameter $\nu^E_{P}(\mathbf{r})$ with $P_\pm=(\frac{2\pi}{8},\pm p)$ following Eq. \[eq:local\_order\]. It turns out that $P=(\frac{2\pi}{8},p)$ has an anti-vortex while $P=(\frac{2\pi}{8},-p)$ has a vortex, as shown in Fig. \[fig:vortex\_canted\]. [0.22]{} ![$\arg \nu^E_P(\mathbf{r})$ for canted PDW in unit of $\pi$.[]{data-label="fig:vortex_canted"}](image/shifted_x_phase.png "fig:"){width="\textwidth"} [0.22]{} ![$\arg \nu^E_P(\mathbf{r})$ for canted PDW in unit of $\pi$.[]{data-label="fig:vortex_canted"}](image/anti-shifted_x_phase.png "fig:"){width="\textwidth"} If momentum resolution is not good enough to decide the value of $p$, we propose to visualize $\nu^E_{P_0}(\mathbf(r))$ with $P_0=(\frac{2\pi}{8},0)$. If it is ordinary PDW-Driven, we get similar plot as in Fig. \[fig:phase\_plot\]. If it is canted PDW-Driven, we will get strange position dependence of $\arg \nu^E_{P_0}(\mathbf r)$ like in Fig. \[a\_canted\_unshift\_plot\]. This is a signature of canted PDW and it’s consistent with the following equation: $$\nu^E_{P_0}(\mathbf r)\sim \cos(\theta-py) \label{eq:simulated_canted}$$ [0.22]{} ![Visualization of $\arg \nu^E_{P_0}(\mathbf r)$ for canted PDW Driven model. $P_0=(\frac{2\pi}{8},0)$ and shifted-momentum is $p=0.032\pi$.[]{data-label="fig:canted-unshift_plot"}](image/simulated_angle.png "fig:"){width="\textwidth"} [0.22]{} ![Visualization of $\arg \nu^E_{P_0}(\mathbf r)$ for canted PDW Driven model. $P_0=(\frac{2\pi}{8},0)$ and shifted-momentum is $p=0.032\pi$.[]{data-label="fig:canted-unshift_plot"}](image/x_phase.png "fig:"){width="\textwidth"} Summary ======= We now summarize some of the conclusions from the discussion in previous sections. The observation of period 8a bidirectional charge order in the vortex halo directly means that there are induced order parameters $\rho_{Q_x/2}, \rho_{Q_y/2}$. In the presence also of a non-zero superconducting order parameter $\Delta_d$ of the usual $d$-wave superconductor, the period-8 charge order necessarily implies that there are also period-8 modulations in the superconducting order parameter $\Delta_{Q_x/2}, \Delta_{Q_y/2}$, [i.e]{}, Pair Density Wave order at the same period. Given this obvious equivalence in the superconductor between charge and pairing modulations, it may seem to be a moot question whether what is observed is primarily charge order or pair order at period-8. Nevertheless we have shown that there are two distinct possibilities for the observed period-8 order which naturally correspond to two distinct driving mechanisms. In the CDW-driven scenario, we simply postulate that there are slow fluctuations of a previously unidentified period-8 CDW in the uniform superconductor. In the vicinity of the vortex the breaking of translational symmetry and the weakening of the superconducting order may then pin the fluctuations of the period-8 CDW and lead to static ordering. Period-4 charge order then appears as a subsidiary order. In this scenario it is natural to expect that the phase of the induced CDW order does not wind on going around the vortex core. In the PDW-driven scenario on the other hand, we postulate that there are slow fluctuations of period-8 PDW that are pinned in the vortex halo. The induced period-8 CDW then will have a strength $\pm 1$ dislocation centered at the vortex core. More precisely the induced period-8 CDW will be a superposition of a configuration with a strength $+1$ dislocation and one with a strength $-1$ dislocation. This leads to a rather different spatial profile for the induced period-8 CDW. A further difference is that there are now two distinct kinds of induced period-4 CDW orders which we have referred to as CDW$_A$ and CDW$_B$. The CDW$_A$ pattern has no winding around the vortex core while the CDW$_B$ pattern is a superposition of strengths-$\pm 2$ dislocations. We discussed the extent to which existing data supports either scenario. In particular in the PDW-driven scenario there is a natural explanation for the absence of peaks at $2\pi \left(\frac{1}{8}, \frac{1}{8}\right)$ as reported in the experiments. It is however important to analyze the data more carefully to clearly establish which of these scenarios is realized, and we described a number of distinguishing features. Most importantly the spatial profile of the induced charge orders due to the dislocation structure in the PDW-driven scenario should be discernible using the methods we describe. Note that within either of these scenarios there is no general reason for a predominantly $d$-form factor period-$8$ charge order to induce only an $s$-form factor period-4 charge order [@Note1]. From our numerical simulation of $d$ wave PDW coexistence with uniform $d$ wave superconductor, period $8$ CDW we get is actually dominated by $d$ wave, instead of $s$ wave from naive expectation. Thus we do not have a natural explanation of the observations on form factors in the experiments. A further question that one can ask is whether the fluctuation order that is pinned on the halo is unidirectional or bidirectional. The observed period-$8$ modulations are apparently bidirectional. The simplest explanation therefore is that the “parent’ order is also bidirectional. However one may postulate that there are domains of different unidirectional patches within the vortex halo. This may be easy to check in the STM data. Finally an important question is whether the period-8 PDW (if it is really the driver) is merely a competing/intertwined order with the standard $d$-wave superconductor or whether it is a “mother" state with a very large amplitude that controls the physics up to a much larger energy scale than the standard $d$-wave order itself. Just based on the STM experiments alone there does not seem to be any clear way to answer this question. However in the following section, by combining with information from other existing experiments, we will provide suggestive arguments in favor of a mother PDW state. A broader perspective on PDW and its relation to the pseudogap state of the Cuprates ==================================================================================== In this section we take a broader perspective and ask whether the message learned from the STM data on Bi-2212 can inform us on anomalies observed in other cuprates and more generally on the pseudo-gap itself. We shall assume the that the data are described by the fluctuating PDW (“mother state”) scenario and we shall assume that scenario continues to hold in other under-doped Cuprates. We focus our attention on YBCO where extensive data on the CDW up to high magnetic field is available[@changNatureComm72016magnetic; @ZX1science350949gerber2015three; @ZX2PNAS11314647jang2016ideal]. The picture that emerges from these studies is that SRO CDW appears below about 150K over a doping range between x=0.09 and 0.16[@BlancoPhysRevB.90.054513]. This SRO CDW has very weak interlayer ordering centered around L=1/2 where L is the c axis wave-vector in reciprocal lattice unit. These peaks grow with decreasing temperature but their strength weaken and their in plane linewidth broaden below $\text{T}_c$. These peaks occur along both a and b axes. Above a field of 15 to 20T, a uni-directional CDW emerges and rapidly becomes long range along the b axis. The onset of long range ordered CDW is consistent with earlier NMR data.[@wu2013emergence; @wu2015incipient] At the same time, the SRO CDW remains along both a and b axes. Thus the high magnetic field data shows that there are two kinds of CDW with the same incommensurate period which does not change with magnetic field. As the experimentalists remarked[@ZX1science350949gerber2015three; @ZX2PNAS11314647jang2016ideal], this is very puzzling because having the same incommensurate wave-vector suggests the two kinds of CDW share a common origin. If we interpret the observed CDW as subsidiary to a fluctuating PDW, the latter must exist above the CDW onset at 150K and most likely above $T^$ which is taken as the thermodynamic signature of the pseudogap. Similarly we take the viewpoint that quantum oscillations require the existence of bi-directional CDW[@sebastian2012towards],which implies that fluctuating PDW extends to magnetic fields of 100T and beyond. By continuity we expect fluctuating PDW to cover a large segment of the H-T plane, as shown in Fig. \[fig:phase\_diagram\]. The PDW must be strongly fluctuating in time, because there is no sign of superconductivity from transport measurements outside of a limited region near $T_c$ and $\text{H}_{c2}$. However, diamagnetic signals are observed over a much larger regime[@yuPNAS126672016magnetic], a point which we shall return to later. Nevertheless, our picture is that the subsidiary orders such as CDW can be more robust and make their presence felt. This is particularly true of $CDW_A$ (see Eq. \[Eq: CDWA\]) which does not require d wave pairing for its presence. So we assign $CDW_A$ to be the SRO CDW which onsets below 150K, as shown by the dashed line in Fig. \[fig:phase\_diagram\]. Below $T_c$ the phase stiffness of the LRO d wave robs oscillator strength from the PDW, diminishing its already weak phase stiffness even further. This explains the reduction of the CDW strength below $T_c$. On the other hand, we saw in section III that in a magnetic field a vortex can pin the PDW to form a static but short range halo around the core. This in turn induces CDW at wave-vector Q/2 and its harmonic $CDW_B$. All these states are located roughly inside the superconducting region as indicated in Fig. \[fig:phase\_diagram\]. Of course being tied to the vortices mean that the strengths of these states are proportional to the magnetic field. Note that we expect the d wave phase stiffness to be reduced inside the halo while that of the PDW to be strengthened. We define the field $H_0$ as H\_0=\_0/(2\^2\_P) where $\phi_0 = hc/2e$ is the flux quanta in a superconductor, $\xi_P$ is the correlation length of the pinned PDW. The $2\pi$ in the denominator has been inserted to make this equation resemble the definition of $H_{c2}$ and the exact numerical factor should not be taken seriously. The point is to provide a scale for the field where the pinned PDW starts to strongly overlap. For $H>H_0$, the d wave superconductor is being squeezed out and the PDW phase regains its stiffness. It eventually becomes depinned as the d wave pairing diminishes and resumes its dynamical fluctuation. In this region the $CDW_A$ grows in strength and coherence, recovering the growth with decreasing temperature that was interrupted by the onset of $T_c$ for $H<H_0$. The fact that the LRO CDW is uni-directional even though the PDW is bi-directional can be rationalized by the following argument. There is a term in the Landau free energy $\gamma \|\rho_{Q_x}\|^2 \|\rho_{Q_y}\|^2$ which strongly prefers uni-directional order when $\gamma$ is large and positive. In YBCO the presence of the chain already broke tetragonal symmetry to begin with, making it even more plausible that the order grow strongly in one direction. On the other hand, the term $\Delta_P \Delta_{-P}^* \rho_Q$ is linear in $\rho_Q$, meaning that some SRO is likely generated in the orthogonal direction. We shall return to this point later. Returning to the region below $\text{H}_{c2}$ we expect to find the pinned PDW and the CDW with period Q/2 as static but short range ordered. This is because the static order of Q/2 CDW requires the static order of d wave pairing as well as PDW. The Q/2 CDW should persist to lower field with decreasing amplitude. It may be expected to have correlation length similar to that found in the STM experiment, which we estimate to be about 16 lattice spacings. It will of course be of great interest to search for this by X-ray scattering. On the other hand, the period Q $\text{CDW}_B$ can be thought of as a harmonic of the period Q/2 CDW, but it can exist even in its absence. Thus we expect it to exist up to higher field. We do not know exactly how high a field it can persist to, but it cannot go above the d wave vortex liquid regime. It is worth noting that in practice there can be remnants of static pinned vortices even above $H_c2$. Yu et al[@yuPNAS126672016magnetic] reported hysteretic behavior which extends to very high field at low temperatures, leading them to identify a second vortex solid regime. The existence of some form of bi-directional CDW that persists up to high field at low temperature is important in order to explain the quantum oscillations. We believe the LRO unidirectional CDW cannot by itself give rise to quantum oscillations, but the combination with some SRO CDW in the direction perpendicular to it may be sufficient. This can come from the bi-directional $\text{CDW}_B$ discussed above if it persists to high field, or it is possible that a short range order $\text{CDW}_A$ is generated along direction $a$ at higher field as explained earlier. In support of the picture outlined above, we note that there is extensive NMR data showing that $\text{H}_0$ is typically 5 to 10T below the $\text{H}_{c2}$ as measured by transport[@Julien2arXivzhou2017spin; @wu2013emergence]. Thus there is a close relationship between $\text{H}_{c2}$ and the vortex halo size as defined by the size of the pinned PDW. We also recall that the CDW that we identify as type A in Bi-2212 is uni-directional, which agrees with this assignment for YBCO. We note that the Bi-2212 sample used has a doping of 0.17 which lies on the upper end of the observability of CDW in YBCO samples. The $\text{H}_{c2}$ and corresponding $\text{H}_0$ are expected to be very high. So the 8.25T used in the STM experiment is expected to be far below the regime where $\text{CDW}_A$ can achieve long range order. In Fig. \[fig:phase\_diagram\] we add the line $\text{H}_0$ to a phase diagram in the H-T plane for under-doped Cuprates, following the proposal of Yu et al[@yuPNAS126672016magnetic]. The resistive $\text{H}_{c2}$ is the boundary of the vortex solid and marks the resistive transition. (To avoid cluttering, we did not show the emergence of a second vortex solid regime mentioned earlier that extends to high field at low temperature[@yuPNAS126672016magnetic].) The key point made by Yu et al is that there is a large region of vortex liquid in the phase diagram where there is strong superconducting amplitude. The evidence for this is a strong diamagnetic signal. Given the small size of the true vortex core where the d wave coherence peak is destroyed, it is reasonable to interpret the vortex liquid as a region of strong d-wave superconducting amplitude with dynamical vortices that persists to very high field. It is less certain how high in temperature the d-wave vortex liquid extend. It is possible that the diamagnetic signal may come from PDW fluctuations at high fields[@yuPNAS126672016magnetic; @lee2014amperean]. Thus the location of the dotted line in Fig. \[fig:phase\_diagram\] that indicate the extent of d wave vortex liquid is quite uncertain, especially in the temperature direction. ![$H-T$ Phase Diagram for an underdoped Cuprate. The light blue shading indicates that a fluctuating PDW is pervasive over a large segment of the $H-T$ plane for underdoped Cuprates. Dashed line indicates the onset of short range ordered CDW at wave-vector Q. It is a subsidiary order of the PDW which we refer to as $\text{CDW}_A$. Sold red line marks the magnetic field $H_0$ as defined in Eq.(14) in terms of the coherence length $\xi_P$ of the PDW which marks the size of the vortex halo. It is closely related to the field $H_{c2}$ which marks the onset of a vortex solid phase and LRO superconductivity. Within this phase and inside the vortex halo we expect the pinned static PDW, Q/2 CDW as well as its harmonic, a wave-vector Q CDW which we refer to as $\text{CDW}_B$. The $\text{CDW}_B$ short range order state may extend to higher magnetic field much beyond $H_{c2}$. The dotted red line indicates the onset of a vortex liquid phase. The brown area indicates the appearance of long range ordered type A CDW with wave-vector Q. []{data-label="fig:phase_diagram"}](image/phase_diagram.png){width="0.9\linewidth"} ![Illustration of the Loop Current produced by the canted PDW.[]{data-label="fig:loop_current"}](image/loop_current.png){width="0.9\linewidth"} We should mention that similar CDW has been seen in the Hg-based compound. Here the doping range extends further down to $x$ of order 0.06 and up to about 0.12. Another difference is that there is no clear suppression of the CDW at $\text{T}_c$. Instead its strength seems to saturate. It should be noted that unlike YBCO, this is a tetragonal system. From existing X ray data, it is not known whether the CDW is bi-directional or uni-directional. Apart from these differences, the observations seem to fit into the same phase diagram shown in Fig. \[fig:phase\_diagram\]. Finally we comment on the symmetry breaking observed at the $T^$ lines which lies at a temperature above the onset of SRO CDW. This seems to be associated with breaking a lattice symmetry, perhaps a kind of nematic order. Importantly, a recent experiment on the anisotropy of the spin susceptibility[@Matsuda2unpublished] found the nematic axis to be along the diagonal in a single layered Hg-based compound, while it is along the bond direction in YBCO[@MatsudaNaturePhysicssato2017thermodynamic]. This would rule out nematicity based on CDW which should be along the bond direction in a single layer tetragonal system. The observation in YBCO can be understood from the stacking of two orthogonal directions of diagonal nematicity in each layer. Such nematicity agrees with the symmetry of the orbital current model[@aji2009quantum]. As mentioned earlier, in the PDW model it was pointed out by Agterberg et al.[@Agt2PhysRevB.91.054502] that adding canting to the PDW model as described earlier has the same symmetry as the orbital current model. The four different combinations of (p1,p2) give rise to a 4 state clock model. Fluctuations between (1,1) and (-1,-1) restores time reversal symmetry but gives rise to a diagonal breaking of nematic symmetry, just like the orbital current model. Indeed a canted PDW model will carry intra-cell currents as shown in Fig. \[fig:loop\_current\], which is the closest we can get to Varma’s model in a single band model. As seen in this figure, the current can be understood as supercurrent running along x and y, with a return current along one of the diagonal bond. In fact we find that such a current pattern emerges from the PDW model. Without self-consistent determination of the mean field ground state, there is a net current along x and y, which presumably will be fixed by a proper return current in a self-consistent mean field theory. However, the current we find is very small, on the order of $10^{-3} t$ on each bond. This gives rise to a moment of about $10^{-3} \mu_B$ which is too small compared with the 0.1 $\mu_B$ reported by neutron scattering. We note on general ground that the orbital current in the PDW model must be small. Let us define the canted component of the wave-vector as $p = (P + P')/2$. The supercurrent can be estimated from the product of the phase gradient which is $p$ and the spectral weight, which is $x/m$ where $1/m$ is proportional to $ta^2$. Thus we expect the maximal supercurrent to be $x\|p\|t$ where $p$ is in reciprocal lattice units. Since $\|p\|$ should be less than $\|P\|$, we expect $x\|P\|$ to be less than $10^{-2}$ and similarly for the moment in units of $\mu_B$. Thus it is unlikely that the canted PDW model can account for the orbital current observed by neutron. However, it potentially can explain the onset of diagonal nematicity at $T^$. Finally we call attention to the most interesting part of the phase diagram, the region at zero temperature and above $H_{c2}$. In our picture this is a ground state consisting of a PDW which does not order due to quantum fluctuations. This state is metallic with some combination of long range and short range CDW order, sufficient to form pockets visible by quantum oscillations. What is the nature of this state? Is it a Fermi liquid? Is the dissipation due to the metallic state responsible for quantum disordering the PDW? These are fascinating questions that are beyond the purview of the present phenomenology oriented paper. Conclusion ========== Based on our analysis, we come to the following conclusions: 1\. It is likely that the 8a CDW observed in the STM experiment has its origin in a period 8 PDW which is pinned to be static near the vortex core. The main evidence based on the currently available data is the absence of a peak at (1/8,1/8) which would be expected if the 8a CDW were primary. We propose further analyses of the data which can nail down this conclusion. The main point is that the winding of the d wave superconducting phase around the vortex core imprints a very special signature on the period 8 CDW which is visible either as a splitting of the Fourier transform peak or a sign change across an oriented line in the Fourier filtered data. 2\. We think it is likely that the PDW pinned near the vortex core is bi-directional, because both the 8a and 4a CDW observed there appears to be bi-diagonal. A bi-diagonal PDW can generate uni-directional CDW but the converse is not true: a uni-directional PDW may be able to generate checkerboard patterns made up of patches of uni-directional stripe CDW, but that distinction should be amenable to experimental test. 3\. The naive expectation that the subsidiary 4a order has local $s$ symmetry is not generally correct, given the definition of the form factor used in the STM experiments[@Note1]. In fact, in our microscopic mean field model, we find these to have mainly $d$ symmetry. The local symmetry depends on the microscopic detail and it is no surprise that it is not captured by our simple mean-field theory, but we want to convey the message that a $d$ symmetry subsidiary order can readily be generated. Thus the observed $d$ symmetry CDW that is already present at zero field may also be a subsidiary order due to PDW. For Bi-2201 the CDW is close to commensurate with period 4 and we cannot rule out that this CDW is not simply an independent order, as advocated in a recent preprint. [@wang2018] On the other hand the idea of independent order is difficult to justify for YBCO, where two different CDW seem to co-exist with the same incommensurate period. We discuss a scenario where both CDW’s are subsidiary to the same PDW. 4\. Up to now the notion of a halo around a vortex core is not a well-defined one. The coherence peak associated with d wave superconductivity is killed only inside the true core, which has a radius of 2 or 3 lattice spacing. The coherence peak remains visible throughout the halo region, indicating that d wave order is not fully destroyed. We propose that the size of the pinned PDW provides a way to define the halo radius and we introduce a magnetic field scale $H_0$ associated with this length scale. We relate this field scale to the growth of the 4a CDW observed in underdoped YBCO samples and with $H_{c2}$. 5\. A canted PDW is an attractive scenario that can unify the pseudo phenomenology with the nematic transition observed at $T^$. The STM data offers a way to search for this kind of order, even though the required resolution may be challenging. In summary we answer the question we first posed in the introduction: we think that that the observed period 8 CDW is opening a new window into the world of underdoped cuprates and pseudogap physics. Much exciting further developments are sure to come. Acknowledgement =============== We thank J. C. Davis and M. Hamidian for sharing with us their data prior to publication and for very helpful discussions. PAL acknowledges the support of NSF under DMR-1522575. TS is supported by a US Department of Energy grant DE-SC0008739, and in part by a Simons Investigator award from the Simons Foundation. We thank the Moore Foundation EPiQS program for facilitating our interaction with J. C. Davis. TS thanks the conference on High Temperature Superconductivity at the Aspen Center for Physics, which is supported by NSF grant PHY-1607611, for enabling a part of this work. Numerical calculation of band structure of PDW state ==================================================== For uniform PDW state, we calculate the band structure by diagonalize a BdG Hamiltonian $H(k)$ for each momentum $k$. At each $k$, we need to use a $812=162$ basis:$\Psi_k=(\psi_\uparrow(k),\psi_\downarrow^\dagger(-k))$. $\psi_\sigma(k)$ is a collection of $9\times9=81$ electron annihilation operators: $c_{k'}$ with momenta $k'=k+m \mathbf{P_x}+n\mathbf{P_y}$ where $\mathbf{P_x}\approx(\frac{2\pi}{8},0)$ and $\mathbf{P_y}\approx(0,\frac{2\pi}{8})$, $m,n=-4,-3,-2,-1,0,1,2,3,4$. In Sec. \[Sec: PDW with long range order\] we use $\mathbf{P_x}\approx(0.14\times(2\pi),0)$ and $\mathbf{P_x}\approx(0,0.14\times(2\pi))$. We set a large truncation for m and n to better capture the effect of subsidiary CDW generated by PDW. In this basis, we rewrite the mean field Hamiltonian in Eq. \[Eq: long range PDW mean field\] at momentum $k$ as H\_k &=& \_[m,n]{}\_[k+m +n]{} c\^\_[k+m +n,]{}c\_[k+m +n,]{}\ &-&\_[m,n]{}\_[-k-m -n]{} c\_[-k-m -n,]{}c\^\_[-k-m -n,]{}\ &+&\_[m,n]{}2((k\_x+m P\_x+nP\_y - P\_x/2) - (k\_y+m P\_x+n P\_y)) c\_[k+m +n,]{}c\_[-k -m -n + ,]{}\ &+&\_[m,n]{}2((k\_x+m P\_x+nP\_y + P\_x/2) - (k\_y+m P\_x+nP\_y)) c\_[k+m +n,]{}c\_[-k -m -n - ,]{}\ &+&\_[m,n]{}2((k\_x+m P\_x+nP\_y) - (k\_y+m P\_x+n P\_y- P\_y/2)) c\_[k+m +n,]{}c\_[-k -m -n + ,]{}\ &+&\_[m,n]{}2((k\_x+m P\_x+nP\_y) - (k\_y+m P\_x+nP\_y+ P\_y/2)) c\_[k+m +n,]{}c\_[-k -m -n - ,]{}\ &+& h.c., where $\D = 45$meV. For the bare band dispersion $\e_k$, we use a tight banding model on square lattice with nearest neighbor hopping $t=0.21$eV, second neighbor hopping $t_p=-0.047$eV, third neighbor hopping $t_{pp}=0.04$eV and fourth neighbor hopping $t_{ppp}=-0.01$eV. \_k = -2t((k\_x)+(k\_y)) - 4t\_p(k\_x)(k\_y) - 2t\_[pp]{}((2k\_x)+(2k\_y))\ - 4t\_[ppp]{}((2k\_x)(k\_y) + (k\_x)(2k\_y)) - \_0 We fix the chemical potential $\e_0$ self-consistently to match the hole doping. Numerical simulation of d wave vortex halo ========================================== We did exact diagonalization to simulate Local Density of State(LDoS) inside Vortex Halo. Our Hamiltonian for PDW-Driven Model is: $$H_P=H_0+\sum_{\mathbf x,\mathbf{\mu}}F_d(\mu)\left(\|\Delta_D\|e^{i\theta_d+i\theta}+\left(\sum_a\|\Delta_{P_a}\|e^{i\theta_a+i\theta_d}\sin(\frac{1}{2}\mathbf{Q_a}\cdot (\mathbf{x+\frac{\mu}{2}}))\right)\right)c^\dagger_\uparrow(\mathbf x)c^\dagger_\downarrow(\mathbf{x+\mu})+h.c. \label{eq:pdw_real}$$ where $\mathbf \mu=\hat{x}$ or $\hat{y}$ labels two different kinds of nearest neighbor bond. $F_d(\hat x)=1$ and $F_d(\hat y)=-1$. $a$ means $x$ or $y$. We used $\|\Delta_{P_x}\|=\|\Delta_{P_y}\|=30$meV at vortex center in our calculation, away from vortex center the PDW profile is $$\Delta_P(r)=30e^{1-\sqrt{r^2+\xi^2}/\xi} meV$$ with $\xi=15$ Our Hamiltonian for CDW-Driven Model is: $$H_C=H_0+\sum_{\mathbf x,\mathbf{\mu}}F_d(\mu)\|\Delta_D\|e^{i\theta_d+i\theta}c^\dagger_\uparrow(\mathbf x)c^\dagger_\downarrow(\mathbf{x+\mu})+\sum_{\mathbf x,\mathbf{\mu}}F_s(\mu)\left(\sum_a\|\Delta_{C_a}\|e^{i\theta_a}\sin(\frac{1}{2}\mathbf{Q_a}\cdot (\mathbf{x+\frac{\mu}{2}}))\right)\sum_\sigma c^\dagger_\sigma(\mathbf x)c_\sigma(\mathbf{x+\mu})+h.c.$$ where $F_s(\hat x)=F_s(\hat y)=1$ is a $s$ wave form factor. We used $\|\Delta_{C_x}\|=\|\Delta_{C_y}\|=30$meV at vortex center in our calculation. Away from vortex center CDW has a profile similar to PDW-Driven model: $$\Delta_C(r)=30e^{1-\sqrt{r^2+\xi^2}/\xi} meV$$ For both PDW-Driven and CDW-Driven model, we use $\|\Delta_D\|=20$meV far away from vortex core and $\Delta_D(r,\theta)=20 \frac{r}{\sqrt{r^2+r_0^2}}$ meV near vortex core. We add one d-wave vortex to a $100 a \times 100 a$ square lattice with open boundary condition. $\mathbf{Q_x/2}=(\frac{2\pi}{8},0)$ and $\mathbf{Q_y/2}=(0,\frac{2\pi}{8})$. After Exact Diagonalization, we can easily get on-site LDoS at any energy: $$\rho(\mathbf x, \omega)=\sum_{E,\sigma} \delta(\omega-E)\psi^_E(\mathbf x;\sigma)\psi_E(\mathbf x;\sigma)$$ where $E$ labels all energy levels and $\psi_E(x;\sigma)$ is the wavefunction for $\mathbf x$ site and spin $\sigma$ at energy level $E$. For STM experiment, LDoS at Oxygen site is actually more important. In our simple one band model, we can define bond LDoS: $$\rho_{\mu}(\mathbf x, \omega)=\sum_{E,\sigma} \delta(\omega-E)\left(\psi^_E(\mathbf x;\sigma)\psi_E(\mathbf {x+\mu};\sigma)+\psi^*_E(\mathbf {x+\mu};\sigma)\psi_E(\mathbf x;\sigma)\right)$$ where $\mu=\hat x$ or $\hat y$. It’s then easy to define $s$ wave Bond LDoS as $$\rho_d(\mathbf x,\omega)=\rho_{\hat x}(\mathbf x,\omega)+\rho_{\hat y}(\mathbf x, \omega)$$ and $d$ wave Bond LDoS as $$\rho_s(\mathbf x,\omega)=\rho_{\hat x}(\mathbf x,\omega)-\rho_{\hat y}(\mathbf x, \omega)$$ For PDW-Driven model, we found $\rho_d$ dominates and therefore we only show $d$ wave Bond DoS in the main text. For our CDW-Driven model, it’s dominated by $s$ wave CDW as an input and we show $s$ wave CDW in the main text. [^1]: These two authors contributed equally [^2]: These two authors contributed equally [^3]: In momentum space, there are two amplitudes $A^x_a$ and $A^y_a$ at momentum $\mathbf{Q_a/2}$ which correspond to density waves in $x$ bond and $y$ bond. Here $a$ denotes $x$ or $y$: $\mathbf{Q_x/2}=(\frac{2\pi}{8},0)$ and $\mathbf{Q_y/2}=(\frac{2\pi}{8},0)$. The definition currently used by the community is to define $A^x_a\pm A^y_a$ as the s/d wave component. However, under $C_4$ rotation, $A^x_x$ transforms to $A^y_y$. Therefore the current definiton of s/d wave form factor is not related to symmetry and generallly they should be mixed. An alternative definition of s vs d wave component is $A^x_x \pm A^y_y$, which is related to the C4 rotation around a particular reference point. However, this definition may not be very useful because if we shift the reference point by half of the period in one direction, what we would define as d wave would becaome s wave. [^4]: There is a redundancy in this definition: we can shift $\theta_x$ and $\phi_x$ ($\theta_y$ and $\phi_y$) both by $\pi$ without changing any physical order parameter. Thus, $\phi_x$ ($\phi_y$) is determined only up to $\pi$ without reference to the choice of $\theta_x$ ($\theta_y$). [^5]: Physically, there is only one electron pocket, the 4 pockets shown in Fig. \[Fig: PDW band stucture, pocket\](c) are copies of the same pocket shifted in momentum, as a consequence of B.Z. folding.	ArXiv
--- abstract: 'This paper proposes a novel approach to regularize the ill-posed blind image deconvolution (blind image deblurring) problem using deep generative networks. We employ two separate deep generative models — one trained to produce sharp images while the other trained to generate blur kernels from lower-dimensional parameters. To deblur, we propose an alternating gradient descent scheme operating in the latent lower-dimensional space of each of the pretrained generative models. Our experiments show excellent deblurring results even under large blurs and heavy noise. To improve the performance on rich image datasets not well learned by the generative networks, we present a modification of the proposed scheme that governs the deblurring process under both generative and classical priors.' bibliography: - 'egbib.bib' title: Blind Image Deconvolution using Pretrained Generative Priors --- Introduction and Related Work {#sec:introduction} ============================= Blind image deblurring aims to recover a true image $i$ and a blur kernel $k$ from blurry and possibly noisy observation $y$. For a uniform and spatially invariant blur, it can be mathematically formulated as $$\label{eq:bid} y = i \otimes k + n,$$ where $\otimes $ is a convolution operator and $n$ is an additive Gaussian noise. In its full generality, the inverse problem is severely ill-posed as many different instances of $i$, and $k$ fit the observation $y$ [@campisi2016blind; @kundur1996blind]. To resolve between multiple instances, priors are introduced on images and/or blur kernels in the image deblurring algorithms. Priors assume an a priori model on the true image/blur kernel or both. These natural structures expect images or blur kernels to be sparse in some transform domain; see, for example, [@chan1998total; @fergus2006removing; @levin2009understanding; @hu2010single; @zhang2011sparse; @cai2009blind]. Some of the other penalty functions to improve the conditioning of the blind image deblurring problem are low-rank [@ren2016image], and total variation based priors [@pan2014motion]. A recently introduced dark channel prior [@pan2016blind] also shows promising results; it assumes a sparse structure on the dark channel of the image, and exploits this structure in an optimization program [@xu2011image] to solve the blind image deblurring problem. Other works include extreme channel priors [@yan2017image], outlier robust deblurring [@dong2017blind], learned data fitting [@pan2017learning], and discriminative prior based blind image deblurring approaches [@li2018learning]. Although generic and applicable to multiple applications, these engineered models are not very effective as many unrealistic images also fit the prior model [@hand2017global]. \ \ Recently deep learning based blind image deblurring approaches have shown impressive results due to their power of learning from large training data [@hradivs2015convolutional; @nah2016deep; @schuler2016learning; @xu2017learning; @nimisha2017blur; @kupyn2017deblurgan]. Generally, these deep learning based approaches invert the forward acquisition model of blind image deblurring via end-to-end training of deep neural networks in a supervised manner. The main drawback of this end-to-end deep learning approach is that it does not explicitly take into account the knowledge of forward map , but rather learns implicitly from training data. Consequently, the deblurring is more sensitive to changes in the blur kernels, images, or noise distributions in the test set that are not representative of the training data, and often requires expensive retraining of the network for a competitive performance [@lucas2018using]. Meanwhile, neural network based implicit generative models such as generative adversarial networks (GANs) [@goodfellow2014generative] and variational autoencoders (VAEs) [@kingma2013auto] have found much success in modeling complex data distributions especially that of images. Recently, GANs and VAEs have been used for blind image deblurring but only in an end-to end manner [@xu2017learning; @nimisha2017blur; @kupyn2017deblurgan] , which is completely different from our approach as will be discussed in detail. These methods show competitive performance, but since these generative model based approaches are end-to-end they suffer from the same draw backs as other deep learning based debluring approaches. On the other hand, pretrained generative models have recently been employed as regularizers to solve inverse problems in imaging including compressed sensing [@bora2017compressed; @shah2018solving], image inpainting [@yeh2017semantic], Fourier ptychography [@shamshad2018deep], and phase retrieval [@shamshad2018robust; @hand2018phase]. However the applicability of these pretrained generative models in blind image deblurring is relatively unexplored. Recently [@gandelsman2018double] employ a combination of multiple untrained deep generative models and show their effectiveness on various image layer decomposition tasks including image water mark removal, image dehazing, image segmentation, and transparency separation in images and videos. Different from their approach, we show the effectiveness of our blind image deblurring method by leveraging trained generative models for images and blurs. In this work, we use the expressive power of pretrained GANs and VAEs to tackle the challenging problem of blind image deblurring. Our experiments in Figure \[fig:intro-results\] confirm that integrating deep generative priors in the image deblurring problem enables a far more effective regularization yielding sharper and visually appealing deblurred images. Specifically, our main contributions are - To the best of our knowledge, this is the first instance of utilizing pretrained generative models for tackling challenging problem of blind image deblurring. - We show that simple gradient descent approach assisted with generative priors is able to recover true image and blur kernel, to with in the range of respective generative models, from blurry image. - We investigate a modification of the loss function to allow the recovered image some leverage/slack to deviate from the range of the image generator. This modification effectively addresses the performance limitation due to the range of the generator. - Our experiments demonstrate that our approach produce superior results when compared with traditional image priors and unlike deep learning based approaches does not require expensive retraining for different noise levels. Problem Formulation and Proposed Solution {#sec:Problem-Formulation} ========================================= We assume the image $i \in \R^n$ and blur kernel $k \in \R^n$ in are members of some structured classes $\mathcal{I}$ of images, and $\mathcal{K}$ of blurs, respectively. For example, $\setI$ may be a set of celebrity faces and $\setK$ comprises of motion blurs. A representative sample set from both classes $\setI$ and $\setK$ is employed to train a generative model for each class. We denote $G_{\mathcal{I}}: \mathbb{R}^l \rightarrow \mathbb{R}^n$ and $G_{\mathcal{K}}: \mathbb{R}^m \rightarrow \mathbb{R}^n$ as the generators for class $\setI$, and $\setK$, respectively. Given low-dimensional inputs $z_i \in \mathbb{R}^l$, and $z_k \in \mathbb{R}^m$, the pretrained generators $G_{\setI}$ and $G_{\setK}$ generate new samples $G_{\setI}(z_i)$, and $G_{\setK}(z_k)$ that are representative of the classes $\setI$ and $\setK$, respectively. Once trained, the weights of the generators are fixed. To recover the sharp image and blur kernel $(i,k)$ from the blurred image $y$ in , we propose minimizing the following objective function $$\begin{aligned} \label{eq:Optimization-Ambient} (\hat{i},\hat{k}) := \underset{\substack{i \in\text{Range}(G_{\setI}) \\k \in\text{Range}(G_{\setK})}}{\text{argmin}} \ \\|y - i \otimes k \\|^2, \end{aligned}$$ where $\\|\cdot\\|$ is the $\ell_2$-distance, Range($G_\setI$) and Range($G_\setK$) is the set of all the images and blurs that can be generated by $G_\setI$ and $G_\setK$, respectively. In words, we want to find an image $i$ and a blur kernel $k$ in the range of their respective generators, that best explain the forward model . Ideally, the range of a pretrained generator comprises of only the samples drawn from the probability distribution of the training image or blur class. Constraining the solution $(\hat{i},\hat{k})$ to lie only in generator ranges forces the solution to be the members of classes $\setI$ and $\setK$. The minimization program in can be equivalently formulated in the lower dimensional, latent representation space as follows: $$\begin{aligned} \label{eq:Optimization-latent} (\hat{z}_i, \hat{z}_k) = \underset{z_i \in \R^l, z_k \in \R^m}{\text{argmin}} \ \\| y - G_{\mathcal{I}}(z_i) \otimes G_\setK(z_k) \\|^2.\end{aligned}$$ This optimization program can be thought of as tweaking the latent representation vectors $z_i$ and $z_k$, (input to the generators $G_{\setI}$, and $G_{\setK}$, respectively) until these generators generate an image $i$ and blur kernel $k$ whose convolution comes as close to $y$ as possible. Incorporating the fact that latent representation vectors $z_i$, and $z_k$ are assumed to be coming from standard Gaussian distributions, we further augment the measurement loss in with $\ell_2$ penalty terms on the latent representations. The resultant optimization program is then $$\label{eq:regularized-program} \underset{z_i \in \R^l, z_k \in \R^m}{\text{argmin}}\ \\| y - G_\setI(z_i) \otimes G_\setK(z_k) \\|^2+ \gamma\\| z_i \\|^2 + \lambda\\| z_k \\|^2,$$ where $\gamma$ and $\lambda$ are free scalar parameters. For brevity, we denote the objective function above by $\setL(z_i,z_k)$. Importantly, the weights of the generators are always fixed as they enter into this algorithm as pretrained models. To minimize this non-convex objective, we begin by initializing $z_i$ and $z_k$ by sampling from standard Gaussian distribution, and resort to an alternating gradient descent algorithm by taking a gradient step in one of these while fixing the other to find a minima $(\hat{z}_i, \hat{z}_k)$. To avoid being stuck in a not good enough local minima, we restart the algorithm with a new random initialization (Random Restarts) when the measurement loss in does not reduce sufficiently after reasonably many iterations. We dubbed proposed deblurring algorithm as Deep Deblur and denote blurry image deblurred via Deep Deblur as $\hat{\textit{{i}}}_\text{DD}$.The estimated deblurred image and the blur kernel are acquired by a forward pass of the solutions $\hat{z}_i$ and $\hat{z}_k$ through the generators $G_{\mathcal{I}}$ and $G_{\mathcal{K}}$. Mathematically, $(\hat{i},\hat{k}) = ( G_\mathcal{I}(\hat{z}_i), G_\mathcal{K}(\hat{z}_k))$. Problem Formulation {#sec:Problem-Formulation} =================== We assume the image $i \in \R^n$ and blur kernel $k \in \R^n$ in are members of some structured classes $\mathcal{I}$ of images, and $\mathcal{K}$ of blurs, respectively. For example, $\setI$ may be a set of celebrity faces and $\setK$ comprises of motion blurs. A representative sample set from both classes $\setI$ and $\setK$ is employed to train a generative model for each class. We denote by the mappings $G_{\mathcal{I}}: \mathbb{R}^l \rightarrow \mathbb{R}^n$ and $G_{\mathcal{K}}: \mathbb{R}^m \rightarrow \mathbb{R}^n$, the generators for class $\setI$, and $\setK$, respectively. Given low-dimensional inputs $z_i \in \mathbb{R}^l$, and $z_k \in \mathbb{R}^m$, the pretrained generators $G_{\setI}$, and $G_{\setK}$ generate new samples $G_{\setI}(z_i)$, and $G_{\setK}(z_k)$ that are representative of the classes $\setI$, and $\setK$, respectively. Once trained, the weights of the generators are fixed. To recover the sharp image, and blur kernel $(i,k)$ from the blurred image $y$ in , we propose minimizing the following objective function $$\begin{aligned} \label{eq:Optimization-Ambient} (\hat{i},\hat{k}) := \underset{\substack{i \in\text{Range}(G_{\setI}) \\k \in\text{Range}(G_{\setK})}}{\text{argmin}} \ \\|y - i \otimes k \\|^2, \end{aligned}$$ where Range($G_\setI$) and Range($G_\setK$) is the set of all the images and blurs that can be generated by $G_\setI$ and $G_\setK$, respectively. In words, we want to find an image $i$ and a blur kernel $k$ in the range of their respective generators, that best explain the forward model . Ideally, the range of a pretrained generator comprises of only the samples drawn from the distribution of the image or blur class. Constraining the solution $(\hat{i},\hat{k})$ to lie only in generator ranges, therefore, implicitly reduces the solution ambiguities inherent to the ill-posed blind deconvolution problem, and forces the solution to be the members of classes $\setI$, and $\setK$. The minimization program in can be equivalently formulated in the lower dimensional, latent representation space as follows $$\begin{aligned} \label{eq:Optimization-latent} (\hat{z}_i, \hat{z}_k) = \underset{z_i \in \R^l, z_k \in \R^m}{\text{argmin}} \ \\| y - G_{\mathcal{I}}(z_i) \otimes G_\setK(z_k) \\|^2.\end{aligned}$$ This optimization program can be thought of as tweaking the latent representation vectors $z_i$, and $z_k$, (input to the generators $G_{\setI}$, and $G_{\setK}$, respectively) until these generators generate an image $i$ and blur kernel $k$ whose convolution comes as close to $y$ as possible. The optimization program in is obviously non-convex owing to the bilinear convolution operator, and non-linear deep generative models. We resort to an alternating gradient descent algorithm to find a local minima $(\hat{z}_i, \hat{z}_k)$. Importantly, the weights of the generators are always fixed as they enter into this algorithm as pretrained models. At a given iteration, we fix $z_i$ and take a descent step in $z_k$, and vice verse. The gradient step in each variable involves a forward and backward pass through the generator networks. Section \[proposedapproach\] talks about the back propagation, and gives explicit gradient forms for descent in each $z_i$ and $z_k$ for this particular algorithm. The estimated deblurred image and the blur kernel are acquired by a forward pass of the solutions $\hat{z}_i$ and $\hat{z}_k$ through the generators $G_{\mathcal{I}}$ and $G_{\mathcal{K}}$. Mathematically, $(\hat{i},\hat{k}) = ( G_\mathcal{I}(\hat{z}_i), G_\mathcal{K}(\hat{z}_k))$. Image Deblurring Algorithm {#sec:proposedapp} ========================== Our approach requires pretrained generative models $G_\setI$ and $G_\setK$ for classes $\setI$ and $\setK$, respectively. We use both GANs and VAEs as generative models on the clean images and blur kernels. ![image](./figures/block_diag.png){width="75.00000%"} Naive Deblurring {#backprop} ---------------- To deblur an image $y$, a simplest possible strategy is to find an image closest to $y$ in the range of the given generator $G_{\setI}$ of clean images. Mathematically, this amounts to solving the following optimization program $$\begin{aligned} \label{eq:naive-deblur} \underset{z_i \in \R^l}{\text{argmin}} \ \\| y - G_\setI(z_i) \\|, \quad \quad \hat{i} = G_\setI(\hat{z}_i), \end{aligned}$$ where we emphasize again that in the optimization program above, the weights of the generator $G_\setI$ are fixed (pretrained). Although non-convex, a local minima $\hat{z}_i$ can be achieved via gradient descent implemented using the back propagation algorithm. The recovered image $\hat{i}$ is obtained by a forward pass of $\hat{z}_i$ through the generative model $G_{\setI}$. Expectedly, this approach fails to produce reasonable recovery results; see Figure \[fig:Back-Propagation\]. The reason being that this back projection approach completely ignores the knowledge of the forward blur model in . We now address this shortcoming by including the forward model and blur kernel in the objective (\[eq:naive-deblur\]). Deconvolution using Deep Generative Priors {#proposedapproach} ------------------------------------------ We discovered in the previous section that simply finding a clean image close to the blurred one in the range of the image generator $G_{\setI}$ is not good enough. A more natural and effective strategy is to instead find a pair consisting of a clean image and a blur kernel in the range of $G_{\setI}$, and $G_{\setK}$, respectively, whose convolution comes as close to the blurred image $y$ as possible. As outlined in Section \[sec:Problem-Formulation\], this amounts to minimizing the measurement loss $$\label{eq:measurement-loss} \\| y - G_\setI(z_i) \otimes G_\setK(z_k) \\|^2,$$ over both $z_i$, and $z_k$, where $\otimes$ is the convolution operator. Incorporating the fact that latent representation vectors $z_i$, and $z_k$ are assumed to be coming from standard Gaussian distributions in both the adversarial learning and variational inference framework, outlined in Section \[generative\], we further augment the measurement loss in with $\ell_2$ penalty terms on the latent representations. The resultant optimization program is then $$\label{eq:regularized-program} \underset{z_i \in \R^l, z_k \in \R^m}{\text{argmin}}\ \\| y - G_\setI(z_i) \otimes G_\setK(z_k) \\|^2+ \gamma\\| z_i \\|^2 + \lambda\\| z_k \\|^2,$$ where $\lambda$, and $\gamma$ are free scalar parameters. For brevity, we denote the objective function above by $\setL(z_i,z_k)$. To minimize this non-convex objective, we begin by initializing $z_i$, and $z_k$ as standard Gaussian vectors, and then take a gradient step in one of these while fixing the other. To avoid being stuck in a not good enough local minima, we may restart the algorithm with a new random initialization (Random Restarts) when the measurement loss in does not reduce sufficiently after reasonably many iterations. Algorithm \[alg:generative-prior-deblurring\] formally introduces the proposed alternating gradient descent scheme. Henceforth, we will denote the image deblurred using Algorithm \[alg:generative-prior-deblurring\] by $\hat{i}_\text{DD}$. Beyond the Range of Generator {#sec:proposed-method2} ----------------------------- As described earlier, the optimization program implicitly constrains the deblurred image to lie in the range of the generator $G_{\setI}$. This may leads to some artifacts in the deblurred images when the generator range does not completely span the set $\setI$. This inability of the generator to completely learn the image distribution is often evident in case of more rich and complex natural images. In such cases, it makes more sense to not strictly constrain the recovered image to come from the range of the generator, and rather also explore images a bit outside the range. To accomplish this, we propose minimizing the measurement loss of images inside the range exactly as in together with the measurement loss $\\| y - i \otimes G_\setK(z_k) \\|^2$ of images not necessarily within the range. The in-range image $G_{\setI}(z_i)$, and the out-range image $i$ are then tied together by minimizing an additional penalty term, $\text{Range Error(i)} := \\| i - G_\setI(z_i) \\|^2$. The idea is to strictly minimize the range error when pretrained generator has effectively learned the image distribution, and afford some slack when it is not the case. The amount of slack can be controlled by tweaking the weights attached with each loss term in the final objective. Finally, to guide the search of a best deblurred image beyond the range of the generator, one of the conventional image priors such as total variation measure $\\|\cdot\\|_{\text{tv}}$ is also introduced. This leads to the following optimization program $$\begin{aligned} \label{eq:opt-admm} \underset{i, z_i, z_k}{\text{argmin}} \ & \\| y - i \otimes G_\setK(z_k) \\|^2 + \tau \\| i - G_\setI(z_i) \\|^2 + \zeta\\| y - G_\setI(z_i) \otimes G_\setK(z_k) \\|^2 + \rho\\|i\\|_{\text{tv}}.\end{aligned}$$ All of the variables are randomly initialized, and the objective is minimized using gradient step in each of the unknowns, while fixing the others. The computations of gradients is very similar to the steps outlined in Section \[proposedapproach\]. We take the solution $\hat{i}$, and $G(z_k)$ as the deblurred image, and the recovered blur kernel. The iterative scheme is formally given in Algorithm \[alg:generative+classical-prior-deblurring\]. For future references, we will denote the recovered image using Algorithm \[alg:generative+classical-prior-deblurring\] by $\hat{i}_\text{DDS}$. ![[]{data-label="fig:proposed_approach"}](./figures/block_diag.png){width="90.00000%"} A more natural and effective strategy is to instead find a pair consisting of a clean image and a blur kernel in the range of $G_{\setI}$, and $G_{\setK}$, respectively, whose convolution comes as close to the blurred image $y$ as possible; see Figure \[fig:proposed\_approach\]. As outlined in Section \[sec:Problem-Formulation\], this amounts to minimizing the measurement loss $$\label{eq:measurement-loss} \\| y - G_\setI(z_i) \otimes G_\setK(z_k) \\|^2,$$ over both $z_i$ and $z_k$. Incorporating the fact that latent representation vectors $z_i$, and $z_k$ are assumed to be coming from standard Gaussian distributions in both the adversarial learning and variational inference framework, we further augment the measurement loss in with $\ell_2$ penalty terms on the latent representations. The resultant optimization program is then $$\label{eq:regularized-program} \underset{z_i \in \R^l, z_k \in \R^m}{\text{argmin}}\ \\| y - G_\setI(z_i) \otimes G_\setK(z_k) \\|^2+ \gamma\\| z_i \\|^2 + \lambda\\| z_k \\|^2,$$ where $\gamma$ and $\lambda$ are free scalar parameters. For brevity, we denote the objective function above by $\setL(z_i,z_k)$. Importantly, the weights of the generators are always fixed as they enter into this algorithm as pretrained models. To minimize this non-convex objective, we begin by initializing $z_i$ and $z_k$ as standard Gaussian vectors, and then take a gradient step in one of these while fixing the other. To avoid being stuck in a not good enough local minima, we may restart the algorithm with a new random initialization (Random Restarts) when the measurement loss in does not reduce sufficiently after reasonably many iterations. Algorithm \[alg:generative-prior-deblurring\] formally introduces the proposed alternating gradient descent scheme. Henceforth, we will denote the image deblurred using Algorithm \[alg:generative-prior-deblurring\] by $\hat{i}_1$. \[alg:AltGradDescent\] Input: $y$, $G_\setI$ and $G_\setK$\ Output: Estimates $\hat{i}_1$ and $\hat{k}$\ Initialize:\ $z_i^{(0)} \sim \setN(0,I), ~~ z_k^{(0)} \sim \setN(0,I)$ (\[eq:regularized-program\])\ (\[eq:regularized-program\])\ $\hat{i}_1 \leftarrow G_\setI(z^{(T)}_i), \hat{k} \leftarrow G_\setK(z^{(T)}_k)$ \[alg:generative-prior-deblurring\] Beyond the Range of Generator {#sec:proposed-method2} ----------------------------- \[alg:admm\] Input: $y$, $G_\setI$ and $G_\setK$\ Output: Estimates $\hat{i}_2$ and $\hat{k}$\ Initialize:\ $z_i^{(0)} \sim \setN(0,I), z_k^{(0)} \sim \setN(0,I), i^{(0)} \sim \setN(0.5,10^{-2}I)$ (\[eq:opt-admm\])\ (\[eq:opt-admm\])\ (\[eq:opt-admm\])\ $\hat{i}_2 \leftarrow {i^{(T)}}, \hat{k} \leftarrow G_\setK(z_k^{(T)})$ \[alg:generative+classical-prior-deblurring\] \[algorithm:admm\] Beyond the Range of Generator {#sec:proposed-method2} ----------------------------- As described earlier, the optimization program implicitly constrains the deblurred image to lie in the range of the generator $G_{\setI}$. This may lead to some artifacts in the deblurred images when the generator range does not completely span the set $\setI$. In such case, it makes more sense to not strictly constrain the recovered image to come from the range of the generator, and rather also explore images a bit outside the range. To accomplish this, we propose minimizing the measurement loss of images inside the range exactly as in together with the measurement loss $\\| y - i \otimes G_\setK(z_k) \\|^2$ of images not necessarily within the range. The in-range image $G_{\setI}(z_i)$ and the out-range image $i$ are then tied together by minimizing an additional penalty term, $\text{Range Error(i)} := \\| i - G_\setI(z_i) \\|^2$. The idea is to strictly minimize the range error when pretrained generator has effectively learned the image distribution, and afford some slack otherwise. Finally, to guide the search of a best deblurred image beyond the range of the generator, one of the conventional image priors such as total variation measure $\\|\cdot\\|_{\text{tv}}$ is also introduced. This leads to the following optimization program $$\begin{aligned} \label{eq:opt-admm} \underset{i, z_i, z_k}{\text{argmin}} \ & \\| y - i \otimes G_\setK(z_k) \\|^2 + \tau \\| i - G_\setI(z_i) \\|^2 +\zeta\\| y - G_\setI(z_i) \otimes G_\setK(z_k) \\|^2 + \rho\\|i\\|_{\text{tv}}\end{aligned}$$ All of the variables are randomly initialized, and the objective is minimized using gradient step in each of the unknowns, while fixing the others. We take the solution $\hat{i}$, and $G(\hat{z}_k)$ as the deblurred image, and the recovered blur kernel. We dubbed this approach as Deep Deblur with Slack (DDS) and the image deblurred using this approach is referred to as $\hat{\textit{i}}_\text{DDS}$. Experimental Results {#experiments} ==================== In this section, we provide a comprehensive set of experiments to evaluate the performance of Deep Deblur and Deep Deblur with Slack against iterative and deep learning based baseline methods. We also evaluate performance under increasing noise and large blurs. In all experiments, we use noisy blurred images, generated by convolving images $i$, and blurs $k$ from their respective test sets and adding 1$\%$ [^1] Gaussian noise (unless stated otherwise). The choice of free parameters in both algorithms for each dataset are provided in the supplementary material. Implementation Details ---------------------- Datasets: We choose three image datasets. First dataset, SVHN, consists of house number images from Google street view. A total of 531K images, each of dimension $32\times32 \times 3$, are available out of which 30K are held out as test set. Second dataset, Shoes [@yu2014fine] consists of 50K RGB examples of shoes, resized to $64 \times 64 \times 3$. We leave $1000$ images for testing and use the rest as training set. Third dataset, CelebA, consists of relatively more complex images of celebrity faces. A total of 200K, each center cropped to dimension $64 \times 64 \times 3$, are available out of which 22K are held out as a test set. A motion blur dataset is generated consisting of small to very large blurs of lengths varying between 5 and 28; following strategy given in [@boracchi2012modeling]. We generate 80K blurs out of which 20K is held out as a test set. Generative Models: We choose VAE as the generative model for SVHN images and motion blurs. For Shoes and CelebA, the generative model $G_{\setI}$ is the default deep convolutional generative adversarial network (DCGAN) [@salimans2016improved]. Further details on architectures of generative models are provided in the supplementary material. Baseline Methods: Among the conventional algorithms using engineered priors, we choose dark prior (DP) [@pan2016blind], extreme channel prior (EP) [@yan2017image], outlier handling (OH) [@dong2017blind], and learned data fitting (DF) [@pan2017learning] based blind deblurring as baseline algorithms. We optimized the parameters of these methods in each experiment to obtain the best possible baseline results. Among driven approaches for deblurring, we choose [@hradivs2015convolutional] that trains a convolutional neural network (CNN) in an end-to-end manner, and [@kupyn2017deblurgan] that trains a neural network (DeblurGAN) in an adversarial manner. Deblurred images from these baseline methods will be referred to as $i_\text{DP}$, $i_\text{EP}$, ${i_\text{OH}}$, ${i_\text{DF}}$, $i_\text{CNN}$ and $i_\text{DeGAN}$. Deblurring Results under Pretrained Generative Priors {#sec:Exps-PretrainedPriors} ----------------------------------------------------- The central limiting factor in the Deep Deblur performance is the ability of the generator to represent the (original, clean) image to be recovered. As pointed out earlier that often the generators are not fully expressive (cannot generate new representative samples) on a rich/complex image class such as face images compared to a compact/simple image class such as numbers. Such a generator mostly cannot adequately represent a new image in its range. Since Deep Deblur strictly constrains the recovered image to lie in the range of image generator, its performance depends on how well the range of the generator spans the image class. Given an arbitrary test image $i_{\text{test}}$ in the set $\setI$, the closest image $i_{\text{range}}$, in the range of the generator, to $i_{\text{test}}$ is computed by solving the following optimization program $$\begin{aligned} z_{\text{test}} := \underset{z} {\text{argmin}} \\|i_{\text{test}}-G_{\setI}(z)\\|^2, \quad i_{\text{range}} = G_{\setI}(z_{\text{test}})\end{aligned}$$ We solve the optimization program by running $10,000(6,000)$ gradient descent steps with a step size of $0.001(0.01)$ for CelebA(SVHN). Parameters for Shoes are the same as CelebA. A more expressive generator leads to a better deblurring performance as it can well represent an arbitrary original (clean) image $i_{\text{test}}$ leading to a smaller mismatch $ \text{range error} := \\|i_{\text{test}} - i_{\text{range}}\\|$ to the corresponding range image $i_{\text{range}}$. ### Impact of Generator Range on Image Deblurring To judge the proposed deblurring algorithms independently of generator range limitations, we present their deblurring performance on range image $i_{\text{range}}$; we do this by generating a blurred image $y = i_{\text{range}} \otimes k + n$ from an image $i_{\text{range}}$ already in the range of the generator; this implicitly removes the range error as now $i_{\text{test}} = i_{\text{range}}$. We call this range image deblurring, where the deblurred image is obtained using Deep Deblur, and is denoted by $\hat{i}_{\text{range}}$. For completeness, we also assess the overall performance of the algorithm by deblurring arbitrary blurred images $y = i_{\text{test}} \otimes k +n$, where $i_{\text{test}}$ is not necessarily in the range of the generator. Unlike above, the overall error in this case accounts for the range error as well. We call this arbitrary image deblurring, and specifically the deblurred image is obtained using Deep Deblur, and is denoted by $\hat{i}_\text{DD}$. Figure \[fig:range-images\] shows a qualitative comparison between $i_{\text{test}}$, $i_{\text{range}}$, and $\hat{i}_\text{DD}$ on CelebA dataset. It is clear that the recovered image $\hat{i}_\text{DD}$ is a good approximation of the range image, $i_{\text{range}}$, indicating the limitation of the image generative network. \ [./figures/CelebA/“deblurring - 1perc noise - 10RR”/deblurring\_19\_x\_hat\_from\_test.png]{} (64,0) [./figures/CelebA/“deblurring - admm - 1perc noise - 10RR”/deblurring\_19\_x\_hat\_from\_test.png]{} (64,0) [./figures/CelebA/“Original Images”/x\_orig\_19.png]{} (64,0) \ Deep Deblur with Slack mitigates the range error by not strictly constraining the recovered image to lie in the range of the image generator, for details, see Section \[sec:proposed-method2\]. As shown in Figure \[fig:celebA-results\], estimate $\hat{i}_\text{DDS}$ of true image $i_\text{test}$ from blurry observations is close to $i_\text{test}$ instead of $i_\text{range}$, thus mitigating the range issue. \ [./figures/SVHN/“deblurring - 1perc noise - 10RR”/deblurring\_92\_x\_hat\_from\_test.png]{} (55,0) [./figures/SVHN/“Original Images”/x\_orig\_92.png]{} (55,0) \ \ \ \ ### Deblurring Results on CelebA, Shoes and SVHN Qualitative results on CelebA: Figure \[fig:celebA-results\] gives a qualitative comparison between $i$, $i_{\text{range}}$, $\hat{i}_\text{DD}$, $\hat{i}_\text{DDS}$, and baseline approaches on CelebA and Shoes dataset. The deblurred images under engineered priors are qualitatively a lot inferior than the deblurred images $\hat{i}_\text{DD}$, and $\hat{i}_\text{DDS}$ under the proposed generative priors, especially under large blurs. On the other hand, the end-to-end training based approaches CNN, and DeblurGAN perform relatively better, however, the well reputed CNN is still displaying over smoothed images with missing edge details, etc compared to our results $\hat{i}_\text{DDS}$. DeblurGAN, though competitive, is outperformed by the proposed Deep Deblur with Slack by more than 1.5dB as shown in Table \[table:psnr-ssim-results\]. The images $\hat{i}_\text{DD}$ are sharp and with well defined facial boundaries and markers owing to the fact they strictly come from the range of the generator, however, in doing so these images might end up changing some image features such as expressions, nose, etc. On a close inspection, it becomes clear that how well $\hat{i}_\text{DD}$ approximates $i_{\text{test}}$ roughly depends (see, images specifically in Figure \[fig:range-images\]) on how close $i_{\text{range}}$ is to $i_{\text{test}}$ exactly. While as $\hat{i}_\text{DDS}$ are allowed some leverage, and are not strictly confined to the range of the generator, they tend to agree more closely with the ground truth. We go on further by utilizing pretrained PGGAN [@karras2017progressive] in Deep Deblur by convolving sampled images with large blurs ($30 \times 30$); see Figure \[fig:pg-gan-results\]. It has been observed that pre-trained generators struggle at higher resolutions [@athar2018latent], so we restrict our results at $128 \times 128$ resolution. In Figure \[fig:pg-gan-results\], it can be seen that under expressive generative priors our approach exceeds all other baseline methods recovering fine details from extremely blurry images. Qualitative Results on SVHN: Figure \[fig:svhn-results\] gives qualitative comparison between proposed and baseline methods on SVHN dataset. Here the deblurring under classical priors again clearly under performs compared to the proposed image deblurring results $\hat{i}_\text{DD}$. CNN also continues to be inferior, and the DeblurGAN also shows artifacts. We do not include the results $\hat{i}_\text{DDS}$ in these comparison as $\hat{i}_\text{DD}$ already comprehensively outperform the other techniques on this dataset. The convincing results $\hat{i}_\text{DD}$ are a manifestation of the fact that unlike the relatively complex CelebA and Shoes datasets, the simpler image dataset SVHN is effectively spanned by the range of the image generator. \ Quantitative Results: Quantitative results for CelebA, Shoes[^2] and SVHN using peak-signal-to-noise ratio (PSNR) and structural-similarity index (SSIM) [@wang2004image], averaged over 80 respective test set images, are given in Table \[table:psnr-ssim-results\]. On CelebA and Shoes, the results clearly show a better performance of Deep Deblur with Slack, on average, compared to all baseline methods. On SVHN, the results show that Deep Deblur outperforms all competitors. The fact that Deep Deblur performs more convincingly on SVHN is explained by observing that the range images $i_{\text{range}}$ in SVHN are quantitatively much better compared to range images of CelebA and Shoes. ### Robustness against Noise and Large Blurs Robustness against Noise: Figure \[fig:psnr-ssim-blursize\] gives a quantitative comparison of the deblurring obtained via Deep Deblur (the free parameters $\lambda$, $\gamma$ and random restarts in the algorithm are fixed as before), and baseline methods CNN, DeblurGAN (trained on fixed 1% noise level and on varying 1-10% noise levels) in the presence of Gaussian noise. We also include the performance of deblurred range images $\hat{i}_{\text{range}}$, introduced in Section \[sec:Exps-PretrainedPriors\], as a benchmark. Conventional prior based approaches are not included as their performance substantially suffers on noise compared to other approaches. On the vertical axis, we plot the PSNR and on the horizontal axis, we vary the noise strength from 1 to 10%. In general, the quality of deblurred range images (expressible by the generators) $\hat{i}_{\text{range}}$ under generative priors surpasses other algorithms on both CelebA, and SVHN. This in a way manifests that under expressive generative priors, the performance of our approach is far superior. The quality of deblurred images $\hat{i}_\text{DD}$ under generative priors with arbitrary (not necessarily in the range of the generator) input images is the second best on SVHN, however, it under performs on the CelebA dataset; the most convincing explanation of this performance deficit is the range error (not as expressive generator) on the relatively complex/rich images of CelebA. The end-to-end approaches trained on fixed 1% noise level display a rapid deterioration on other noise levels. Comparatively, the ones trained on 1-10% noise level, expectedly, show a more graceful performance. Qualitative results under heavy noise are depicted in Figure \[fig:results-heavynoise\]. Our deblurred image $\hat{i}_\text{DD}$ visually agrees better with $i_{\text{test}}$ than other methods. Robustness against Large Blurs: Figure \[fig:results-largeblur\] shows the deblurred images obtained from a very blurry face image. The deblurred image $\hat{i}_\text{DDS}$ using Deep Deblur with Slack is able to recover the true face from a completely unrecognizable face. The classical baseline algorithms totally succumb to such large blurs. The quantitative comparison against end-to-end neural network based methods CNN, and DeblurGAN is given in Figure \[fig:psnr-ssim-blursize\]. We plot the blur size against the average PSNR for both Shoes, and CelebA datasets. On both datasets, deblurred images $\hat{i}_\text{DDS}$ convincingly outperforms all other techniques. For comparison, we also add the performance of $\hat{i}_{\text{range}}$. Excellent deblurring under large blurs can also be seen in Figure \[fig:pg-gan-results\] for PGGAN. To summarize, the end-to-end approaches begin to lag a lot behind our proposed algorithms when the blur size increases. This is owing to the firm control induced by the powerful generative priors on the deblurring process in our newly proposed algorithms. Conclusion {#sec:conc} ========== This paper proposes a novel framework for blind image deblurring that uses deep generative networks as priors rather than in a conventional end-to-end manner. We report convincing deblurring results under the generative priors in comparison to the existing methods. A thorough discussion on the possible limitations of this approach on more complex images is presented along with an effective remedy to address these shortcomings. Our main contribution lies in introducing pretrained generative model in solving blind deconvolution. Introducing more expressive generative models with new novel architectures would improve the performance. We leave these exciting directions for future work. Supplementary Stuff =================== Algorithm Parameters -------------------- The choice of free parameters for SVHN, Shoes and CelebA of Deep Deblur and Deep Deblur with Slack are given in Table \[table:alg1-param\] and \[table:alg2-param\]. Both algorithms were implemented in Tensorflow and the code will be made publicly available. Generative Models ================= The generative model of SVHN images is a trained VAE with the network architecture described in Table \[table:vae-architectures\]. The dimension of the latent space of VAE is fixed at 100, and training is carried out on SVHN with a batch size of 1500, and a learning rate of $10^{-5}$ using the Adam optimizer. After training, the decoder part is extracted as the desired generative model $G_{\setI}$. For CelebA and Shoes dataset, the generative model $G_{\setI}$ is the default deep convolutional generative adversarial network (DCGAN)[@salimans2016improved]. The generative model of motion blur dataset is a trained VAE with the network architecture given in Table \[table:vae-architectures\]. This VAE is trained using Adam optimizer with latent dimension 50, batch size 5, and learning rate $10^{-5}$. After training, the decoder part is extracted as the desired generative model $G_{\setK}$. [./figures/CelebA/“deblurring - 1perc noise - 10RR”/deblurring\_10\_x\_hat\_from\_test.png]{} [./figures/CelebA/“deblurring - admm - 1perc noise - 10RR”/deblurring\_10\_x\_hat\_from\_test.png]{} [./figures/CelebA/“Original Images”/x\_orig\_10.png]{} \ [./figures/CelebA/“deblurring - 1perc noise - 10RR”/deblurring\_18\_x\_hat\_from\_test.png]{} [./figures/CelebA/“deblurring - admm - 1perc noise - 10RR”/deblurring\_18\_x\_hat\_from\_test.png]{} [./figures/CelebA/“Original Images”/x\_orig\_18.png]{} \ [./figures/CelebA/“deblurring - 1perc noise - 10RR”/deblurring\_26\_x\_hat\_from\_test.png]{} [./figures/CelebA/“deblurring - admm - 1perc noise - 10RR”/deblurring\_26\_x\_hat\_from\_test.png]{} [./figures/CelebA/“Original Images”/x\_orig\_26.png]{} \ [./figures/CelebA/“deblurring - 1perc noise - 10RR”/deblurring\_48\_x\_hat\_from\_test.png]{} [./figures/CelebA/“deblurring - admm - 1perc noise - 10RR”/deblurring\_48\_x\_hat\_from\_test.png]{} [./figures/CelebA/“Original Images”/x\_orig\_48.png]{} \ [./figures/CelebA/“deblurring - 1perc noise - 10RR”/deblurring\_73\_x\_hat\_from\_test.png]{} [./figures/CelebA/“deblurring - admm - 1perc noise - 10RR”/deblurring\_73\_x\_hat\_from\_test.png]{} [./figures/CelebA/“Original Images”/x\_orig\_73.png]{} \ [./figures/CelebA/“deblurring - 1perc noise - 10RR”/deblurring\_53\_x\_hat\_from\_test.png]{} [./figures/CelebA/“deblurring - admm - 1perc noise - 10RR”/deblurring\_53\_x\_hat\_from\_test.png]{} [./figures/CelebA/“Original Images”/x\_orig\_53.png]{} \ [./figures/CelebA/“deblurring - 1perc noise - 10RR”/deblurring\_35\_x\_hat\_from\_test.png]{} [./figures/CelebA/“deblurring - admm - 1perc noise - 10RR”/deblurring\_35\_x\_hat\_from\_test.png]{} [./figures/CelebA/“Original Images”/x\_orig\_35.png]{} \ [./figures/CelebA/“deblurring - 1perc noise - 10RR”/deblurring\_02\_x\_hat\_from\_test.png]{} [./figures/CelebA/“deblurring - admm - 1perc noise - 10RR”/deblurring\_02\_x\_hat\_from\_test.png]{} [./figures/CelebA/“Original Images”/x\_orig\_02.png]{} \ [./figures/CelebA/“deblurring - 1perc noise - 10RR”/deblurring\_79\_x\_hat\_from\_test.png]{} [./figures/CelebA/“deblurring - admm - 1perc noise - 10RR”/deblurring\_79\_x\_hat\_from\_test.png]{} [./figures/CelebA/“Original Images”/x\_orig\_79.png]{} \ [./figures/CelebA/“deblurring - 1perc noise - 10RR”/deblurring\_06\_x\_hat\_from\_test.png]{} [./figures/CelebA/“deblurring - admm - 1perc noise - 10RR”/deblurring\_06\_x\_hat\_from\_test.png]{} [./figures/CelebA/“Original Images”/x\_orig\_06.png]{} \ [./figures/CelebA/“deblurring - 1perc noise - 10RR”/deblurring\_40\_x\_hat\_from\_test.png]{} [./figures/CelebA/“deblurring - admm - 1perc noise - 10RR”/deblurring\_40\_x\_hat\_from\_test.png]{} [./figures/CelebA/“Original Images”/x\_orig\_40.png]{} \ [./figures/CelebA/“deblurring - 1perc noise - 10RR”/deblurring\_63\_x\_hat\_from\_test.png]{} [./figures/CelebA/“deblurring - admm - 1perc noise - 10RR”/deblurring\_63\_x\_hat\_from\_test.png]{} [./figures/CelebA/“Original Images”/x\_orig\_63.png]{} \ [./figures/CelebA/“deblurring - 1perc noise - 10RR”/deblurring\_72\_x\_hat\_from\_test.png]{} [./figures/CelebA/“deblurring - admm - 1perc noise - 10RR”/deblurring\_72\_x\_hat\_from\_test.png]{} [./figures/CelebA/“Original Images”/x\_orig\_72.png]{} \ [./figures/CelebA/“deblurring - 1perc noise - 10RR”/deblurring\_39\_x\_hat\_from\_test.png]{} [./figures/CelebA/“deblurring - admm - 1perc noise - 10RR”/deblurring\_39\_x\_hat\_from\_test.png]{} [./figures/CelebA/“Original Images”/x\_orig\_39.png]{} \ [./figures/CelebA/“deblurring - 1perc noise - 10RR”/deblurring\_00\_x\_hat\_from\_test.png]{} [./figures/CelebA/“deblurring - admm - 1perc noise - 10RR”/deblurring\_00\_x\_hat\_from\_test.png]{} [./figures/CelebA/“Original Images”/x\_orig\_00.png]{} \ [./figures/CelebA/“deblurring - 1perc noise - 10RR”/deblurring\_07\_x\_hat\_from\_test.png]{} [./figures/CelebA/“deblurring - admm - 1perc noise - 10RR”/deblurring\_07\_x\_hat\_from\_test.png]{} [./figures/CelebA/“Original Images”/x\_orig\_07.png]{} \ [./figures/SVHN/“deblurring - 1perc noise - 10RR”/deblurring\_03\_x\_hat\_from\_test.png]{} [./figures/SVHN/“Original Images”/x\_orig\_03.png]{} \ [./figures/SVHN/“deblurring - 1perc noise - 10RR”/deblurring\_04\_x\_hat\_from\_test.png]{} [./figures/SVHN/“Original Images”/x\_orig\_04.png]{} \ [./figures/SVHN/“deblurring - 1perc noise - 10RR”/deblurring\_07\_x\_hat\_from\_test.png]{} [./figures/SVHN/“Original Images”/x\_orig\_07.png]{} \ [./figures/SVHN/“deblurring - 1perc noise - 10RR”/deblurring\_10\_x\_hat\_from\_test.png]{} [./figures/SVHN/“Original Images”/x\_orig\_10.png]{} \ [./figures/SVHN/“deblurring - 1perc noise - 10RR”/deblurring\_17\_x\_hat\_from\_test.png]{} [./figures/SVHN/“Original Images”/x\_orig\_17.png]{} \ [./figures/SVHN/“deblurring - 1perc noise - 10RR”/deblurring\_31\_x\_hat\_from\_test.png]{} [./figures/SVHN/“Original Images”/x\_orig\_31.png]{} \ [./figures/SVHN/“deblurring - 1perc noise - 10RR”/deblurring\_37\_x\_hat\_from\_test.png]{} [./figures/SVHN/“Original Images”/x\_orig\_37.png]{} \ [./figures/SVHN/“deblurring - 1perc noise - 10RR”/deblurring\_44\_x\_hat\_from\_test.png]{} [./figures/SVHN/“Original Images”/x\_orig\_44.png]{} \ [./figures/SVHN/“deblurring - 1perc noise - 10RR”/deblurring\_57\_x\_hat\_from\_test.png]{} [./figures/SVHN/“Original Images”/x\_orig\_57.png]{} \ [./figures/SVHN/“deblurring - 1perc noise - 10RR”/deblurring\_61\_x\_hat\_from\_test.png]{} [./figures/SVHN/“Original Images”/x\_orig\_61.png]{} \ [./figures/SVHN/“deblurring - 1perc noise - 10RR”/deblurring\_68\_x\_hat\_from\_test.png]{} [./figures/SVHN/“Original Images”/x\_orig\_68.png]{} \ [./figures/SVHN/“deblurring - 1perc noise - 10RR”/deblurring\_76\_x\_hat\_from\_test.png]{} [./figures/SVHN/“Original Images”/x\_orig\_76.png]{} \ [./figures/Shoes/“deblurring - 1perc noise - 10RR”/deblurring\_12\_x\_hat\_from\_test.png]{} [./figures/Shoes/“deblurring - admm - 1perc noise - 10RR”/deblurring\_12\_x\_hat\_from\_test.png]{} [./figures/Shoes/“Original Images”/x\_orig\_12.png]{} \ [./figures/Shoes/“deblurring - 1perc noise - 10RR”/deblurring\_24\_x\_hat\_from\_test.png]{} [./figures/Shoes/“deblurring - admm - 1perc noise - 10RR”/deblurring\_24\_x\_hat\_from\_test.png]{} [./figures/Shoes/“Original Images”/x\_orig\_24.png]{} \ [./figures/Shoes/“deblurring - 1perc noise - 10RR”/deblurring\_26\_x\_hat\_from\_test.png]{} [./figures/Shoes/“deblurring - admm - 1perc noise - 10RR”/deblurring\_26\_x\_hat\_from\_test.png]{} [./figures/Shoes/“Original Images”/x\_orig\_26.png]{} \ [./figures/Shoes/“deblurring - 1perc noise - 10RR”/deblurring\_31\_x\_hat\_from\_test.png]{} [./figures/Shoes/“deblurring - admm - 1perc noise - 10RR”/deblurring\_31\_x\_hat\_from\_test.png]{} [./figures/Shoes/“Original Images”/x\_orig\_31.png]{} \ [./figures/Shoes/“deblurring - 1perc noise - 10RR”/deblurring\_33\_x\_hat\_from\_test.png]{} [./figures/Shoes/“deblurring - admm - 1perc noise - 10RR”/deblurring\_33\_x\_hat\_from\_test.png]{} [./figures/Shoes/“Original Images”/x\_orig\_33.png]{} \ [./figures/Shoes/“deblurring - 1perc noise - 10RR”/deblurring\_40\_x\_hat\_from\_test.png]{} [./figures/Shoes/“deblurring - admm - 1perc noise - 10RR”/deblurring\_40\_x\_hat\_from\_test.png]{} [./figures/Shoes/“Original Images”/x\_orig\_40.png]{} \ [./figures/Shoes/“deblurring - 1perc noise - 10RR”/deblurring\_35\_x\_hat\_from\_test.png]{} [./figures/Shoes/“deblurring - admm - 1perc noise - 10RR”/deblurring\_35\_x\_hat\_from\_test.png]{} [./figures/Shoes/“Original Images”/x\_orig\_35.png]{} \ [./figures/Shoes/“deblurring - 1perc noise - 10RR”/deblurring\_37\_x\_hat\_from\_test.png]{} [./figures/Shoes/“deblurring - admm - 1perc noise - 10RR”/deblurring\_37\_x\_hat\_from\_test.png]{} [./figures/Shoes/“Original Images”/x\_orig\_37.png]{} \ \ \ \ [^1]: For an image scaled between 0 and 1, Gaussian noise of $1\%$ translates to Gaussian noise with standard deviation $\sigma = 0.01$ and mean $\mu=0$. [^2]: For qualitative results, see supplementary material.	ArXiv
--- abstract: \| A system of three particles undergoing inelastic collisions in arbitrary spatial dimensions is studied with the aim of establishing the domain of “inelastic collapse”—an infinite number of collisions which take place in a finite time. Analytic and simulation results show that for a sufficiently small restitution coefficient, $0\leq r<7-4\sqrt{3}\approx 0.072$, collapse can occur. In one dimension, such a collapse is stable against small perturbations within this entire range. In higher dimensions, the collapse can be stable against small variations of initial conditions, within a smaller $r$ range, $0\leq r<9-4\sqrt{5}\approx 0.056$. PACS: 47.50.+d, 05.20.Dd author: - \| Tong Zhou[^1] and Leo P. Kadanoff[^2]\ The James Franck Institute\ The University of Chicago\ Chicago, IL 60637 title: Inelastic Collapse of Three Particles --- [2]{} Introduction ============ A system of particles interacting only through inelastic collisions is a useful idealization of granular materials, and has been much investigated recently[@1; @2; @3; @4; @5; @6; @7; @8; @9; @10]. Inelasticity can make such a system evolve into a collapse state, in which several of the particles collide an infinite number of times in a finite time interval. Inelastic collapse in one dimension is well understood[@1; @2; @3; @4]. In two dimensions, McNamara and Young carried out numerical investigations and found some evidence for inelastic collapses of three particles[@5]. To understand the collapse mechanism in higher dimensions, we study the behavior of three particles in a particular model. Our model involves collisions which preserve the total momentum and the components of the momentum perpendicular to the line of centers. The component of the relative velocity along the line of centers is reversed (as in an elastic collision) and reduced by the restitution coefficient, $0\le r \le 1$. We look at a situation in which one particle (labeled zero) takes part in all collisions. The other two particles (labeled one and two) are alternatively a collider and a spectator. We assume that all particles have the same mass, and that particles one and two have identical radii. There are two possible reasons that previous numerical studies might have shown collapse. One scenario is that the collapsed state is represented by one or many attractive fixed points, so that the collapsing orbit can be stable against small variations in initial data. The other scenario is that each orbit is unstable but that the infinity of collapsing orbits produces an observable collapse probability. For the specific example of three particles, we find an attractive fixed point for all dimensions and sufficiently small coefficient of restitution, $r$. Thus, we establish the possibility of the first scenario. The second scenario is still possible, but we have seen no evidence for it. For a larger $r$, there is an interval in which the fixed point is unstable against changes in the initial conditions. For dimensions greater than one, after the collapse has occurred, the particles can separate from one another. Thus beyond $d=1$, inelastic collapse is an event, not an end-point, in the “lives” of the particles. To study the collapse, we use the methods of dynamical systems theory. Specifically, we examine the situation in which particles one and two are very close to particle zero and aimed so that the system is very close to the collapsing fixed point. Figure 1 shows a typical configuration. Particle $0$ keeps colliding with particle $1$ and particle $2$ repeatedly, and an inelastic collapse may occur. After many collisions, the distances between the particles become small, and the remaining collisions take place so rapidly that the relative motion of the particles is small. Therefore, in the inelastic collapse, the angle $\theta$ approaches a limiting value as the number of collision goes to infinity. We identify the constant-$\theta$ fixed point and then investigate its stability. We find that a fixed point of an inelastic collapse exists only when the final $\theta$ obeys: $$\cos\theta \geq \frac{4\sqrt{r}}{1+r}. \label{eq:A}$$ A collapse state will occur whenever this criterion holds and also steric effect do not block off the required collisions. (For example such a blockage will always occur at $\theta = \pi$.) The stability analysis implies that for the collapse to be stable against small perturbations in the initial velocities, a stronger condition is required, namely $$\cos\theta >\frac{2\root 3 \of r (1+\root 3 \of r)}{1+r}. \label{eq:B}$$ The collision model =================== We use the standard model of inelastic collision: due to a collision the component of the relative velocity of the colliders along the line of centers, changes by a factor of $-r$. We denote by $\vec{u}_j$ and $\vec{x}_j$ the velocity and the position of the $j$th particle at the instant before a collision occurs. Let us consider a collision between particles $1$ and $0$. In the course of the collision, the velocities of the particles change to: $$\begin{aligned} \vec{u}_1' & = & \vec{u}_1 - \vec{\Delta}, \nonumber \\ \vec{u}_0' & = & \vec{u}_0 + \vec{\Delta}, \nonumber \\ \vec{u}_2' & = & \vec{u}_2. \label{eq:1}\end{aligned}$$ Here, the momentum transfer is $\vec{\Delta}$. It must point in the direction of the line of centers. In terms of the coefficient of restitution, $r$, this transfer is given by the expression $$\vec{\Delta} = \frac{1+r}{2} (\vec{x}_1-\vec{x}_0) [(\vec{x}_1-\vec{x}_0) \cdot (\vec{u}_1-\vec{u}_0)].$$ Here we have assumed that the radii of the colliding particles sum to unity so that, at the point of collision: $$(\vec{x}_1-\vec{x}_0)^2 = 1. \label{eq:3}$$ To do dynamical systems theory, we wish to look at the very same process repeatedly. Therefore, we introduce the superscripts, $c$, denoting the collider and $s$ denoting the spectator particle, as well as a subscript, $n$, to denote the instant before the $n$th collision occurs (Figure 1). For simplicity, we take the velocity and the position of particle $0$ to be zero. In order to make sure that the velocity of particle $0$ continues to vanish after the collision, we view the post-collision system from a frame moving with velocity $\vec{u}_0'= \vec{\Delta}$. Then equations (\[eq:1\]) - (\[eq:3\]) read $$\begin{aligned} \vec{u}_{n+1}^s & = & \vec{u}_{n}^c - 2 \vec{\Delta}, \nonumber \\ \vec{u}_{n+1}^c & = & \vec{u}_{n}^s-\vec{\Delta}. \label{eq:3a}\end{aligned}$$ These equations are supplemented by the conditions: $$\vec{\Delta} = \frac{1+r}{2} (\vec{x}_{n}^c) (\vec{x}_{n}^c \cdot \vec{u}_{n}^c),$$ $$(\vec{x}_{n}^c)^2 = 1.$$ Additionally, both the velocity and the position of particle $0$ vanish. Notice that a collider becomes a spectator immediately following a collision. The above equations are complemented by the equations corresponding to the positions of the particles at the next collision, $$\begin{aligned} \vec{x}_{n+1}^s & = & \vec{x}_{n}^c + t_n \vec{u}_{n+1}^s, \nonumber \\ \vec{x}_{n+1}^c & = & \vec{x}_{n}^s + t_n \vec{u}_{n+1}^c. \label{eq:5}\end{aligned}$$ The time interval between the $n$th and the $(n+1)$th collisions, $t_n$, is such that the magnitude of $\vec{x}_{n+1}^c$ is unity. Flat Surface Approximation ========================== We now seek fixed points in these equations. We assume that the time between collisions is sufficiently small so that the $t_n$ terms in equations (\[eq:5\]) are negligible and consequently $$\begin{aligned} \vec{x}_{n+1}^s & = & \vec{x}_{n}^c, \nonumber\\ \vec{x}_{n+1}^c & = & \vec{x}_{n}^s, \label{eq:apr}\end{aligned}$$ during the approach to the fixed point. We wish to find a fixed point in the components of the velocity in the direction of the lines of centers. Specifically, we would like to investigate how this component decreases in each iteration. We can define: $$\vec{x}_{n+1}^c \cdot \vec{u}_{n+1}^c \equiv k_n \vec{x}_{n}^c \cdot \vec{u}_{n}^c. \label{eq:8}$$ Taking the dot product of equations (\[eq:3a\]) respectively into $\vec{x}_{n}^s$ and $\vec{x}_{n}^c$, and using equations (\[eq:apr\]) gives $$\vec{x}_{n+1}^s \cdot \vec{u}_{n+1}^s = -r \vec{x}_{n}^c \cdot \vec{u}_{n}^c, \label{eq:9a}$$ $$\vec{x}_{n+2}^c \cdot \vec{u}_{n+2}^c = \vec{x}_{n+1}^s \cdot \vec{u}_{n+1}^s+\frac{1+r}{2} \vec{x}_{n+1}^c \cdot \vec{u}_{n+1}^c\cos\theta. \label{eq:9}$$ Equations (\[eq:8\])-(\[eq:9\]) thus imply the recursion satisfied by the scaling factor $k_n$ $$k_{n+1}=-\frac{r}{k_n}+\frac{1+r}{2}\cos\theta. \label{eq:rok}$$ Fixed points can be found by setting $k_{n+1}=k_n$ in (\[eq:rok\]), $$k^2 - k \frac{1+r}{2}\cos\theta +r=0.$$ As a result, the fixed point of the scaling factor has two possible values $$k_\pm =\frac{1+r}{4}\cos\theta \pm \sqrt{\left(\frac{1+r}{4}\cos\theta\right)^2-r}. \label{eq:11}$$ Equation (\[eq:11\]) is one of the major results of our study. In every collision, the colliding particle must approach particle $0$. Hence, $\vec{x}_c \cdot \vec{u}_c$ must be negative in every iteration. This is possible only if $k$ is a positive real number. One kind of failure arises when $k$ is complex. Then the real part of the dot product will change sign infinitely often and no fixed point can be reached. Thus, for inelastic collapse to occur, the quantity under the square root in equation (\[eq:11\]) must be positive. This positivity still permits both signs of $\cos\theta$. However, if the roots are real and the cosine is negative, both roots will be negative. Hence neither is a possible solution for inelastic collapse. The only remaining possibility is that inelastic collapse may occur under the condition on the cosine given by equation (\[eq:A\]). According to that statement, when $r\rightarrow 0$, $\theta$ can have a value between $0$ and $\frac{\pi}{2}$. On the other hand, when $\theta =0$, the well-known one-dimensional result is recovered [@2], [i.e.]{}, inelastic collapse is possible for $0\le r< 7-4\sqrt{3}$. Regions (a) and (b) in figure 2 are the region of $r$ and $\theta$ for which we may have inelastic collapse. We now consider the stability of the above fixed points. Stability will imply that a small change in the initial conditions will leave the system in a collapse state, or in other words, changes will still permit an infinite number of collisions. There are two collapse fixed points distinguished by the values of the two multipliers $k=k_+$ and $k=k_-$. Subtracting $k_\pm$ from both sides of (\[eq:rok\]) yields $$\frac{k_{n+1}-k_\pm }{k_n-k_\pm }=\frac{k_\mp}{k_n}.$$ It is seen that $k_-$ corresponds to an unstable fixed point, while $k_+$ corresponds to a stable one. Henceforth, we use $k$ to denote the stable fixed point $k\equiv k_+$. Next, we investigate the time interval between successive collisions. Assume that relative motion of the two colliding particles between each pair of collisions covers a distance which is very small in comparison to their radii. Then, we can think of the surface of the particles as flat. After the $n$th collision, the colliding particle moves away from the surface and covers a distance $t_n \vec{x}_{n+1}^s \cdot \vec{u}_{n+1}^s$. In the next step, this particle moves back over the same distance, which is given by -$t_{n+1} \vec{x}_{n+2}^c \cdot \vec{u}_{n+2}^c $ and reaches the surface. Thus, the ratio of times is $$\frac{t_{n+1}}{t_n}= \frac{-\vec{x}_{n+1}^s \cdot \vec{u}_{n+1}^s} {\vec{x}_{n+2}^c \cdot \vec{u}_{n+2}^c}.$$ This result may be simplified with the aid of equations (\[eq:8\])-(\[eq:9a\]) to give $$\frac{t_{n+1}}{t_n}=\frac{r}{k^2}. \label{eq:14}$$ Let $d$ denote the shortest distance between two particles. The distance ratio equals the product of the time ratio and the velocity ratio, $k$, $$\frac{d_{n+1}}{d_n}=\frac{r}{k}. \label{eq:15}$$ We performed numerical simulations of the collision process by considering three inelastic particles moving in two-dimension with random initial conditions. When collapse happens, we compared the ratios calculated from simulations with the predictions of equations (\[eq:8\]), (\[eq:14\]), (\[eq:15\]). We found excellent agreement (Figure 3), indicating that the fixed points are attractive and indeed correspond to collapse Validation of the Flat Surface Approximation ============================================ As observed in the last section, making the approximation (\[eq:apr\]) simplifies dramatically the original system and it can be described by a single ratio $k_n$. It is as if the particles have flat surfaces, so that the effect of the tangential components of the velocities of particles $1$ and $2$ can be ignored. This is true only if the time intervals are negligible, so that (\[eq:apr\]) can be obtained from (\[eq:5\]). We will see when such a simplification is valid, and we will set a criterion for the range of validity of the approximation. In our way to equations (\[eq:9a\])-(\[eq:9\]), we neglected the terms like $t_n (u_{n}^c)^2$ in comparison to $\vec{x}_{n}^c \cdot \vec{u}_{n}^c$, by using the approximation (\[eq:apr\]). As noted above, the former terms decrease as $(\frac{r}{k^2})^n$, while the latter terms decrease as $k^n$. Thus, the flat surface approximation is reliable only when $r<k^3$ so that terms proportional to $t_n$ can be safely ignored. This condition can be explicitly written as, $$r<\left(\frac{1+r}{4}\cos\theta +\sqrt{\left(\frac{1+r}{4}\cos\theta\right)^2-r}\right)^3,$$ which can then be simplified into the form of condition (\[eq:B\]). The region of stability determined by this condition is region (a) in Figure 2. The maximum possible value of $r$ for stable behavior is $r_c=9-4\sqrt{5}$. To this point, our calculation did not rely on the circular geometry of the particles. The name “flat surface” suggests that when criterion (\[eq:B\]) is satisfied, particles do not experience the curvature of their surfaces, and collide as if they are flat. This calculation is valid for arbitrary particle shapes when (\[eq:B\]) is satisfied. We also observe that when criterion (\[eq:B\]) is satisfied, the time interval $t_n$ decreases faster than the radial component of the velocities. In such a situation, collapse happens so quickly that all other effects, external or internal, have no essential influences to the process. One can further consider arbitrary interactions between the particles as well as arbitrary external fields, as long as all the interactions depends only on the relative positions. Since the particles’ relative positions only change very little during the process of collapse, all the effects of the interactions on, say, particle $1$ can be replaced by a constant total force acting on it which induces an constant acceleration. This acceleration has very little effect in the tangential direction since time interval is too small for it to change the tangential component of the velocity. When the flat surface approximation is valid, the time interval is even too small to change the radial velocity component. We conclude that the the previously obtained fixed points are unchanged. Circular Geometry ================= In the previous sections, the calculation were performed by neglecting the $t_n$ terms completely out of equations (\[eq:5\]) and the fixed points for inelastic collapse were found when the final state satisfied condition (\[eq:B\]). After understanding the characteristic behavior of the collapse, we can do a more rigorous calculation to investigate how the system behaves outside the region satisfying (\[eq:B\]). We now see that during the collapse process, the radial velocity components of particles $1$ and $2$ monotonically decrease till they vanish at the moment of singularity, while their tangential components approach limiting values as the number of collision diverges. Hence, we take those tangential components as constants, and concentrate on the radial components in the calculation of fixed points. For simplicity we study in detail one collision in the situation where particle $1$ has a zero radius while particle $0$ has a unit radius. Since the theory depends only upon the sum of the two radii, this case subsumes all others. Here we denote the initial instant with a subscript $i$, and the instant before the collision with $f$. We drop the superscripts since only particle $1$ is considered. We also assume the particles are very close, $d\ll 1$, and $-\vec{u}\cdot \vec{x}_i \ll u_t$, where $u_t$ is the magnitude of the tangential velocity component of particle $1$. The collision time is $$t=\frac{1}{u_t}\left(\frac{-\vec{u}\cdot \vec{x}_i }{u_t}-\sqrt{\left(\frac{\vec{u}\cdot \vec{x}_i}{u_t}\right)^2-2d}\right). \label{eq:19}$$ Immediately before the collision, the radial velocity component of particle $1$ equals $$-\vec{u}\cdot \vec{x}_f =\sqrt{(\vec{u}\cdot \vec{x}_i )^2-2du_t^2}. \label{eq:20}$$ Equation (\[eq:19\]) gives the quantity needed to complete the equation set (\[eq:3a\])-(\[eq:5\]). In order to abbreviate this calculation, we introduce an effective centrifugal acceleration. In the above calculation, if we view the situation in a frame rotating with an angular velocity of $u_t$, around an axis perpendicular to both $\vec{u}$ and the line of centers, passing through the center of particle $0$, then particle $1$ has zero tangential velocity, and the effect of the tangential velocity can be represented by a centrifugal acceleration $a_1=u_t^2$. This substitution is justified by noticing that we can get exactly the same expressions as (\[eq:19\]) and (\[eq:20\]) by using this acceleration. We do not need really use such a rotating frame. We use $a_1$ as an effective centrifugal acceleration to replace the tangential component of the velocity with the same effects. After these preparations, we are in a much clearer position. Particle $1$ and particle $2$ respectively have centrifugal accelerations $a_1$ and $a_2$, which are all in the radial directions. Equation (\[eq:apr\]) is again a good approximation. Particle $1$ is moving on a line, and so is particle $2$. We can further drop the vector notation. In the following, we use $u$ to denote the radial component of the velocity of particles $1$ and $2$ immediately before a collision, with its positive direction pointing towards the center of particle $0$. Consequently, $a_1=-u_{1t}^2$ and $a_2=-u_{2t}^2$. The equation set (\[eq:3a\])-(\[eq:5\]) reduces to $$\begin{aligned} u_{n+1}^s & = & -ru_n^c+a_{n+1}^st_n \nonumber \\ u_{n+1}^c & = & u_n^s+\frac{1+r}{2}\cos\theta\cdot u_n^c +a_{n+1}^ct_n \nonumber \\ d_n & = & u_{n+1}^c\cdot t_n-\frac{1}{2}a_{n+1}^ct_n^2 \nonumber \\ -d_{n+1} & = & -ru_n^c \cdot t_n+\frac{1}{2}a_{n+1}^st_n^2 \nonumber\\ a_{n+1}^c & = & a_n^s \nonumber \\ a_{n+1}^s & = & a_n^c \label{eq:set}\end{aligned}$$ where $d_n$ is the distance between the spectator and particle $0$ at the instant of the $n$th collision. Recall that $t_n$ is the time interval between the $n$th and $(n+1)$th collisions. One simple case can be fully carried through, the case $a_1=a_2\equiv a$. In this case, particle $1$ and particle $2$ are in a symmetrical situation so that the recursion relation of the system can be obtained after a single collision. As we did before, we use two nondimensional numbers $k_n$ and $\alpha_n$ to describe the evolution of the system: $$k_n\equiv\frac{u_{n+1}^c}{u_n^c}\quad\mbox{and}\quad\alpha_n\equiv\frac{-a\cdot t_n}{u_n^c}.$$ Starting from (\[eq:set\]), after some straightforward calculation, we get following recursion relation: $$k_{n+1}=\sqrt{\left(\frac{1+r}{2}\cos\theta -\frac{r}{k_n}\right)^2-(1+r)\cos\theta \frac{\alpha_n}{k_n}}, \label{eq:27}$$ and $$\begin{aligned} \alpha_{n+1} & = & \frac{1+r}{2}\cos\theta -\frac{r}{k_n} -\frac{\alpha_n}{k_n}\nonumber\\ & - & \sqrt{\left(\frac{1+r}{2}\cos\theta -\frac{r}{k_n}\right)^2-(1+r)\cos\theta \frac{\alpha_n}{k_n}}. \label{eq:28}\end{aligned}$$ Suppose the fixed point is $(\alpha,k)$, then from (\[eq:28\]), we have $$\alpha =\frac{-k^2+\frac{1+r}{2}\cos\theta\cdot k-r}{k+1}. \label{eq:29}$$ Since $\alpha\ge 0$ from its definition, the right hand side of the above equality must be nonegative. Thus, there must exist two real solutions of $k$ satisfying r.h.s.$=0$. And we readily recover the condition (\[eq:A\]). Substituting (\[eq:29\]) into (\[eq:27\]), we have $$\begin{aligned} k^4 & = & \left(\frac{1+r}{2}\cos\theta\cdot k-r\right)^2\nonumber\\ & & -(1+r)\cos\theta\cdot k\frac{-k^2+\frac{1+r}{2}\cos\theta\cdot k-r}{k+1}. \label{eq:31}\end{aligned}$$ Of course, $k_\pm$, which appeared before, are solutions of (\[eq:31\]). From (\[eq:29\]) we learn that $(0,k_\pm )$ are fixed points. Let us look at other solutions of equation (\[eq:31\]) which satisfy $$\begin{aligned} k^3+\left(1+\frac{1+r}{2}\cos\theta\right)k^2-\left(r+\frac{1+r}{2}\cos\theta\right)k\nonumber \\ -r=0.\end{aligned}$$ This equation has one and only one solution of $k$ in the interval $[0:1]$. We denote the corresponding fixed point as $(\alpha_0,k_0)$. Of the three relevant fixed points, $(0,k_-)$ is unstable, and we concentrate on the stability condition for the other two fixed points. Suppose that $(\alpha_n,k_n)$ has a small deviation from the fixed point $(\alpha, k)$. then $$\left( \begin{array}{c} \delta k_{n+1}\\ \delta\alpha_{n+1}\end{array} \right) =\left( \begin{array}{cc} A_{11} & A_{12}\\A_{21} & A_{22} \end{array} \right) \left( \begin{array}{c} \delta k_n\\ \delta\alpha_n\end{array} \right)$$ where $$\begin{aligned} A_{11} & = & \frac{1}{2k^3}\left[ 2\left(\frac{1+r}{2}\cos\theta -\frac{r}{k}\right)r+(1+r)\cos\theta\cdot\alpha\right] \nonumber \\ A_{12} & = & -\frac{(1+r)\cos\theta}{2k^2} \nonumber \\ A_{21} & = & \frac{\alpha +r}{k^2}-A_{11} \nonumber \\ A_{22} & = & \frac{(1+r)\cos\theta-2k}{2k^2} \nonumber\end{aligned}$$ Let us denote by $\lambda$ the eigenvalue of matrix $A$, hence we have $$\lambda^2-b\lambda+c=0 \label{eq:39}$$ where $$\begin{aligned} b & = & \frac{1}{2}-\frac{1}{k}+\left[(1+r)\cos\theta-\left(\frac{1+r}{2}\cos\theta\right)^2\right]\frac{1}{2k^2}\nonumber\\ & & +r(1+r)\cos\theta\cdot\frac{1}{k^3}-\frac{3r^2}{2k^4},\nonumber\\ c & = & \frac{r^2}{k^5}.\nonumber\end{aligned}$$ For the fixed point $(0,k_+)$, $$b=\frac{r}{k_+^3}(1+k_+)\quad\mbox{and}\quad c=\frac{r^2}{k_+^5}.$$ So $$\lambda_1=\frac{r}{k_+^2}\quad\mbox{and}\quad \lambda_2=\frac{r}{k_+^3}. \label{eq:eig}$$ The point $(0,k_+)$ is an attractive fixed point if and only if $r<k_+^3$. Hence we recover the condition of (\[eq:B\]). If we only require $\lambda_1<1$, we can recover condition (\[eq:A\]). For the fixed point $(\alpha_0,k_0)$, we find that $\alpha_0>0$ if and only if the condition of (\[eq:B\]) is satisfied, but that condition also decides the range of $(r,\theta)$ inside which we will have $$1-b+c<0$$ From equation (\[eq:39\]) we know that one eigenvalue of matrix $A$ is larger than $1$, implying that the corresponding fixed point is unstable. We conclude that there are no additional stable fixed points beyond those which satisfy condition (\[eq:B\]), and there are no stable collapses outside that range in the case $a_1=a_2$. We believe this is true for the general situation of $a_1\ne a_2$. When condition (\[eq:B\]) is violated, our simulations show that even though the particles can be very close, they will get apart before having collided an infinite number of times. From the above calculation, and specifically equation (\[eq:eig\]), we can see the parameter space $(r,\theta)$ can be divided into three regions (Figure 2): a\) When condition (\[eq:B\]) is satisfied, both the eigenvalues of matrix $A$ are smaller than unity. The fixed point is stable in all directions in space $(\alpha, k)$. It is the collapse region. b\) When condition (\[eq:B\]) is violated but the condition (\[eq:A\]) is satisfied, the eigenvalue $\lambda_2>1$, while $\lambda_1<1$. In this region, particles can have any number of collisions before they might eventually separate. c\) When the condition (\[eq:A\]) is violated, both eigenvalues of matrix $A$ are larger than unity. Collapse does not occur. The above calculation is independent of the sources of the accelerations, which could be the interactions between particles. The calculation confirms our previous argument that interactions are irrelevant in the process of collapse[^3]. We also observe that in the above calculation, the circular geometry of the particles is not essential. The calculation is also valid for arbitrary shape of the particles, with the corresponding radius of curvature replacing the radii used. Even though the radius of curvature is not relevant in deciding the behavior of collapse process—the radii do not show up in the expression of the stable fixed point or condition (\[eq:B\]), it does have some effects. The centrifugal acceleration, which obviously is important in deciding the probability of collapse, is inversely proportional to the radius. So chances are larger for collapse to happen when the colliding point is at a position on the surface with a larger radius of curvature. Conclusion ========== We demonstrated analytically the existence of inelastic collapse for three particles in all dimensions. At the last moment of collapse, the three particles have a cyclic behavior, which is characterized by a fixed point. We have established the range of the parameters for which the fixed point exists and the range for which it is stable. Acknowledgments {#acknowledgments .unnumbered} --------------- We would like to express our thanks to Y.Du, S.Esipov, H.M.Jaeger, M.Mungan, S.R.Nagel, N.Schörghofer and W.R.Young for very helpful discussions and especially to Eli Ben-Naim for continued advice and discussions. This work was supported in part by NSF-DMR and in part by DOE. [99]{} B. Bernu and R. Mazighi, [J. Phys. A]{} [23]{}, 5745 (1990). S. McNamara and W. R. Young, [Phy. Fluids A]{} [4]{}, 496 (1992). E. Clement, S. Luding, A. Blumen, J. Rajchenbach and J. Duran, [Int. J. Mod. Phys. B]{} [7]{}, 1807 (1993). P. Constantin, E. Grossman and M. Mungan, [Physica D]{} [83]{}, 409 (1995). S. McNamara and W. R. Young, [Phy. Rew. E]{} [50]{}, R28 (1994). P. K. Haff, [J. Fluid Mech.]{} [134]{}, 401 (1983). M. A. Hopkins and M. Y. Louge, [Phy. Fluids A]{} [3]{}, 4 (1990). I. Goldhirsch and G. Zanetti, [Phys. Rev. Lett.]{} [70]{}, 1619 (1993). Y. Du, H. Li and L. P. Kadanoff, [Phys. Rev. Lett.]{} [74]{}, 1268 (1995). H. M. Jaeger, S. R. Nagel and R. P. Behringer, “The Physics of Granular Materials”, preprint. [^1]: Email: [email protected] [^2]: Email: [email protected] [^3]: But when there is a sufficiently strong attraction, e.g. gravity, between particles so that the directions of the accelerations are reversed, we want fixed points with $\alpha\le 0$. Then the fixed point $(\alpha_0,k_0)$ becomes stable when condition (\[eq:B\]) is violated. Inelastic collapse can happen in a much larger region of $(r,\theta)$.	ArXiv
--- abstract: 'Polarization-resolved magneto-luminescence, together with simultaneous magneto-transport measurements, have been performed on a two-dimensional electron gas (2DEG) confined in CdTe quantum well in order to determine the spin-splitting of fully occupied electronic Landau levels, as a function of the magnetic field (arbitrary Landau level filling factors) and temperature. The spin splitting, extracted from the energy separation of the $ \sigma^+$ and $\sigma^-$ transitions, is composed of the ordinary Zeeman term and a many-body contribution which is shown to be driven by the spin-polarization of the 2DEG. It is argued that both these contributions result in a simple, rigid shift of Landau level ladders with opposite spins.' author: - 'J.' - 'K.' - 'F. J.' - 'P.' - 'B. A.' - 'D. K.' - 'M.' - 'V.' - 'G.' - 'T.' title: 'Enhancement of the spin-gap in fully occupied two-dimensional Landau levels' --- A number of experiments on two-dimensional electron gases (2DEGs) [@Nicholas88; @Usher90; @Leadley98; @Maude98] clearly show that the thermal activation of carriers across the Fermi energy, located between the spin split Landau levels at odd integer filling factors ($\nu$), is governed by a gap which can significantly surpass the single particle Zeeman energy included in band structure models. This phenomenon, referred to in the literature as $g$ factor or spin gap enhancement, [@Fang68; @Janak69; @Ando74] is thought to be driven by the spin polarization of a 2DEG and is a primary manifestation of the interactions between two-dimensional electrons in the integer quantum Hall effect (QHE) regime. It is a result of the specific character of the spin-excitation spectra of a 2DEG at odd integer $\nu$-QHE states [@Bychkov81; @Kallin84]. It can be seen as arising from the contribution of Coulomb interactions (including exchange terms) to the energy which is required to remove, or inject, an electron from, or to, a given spin resolved Landau level (LL). To date, the effect of the spin-gap enhancement has been generally limited to experiments [@Nicholas88; @Usher90; @Maude98; @Dolgopolov97; @Wang92; @Wiegers97] which probe the spin splitting at the Fermi level, for QHE states at exactly odd filling factors. This limitation has been thought to be overcome with spectroscopic methods such as, for example, interband optics [@Kukushkin93; @Kukushkin96; @Potemski98] or tunneling experiments [@Dial07], which, within their trivial description, permit to investigate the processes of removing/adding an electron from/to a 2DEG, at arbitrary energy, filling factor and temperature. Among the different spectroscopic methods, magneto-luminescence measurements has been widely invoked to investigate electron-electron correlation in the QHE regime, however, measurements to probe the spin-gap enhancement are rather scarce [@Kukushkin93; @Dial07]. Here, we report on magneto-photoluminescence studies of a 2DEG confined in a high quality CdTe quantum well, and, show that the enhancement of the spin splitting is not only a property of spin excitations at the Fermi level, but that it is also relevant for fully occupied spin Landau levels, located well below the Fermi energy. We have measured the many body contribution to the spin gap for fully populated spin Landau levels over a wide range of filling factors and temperatures, and show that it is driven by Coulomb interaction, apparent via the spin polarization of the investigated 2DEG with its relatively large bare Zeeman splitting. The increasingly high quality of GaAs/GaAlAs structures has been driving advances in the physics of interacting 2D electrons. Notably, 2D electrons in a GaAs matrix are characterized by a relatively small bare $g$ factor (-0.44) and therefore by a small value of the interaction parameter $\eta=E_z/\mathcal{D}$, where $E_z=g\mu_BB$, $\mathcal{D}=e^2/ \epsilon l_B$, and, $l_B=\sqrt{\hbar/eB}$ is the magnetic length. The small value of $\eta$ is responsible for the rich physics exhibited by interacting 2D electrons in the QHE regime, for example the occurrence of competing spin polarized/unpolarized many body ground states [@Clark89] or Skyrmion-type spin texture excitations [@Sondhi93; @Schmeller95; @Maude96]. However, this complex physics often masks the appearance of simpler and basic many body effects, which should emerge more clearly when $\eta$ is sufficiently large. Disorder is an additional source of complications in ascertaining the spin polarization in systems with small $g$ factors. While high electron mobilities are obviously advantageous, GaAs-based structures are also rather fragile, displaying, for example, metastable effects upon illumination, with an associated decrease in mobility and homogeneity, which frequently prevents the simultaneous basic characterization of such structures using magneto-optics and magneto-transport. A 2DEG in a CdTe matrix [@Karczewski98], used in our experiments, is characterized by relatively large (bare) $g$ factor (-1.6) and the $\eta$-parameter in this system exceeds by a factor of $\approx 3$ its value in GaAs structures (the dielectric screening $\epsilon=10$ is slightly less efficient in CdTe). CdTe, which has a conduction band as simple as the one in GaAs, appears to be an almost ideal model system to study the QHE physics of the primary spin-polarized states. The significant progress in the crystal growth of CdTe quantum wells permits nowadays to attain a 2DEG with reasonably high mobilities. As shown in Fig. \[Fig1\], the sample studied here shows a well pronounced fractional QHE and permits a trouble-free, simultaneous measurement of high quality magneto-photoluminescence and magneto-transport. The active part of the investigated structure consist of a $20$ nm-wide CdTe quantum well (QW), modulation doped on one side with iodine, and embedded between Cd$_{0.74}$Mg$_{0.26}$Te barriers. The sample, in form of 1.5$\times$6mm rectangle, was equipped with electrical contacts in a Hall bar configuration to permit simultaneous optical and electric measurements. Experiments have been carried out using either a $^3$He/$^4$He dilution refrigerator or a variable temperature $^4$He cryostat, in magnetic fields supplied by a resistive (28 T) or superconducting (11 T) magnets. A standard, low frequency ($\approx 10$ Hz) lock-in technique has been applied for the resistance measurements. Polarization resolved, $\sigma^{+}$ and $\sigma^{-}$ photoluminescence (PL) spectra have been measured using a single 600 $\mu$m-diameter optical fiber to transmit the excitation beam (514 nm-line of Ar$^+$ laser) and to collect the photoluminescence signal for the spectrometer (spectral resolution $\approx100\mu eV$) equipped with a CCD camera. An appropriate linear polarizer and $\lambda$/4-plate were placed directly between the end of the fiber and the sample. The $\sigma^{+}$ and $\sigma^{-}$ PL components were measured by reversing the polarity of the magnetic field. Special attention has been paid to assure a low level of laser excitation ($\approx50$ $\mu$W/cm$^2$), to precisely calibrate the magnetic field, and to measure the spectra at small intervals (down to 5 mT) of the magnetic field. Under our experimental conditions (continuous laser illumination), the 2DEG density of $\approx4.5\times 10^{11}$ cm$^{-2}$ and mobility of $\mu=2.6 \times 10^{5}$ cm$^2$/Vs were well reproduced in different experimental runs. The representative results of simultaneous magneto-PL and magneto-resistivity measurements of our sample are shown in Fig. \[Fig1\]. As can be seen in Fig. \[Fig1\](b), the investigated 2DEG shows all typical attributes of the QHE in a system with fairly high mobility and relatively high electron concentration; well developed integer QHE states and the appearance of $5/3$, $4/3$ and $2/3$ fractional states (which will be discussed elsewhere). From the field at which the Shubnikov de Haas (SdH) oscillations ($B_{1}\approx94$ mT), and spin-splitting appears ($B_{2}\approx0.51$ T), we obtain a first estimate of the enhanced $g$ factor, $g^\approx3.7$ using the condition ($\hbar eB_1/m^{} \approx g^{}\mu _{B}B_2$) where the electron effective mass $m^{}=0.1m_{e}$ was derived from cyclotron resonance absorption measured on a parent sample. A Dingle analysis of the SdH oscillations gives a quantum lifetime $\tau_q=\hbar/2\Gamma=(3.0\pm0.3)$ ps (broadening of Lorentzian Landau levels $\Gamma\approx110~\mu$eV) as compared to the transport lifetime $\tau_{\tau}\approx15$ ps (derived from the measured mobility). The evolution of the PL with the magnetic field (Fig. \[Fig1\]) resembles spectra reported in numerous PL investigations, widely applied in the past to GaAs-based structures [@Asano02]. Peaks in the magneto-PL spectra are due to the recombination of electrons from occupied conduction band LLs ($L_{N}, E_{N}=(N+1/2)\hbar\omega_{c}, N=0,1,..$) with photo-excited holes from valence band LLs ($L^{h}_{N}, E^{h}_{N}=(N+1/2)\hbar\omega^{h}_{c}, N=0,1,...)$, where $\omega_{c}$ and $\omega^{h}_{c}$ is the cyclotron frequency of the electrons and holes respectively. The energy of the main peaks, due to $L_{N} \rightarrow L^{h}_{N}$ ($N_{e}-N_{h}=0$) transitions which scale as $E_{0}+(N+1/2)(\hbar\omega_{c}+\hbar\omega^{h}_{c})$, are shown as black dots in Fig. \[Fig1\]. Since $\hbar\omega^{h}_{c}/\hbar\omega_{c}\approx5$ [@Romestain80] the magneto-PL spectra reflect largely the characteristic fan chart of electronic LLs (with respect to band-edge energy, $E_{0}$) including their occupation factor. Opposite LL spin components are resolved in the $\sigma^{+}$ and $\sigma^{-}$ spectra. The exchange of the intensity between the $\sigma^{+}$ and $\sigma^{-}$ PL when sweeping through filling factor $\nu=1$ is typical of the 2DEG studied here and results from the selection rules which are specific to CdTe (see Fig. \[Fig1\](d)). The non-monotonic variation, with magnetic field, of the transition energies and intensities (oscillations which correlate with filling factor) and possible appearance of line splitting (see Fig. \[Fig1\]) are other common features of magneto-PL investigations of a 2DEG. Electron-electron interactions, combined with different perturbations induced by the presence of the valence band hole, are almost certainly at the origin of these features [@Asano02]. The understanding these features is far from universal and a detailed analysis of the energy and intensity of each individual magneto-PL transitions is beyond the scope of our paper. We have found, however, that information on the effects of electron-electron interactions can be extracted from the relative positions of polarization-resolved PL peaks arising from different LL spin components. We focus our attention on the two lowest energy $\sigma^{+}$ and $\sigma^{-}$ magneto-PL transitions (Fig. \[Fig2\]) which are due to electrons, with different spins, recombining from the fully populated ($L_{0}$) LL. While the energy of each of these peaks displays a non-trivial dependence on the magnetic field, here we focus on the evolution of the energy separation $\Delta E$ between the $\sigma ^{+}$ and $\sigma ^{-}$ transitions plotted in Fig. \[Fig2\](c). The splitting $\Delta E$ does not follow a linear field dependence which is expected for the case of an ordinary Zeeman effect. This can be even seen in the raw data in Fig. \[Fig2\](a-b); The splitting $\Delta E$ observed at higher field $B=4.63$ T is clearly smaller than the splitting at lower fields $B=3.7$ T. The $\Delta E$ versus $B$ dependence in Fig. \[Fig2\](c) naturally suggests that this dependence is composed of two terms; a Zeeman term ($\Delta E _{Z}$) linear with $B$ and a many body term ($\Delta E _{\uparrow\downarrow}$), which is non-monotonic with $B$, having maxima at odd integer $\nu$ and zeros at even integer $\nu$. The linear term can be extracted from the splitting at even integer $\nu$ and it is in agreement with the ordinary Zeeman effect expected in our structure. Taking into account the selection rules depicted in Fig. \[Fig1\](d), the splitting $\Delta E _{Z}=(\|g\|-\|g_{h}\|)\mu_{B}B=g_{eff}\mu_{B}B$, which requires $g_{eff}=1.1$ to fit the data in agreement with the reported values of $g=-1.6$ and $g_{h}\approx0.5$, for electronic and valence hole $g$ factors in CdTe QWs [@Zhao96]. To further clarify the origin of the $\Delta E_{\uparrow\downarrow}$ term, we plot this term as a function of filling factor and show its characteristic evolution with temperature (Fig. \[Fig3\]). We have extracted $\Delta E_{\uparrow\downarrow}$ from different experimental runs, by subtracting the ordinary Zeeman term which is assumed to be temperature independent. The electron concentration (filling factor scale) was determined using simultaneous magneto-resistance measurements. An inspection of the results presented in Fig. \[Fig3\] strongly suggest that $\Delta E_{\uparrow\downarrow}$ is ruled by the spin polarization $\mathcal{P}=\frac{n_\downarrow-n_\uparrow}{n_\downarrow+n_\uparrow}$ of the 2DEG. A quantitative verification of this hypothesis is provided by the following simple model. We consider the ideal case of a 2DEG with discrete Landau levels separated by $\hbar\omega_{c}$ and spin split by $$\label{Spingap} \Delta_{s}=\|g\|\mu_{B}B + \Delta E_{\uparrow\downarrow}=\|g\|\mu_{B}B+ \Delta_{0}'\varphi(B)\cdot\frac{n_\downarrow-n_\uparrow}{n_\downarrow+n_\uparrow}.$$ In particular, we assume that the enhanced part ($\Delta E _{\uparrow\downarrow}$) of the spin splitting is common for all Landau levels, including the lowest LL ($L_{0}$) which we probe with PL and the LL in the vicinity of the Fermi energy, the occupation of which determines the spin polarization. Furthermore, we suppose that $\varphi(B)=\sqrt[4]{B^{2}+B^{2} _{0}}$ in order to phenomenologically account for the expected behavior of $\Delta E_{\uparrow\downarrow}$ in the limit of high magnetic fields ($\varphi(B) \sim \sqrt{B}$) and when $B$ tends to zero (($\varphi(B)$=constant) [@NOTE]. Finally, we self-consistently calculate $\Delta E _{\uparrow\downarrow}$ (and $\mathcal{P}$) and obtain agreement with the data by adjusting the two fitting parameters, $\Delta_0=\Delta_0'\sqrt{B_{0}}=2.1$ meV and $B_0=3.7$ T. Despite the rather crude approximations, the calculations well reproduce the experimental data (Fig. \[Fig3\]) over a wide range of filling factors ($4\leq \nu \leq 10$) and for different temperatures up to the temperature for which $E_{\uparrow\downarrow}$ (and $\mathcal{P}$) vanishes. The agreement is less satisfactory in the vicinity of $\nu=3$, and completely fails around $\nu=1$ where difference between $\sigma^+$ and $\sigma^-$ peaks shows almost no enhancement effect. These discrepancies are due to the fact that the physics of PL processes for a 2DEG at low filling factors is far more complex [@Asano02] compared to our temptingly simple picture of electrons which recombine (are extracted) from the homogenous Fermi sea of a 2DEG. The use of discrete LLs in our calculations is justified by the large bare Zeeman energy which exceeds the LL width (110 $\mu$eV, extracted from low field transport data) already at fields of $\sim2$ T ($\nu \sim 9$). The assumption that $E_{\uparrow\downarrow}$ does not depend on LL index is probably also realistic. When modelling the data, we have investigated various scenarios for a LL index dependence of the spin-gap enhancement but found that a constant value reproduces the data fairly well. Although it is more difficult to accurately determine the energy of the weak magneto-PL peaks for the higher $N>0$ LLs, it is possible to follow the separation between $\sigma+$ and $\sigma-$ transitions associated with the $L_{1}$-level in the vicinity of $\nu=5$. As shown in Fig. \[Fig3\] (right panel), the extracted enhancement of the spin gap in the $L_{1}$-level is practically the same as in the $L_{1}$-level. Moreover, we find a fair agreement between the spin gaps $\|g\|\mu_{B}B + \Delta E_{\uparrow\downarrow}$, extracted from PL, for fully populated LLs and the activation gaps, of $0.95$, $0.63$ and $0.36$ meV, for spin excitations across the Fermi energy, which we have estimated from resistance measurements at filling factors $\nu=5,7$, and $9$, respectively. Finally, let us speculate about a possible extension of the assumed model to the limit of low magnetic fields and to the particular case of $\nu=1$. When $B\rightarrow0$, the extrapolation of Eq. \[Spingap\] (at $T=0$ K) yields a linear $\Delta_{S}$ versus $B$ dependence; $\Delta_{S}=\|g\|\mu_{B}B+\Delta_{0}/\nu=(\|g\|+\Delta_{0}/\mu_{B}B_{\nu=1}) \mu_{B}B =g^{}\mu_{B}B$, where $B_{\nu=1}$ corresponds to the magnetic field for $\nu=1$. With $B_{\nu=1}=18.5$ T ($n=4.5\times 10^{11}$ cm$^{-2}$) and $\Delta_{0}=2.1$ meV we extract $g^{}=3.6$ for the enhanced $g$ factor in good agreement with the estimation of $g^{}\sim3.7$ from the low field onset of spin splitting in the SdH oscillations. Setting $\nu=1$ (and $\mathcal{P}=1$) in Eq. ( \[Spingap\]) we extrapolate $\Delta_{S}=\|g\|\mu_{B}B+\Delta_{0}\sqrt[4]{1+B^{2} _{\nu=1}/ B^{2} _{0}}$ and calculate $\Delta_{S}=6.4$ meV. This value is a factor of $\sim4$ smaller than its ultimate limit of $\sqrt{\pi/2}e^2/\epsilon l_B$ [@Bychkov81; @Kallin84] but in good agreement with the reported values in GaAs structures from optical and capacitance measurements [@Kukushkin96; @Dolgopolov97]. In conclusion, spectroscopic polarization-resolved magneto-PL studies of a 2DEG confined in CdTe quantum well reveal the many-body enhancement of the spin-splitting of fully occupied 2D Landau levels well below the Fermi energy. The enhancement is mainly determined by the spin polarization of the 2DEG, since the spin gap is maximized at odd filling factors, but vanishes at even filling factors or high temperatures. We argue that the spin polarization simply induces, in addition to the ordinary Zeeman splitting, a rigid shift of the spin up Landau levels with respect to spin-down Landau levels. This simple picture for the many body spin-gap enhancement emerges from magneto-PL studies of a 2DEG with relatively large (single particle) $g$ factor. [25]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , *, (). , , (). , , (). , , (). , , (). , , (). , , (). , , (). , , (). , , (). , , (). , , (). , , (). , , (). , , (). , , (). , , (). , , (). , , (). , , (). , , (). , , (), . , , (). , **, ().	ArXiv
--- abstract: 'The description of the proximity effect in superconducting/ferromagnetic heterostructures requires to use spin-dependent boundary conditions. Such boundary conditions must take into account the spin dependence of the phase shifts acquired by electrons upon scattering on the boundaries of ferromagnets. The present article shows that this property can strongly affect the critical temperature and the energy dependence of the density of states of diffusive heterostructures. These effects should allow a better caracterisation of diffusive superconductor/ferromagnet interfaces.' author: - Audrey Cottet title: 'Spectroscopy and critical temperature of diffusive superconducting/ferromagnetic hybrid structures with spin-active interfaces' --- Introduction ============ When a ferromagnetic metal ($F$) with uniform magnetization is connected to a BCS superconductor ($S$), the singlet electronic correlations characteristic of the $S$ phase can propagate into $F$ because electrons and holes with opposite spins and excitation energies are coupled coherently by Andreev reflections occurring at the $S/F$ interface. Remarkably, the ferromagnetic exchange field induces an energy shift between the coupled electrons and holes, which leads to spatial oscillations of the superconducting order parameter in $F$ [@Buzdin1982; @Golubov]. This effect has been observed experimentally through oscillations of the density of states (DOS) in $F$ with the thickness of $F$ [@TakisN], or oscillations of the critical current $I_{0}$ through $S/F/S$ structures [@Ryazanov; @TakisI; @SellierPRB; @Blum], with the thickness of $F$ or the temperature. The oscillations of $I_{0}$ have allowed to obtain $\pi$-junctions[@Guichard], i.e. Josephson junctions with $I_{0}<0$, which could be useful in the field of superconducting circuits [@Ioffe; @Taro]. A reentrant behavior of the superconducting critical temperature of $S/F$ bilayers with the thickness of $F$ has also been observed [@TcSF]. At last, some $F/S/F$ trilayers have shown a lower critical temperature for an antiparallel alignment of the magnetizations in the two $F$ layers as compared with the parallel alignment[@SSspinswitch], which should offer the possibility of realizing a superconducting spin-switch[@DeGennes; @TcFSFth]. ![a. Diffusive $F/S/F$[ ]{}trilayer consisting of a BCS superconductor $S$ with thickness $d_{S}$ placed between two ferromagnetic electrodes $F_{1}$ and $F_{2}$ with thickness $d_{F}$. In this picture, the directions of the magnetic polarizations in $F_{1}$ and $F_{2}$ are parallel \[antiparallel\], which corresponds to the configuration $\mathcal{C}=P$ $[AP]$. b. $S/F$[ ]{}bilayer consisting of a BCS superconductor $S$ with thickness $d_{S}/2$ contacted to a ferromagnetic electrode $F$ with thickness $d_{F}$.](structure.eps){width="0.7\linewidth"} For a theoretical understanding of the behavior of $S/F$ hybrid circuits, a proper description of the interfaces between the different materials is crucial. For a long time, the only boundary conditions available in the diffusive case were spin-independent boundary conditions derived for $S/$normal metal interfaces[@Kuprianov]. Recently, spin-dependent boundary conditions have been introduced for describing hybrid diffusive circuits combining BCS superconductors, normal metals and ferromagnetic insulators [@condmatHuertas]. These boundary conditions take into account the spin-polarization of the electronic transmission probabilities through the interface considered, but also the spin-dependence of the phase shifts acquired by electrons upon transmission or reflection by the interface. The first property generates widely known magnetoresistance effects[@magn]. The second property is less commonly taken into account. However, the Spin-Dependence of Interfacial Phase Shifts (SDIPS) can modify the behavior of many different types of mesoscopic circuits with ferromagnetic elements, like those including a diffusive normal metal island [@FNF], a resonant system[@CottetEurophys; @SST], a Coulomb blockade system[@CB; @Cottet06; @SST], or a Luttinger liquid[@Luttinger]. It has also been shown that the SDIPS has physical consequences in $S/F$ hybrid systems[@Tokuyasu; @otherBC; @mixing; @condmatHuertas]. One can note that, in some references, the SDIPS is called ”spin-mixing angle” or ”spin-rotation angle” (see e.g. Refs. ). In the diffusive $S/F$ case, the spin-dependent boundary conditions of Ref. have been applied to different circuit geometries[@demoBC; @applications; @Morten; @cottet05; @Braude] but the only comparison to experimental data has been performed in Ref. . The authors of this Ref. have generalized the boundary conditions of Ref. to the case of metallic $S/F$ interfaces with a superconducting proximity effect in $F$. They have showed that the SDIPS can induce a shift in the oscillations of the critical current of a $S/F/S$ Josephson junction or of the DOS of a $S/F$ bilayer with the thickness of $F$. Signatures of this effect have been identified in the hybrid structures of Refs. . Nevertheless, the problem of characterizing the SDIPS of diffusive $S/F$ interfaces has raised little attention so far, in spite of the numerous experiments performed. A good characterization of the properties of diffusive $S/F$ interfaces would be necessary for a better control of the superconducting proximity effect in diffusive heterostructures. The present article presents other consequences of the SDIPS than that studied in Ref. , which could be useful in this context. In particular, the SDIPS can generate an effective magnetic field in a diffusive $S$ in contact with a diffusive $F $, like found for a ballistic $S$ in contact with a ferromagnetic insulator [@Tokuyasu]. This effective field can be detected, in particular, through the DOS of the diffusive $F$ layer, with a visibility which depends on the thickness of $F$. A strong modification of the variations of the critical temperature of diffusive $S/F$ structures with the thickness of $F$ is also found. These effects should allow to characterize the SDIPS of diffusive $S/F$ interfaces through DOS and critical temperature measurements, by using the heterostructures currently fabricated. The calculations reported in this paper are also appropriate to the case of a diffusive $S$ layer contacted to a ferromagnetic insulator ($FI $). This paper is organized as follows: Section II presents the initial set of equations used to describe the heterostructures considered. The case of $F/S/F$ trilayers is mainly addressed, but the case of $S/F$ (or $S/FI$) bilayers follows straightforwardly. Section III specializes to the case of a weak proximity effect in $F$ and a superconducting layer with a relatively low thickness $d_{S}\leq\xi_{S}$, with $\xi_{S}$ the superconducting coherence length in $S$. The spatial evolution of the electronic correlations in the $S$ and $F$ layers is studied in Section III.A. The energy-dependent DOS of $S/F$ heterostructures is calculated in Section III.B. Section III.C considers briefly the limit of $S/FI$ bilayers. Section III.D discusses SDIPS-induced effective field effects in other types of systems. Section III.E compares the present work to other DOS calculations for data interpretation in $S/F$ heterostructures. Critical temperatures of $S/F$ circuits are calculated and discussed in Section III.F. Conclusions are presented in Section IV. Throughout the paper, I consider conventional BCS superconductors with a s-wave symmetry. Initial description of the problem ================================== This article mainly considers a diffusive $F/S/F$[ ]{}trilayer consisting of a BCS superconductor $S$ for $-d_{S}/2<x<d_{S}/2$, and ferromagnetic electrodes $F_{1}$ for $x\in\{-d_{S}/2-d_{F},-d_{S}/2\}$ and $F_{2}$ for $x\in\{d_{S}/2,d_{S}/2+d_{F}\}$ (see Figure 1.a). The magnetic polarization of the two ferromagnets can be parallel (configuration $\mathcal{C}=P$) or anti-parallel (configuration $\mathcal{C}=AP$), but the modulus $\left\| E_{ex}\right\| $ of the ferromagnetic exchange field is assumed to be the same in $F_{1}$ and $F_{2}$. Throughout the structure, the normal quasiparticle excitations and the superconducting condensate of pairs can be characterized with Usadel normal and anomalous Green’s functions $G_{n,\sigma}=\mathrm{sgn}(\omega_{n})\cos(\theta_{n,\sigma})$ and $F_{n,\sigma}=\sin(\theta_{n,\sigma})$, with $\theta_{n,\sigma}(x)$ the superconducting pairing angle, which depends on the spin direction $\sigma \in\{\uparrow,\downarrow\}$, the Matsubara frequency $\omega_{n}(T)=(2n+1)\pi k_{B}T$, and the coordinate $x$ (see e.g. Ref. ). The Usadel equation describing the spatial evolution of $\theta_{n,\sigma}$ writes $$\frac{\hbar D_{S}}{2}\frac{\partial^{2}\theta_{n,\sigma}}{\partial x^{2}% }=\left\| \omega_{n}\right\| \sin(\theta_{n,\sigma})-\Delta(x)\cos (\theta_{n,\sigma}) \label{UsadelS}%$$ in $S$ and $$\frac{\hbar D_{F}}{2}\frac{\partial^{2}\theta_{n,\sigma}}{\partial x^{2}% }=\left( \left\| \omega_{n}\right\| +iE_{ex}\sigma\mathrm{sgn}(\omega _{n})\right) \sin(\theta_{n,\sigma}) \label{UsadelF2}%$$ in $F_{1}$ and $F_{2}$, with $D_{F}$ the diffusion constant of the ferromagnets and $D_{S}$ the diffusion constant of $S$. The self-consistent superconducting gap $\Delta(x)$ occurring in (\[UsadelS\]) can be expressed as $$\Delta(x)=\frac{\pi k_{B}T\lambda}{2}\sum\limits_{\substack{\sigma \in\{\uparrow,\downarrow\}\\\omega_{n}(T)\in\{-\Omega_{D},\Omega_{D}\}}% }\sin(\theta_{n,\sigma}) \label{Delta}%$$ with $\Omega_{D}$ the Debye frequency of $S$, $\lambda^{-1}=2\pi k_{B}T_{_{c}% }^{BCS}\sum\nolimits_{\omega_{n}(T_{_{c}}^{BCS})\in\{0,\Omega_{D}\}}\omega _{n}^{-1}$ the BCS coupling constant and $T_{_{c}}^{BCS}$ the bulk transition temperature of $S$. I assume $\Delta=0$ in $F_{1}$ and $F_{2}$. The above equations must be supplemented with boundary conditions describing the interfaces between the different materials. First, one can use $$\left. \frac{\partial\theta_{n,\sigma}}{\partial x}\right\| _{x=\pm (d_{S}/2+d_{F})}=0 \label{derZero}%$$ for the external sides of the structure. Secondly, the boundary conditions at the $S/F$ interfaces can be calculated by assuming that the interface potential locally dominates the Hamiltonian, i.e. at a short distance it causes only ordinary scattering (with no particle-hole mixing) (see e.g. Ref. ). This ordinary scattering can be described with transmission and reflection amplitudes $t_{n,\sigma}^{S(F)}$ and $r_{n,\sigma }^{S(F)}$ for electrons coming from the $S$($F$) side of the interface in channel $n$ with a spin direction $\sigma$. The phases of $t_{n,\sigma}% ^{S(F)}$ and $r_{n,\sigma}^{S(F)}$ can be spin-dependent due to the exchange field $E_{ex}$ in $F_{1(2)}$ and a possible spin-dependence of the barrier potential between $S$ and $F_{1(2)}$. Boundary conditions taking into account this so-called Spin-Dependence of Interfacial Phase Shifts (SDIPS) have been derived for $\|t_{n,\uparrow}^{S}\|^{2},\|t_{n,\downarrow}^{S}\|^{2}\ll1$ and a weakly polarized $F$[@condmatHuertas; @cottet05]. When there is no SDIPS, the boundary conditions involve the tunnel conductance $G_{T}=G_{Q}% \sum\nolimits_{n}T_{n}$ and the magnetoconductance $G_{MR}=G_{Q}% \sum\nolimits_{n}(\|t_{n,\uparrow}^{S}\|^{2}-\|t_{n,\downarrow}^{S}\|^{2})$, with $\uparrow(\downarrow)$ the majority(minority) spin direction in the $F$ electrode considered, $G_{Q}=e^{2}/h$, and $T_{n}=\|t_{n,\uparrow}^{S}% \|^{2}+\|t_{n,\downarrow}^{S}\|^{2}$. In the case of a finite SDIPS, one must also use the conductances $G_{\phi}^{F(S)}=2G_{Q}\sum\nolimits_{n}(\rho _{n}^{F(S)}-4[\tau_{n}^{S(F)}/T_{n}])$, $G_{\xi}^{F(S)}=-G_{Q}\sum \nolimits_{n}\tau_{n}^{S(F)}$ and $G_{\chi}^{F(S)}=G_{Q}\sum\nolimits_{n}% T_{n}(\rho_{n}^{F(S)}+\tau_{n}^{S(F)})/4$, with $\rho_{n}^{m}% =\operatorname{Im}[r_{n,\uparrow}^{m}r_{n,\downarrow}^{m~\ast}]$ and $\tau _{n}^{m}=\operatorname{Im}[t_{n,\uparrow}^{m}t_{n,\downarrow}^{m~\ast}]$ for $m\in\{S,F\}$. In the following, I will focus on the effects of $G_{\phi}^{F}$ and $G_{\phi}^{S}$, and I will assume $G_{MR}$, $G_{\xi}^{F(S)}$ and $G_{\chi }^{F(S)}$ to be negligible, like found with a simple barrier model in the limit $T_{n}\ll1$ and $E_{ex}\ll E_{F}$[@cottet05]. In this case, one finds that the boundary conditions for the $S/F$ interface located at $x=x_{j}=(-1)^{j}d_{s}/2$, with$\ j\in\{1,2\}$, write $$\xi_{F}\left. \frac{\partial\theta_{n,\sigma}^{F}}{\partial x}\right\| _{x_{j}}=(-1)^{j}\gamma_{T}\sin[\theta_{n,\sigma}^{F}-\theta_{n,\sigma}% ^{S}]+i\gamma_{\phi}^{F}\epsilon_{n,\sigma}^{\mathcal{C},j}\sin[\theta _{n,\sigma}^{F}] \label{BCright}%$$ and $$\xi_{F}\left. \frac{\partial\theta_{n,\sigma}^{F}}{\partial x}\right\| _{x_{j}}-\frac{\xi_{S}}{\gamma}\left. \frac{\partial\theta_{n,\sigma}^{S}% }{\partial x}\right\| _{x_{j}}=\sum\limits_{m\in\{F,S\}}i\gamma_{\phi}% ^{m}\epsilon_{n,\sigma}^{\mathcal{C},j}\sin[\theta_{n,\sigma}^{m}] \label{BCleft}%$$ where the indices $S$ and $F$ indicate whether $\theta_{n,\sigma}$ and its derivative are taken at the $S$ or $F$ side of the interface. These equations involve the reduced conductances $\gamma_{T}=G_{T}\xi_{F}/A\sigma_{F}$ and $\gamma_{\phi}^{F(S)}=G_{\phi}^{F(S)}\xi_{F}/A\sigma_{F}$, the barrier asymmetry coefficient $\gamma=\xi_{S}\sigma_{F}/\xi_{F}\sigma_{S}$, the superconducting coherence lengthscale $\xi_{S}=(\hbar D_{S}/2\Delta _{BCS})^{1/2}$, the magnetic coherence lengthscale $\xi_{F}=(\hbar D_{F}/\left\| E_{ex}\right\| )^{1/2}$, the gap $\Delta_{BCS}$ for a bulk $S$, the normal state conductivity $\sigma_{F(S)}$ of the $F(S)$ material, and the junction area $A$. The coefficient $\epsilon_{n,j}^{\mathcal{C}}$ takes into account the direction of the ferromagnetic polarization of electrode $F_{j}$ in configuration $\mathcal{C}\in\{P,AP\}$. One can use the convention $\epsilon_{n,\sigma}^{P,j}=(-1)^{j}\sigma\mathrm{sgn}(\omega_{n})$ and $\epsilon_{n,\sigma}^{AP,j}=\sigma\mathrm{sgn}(\omega_{n})$, in which the factor $\mathrm{sgn}(\omega_{n})$ arising from the definition of $\theta_{n,\sigma}$ and the terms $(-1)^{j}$ and $\sigma$ arising from the boundary conditions have been included for compactness of the expressions. Note that in the presence of a finite SDIPS i.e. $\gamma_{\phi}^{F(S)}\neq0$, the right hand side of equation (\[BCleft\]) is not zero contrarily to what found in the spin-degenerate case [@Kuprianov]. In the general case, $\gamma_{\phi}^{F}$ and $\gamma_{\phi}^{S}$ are different (see e.g. Appendix A). This implies that, with the present approximations, each interface is characterized by three parameters: $\gamma_{T}$, $\gamma_{\phi}^{F}$ and $\gamma_{\phi}^{S}$. For the sake of simplicity, symmetric $F/S/F$ trilayers are considered, so that $\gamma_{T}$, $\gamma_{\phi}^{F}$ and $\gamma_{\phi }^{S}$ are the same for the two $S/F$ interfaces. Before working out the above system of equations, it is interesting to note that the angle $\theta_{n,\sigma}$ calculated in the parallel configuration $\mathcal{C}=P$ for $x>0$ also corresponds to the angle $\theta_{n,\sigma}$ expected for a $S/F$ bilayer consisting of a superconductor $S$ for $0<x<d_{S}/2$, and a ferromagnetic electrode $F$ for $x\in\{d_{S}% /2,d_{S}/2+d_{F}\}$ (Figure 1.b). In practice, using a $F/S/F$ geometry can allow one to obtain more information on spin effects, as shown below. Case of a thin superconductor and a weak proximity effect in $F$ ================================================================ Spatial variations of the pairing angle --------------------------------------- I will assume that the amplitude of the superconducting correlations in $F_{1(2)}$ is weak, i.e. $\left\| \theta_{n,\sigma}\right\| \ll1$ for $x\in\{-d_{S}/2-d_{F},-d_{S}/2\}$ and $x\in\{d_{S}/2,d_{S}/2+d_{F}\}$ (hypothesis 1) so that one can develop the Usadel equation (\[UsadelF2\]) at first order in $\theta_{n,\sigma}$. This leads to $$\frac{\partial^{2}\theta_{n,\sigma}}{\partial x^{2}}-\left( \frac {k_{n,\sigma}^{\mathcal{C},j}}{\xi_{F}}\right) ^{2}\theta_{n,\sigma}=0 \label{UsadelF}%$$ in the ferromagnet $F_{j}$, with $j\in\{1,2\}$, $k_{n,\sigma}^{\mathcal{C}% ,j}=(2[i\eta_{n,\sigma}^{\mathcal{C},j}+\left\| \omega_{n}/E_{ex}\right\| )])^{1/2}$ and $\eta_{n,\sigma}^{\mathcal{C},j}=(-1)^{j}\epsilon_{n,\sigma }^{\mathcal{C},j}$. Combining Eqs. (\[derZero\]) and (\[UsadelF\]), one finds in $F_{j}$ $$\theta_{n,\sigma}(x)=\theta_{n,\sigma}^{F}(x_{j})\frac{\cosh\left( \left[ x-(-1)^{j}\left( d_{F}+\frac{d_{S}}{2}\right) \right] \frac{k_{n,\sigma }^{\mathcal{C},j}}{\xi_{F}}\right) }{\cosh\left( d_{F}\frac{k_{n,\sigma }^{\mathcal{C},j}}{\xi_{F}}\right) } \label{theta}%$$ This result together with boundary condition (\[BCright\]) leads to $$\theta_{n,\sigma}^{F}(x_{j})=\frac{\gamma_{T}\sin(\theta_{n,\sigma}^{S}% (x_{j}))}{\gamma_{T}\cos(\theta_{n,\sigma}^{S}(x_{j}))+i\gamma_{\phi}^{F}% \eta_{n,\sigma}^{\mathcal{C},j}+B_{n,\sigma}^{\mathcal{C},j}} \label{thetaF}%$$ with $B_{n,\sigma}^{\mathcal{C},j}=k_{n,\sigma}^{\mathcal{C},j}\tanh [d_{F}k_{n,\sigma}^{\mathcal{C},j}/\xi_{F}]$. This allows to rewrite the boundary condition (\[BCleft\]) in closed form with respect to $\theta_{n,\sigma}^{S}$, i.e. $$\xi_{S}\left. \frac{\partial\theta_{n,\sigma}^{S}}{\partial x}\right\| _{x_{j}}=(-1)^{j+1}\mathcal{L}_{j,n,\sigma}^{\mathcal{C}}\sin[\theta _{n,\sigma}^{S}(x_{j})]$$ for the $S/F$ interface located at $x=x_{j}$, with $$\frac{\mathcal{L}_{j,n,\sigma}^{\mathcal{C}}}{\gamma}=\frac{\gamma _{T}(B_{n,\sigma}^{\mathcal{C},j}+i\gamma_{\phi}^{F}\eta_{n,\sigma }^{\mathcal{C},j})}{\gamma_{T}\cos(\theta_{n,\sigma}^{S})+B_{n,\sigma }^{\mathcal{C},j}+i\gamma_{\phi}^{F}\eta_{n,\sigma}^{\mathcal{C},j}}% +i\gamma_{\phi}^{S}\eta_{n,\sigma}^{\mathcal{C},j}%$$ In the following, I will assume $\left\| E_{ex}\right\| \gg\Delta_{BCS}$ like in most experiments, so that $k_{n,\sigma}^{\mathcal{C},j}=1+i\eta_{n,\sigma }^{\mathcal{C},j}$. I will also assume $d_{S}/\xi_{S}\leq1$, so that one can use, for the $\mathcal{C}$ configuration and $-d_{S}/2<x<d_{S}/2$, $$\theta_{n,\sigma}(x)=\widetilde{\theta}_{n,\sigma}^{\mathcal{C}}% -\alpha_{n,\sigma}^{\mathcal{C}}(x/\xi_{S})-\beta_{n,\sigma}^{\mathcal{C}% }(x/\xi_{S})^{2} \label{thetaQuad}%$$ with $\left\| \theta_{n,\sigma}(x)-\widetilde{\theta}_{n,\sigma}^{\mathcal{C}% }\right\| \ll1$ (hypothesis 2). Note that although experiments are often performed in the limit of thick superconducting layers $d_{S}>\xi_{S}$, assuming $d_{S}\leq\xi_{S}$ is not unrealistic since one can obtain diffusive superconducting layers with a thickness $d_{S}\sim\xi_{S}$ (see e.g. Ref. ). Furthermore, using relatively low values of $d_{S}$ is more favorable for obtaining efficient superconducting spin-switches[@Baladie]. Hypothesis 2 allows one to develop $\sin (\theta_{n,\sigma})$ and $\cos(\theta_{n,\sigma})$ at first order with respect to $\theta_{n,\sigma}-\widetilde{\theta}_{n,\sigma}^{\mathcal{C}}$ in $S$. Accordingly, I will neglect the space-dependence of $\Delta(x)$ and assign to it the value $\Delta^{\mathcal{C}}$ in configuration $\mathcal{C}$. Note that I do not make any assumption on the value of the angle $\widetilde{\theta}_{n,\sigma}^{\mathcal{C}}$, which is not necessarily close to the bulk BCS value. The coefficient $\mathcal{L}_{j,n,\sigma}% ^{\mathcal{C}}$ is transformed into its conjugate when the magnetic polarization of electrode $F_{j}$ is reversed. Therefore, I will note $\mathcal{L}_{1,n,\sigma}^{P}=\mathcal{L}_{2,n,\sigma}^{P(AP)}=\mathcal{L}% _{n}^{\sigma}$ and $\mathcal{L}_{1,n,\sigma}^{AP}=(\mathcal{L}_{n}^{\sigma })^{\ast}$. The above assumptions lead to $\alpha_{n,\sigma}^{P}=0$, $$\beta_{n,\sigma}^{P}=\frac{4\mathcal{L}_{n}^{\sigma}\sin(\widetilde{\theta }_{n,\sigma}^{\mathcal{C}})}{4\delta_{S}+\cos(\widetilde{\theta}_{n,\sigma }^{\mathcal{C}})\delta_{S}^{2}\mathcal{L}_{n}^{\sigma}} \label{bP}%$$$$\alpha_{n,\sigma}^{AP}=i\operatorname{Im}[\mathcal{L}_{n}^{\sigma}]\frac {4\sin(\widetilde{\theta}_{n,\sigma}^{\mathcal{C}})-\cos(\widetilde{\theta }_{n,\sigma}^{\mathcal{C}})\delta_{S}^{2}\beta_{AP}}{4+2\operatorname{Re}% [\mathcal{L}_{n}^{\sigma}]\cos(\widetilde{\theta}_{n,\sigma}^{\mathcal{C}% })\delta_{S}}%$$ and $$\beta_{n,\sigma}^{AP}=\frac{\left( 8\operatorname{Re}[\mathcal{L}_{n}% ^{\sigma}]+4\left\| \mathcal{L}_{n}^{\sigma}\right\| ^{2}\cos(\widetilde {\theta}_{n,\sigma}^{\mathcal{C}})\delta_{S}\right) \sin(\widetilde{\theta }_{n,\sigma}^{\mathcal{C}})}{8\delta_{S}+6\operatorname{Re}[\mathcal{L}% _{n}^{\sigma}]\cos(\widetilde{\theta}_{n,\sigma}^{\mathcal{C}})\delta_{S}% ^{2}+\left\| \mathcal{L}_{n}^{\sigma}\right\| ^{2}\cos^{2}(\widetilde{\theta }_{n,\sigma}^{\mathcal{C}})\delta_{S}^{3}} \label{bAP}%$$ with $\delta_{S}=d_{S}/\xi_{S}$. On the other hand, from (\[UsadelS\]), one finds $$\beta_{n,\sigma}^{\mathcal{C}}=\frac{\Delta^{\mathcal{C}}\cos(\widetilde {\theta}_{n,\sigma}^{\mathcal{C}})-\left\| \omega_{n}\right\| \sin (\widetilde{\theta}_{n,\sigma}^{\mathcal{C}})}{2\Delta_{BCS}} \label{bc}%$$ The comparison between equations (\[bP\]), (\[bAP\]) and (\[bc\]) allows one to find $\widetilde{\theta}_{n,\sigma}^{\mathcal{C}}$ as a function of $\Delta^{\mathcal{C}}$. Then, one has to calculate $\Delta^{\mathcal{C}}$ by using the self-consistency relation (\[Delta\]). I will study below the DOS and the critical temperature following from these Eqs., in a limit which leads to simple analytical expressions. Low-temperature density of states of $S/F$ heterostructures ----------------------------------------------------------- The DOS of the ferromagnets $F_{1}$ and $F_{2}$ of Figure 1.a can be probed at $x=\pm(d_{F}+d_{S}/2)$ by performing tunnelling spectroscopy through an insulating layer. So far, this quantity has been less measured[@TakisN; @Kontos04; @Courtois; @Reymond] than critical temperatures or supercurrents. However, this way of probing the superconducting proximity effect is very interesting because it allows one to obtain spectroscopic information. It has been shown that the zero-energy DOS of a $F$ layer in contact with a superconductor oscillates with the thickness of $F$. For certain thicknesses, this zero-energy DOS can even become higher than its normal state value $N_{0}$ [@theoDOS; @Zareyan; @Bergeret; @yokohama], as shown experimentally in Ref. . Remarkably, the SDIPS can shift these oscillations[@cottet05]. Although the energy dependence of the DOS of diffusive $S/F$ structures has raised some theoretical and experimental interest, the effect of the SDIPS on this energy dependence has not been investigated so far. For calculating analytically the low-temperature DOS of the structure of Fig. 1.a., one can assume $\gamma_{T}^{2}\cos(\theta_{n,\sigma}^{S}(x_{j}))/\left\| (\right. \gamma_{T}\cos(\theta_{n,\sigma}^{S}(x_{j}))+i\gamma_{\phi}^{F}% \eta_{n,\sigma}^{P,2}+B_{n,\sigma}^{P,2})(\gamma_{T}+i\gamma_{\phi}^{S}% \eta_{n,\sigma}^{P,2}\left. )\right\| \ll1$ (hypothesis 3), which leads to $\mathcal{L}_{n}^{\sigma}=\gamma\left( \gamma_{T}+i\gamma_{\phi}% ^{S}\sigma\mathrm{sgn}(\omega_{n})\right) $. This hypothesis is e.g. valid for $d_{F}\geq\xi_{F}$ and any value of $\gamma_{\phi}^{S(F)}$ if $\gamma_{T}$ is relatively small (see e.g. Fig. \[DOS1\]). I will also assume that the lowest order terms in $\delta_{S}$ prevail in the numerators and denominators of expressions (\[bP\]) and (\[bAP\]), i.e. $\beta_{n,\sigma}^{P}% \sim\mathcal{L}_{n}^{\sigma}\sin(\widetilde{\theta}_{n,\sigma}^{\mathcal{C}% })\delta_{S}^{-1}$ and $\beta_{n,\sigma}^{AP}\sim\operatorname{Re}% [\mathcal{L}_{n}^{\sigma}]\sin(\widetilde{\theta}_{n,\sigma}^{\mathcal{C}% })\delta_{S}^{-1}$ (hypothesis 4). Taking into account hypothesis 3 and $\gamma\sim1$, hypothesis 4 is valid provided $\gamma_{T}$ and $\left\| \gamma_{\phi}^{S}\right\| $ are relatively small compared to $1$. Importantly, hypotheses 3 and 4 are less restrictive regarding the value of $\gamma_{\phi }^{F}$. Accordingly, I will often use $\gamma_{T},\left\| \gamma_{\phi}% ^{S}\right\| \ll\left\| \gamma_{\phi}^{F}\right\| $ in the Figs. of this paper. Hypotheses 3 and 4 lead to $$\widetilde{\theta}_{n,\sigma}^{\mathcal{C}}=\mathrm{\arctan}\left( \frac{\Delta^{\mathcal{C}}}{\left\| \omega_{n}\right\| +\Omega_{n,\sigma }^{\mathcal{C}}}\right) \label{thetaS}%$$ with $$\Omega_{n,\sigma}^{P}=2\Delta_{BCS}\gamma(\gamma_{T}+i\gamma_{\phi}^{S}% \sigma\mathrm{sgn}(\omega_{n}))\delta_{S}^{-1} \label{omegaa}%$$ and $$\Omega_{n,\sigma}^{AP}=2\Delta_{BCS}\gamma\gamma_{T}\delta_{S}^{-1} \label{Om2}%$$ ![Zero energy density of states $N^{F}(\varepsilon=0)$ at $x=d_{F}+d_{S}/2$ as a function of $d_{F}$, for the $F/S/F$ structure of Figure 1.a in the $P$ configuration, with $\gamma=1$, $\gamma_{T}=0.12$, and $d_{S}/\xi_{S}=1$. The different curves correspond to different SDIPS parameters:$\ \gamma_{\phi}^{F}=\gamma_{\phi }^{S}=0$ (black dotted curve), $\{\gamma_{\phi}^{F}=-2.1$, $\gamma_{\phi}% ^{S}=0\}$ (black full curve), and $\{\gamma_{\phi}^{F}=-2.1$, $\gamma_{\phi }^{S}=0.12\}$ (red full curve). The low-temperature self-consistent gap $\Delta^{P}$ found for these three different cases is indicated at the bottom right of the Figure. Using a strong value for $\gamma_{\phi}^{F}$ allows one to change significantly the phase of the oscillations of $N_{\sigma}^{F}(0)$ with $d_{F}$ (effect similar to Ref. ). One can check that the hypotheses 1 to 4 are valid for the parameters used in this Figure. Note that for the different cases considered here, the critical temperature $T_{c}^{P}$ of the structure (see Sec. III.F) is such that $0.615T_{c}^{BCS}<T_{c}^{P}<0.635T_{c}^{BCS}$. The four red points correspond to the red full curves shown in Fig. \[DOS2\].[]{data-label="DOS1"}](DOS1.eps){width="0.8\linewidth"} Equations (\[thetaS\]) and (\[omegaa\]) show that an effective magnetic field $H_{eff}$ appears in the $S$ layer in the $P$ configuration, due to $\gamma_{\phi}^{S}\neq0$. From Eqs. (\[thetaS\]) and (\[omegaa\]), $H_{eff}$ can be expressed as $$g\mu_{B}H_{eff}=\frac{\hbar v_{F}^{S}}{d_{s}}\frac{2G_{\phi}^{S}\ell_{e}^{S}% }{3\sigma_{S}A}=2E_{TH}^{S}\frac{G_{\phi}^{S}}{G_{S}} \label{heff}%$$ Here, $v_{F}^{S}$ and $\ell_{e}^{S}$ denote the Fermi velocity and mean free path in $S$, and $G_{S}=\sigma_{S}A/d_{S}$ and $E_{TH}^{S}=\hbar D_{S}% /d_{S}^{2}$ denote the normal state conductance and the Thouless energy of the $S$ layer of Fig. 1.a. From Equations (\[thetaS\]) and (\[Om2\]), the effective field effect disappears in the $AP$ configuration because the two contacts are assumed to be symmetric, and therefore, their contributions to $H_{eff}$ compensate each other in the $AP$ case. Note that, in principle, the $\gamma_{\phi}^{F}$ term can induce an effective field analogue to $H_{eff}$ in the $F$ layer, but this effect is not relevant in the regime studied in this paper (see Appendix B). The effects of $H_{eff}$ on the DOS of the structure will be investigated in next paragraphs. In order to calculate $\Delta^{\mathcal{C}}$, one has to combine the self-consistency relation (\[Delta\]) with Eq. (\[thetaS\]), which gives, at low temperatures, $$\operatorname{Re}[\log(\frac{\Omega_{n,\sigma}^{\mathcal{C}}+\sqrt{\left( \Omega_{n,\sigma}^{\mathcal{C}}\right) ^{2}+\left( \Delta^{\mathcal{C}% }\right) ^{2}}}{\Delta_{BCS}})]=0 \label{DeltaEq}%$$ This equation can be solved numerically. The resulting $\Delta^{\mathcal{C}}$ is independent from the values of $n$ and $\sigma$ used in Eq. (\[DeltaEq\]). Then, the value of the pairing angle $\theta_{n,\sigma}$ in the ferromagnets can be found by using Eqs. (\[theta\]), (\[thetaF\]), (\[thetaQuad\]) and (\[thetaS\]). Note that for $\gamma\ll1$, the above Eqs. are in agreement with formula (5) of Ref. , obtained with rigid boundary conditions, i.e. $\theta_{n,\sigma}$ equal to its bulk value at the $S$ side. The energy dependence of $\theta_{n,\sigma}$ can be found by performing the analytic continuation $\omega_{n}=-i\varepsilon +\Gamma$ and $\mathrm{sgn}(\omega_{n})=1$ in the above equations. The rate $\Gamma=0.05$ is used to account for inelastic processes [@theseW]. At last, the density of states $N_{\sigma}(x,\varepsilon)$ at position $x$ for the spin direction $\sigma\in\{\uparrow,\downarrow\}$ can be calculated by using $N_{\sigma}(x,\varepsilon)=\left( N_{0}/2\right) \operatorname{Re}% [\cos[\theta_{n,\sigma}(x)]]$, where $N_{0}/2$ is the normal density of states per spin direction. ![Density of states $N^{F}(\varepsilon)$ at $x=d_{F}+d_{S}/2$ as a function of the energy $\varepsilon$, for the $F/S/F$ structure of Figure 1.a with $\gamma=1$, $\gamma_{T}=0.12$, $d_{S}/\xi_{S}=1$, $\gamma_{\phi}^{F}=-2.1$ and $\gamma_{\phi}^{S}=0.12$. The different panels correspond to different values of $d_{F}/\xi_{F}$. The red full curves correspond to the $P$ configuration and the blue dashed curves to the $AP$ configuration. For finite values of $\gamma_{\phi}^{S}$, some characteristic ”double-gap” structures appear in $N_{\sigma}^{F}(\varepsilon)$ in the $P$ case, due to a SDIPS-induced effective field appearing in $S$. The visibility of this effect strongly depends on the thickness $d_{F}$ of the $F$ layers. For the different cases considered here, one can check that hypotheses 1 to 4 are valid and that the critical temperature $T_{c}^{\mathcal{C}}$ of the structure (see Sec. III.F) is such that $0.62T_{c}^{BCS}<T_{c}^{\mathcal{C}}<0.67T_{c}^{BCS}$. Note that for the above parameters and $\gamma_{\phi}^{S}=0$, the curves obtained in the $P$ and $AP$ case would be identical and very close to the $AP$ curves shown here. []{data-label="DOS2"}](DOS2.eps){width="0.85\linewidth"} In the following, I will mainly focus on $N^{F}(\varepsilon)=\sum \nolimits_{\sigma\in\{\uparrow,\downarrow\}}N_{\sigma}(x=d_{F},\varepsilon)$. Figure \[DOS1\] shows the variations of the zero energy density of states $N^{F}(\varepsilon=0)$ as a function of $d_{F}$, for interface parameters $\gamma_{T}=\gamma_{\phi}^{S}=0.12$ and $\gamma_{\phi}^{F}=-2.1$. Importantly, the value $\gamma_{T}=0.12$ seems realistic, at least for the weakly polarized $_{0.9}$$_{0.1}$ bilayers used in Ref. , for which one finds $\gamma_{T}\sim0.15$ (see Ref. ). In addition, a simple barrier model suggests that the situation $\left\| G_{\phi}^{F}\right\| \gg G_{T}$, with $G_{\phi}^{F}<0$ can happen (see Appendix A). The value $\gamma_{\phi}^{F}=-2.1$ used in Fig. \[DOS1\] thus seems possible. One can see that $\gamma_{\phi}^{F}$ can change significantly the phase of the oscillations of $N^{F}(\varepsilon=0)$ with $d_{F}$ (this effect has already been studied in Ref. in the case of rigid boundary conditions but I recall it here for the sake of completeness). In Fig. \[DOS1\], using for $\gamma_{\phi}^{F}$ a strong negative value allows to get $N^{F}(0)<N_{0}$ for the lowest values of $d_{F}% $, like often found in experiments. Note that from Fig. \[DOS1\], $\gamma_{\phi}^{S}$ can also shift the oscillations of $N^{F}(\varepsilon=0)$ with $d_{F}$. Here, this effect is much weaker than that of $\gamma_{\phi}% ^{F}$, but one has to keep in mind that the limit $\gamma_{\phi}^{S}\ll \gamma_{\phi}^{F}$ is considered. Equation (\[heff\]) shows that a measurement of $H_{eff}$ should allow to determine the conductance $G_{\phi}^{S}$ of a diffusive $S/F$ interface. In this context, studying the energy dependence of $N^{F}(\varepsilon)$ is very interesting, because it can allow to see clear signatures of $H_{eff}$, as shown below. Figure \[DOS2\] shows the energy dependence of $N^{F}% (\varepsilon)$ in the $P$ and $AP$ configurations, for a finite value of $\gamma_{\phi}^{S}$ and different values of $d_{F}$. For $\mathcal{C}=P$, $N^{F}(\varepsilon)$ shows some ”double-gap” structures which disappear if the device is switched to the $AP$ configuration. These double structures are an indirect manifestation of the effective magnetic field $H_{eff}$ occurring in $S$ in the $P$ configuration, due to $\gamma_{\phi}^{S}\neq0$. Although $H_{eff}$ is localized in the $S$ layer, the double-gap structure that this field produces in the DOS of $S$ is transmitted to the DOS of $F$ due to Andreev reflections occurring at the $S/F$ interfaces, as shown by Eq. (\[thetaF\]). Interestingly, Rowell and McMillan have already observed that an internal property of a $S$ layer can be seen through the superconducting proximity effect occurring in a nearby normal layer. More precisely, these authors have found that the DOS of an layer can reveal the phonon spectrum of an adjacent superconducting layer[@Rowell]. Remarkably, the visibility of $H_{eff}$ in $N^{F}(\varepsilon)$ is modulated by quantum interferences occurring in $F$. Indeed, $H_{eff}$ is more visible for certain values of $d_{F}$ (e.g. $d_{F}/\xi_{F}=1.0$ or $1.2$ in Fig. \[DOS2\]) than others (e.g. $d_{F}/\xi_{F}=2.1$ in Fig. \[DOS2\]), due to the $d_{F}$-dependence of Eq.(\[thetaF\]). It is useful to note that the SDIPS-induced effective field $H_{eff}$ should also occur in the $S/F$ bilayer of Figure 1.b. In this case, the Thouless energy and normal state conductance of the $S$ layer correspond to $\widetilde{E}_{TH}^{S}=4E_{TH}^{S}$ and $\widetilde{G}_{S}=2G_{S}$ respectively, so that one finds $g\mu_{B}H_{eff}=\widetilde{E}_{TH}^{S}% G_{\phi}^{S}/\widetilde{G}_{S}$. Double gap structures strikingly similar to those shown in Figure \[DOS2\] were indeed measured very recently by P. SanGiorgio et al., at the ferromagnetic side of diffusive bilayers, in the absence of any external magnetic field[@SanGiorgio]. Remarkably, the visibility of the observed double structures varies with $d_{F}$, as predicted above. Note that in $S/F$ bilayers, the field $H_{eff}$ should also be observable directly at the $S$ side by measuring $N^{S}% (\varepsilon)=\sum\nolimits_{\sigma\in\{\uparrow,\downarrow\}}N_{\sigma }(x=0,\varepsilon)$. However, for parameters comparable to those of Figure \[DOS2\], this should not enhance the resolution on $H_{eff}$ (see figure \[DOS3\], left). For certain values of $d_{F}$, $H_{eff}$ is even more visible in $N^{F}(\varepsilon)$ than in $N^{S}(\varepsilon)$(see Figure \[DOS2\]). Before concluding this section, I would like to emphasize that from Eqs. (\[thetaS\]) and (\[Om2\]), in the $AP$ configuration, the SDIPS-induced effective field $H_{eff}$ disappears for the $F/S/F$ structure considered in this paper because the two contacts are assumed to be symmetric and have thus opposite contributions to $H_{eff}$ in the $AP$ case. In the case of a dissymmetric structure, this should not be true anymore, but the SDIPS-induced effective field should nevertheless vary from the $P$ to the $AP$ case. This is one practical advantage of working with $F/S/F$ trilayers instead of $S/F$ bilayers. ![Energy dependent density of states $N^{S}(\varepsilon)$ measurable at the S side ($x=0$) of the $S/F$ structure of Figure 1.b. The left panel corresponds to the case of a metallic $F$ contact with parameters corresponding to that of Figs. \[DOS1\] and \[DOS2\]. In this case, the DOS measured at the $S$ side does not allow to resolve the SDIPS-induced effective field $H_{eff}$ better than a DOS measurement at the $F$ side of the structure, as can be seen from a comparison with Fig. \[DOS2\]. For certain values of $d_{F}$, $H_{eff}$ is even more visible in $N^{F}(\varepsilon)$ than $N^{S}(\varepsilon)$. The right panel corresponds to the case in which $F$ is not a metal but an insulating ferromagnet, i.e. $\gamma_{T}=0$ . In this case, one can use the reduced SDIPS parameter $\lambda_{\phi}^{S}=G_{\phi}^{S}\xi_{S}/A\sigma_{S}$. For $\lambda_{\phi}^{S}\neq0$, $N^{S}(\varepsilon)$ shows strong signatures of the effective field $H_{eff}$ $\ $induced by the $FI$ layer in $S$. One can check that the hypotheses 1 to 4 are valid for the parameters used in this Figure.[]{data-label="DOS3"}](DOS3.eps){width="1\linewidth"} Low-temperature density of states of $S/FI$ bilayers ---------------------------------------------------- Twenty years ago, internal Zeeman fields were observed in superconducting Al layers contacted to different types of ferromagnetic insulators ($FI$) (see Refs. ). Using a ballistic $S/FI$ bilayer model, Ref. suggested that the observed internal fields could be induced by the SDIPS[@Tokuyasu]. However, the inadequacy of this ballistic approach for modeling the actual experiments was pointed out in Ref. . Most of the experiments on Al$/FI$ interfaces were interpreted by their authors in terms of a diffusive approach with no SDIPS, and an internal Zeeman field added arbitrarily in the Al layer (see Refs. ). The calculations of Section III.B. provide a microscopic justification for the use of such an internal field in the diffusive model. Indeed, using $\gamma_{T}=0$ in the above calculations allows one to address the case of diffusive $S/FI$ bilayers. One finds that the SDIPS-induced effective field $H_{eff}$ of Eq. (\[heff\]) can occur in a thin diffusive $S$ layer contacted to a $FI$ layer. This effective field effect can be seen e.g. in the density of states $N^{S}(\varepsilon)$ of the $S$ layer at $x=0$ (see Figure \[DOS3\], right). Remarkably, it was found experimentally[@Hao] that $H_{eff}$ scales with $d_{s}^{-1}$, in agreement with Eq. (\[heff\])[@NoteDG]. SDIPS-induced effective fields in other types of system ------------------------------------------------------- Interestingly, the SDIPS can induce effective field effects in other types of systems. First, the case of $S/N/FI$ trilayers with a thickness $d_{N}$ of normal metal $N$ has been studied theoretically[@demoBC; @applications]. In this case, a conductance $G_{\phi}^{N}$ similar to $G_{\phi}^{S}$ can be introduced to take into account the SDIPS for electrons reflected by the $FI$ layer. The $N$ layer is subject to an effective field $g\mu_{B}H_{eff}% ^{\prime}=E_{TH}G_{\phi}^{N}/G_{N}$ with $G_{N}$ the conductance of $N$. The expression of $H_{eff}^{\prime}$ is analogue to that of $H_{eff}$ (see Eq. \[heff\]), up to a factor 2 which accounts for the symmetry of the $F/S/F$ structure with respect to $x=0$ in the $P$ configuration. Secondly, an effective field $H_{eff}^{^{\prime\prime}}$ defined by $g\mu_{B}% H_{eff}^{^{\prime\prime}}=(\hbar v_{F}^{W}/2L)(\varphi_{L}^{\uparrow}% +\varphi_{R}^{\uparrow}-\varphi_{L}^{\downarrow}-\varphi_{R}^{\downarrow})$ has been predicted for a resonant single-channel ballistic wire with length $L$ placed between two ferromagnetic contacts[@CottetEurophys], with $\varphi_{L(R)}^{\sigma}$ the reflection phase of electrons with spin $\sigma$ incident from the wire onto the left(right) contact, and $v_{F}^{W}$ is the Fermi velocity in the wire. The expression of $H_{eff}^{^{\prime\prime}}$ also shows strong similarities with that of $H_{eff}$ (see Eq. \[heff\], middle term). In practice, signatures of the field $H_{eff}^{^{\prime\prime}}$ could be identified in a carbon nanotube contacted with two ferromagnetic contacts [@Cottet06; @Sahoo]. The fields $H_{eff}$, $H_{eff}^{^{\prime}}$ and $H_{eff}^{^{\prime\prime}}$ have the same physical origin: the energies of the states localized in the central conductor of the structure depend on spin due to the spin-dependent phase shifts acquired by electrons at the boundaries of this conductor. In all cases, the DOS of the central conductor reveals the existence of the SDIPS-induced effective field only if it already presents a strong energy dependence near the Fermi energy in the absence of a SDIPS. In the $F/S/F$ case, this energy dependence is provided by the existence of the superconducting gap in $S$[@cond]. In the $S/N/FI $ case, it is provided by the existence of a superconducting minigap in $N$. At last, in the case of the ballistic wire, it is provided by the existence of resonant states in the wire. Comparison between the present work and other models for data interpretation in $S/F$ heterostructures ------------------------------------------------------------------------------------------------------ For characterizing the properties of $S/F$ interfaces, one has to interpret the experimental data showing the oscillations of the density of states $N^{F}(\varepsilon)$ in $F$ with the thickness $d_{F}$ of $F$ (or the oscillations of the critical current $I_{0}$ of a $S/F/S$ Josephson with $d_{F}$). However, if one uses a simple description with spin-degenerate boundary conditions, the amplitude and the phase of these signals are not independent, which makes the agreement with experimental results impossible in most cases. The SDIPS concept can solve this problem since it produces a shift of the signals oscillations with respect to the $G_{\phi}^{S(F)}=0$ case. However, in many cases, the observed shifts were attributed to the existence of a magnetically dead layer (MDL) at the $F$ side of the $S/F$ interface (see e.g. Refs. ). In other cases, the discrepancy between the theory and the data was solved by taking into account spin-scattering processes in the $F$ layer (see e.g. Ref. ). In order to have a better insight into superconducting proximity effect experiments, one must stress the importance of estimating experimentally the MDL thickness and the spin-scattering rate. In principle, spin-scattering rates can be estimated experimentally, as was done for instance for the alloy[@Pratt] which is frequently used in proximity effect measurements (see e.g. ). An experimental determination of the MDL thickness has also been performed in a few structures used to measure $T_{c}$ or $I_{0}% $[@Muhge; @Aarts; @Piano2; @Piano; @Bell], but, so far, this parameter has not been used for a real quantitative analysis of the data. In some situations, a model combining the SDIPS with spin-scattering and/or a MDL may be necessary. In any case, it is important to point out that descriptions based on spin-degenerate boundary conditions are, in principle, incomplete since they do not account for the effective field effect described in section III.B. Before concluding this section, it is interesting to note that the effective field effect produced by $G_{\phi}^{S}$ in $S$ or the phase shift of the spatial oscillations of the DOS provided by $G_{\phi}^{F}$ in $F$ will remain qualitatively similar when the signs of $G_{\phi}^{S}$ and $G_{\phi}^{F}$ are changed (not shown). The sign of $G_{\phi}^{F}$ in the experiments of Refs. could be determined from a quantitative comparison between the theory and the data[@cottet05]. Below, we present a study of the critical temperature of $S/F$ structures which can give information on the signs of $G_{\phi}^{S}$ and $G_{\phi}^{F}$ through qualitative signatures. Critical temperature of $S/F$ heterostructures ---------------------------------------------- The critical temperature of $S/F$ hybrid structures has already been the topic of many theoretical (see e.g. Refs. ) and experimental (see e.g. Refs. ) studies, but the effects of the SDIPS on this quantity have raised little attention so far. I show below that the SDIPS can significantly modify the critical temperatures of $S/F$ diffusive structures. Calculating the critical temperature $T_{c}^{\mathcal{C}}$ of the structure of Figure 1.a in configuration $\mathcal{C}$ requires to consider the limit in which superconducting correlations are weak in $S$ as well as in $F$ (hypotheses 1 and 2 are then automatically satisfied). Equations (\[bP\]), (\[bAP\]) and (\[bc\]), then lead to ![Critical temperature $T_{c}^{P}$ for the $F/S/F$ structure of Figure 1.a in the $P$ configuration (top panels) and difference $\Delta T_{c}=T_{c}^{AP}-T_{c}^{P}$ between the critical temperatures in the $P$ and $AP$ configurations (bottom panels), as a function of the thickness $d_{F}$ of the $F$ layers. The left panels show the effect of a finite $\gamma_{\phi}^{S}$ and the right panels the effect of a finite $\gamma_{\phi}^{F}$. The four panels show the case $\gamma_{\phi}% ^{S}=\gamma_{\phi}^{F}=0$ with full red lines, for comparison. The other parameters used in the Figure are $\gamma=1$, $\gamma_{T}=0.12$ and $d_{S}% /\xi_{S}=1$. The SDIPS modifies $T_{c}^{\mathcal{C}}$ and $\Delta T_{c}$ in a quantitative or qualitative way, depending on the case considered. The type of effects produced by the SDIPS on $T_{c}^{\mathcal{C}}$ and $\Delta T_{c}$ strongly depend on the signs of $\gamma_{\phi}^{F}$ and $\gamma_{\phi}^{S}$.[]{data-label="FSFTc"}](FSFTc.eps){width="1\linewidth"} $$\widetilde{\theta}_{n,\sigma}^{\mathcal{C}}=\frac{\Delta^{\mathcal{C}}% }{\left\| \omega_{n}\right\| +2\Delta_{BCS}b_{n,\sigma}^{\mathcal{C}}% \delta_{S}^{-1}}%$$ with $$b_{n,\sigma}^{P}=\frac{4\mathcal{L}_{n,\sigma}^{0}}{4+\delta_{S}% \mathcal{L}_{n,\sigma}^{0}} \label{bp}%$$$$b_{n,\sigma}^{AP}=\frac{8\operatorname{Re}[\mathcal{L}_{n,\sigma}% ^{0}]+4\left\| \mathcal{L}_{n,\sigma}^{0}\right\| ^{2}\delta_{S}% }{8+6\operatorname{Re}[\mathcal{L}_{n,\sigma}^{0}]\delta_{S}+\left\| \mathcal{L}_{n,\sigma}^{0}\right\| ^{2}\delta_{S}^{2}} \label{bap}%$$ and $$\frac{\mathcal{L}_{n,\sigma}^{0}}{\gamma}=\frac{\gamma_{T}(B_{n,\sigma}% ^{P,2}+i\gamma_{\phi}^{F}\sigma\mathrm{sgn}(\omega_{n}))}{\gamma _{T}+B_{n,\sigma}^{P,2}+i\gamma_{\phi}^{F}\sigma\mathrm{sgn}(\omega_{n}% )}+i\gamma_{\phi}^{S}\sigma\mathrm{sgn}(\omega_{n})$$ These Eqs. together with (\[Delta\]) lead to $$\log\left( \frac{T_{c}^{BCS}}{T_{c}^{\mathcal{C}}}\right) =\operatorname{Re}% \left[ \Psi\left( \frac{1}{2}+\frac{b_{n,\sigma}^{\mathcal{C}}}{\delta _{S}\exp(\Gamma)}\frac{T_{c}^{BCS}}{T_{c}^{\mathcal{C}}}\right) \right] -\Psi\left( \frac{1}{2}\right) \label{Tc}%$$ where $\Gamma$ denotes Euler’s constant. The resulting $T_{c}^{\mathcal{C}}$ is independent from the values of $n$ and $\sigma$ used in Eq. (\[Tc\]). Note that in the case $\gamma_{\phi}^{S}=\gamma_{\phi}^{F}=0$ and $\delta _{S}\rightarrow0$, this equation is in agreement with Eqs. (18) and (19) of Ref. . Performing a numerical resolution of Eq. (\[Tc\]) together with (\[bp\]) and (\[bap\]), one obtains the results of Fig. \[FSFTc\], which shows the critical temperature $T_{c}^{P}$ of the structure in the $P$ configuration (top panels) and the difference $\Delta T_{c}=T_{c}^{AP}-T_{c}^{P}$ (bottom panels) as a function of $d_{F}$, for different interface parameters[@approx]. Here, for simplicity, I consider cases where $T_{c}^{\mathcal{C}}$ does not show a strongly reentrant behavior, i.e. a cancellation of $T_{c}^{\mathcal{C}}$ in a certain interval of $d_{F}$. Such a behavior can happen for instance for larger values of $\gamma_{T}$ (see e.g. Ref. ) or $\gamma_{\phi}^{(S)F}$, but this corresponds to a minority[@Tccoupe] of the experimental observations made so far. The four panels of Fig. \[FSFTc\] show the case $\gamma_{\phi}^{S}=\gamma_{\phi }^{F}=0$ with full red lines, for comparison with the other cases, where the SDIPS is finite. I first comment the results obtained for the $T_{c}^{P}% (d_{F})$ curves (top panels of Fig. \[FSFTc\]). In some cases, the SDIPS can modify quantitatively the $T_{c}^{P}(d_{F})$ curves, for instance by amplifying the dip expected in $T_{c}^{P}(d_{F})$ in the absence of a SDIPS (see e.g. dotted curve in the upper left panel, corresponding to $\gamma _{\phi}^{S}>0$). In some other cases the SDIPS can modify qualitatively the $T_{c}^{P}(d_{F})$ curves, for instance by transforming the minimum expected in $T_{c}^{P}(d_{F})$ into a maximum (see e.g. curve with circles in the upper right panel, corresponding to $\gamma_{\phi}^{F}<0$). I now comment the results obtained for the $\Delta T_{c}(d_{F})$ curves (bottom panels of Fig. \[FSFTc\]). In the range of parameters studied in this work, one always finds $\Delta T_{c}>0$ because the effects of the two $F$ layers on $S$ are partially compensated in the $AP$ case. In the absence of a SDIPS and for a low $\gamma_{T}$, the $\Delta T_{c}(d_{F})$ curve presents a maximum at a finite value of $d_{F}$ (see red full lines in bottom panels). In some cases, the SDIPS can modify quantitatively the $\Delta T_{c}(d_{F})$ curves, for instance by increasing the value of this maximum (see dotted curve in the bottom left panel, corresponding to $\gamma_{\phi}^{S}>0$). In other cases, the SDIPS can modify qualitatively the $\Delta T_{c}(d_{F})$ curves, for instance by turning the maximum expected in $\Delta T_{c}(d_{F})$ into a minimum (see dashed curve in the bottom left panel, corresponding to $\gamma_{\phi}^{S}<0$), or by increasing the complexity of the variations of $\Delta T_{c}$ with $d_{F}$ (see curve with circles in the bottom right panel, corresponding to $\gamma_{\phi}^{F}<0$), or by transforming $\Delta T_{c}(d_{F})$ into a monotonically decreasing curve with a reduced amplitude (see dot-dashed curve in bottom right panel, corresponding to $\gamma_{\phi }^{F}>0$). Remarkably, the type of effects produced by the SDIPS on the $T_{c}^{\mathcal{C}}(d_{F})$ and $\Delta T_{c}(d_{F})$ curves depend on the sign of $\gamma_{\phi}^{F}$ and $\gamma_{\phi}^{S}$. Critical temperature measurements can thus be an interesting way to determine the values of $\gamma_{\phi}^{F}$ and $\gamma_{\phi}^{S}$. For simplicity, I have shown here results for $\gamma_{\phi}^{F}\neq0$ and $\gamma_{\phi}^{S}\neq0$ separately. Nevertheless, the behavior predicted for $\gamma_{\phi}^{F}\neq0$ and $\gamma_{\phi}^{S}\neq0$ simultaneously remain highly informative on the values of $\gamma_{\phi}^{F}$ and $\gamma_{\phi}^{S}$. Note that if $\gamma_{\phi}^{F}$ and $\gamma_{\phi}^{S}$ are increased compared to the values used in Fig. \[FSFTc\], the $T_{c}^{\mathcal{C}}(d_{F})$ curve gets some cancellation points like in Ref. , but the behaviors of $T_{c}^{\mathcal{C}}$ and $\Delta T_{c}$ remain, in many cases, qualitatively dependent on the signs of $\gamma_{\phi}^{F}$ and $\gamma_{\phi }^{S}$ (not shown). Before concluding, I note that with the approximations used in the previous section (III.B), the electronic correlations inside $S$ were affected by the presence of the $F$ electrodes through the parameter $\gamma_{\phi}^{S}$ only. Consequently, the self-consistent gap $\Delta^{\mathcal{C}}$ of the $S$ layer was independent from $d_{F}$, and it was furthermore identical for the $P$ and $AP$ configurations for $\gamma_{\phi}^{S}=0$. In the present section, I have not neglected the dependence of $T_{c}^{P}$ and $T_{c}^{AP}$ on $d_{F}$ and $\gamma_{\phi}^{F}$ because I have considered parameters for which hypothesis 3 is not acceptable anymore. In particular, I have used lower values for $d_{F}$[@regimes]. Conclusion ========== This article shows that the Spin-Dependence of Interfacial Phase Shifts (SDIPS) can have a large variety of signatures in diffusive superconducting/ferromagnetic ($S/F$) heterostructures. Ref. had already predicted that the SDIPS produces a phase shifting of the oscillations of the superconducting correlations with the thickness of $F $ layers. This article shows that this is not the only consequence of the SDIPS in $S/F$ circuits. In particular, the SDIPS can produce an effective magnetic field in a diffusive $S$ layer contacted to a diffusive $F$ layer. This effective field can be seen e.g. through the DOS of the diffusive $F$ layer, with a visibility which oscillates with the thickness of $F$. The SDIPS can also modify significantly the variations of the critical temperature of a $S/F$ bilayer or a $F/S/F$ trilayer with the thickness of $F $, either in a quantitative or in a qualitative way, depending on the regime of parameters considered and in particular the sign of the conductances $G_{\phi}^{S}$ and $G_{\phi}^{F}$ used to account for the SDIPS of the $S/F$ interfaces. In the case of a $F/S/F$ spin valve, this last result also holds for the thickness-dependence of the difference between the critical temperatures in the parallel and antiparallel lead configurations. These effects should help to determine the parameters $G_{\phi}^{S}$ and $G_{\phi}^{F}$ of diffusive $S/F$ interfaces. The calculations shown in this paper are also appropriate to the case of thin diffusive $S$ layers contacted to ferromagnetic insulators. I thank P. SanGiorgio for showing me his experimental data prior to publication, which stimulated part III.B of this work. I acknowledge discussions with T. Kontos and W. Belzig. This work was supported by grants from the Swiss National Science Foundation and Région Ile-de-France. Appendix A: Parameters of a $S/F$ interface from a Dirac barrier model ====================================================================== ![Conductances $G_{T}$ (full line), $\left\| G_{\phi}^{S}\right\| $ (dashed lines), and $\left\| G_{\phi}^{F}\right\| $ (dash-dotted lines) reduced by the number of channels $n$, as a function of the spin-averaged barrier strength $Z$ of a $S/F$ interface modeled with a Dirac barrier (see text). The curves were obtained with typical Fermi energies $E_{f}^{F}=0.8~\mathrm{eV}$ and $E_{f}% ^{S}=12~\mathrm{eV}$ in $F$ and $S$ respectively, a spin-polarization $P=0.06$ of the density of states in $F$, and a spin asymmetry $\alpha=0.06$ for the barrier. Note that for the set of parameters used this Figure, one finds $G_{\phi}^{F}<0$ and $G_{\phi}^{S}<0$. However, for other parameters (e.g. by using $\alpha<0$), one can reverse the signs of $G_{\phi}^{F}$ and $G_{\phi }^{S}$, or obtain opposite signs for $G_{\phi}^{F}$ and $G_{\phi}^{S}$ (not shown).[]{data-label="Dirac"}](Dirac.eps){width="0.85\linewidth"} The exact values of the conductances $G_{T}$, $G_{\phi}^{F}$ and $G_{\phi}% ^{S}$ of a $S/F$ interface depend on the details of this interface and on the microscopic structure of the contacted materials. Nevertheless, it is already interesting to study a simplified Dirac barrier model which shows that the parameters regime assumed in this paper is, in principle, possible. Neglecting the transverse part of the electrons motion, one finds $r_{n,\sigma}% ^{S(F)}=(k_{S(F)}^{\sigma}-k_{F(S)}^{\sigma}-iZ^{\sigma})/D^{\sigma}$ and $t_{n,\sigma}^{S(F)}=2(k_{S}^{\sigma}k_{F}^{\sigma})^{1/2}/D^{\sigma}$ with $D^{\sigma}=k_{S}^{\sigma}+k_{F}^{\sigma}+iZ^{\sigma}$. Here, $k_{S(F)}% ^{\sigma}$ is the Fermi electronic wavevector in $S(F)$ and $Z^{\sigma}$ is the strength of the Dirac barrier for electrons with spin $\sigma$. I use $\hbar k_{S}^{\sigma}=(2m_{e}E_{f}^{S})^{1/2}$, with $m_{e}$ the free electron mass and $E_{f}^{S}$ the Fermi level in $S$. For $F$, I use an s-band Stoner model, in which $\hbar k_{F}^{\sigma}=(2m_{e}E_{f}^{F}[1\pm2P/(1+P^{2}% )])^{1/2}$, with $P$ the spin-polarization of the density of states in $F$ and $E_{f}^{F}$ the Fermi level in $F$. I assume that the barrier can have a spin dependence $\alpha=(Z^{\downarrow}-Z^{\uparrow})/(Z^{\uparrow}+Z^{\downarrow })$ due to the ferromagnetic contact material used to form the interface. The results given by this approach are shown in Fig. \[Dirac\]. The conductances $G_{T}$, $\left\| G_{\phi}^{F}\right\| $ and $\left\| G_{\phi}^{S}\right\| $, reduced by the number of channels $n$, are shown as a function of the spin-averaged barrier strength $Z=(Z^{\uparrow}+Z^{\downarrow})/2$. The three types of conductances go to zero when $Z$ goes to infinity with $\alpha$ constant[@lim]. One can see that in the limit $T_{n}\ll1$ (i.e. here $G_{T}h/ne^{2}\ll1$) and $P\ll1$ in which the boundary conditions (\[BCright\],\[BCleft\]) have been derived, $\left\| G_{\phi}^{F}\right\| $ can be significantly stronger than $G_{T}$, as assumed in Figs. \[DOS1\] and \[DOS2\]. Note that for the set of parameters used in Fig. \[Dirac\], one finds $G_{\phi}^{F}<0$ and $G_{\phi}^{S}<0$. However, for other parameters (e.g. by using $\alpha<0$), one can reverse the signs of $G_{\phi}^{F}$ and $G_{\phi}^{S}$, or obtain opposite signs for $G_{\phi}^{F}$ and $G_{\phi}^{S}$ (not shown). This suggests that there is no fundamental constraint on the signs of $G_{\phi}^{F}$ and $G_{\phi}^{S}$ in the general case. Appendix B: Effective field effect in a $F$ layer ================================================= This appendix reconsiders the case of the $S/F$ bilayer of Fig.1.b, in the limit $d_{F}\ll\xi_{F}$ where $\theta_{n,\sigma}(x)$ can be approximated with a quadratic form in the $F$ layer. From Eqs. (\[UsadelF2\]) and (\[BCright\]), one finds, for $x\in\lbrack d_{S}/2,d_{S}/2+d_{F}]$, $$\theta_{n,\sigma}(x)=\widetilde{\theta}_{n,\sigma}^{F}+b_{n,\sigma}^{F}\left[ \frac{d_{S}-2x}{d_{F}}+\left( \frac{d_{S}-2x}{2d_{F}}\right) ^{2}\right]$$ with $$\widetilde{\theta}_{n,\sigma}^{F}=\arctan\left( \frac{\gamma_{T}% \sin(\widetilde{\theta}_{n,\sigma}^{S})}{\gamma_{T}\cos(\widetilde{\theta }_{n,\sigma}^{S})+\frac{2\xi_{F}}{d_{F}}\frac{\Omega_{n}}{E_{TH}^{F}}% +i\gamma_{\phi}^{F}\eta_{n,\sigma}^{P,2}}\right) \label{ThetaF}%$$$$b_{n,\sigma}^{F}=\frac{\Omega_{n}\sin(\widetilde{\theta}_{n,\sigma}^{F}% )}{E_{TH}^{F}}%$$ $\Omega_{n}=\left\| \omega_{n}\right\| +iE_{ex}\eta_{n,\sigma}^{P,2}$ and $\widetilde{\theta}_{n,\sigma}^{S}=\theta_{n,\sigma}^{S}(x=d_{S}/2)$. Here, $G_{F}=\sigma_{F}A/d_{F}$ and $E_{TH}^{F}=\hbar D_{F}/d_{F}^{2}$ denote the conductance and Thouless energy of the $F$ layer (note that $d_{F}\ll\xi _{F}\Longleftrightarrow E_{ex}\ll E_{TH}^{F}$). From completeness, I also give the analogous equations for $x\in\lbrack0,d_{S}/2]$. Assuming that $\theta_{n,\sigma}(x)$ can be approximated with a quadratic form in the $S$ layer, one finds, from Eqs. (\[UsadelS\]) and (\[BCleft\]), $$\theta_{n,\sigma}(x)=\widetilde{\theta}_{n,\sigma}^{S}+b_{n,\sigma}^{S}\left[ \frac{2(2x-d_{S})}{d_{S}}+\left( \frac{2x-d_{S}}{d_{S}}\right) ^{2}\right]$$ with $$\widetilde{\theta}_{n,\sigma}^{S}=\mathrm{\arctan}\left( \frac{\gamma \gamma_{T}\sin(\widetilde{\theta}_{n,\sigma}^{F})+\frac{4\xi_{S}}{d_{S}}% \frac{\Delta}{\widetilde{E}_{TH}^{S}}}{\gamma\gamma_{T}\cos(\widetilde{\theta }_{n,\sigma}^{F})+\frac{4\xi_{S}}{d_{S}}\frac{\left\| \omega_{n}\right\| }{\widetilde{E}_{TH}^{S}}+i\gamma\gamma_{\phi}^{S}\eta_{n,\sigma}^{P,2}% }\right) \label{ThetaS}%$$ and $$b_{n,\sigma}^{S}=\frac{\left\| \omega_{n}\right\| \sin(\widetilde{\theta }_{n,\sigma}^{S})-\Delta\cos(\widetilde{\theta}_{n,\sigma}^{S})}{\widetilde {E}_{TH}^{S}}%$$ The notations used in the above equations are the same as in Section III. In particular, the Thouless energy and the normal state conductance of the $S$ layer with thickness $d_{S}/2$ are denoted $\widetilde{G}_{S}=2\sigma _{S}A/d_{S}$ and $\widetilde{E}_{TH}^{S}=4\hbar D_{S}/d_{S}^{2}$, and one has $\eta_{n,\sigma}^{P,2}=\sigma\mathrm{sgn}(\omega_{n})$. In the limit $\widetilde{\theta}_{n,\sigma}^{F}\ll1$, Eq. (\[ThetaS\]) is in agreement with Eqs. (\[thetaS\]) and (\[omegaa\]). The analytic continuation of Eq. (\[ThetaS\]) shows that the $S$ layer is subject to the effective field $H_{eff}=\widetilde{E}_{TH}^{S}G_{\phi}^{S}/\widetilde{G}_{S}$, in agreement with Eq. (\[heff\]). The analytic continuation of Eq. (\[ThetaF\]) shows that the $F$ layer is subject to an analogue effective field $H_{eff}^{F}$, defined by $$g\mu_{B}H_{eff}^{F}=E_{TH}^{F}\frac{G_{\phi}^{F}}{G_{F}}%$$ It is interesting to compare Eqs. (\[ThetaF\]) and (\[thetaF\]). First, note that in the regime $\left\| \widetilde{\theta}_{n,\sigma}^{F}\right\| \ll1$, Eq. (\[ThetaF\]) can be recovered from the low $d_{F}$-limit of Eq. (\[thetaF\]). The interest of Eq. (\[ThetaF\]) is that it is valid beyond the regime of a weak superconducting proximity effect studied in part III. In particular, Eq. (\[ThetaF\]) indicates that, in principle, the field $E_{TH}^{F}$ can occur in a thin $F$ layer for any value of the tunneling conductance $G_{T}$ of the $S/F$ interface. Interestingly, in the case of a thick $F$ layer $d_{F}\geq\xi_{F}/2$, from Eq. (\[thetaF\]), the contribution of $G_{\phi}^{F}$ to the pairing angle $\theta_{n,\sigma}^{F}(x)$ cannot be put anymore under the form of an effective exchange field, due to the non-linearity of the $B_{n,\sigma}^{\mathcal{C},j}$ term. Thus, strictly speaking, the concept of a SDIPS-induced effective field is valid only in the limit of thin metallic layers. However, it might be possible, in principle, to observe reminiscences of the double-gap structure appearing in $N^{F}% (\varepsilon)$ in the regime of intermediate layer thicknesses $d_{F}\sim \xi_{F}$ at least, due to the continuity of the equations. Then, one can wonder why this effect does not appear in section III. This is due to the regime of parameters chosen: section III assumes $E_{ex}\gg\Delta_{BCS}$ like in most experiments, and it furthermore focuses on the typical energy range probed in superconducting proximity effect measurements, i.e. $\left\| \varepsilon\right\| \lesssim2\Delta_{BCS}$. In such a regime, one can neglect the term $\left\| \omega_{n}\right\| $ compared to $iE_{ex}\sigma \mathrm{sgn}(\omega_{n})$ in Eqs. (\[thetaF\]) and (\[ThetaF\]), so that $H_{eff}^{F}$ does not emerge in the model. Accordingly, signatures of the SDIPS-induced effective field $H_{eff}^{F}$ are not likely to be observed in practice. [99]{} A. I. Buzdin, L. N. Bulaevskii, and S. V. 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B 55, 15174 (1997); M. G. Khusainov and Yu. N. Proshin, Phys. Rev. B 56, R14283 (1997); L. R. Tagirov, Physica C 307, 145 (1998); Yu. N. Proshin and M. G. Khusainov JETP 86, 930 (1998); Ya. V. Fominov, N. M. Chtchelkatchev, and A. A. Golubov, Phys. Rev. B 66, 014507 (2002); A. Bagrets, C. Lacroix and A. Vedyayev, Phys. Rev. B 68, 054532 (2003). C.-Y. You, Ya. B. Bazaliy, J. Y. Gu, S.-J. Oh, L. M. Litvak, and S. D. Bader, Phys. Rev. B 70, 014505 (2004). With the parameters of Fig. \[FSFTc\], the approximations $b_{n,\sigma}^{P}=\mathcal{L}_{n,\sigma}^{0}$ and $b_{n,\sigma}^{AP}% =\operatorname{Re}[\mathcal{L}_{n,\sigma}^{0}]$ used in Ref. would be sufficient for calculating the $T_{c}^{P}$ and $T_{c}^{AP}$ curves, but not the difference $\Delta T_{c}$. Indeed, for the different cases studied in this Figure, the relative difference between the results obtained with these approximations and those given by Eqs. (\[bp\]) and (\[bap\]) is smaller than $\sim0.8\%$ for $T_{c}^{P}$ and $T_{c}^{AP}$, but it can reach $\sim18\%$ for $\Delta T_{c}$. This is why I have used the general equations (\[bp\]) and (\[bap\]) for plotting Figure \[FSFTc\]. L. R. Tagirov, I. A. Garifullin, N. N. Garifyanov, S. Ya. Khlebnikov, D. A. Tikhonov, K. Westerholt, and H. Zabel, J. Magn. Magn. Mater. 240, 577 (2002). Note that both regimes $d_{F}\leq\xi_{F}$ (see e.g. Ref ) and $d_{F}\geq\xi_{F}$ (see e.g. Ref. ) seem to be accessible in practice. For $Z\rightarrow+\infty$, $G_{\phi}^{F(S)}$ vanishes because $\arg(r_{n,\sigma}^{F(S)})\rightarrow\pi$ and $\arg(t_{n,\sigma}% ^{F(S)})\rightarrow-\pi/2$ for $\sigma\in\{\uparrow,\downarrow\}$.	ArXiv
--- author: - title: 'Comparison of potential ASKAP HI survey source finders ' --- Introduction ============ Radio astronomy is facing a new era, acquiring extremely large data volumes with the coming of the Square Kilometre Array (SKA) [@dewdney2009] and precursors such as MeerKAT [@jonas2009] in South Africa, APERTIF [@verheijen2008] in the Netherlands and the Australian SKA Pathfinder (ASKAP) [@deboer2009] in Australia. Various continuum (2D) and spectral line (3D) surveys, which cover large fractions of the sky, will be conducted with these telescopes. The surveys are expected to detect millions of objects, accelerating the need for reliable automated source finders. A good source finder should have high [completeness]{} and high [ reliability]{}, ie. a low rate of false detections. Choosing a suitable trade-off between both parameters is necessary and depends on both the algorithm and the rms uniformity of the data. Detecting objects is relatively easy in the case of (strong) point sources, but becomes more complicated in the case of irregular shapes and diffuse or extended emission in one or more dimensions and at low signal to noise ratios. The work presented in this paper aims to highlight the strengths and weaknesses of potential 3D source finders for the Deep Investigations of Neutral Gas Origins (DINGO) survey [@meyer2009] and the Widefield ASKAP L-band Legacy All-sky Blind Survey (WALLABY) [@koribalski2009]. These are two of the large survey science projects for ASKAP [@johnston2008]. To achieve the respective science goals, we aim to develop source finding algorithms which reliably and efficiently recover 3D sources. We have identified five different source finders that will be subjected to testing and comparison; 1) the [Duchamp]{} source finder [@whiting2011a], 2) the [Gamma-finder]{} [@boyce2003] 3) the CNHI source finder [@jurek2011], 4) the 2D-1D Wavelet Reconstruction source finder [@floer2011] and 5) the [S+C finder]{} [@serra2011a]. Testing of each algorithm was done on the same set of data cubes. The first containing 961 point sources with varying peak flux and a Gaussian velocity profile. The second cube contains 1024 modelled galaxies with more realistic properties such as extended disks, inclinations and rotation profiles. Here we compare their performance in terms of completeness and reliability. In section 2 we briefly summarise the main properties of the source finding algorithms and in section 3 we describe the testing method and the two model cubes that have been used for the testing. The test results are presented in Section 4, followed by a discussion in Section 5. We compare in detail the performance and reliability of the source finders, to understand where the strong and weak points of the different source finders are and to highlight possible improvements. We finish with a short conclusion in the final section. Source Finders ============== Here we provide a short description of the five source finders compared throughout the paper. For a more extended review of the individual algorithm we refer to the reference papers describing each method in detail. Duchamp source finder --------------------- [Duchamp]{} [@whiting2011a] is a source finder designed for 3D data, although it can be used for 2D and even 1D datasets. The source finder has been developed by Matthew Whiting at CSIRO.[^1] [Duchamp]{} identifies sources by simply applying a specified flux or signal-to-noise threshold and searching for signals above that threshold. In a second step, detections are merged or rejected based on several criteria specified by the user. To improve its performance, [Duchamp]{} offers several methods of preconditioning and filtering of the input data, including spatial and spectral smoothing as well as wavelet reconstruction of the entire image or cube. In a final step, [Duchamp]{} measures several basic parameters for each detected source, including position, radial velocity, size, line width, and integrated flux. The performance of the [Duchamp]{} source finder is tested in @westmeier2011. CNHI source finder ------------------ The Characterised Noise (CNHI) source finder [@jurek2011] is being developed as part of the WALLABY design study. The CNHI source finder treats spectral datacubes as a collection of spectra, using the Kuiper test, which is a variant of the Kolmogorov-Smirnov test, to identify regions in each spectrum that do not look like noise. The Kuiper test is used to calculate the probability that the test region and the rest of the spectrum come from the same distribution of voxel flux values. If the probability is sufficiently low, then the test region is flagged as an object section. The probability threshold is specified by the user. Once all of the spectra have been processed, the object sections are combined into objects. Object sections are combined using a variant of Lutz’s one pass algorithm. There are two caveats to using the CNHI source finder. Firstly, the CNHI source finder assumes that each spectrum is dominated by noise. This is a safe assumption as spectral datacubes are generally sparsely populated by sources. The presence of ripples, artifacts and continuum signal will potentially invalidate this assumption though. The second caveat is that the test region needs to be at least four channels wide for the Kuiper test to be reliable. This matches the smallest channel extent expected of WALLABY sources. Spectral datacubes with a poorer velocity resolution than WALLABY will be affected by this. For a more detailed description of the CNHI source finder see @jurek2011. Gamma-finder ------------ [Gamma-finder]{} is a [Java]{} application developed by [@boyce2003] which automatically searches for objects in 3-dimensional data cubes. The searching algorithm of [Gamma-finder]{} is based on the [Gamma-test]{} [@Stefansson1997], and a full description can be found in [@jones2002]. The [Gamma-test]{} is a near-neighbour data analysis routine which estimates the noise variance in a continuous dataset. This estimate is known as the [Gamma Statistic]{}, denoted by $\Gamma$. When using the [Gamma-finder]{} a Gamma signal-to-noise ratio can be defined which is used as a clipping for objects to be qualified as a detection. The output of the [Gamma-finder]{} is limited compared to other source finders (eg. [Duchamp]{} and CNHI), because it does not do any parametrisation, but only gives the three dimensional position of a detection and the sigma level. 2D-1D Wavelet Reconstruction source finder ------------------------------------------ The 2D-1D Wavelet Reconstruction source finder is described in detail in @floer2011, they have adapted a multi-dimensional wavelet denoising scheme first used by @starck2009. It takes into account that 3D data from spectroscopic surveys have two angular dimensions and one spectral dimension, in which the shape of the sources is vastly different than in the angular dimensions. The algorithm therefore performs a two-dimensional wavelet transform in all planes of the cubes and a subsequent one-dimensional wavelet transform along each line of sight, i.e. each pixel. Once the image has been de-noised by thresholding of the wavelet coefficients, reconstructing the data from only the significant coefficients yields a noise-free cube. The latter can be used to create a mask for the sources in the original data. Smooth plus clip (S+C) finder ----------------------------- @serra2011a developed a source finder which uses a limited number of filters in order to optimise the signal-to-noise ratio of objects present in a data cube. For each dataset, the finder looks for sources in the original cube and in the cubes obtained by smoothing the original cube either on the sky, or in velocity, or along all three axes. In this study we use a Gaussian filter of FWHM=60 arcsec for smoothing on the sky, and a box filter of width 2, 4, 8, 16, and 32 channels for smoothing in velocity. For each smoothed cube a mask is built including all voxels brighter (in absolute value) than a chosen threshold. The final mask is the union of all masks (i.e., a voxel is included in the total mask if it is included in at least one of the individual masks), a value of 1 is allocated to all masked voxels and 0 to all unmasked voxels. A size filter is applied to the final binary mask by convolving it with a 30 arcsec Gaussian kernel, equal to the original angular resolution of the cube and to 3 channels in velocity. Subsequently the mask is shrunk again by taking only voxels in the convolved mask brighter than 0.5. This procedure removes a large number of noise peaks included in the mask whose size is of the order of the cube resolution. Testing method ============== When comparing the five 3D source finders, we concentrate on two main parameters, the completeness and the reliability of a source finder. Completeness is defined as the number of detected sources divided by the total number of sources. While this number is known for simulated cubes, in reality we usually have a much harder problem: we neither know the number of detectable sources in a cube, nor their shape, size or velocity extent. There are a few examples of real datacubes where there is a much deeper datacube of the same region of sky, for example the HIPASS region that is covered by the HIDEEP survey [@minchin2003]. The completeness can be given as a single number, but can also be measured as a function of a certain parameter such as integrated flux or velocity-width. Raw reliability is defined as the number of true detections divided by the total number of detections. In a good scenario the number of false detections is very low, so the reliability is close to 100%. We have to stress that although completeness is a general parameter for a simulation, reliability is highly dependent on the size of a cube. When making a cube twice as large but keeping the number of sources constant, the completeness will not change. However as the noise voxels approximately double, so do the number of false detections. In practise this is complicated by the non-linear steps used by some source finders, and the number of false detections does not necessarily scale linearly with the size of a data cube. The reliability of different source finders can only be compared if the finders are applied on exactly the same data sets. In many cases the reliability of a source finder can be improved upon by applying a threshold for one or more measured parameters like integrated flux. We only concentrate on the capability of source finders to determine detections. Not all source finders have the capability of parametrizing detections, this however is a different problem that can be addressed in the post-processing of detections once they have been identified. Input Models ------------ For the testing and comparison of the different source finders we have used 2 data cubes containing: 1) 961 artificial point sources with Gaussian spectra and 2) 1024 artifical model galaxies with a range of orientation parameters. [ASKAP]{}-specific noise has been added to the cubes, which was generated by the [Uvgen]{} task within [Miriad]{} and is based on the [ASKAP]{} telescope configuration, a system temperature of $T_{sys}$=50K and an integration time of 8 hours. The [rms]{} in the cubes is 1.95 mJy/beam ($30''$) per channel (3.9 km/s). The cubes are similar to the cubes that have been used for the testing of the [ Duchamp]{} source finder by [@westmeier2011]. In the first cube with point sources each source was randomly assigned a peak flux in the range of 1 to 20 $\sigma$, spectral line widths (FWHM) range from approximately 0.4 to 40 km s$^{-1}$. While in reality sources with line widths as small as 0.4 km s$^{-1}$ do not occur, they are included to test the performance of source finders on objects that are spectrally unresolved. In the second cube with model galaxies all sources have an infinitely thin discs with varying inclination (0$^{\circ}$ to 89$^{\circ}$), position angle (0$^{\circ}$ to 180$^{\circ}$), and rotation velocity (20 to 300 km s$^{−1}$). For a more detailed description of the cubes and the input parameters we refer to the paper describing the [Duchamp]{} testing [@westmeier2011]. ![image](point_compl_Fint-eps-converted-to.pdf){width="48.00000%"} ![image](point_compl_S2N-eps-converted-to.pdf){width="48.00000%"} ![image](point_compl_Fpeak-eps-converted-to.pdf){width="48.00000%"} ![image](point_compl_W50-eps-converted-to.pdf){width="48.00000%"} Cross-Matching -------------- To properly compare the five source finders, they have to be analysed in exactly the same manner to exclude any discrepancies based on different methods or interpretations. Apart from the [Gamma-Finder]{}, all source finders produce a 3-dimensional mask containing all the voxels that belong to a detection. Although some source finders such as [Duchamp]{} have the capability to determine source parameters, we have chosen to extract the source parameters from the produced masks, using a separate script. In this way the results of all source finders are treated in exactly the same manner and we are able to make an objective comparison of the results. Using the mask, we have merged all detections that were separated by one pixel in the two spatial dimensions and seven channels in the spectral dimension. Furthermore we required detections to be apparent in at least three channels of the cube to reject spurious detections. The way in which detections are merged can effect the results significantly. For example double-horned, unresolved sources are often split up into two separate sources. They can be recovered as one source, however this depends on the scale that is used for the merging, and it is inevitable that in the merging process not all split sources are recovered properly. Some basic object parameters that have been extracted are the position of the source, the velocity width, the peak flux and the integrated flux. Crucial but not trivial is how the cross-matching is done between the implemented input catalogues and the results of the different source finders. Measuring the central position of a source can be difficult, however in the case of the model cube with point sources the position of the objects is very well determined, both in spatial and velocity direction. The list of input objects is compared with the detections of the source finders, and pairs are sought within $\pm$1 pixel in the spatial direction and $\pm$2 pixels in the spectral direction. As the synthesised beam at FWHM of the used models is described by only three pixels, this is a very robust method. For the cube with disk galaxies the measured centre of a certain object is not always trivial to determine as the sources can be very extended. Due to the rotation, for many objects several components are detected, without detecting emission in the actual centre of the object. As the objects can have line widths of up to several hundred km/s, the central velocity is difficult to estimate and might differ significantly amongst the different source finders. To do the cross-matching we have used a Python script that is used and described in the paper on testing of the [Duchamp]{} source finder [@westmeier2011]. We created a three-dimensional mask containing all voxels containing emission from the model galaxies. For each detection we assess whether the central position $\pm$1 pixel overlaps with one of the voxels in the mask and then determine to which object from the input model catalogue it belongs. Results ======= The range of source properties is large and well documented in many published galaxy catalogs (e.g. @koribalski2004, @meyer2004, @springob2005, @haynes2011) as well as catalogs of high velocity clouds (HVCs; e.g. @putman2002) and peculiar features (@hibbard2001, Rogues Gallery). The shape of spectra ranges from simple Gaussian profiles to steep double-horn profiles and almost everything in between. The distribution of in disk galaxies is often symmetric and regular, but many irregular sources exist, from peculiar dwarf galaxies and rings to plumes/filaments and clouds. As typically only the highest column density gas is detected, it is likely that the low column density gas is more pervasive and irregular. In the following we present a comparison of source finding algorithms applied to the two cubes described in Section 3.1. We start with the simple point sources with Gaussian profiles, then progress to extended disks with more complex profiles. Point sources ------------- Point sources with a Gaussian velocity profile are ideal sources in the sense that they do not have any complicated structures and are relatively easy to detect. Fig. \[point\_comp\_flux\] shows the completeness as a function of integrated flux ($F_{int}$), integrated signal-to-noise ratio, peak flux ($F_{peak}$) and 50% velocity width ($W_{50}$). The integrated flux and integrated signal-to-noise ratio are plotted on a logarithmic scale, to highlight the differences between the source finders. All parameters are the true parameters determined from the input models. For $F_{int}$ we use the same definition as [@westmeier2011] (their equation 4). \[duchamp\_setup\] Parameter Value Comment -------------------- -------- --------------------- threshold (test 1) 0.0039 $2\times $[rms]{} threshold (test 2) 0.0029 $1.5\times $[rms]{} minPix 5 minChannels 3 flagAdjacent true flagATrous true Wavelet reconstr. reconDim 3 in 3 dimensions snrRecon 3 scaleMin (test 1) 1 scaleMin (test 2) 2 : [Duchamp]{} input parameters for the data cube with point sources. We have plotted two results for each of the individual source finders on this particular cube, apart from the 2D-1D wavelet reconstruction method which only produced one output. For [Duchamp]{} the input parameters are given in Table. \[duchamp\_setup\]. For the [ Gamma-finder]{} we use a 3$\sigma$ and a 4$\sigma$ clipping threshold and for the CNHI source finder we use a probability of $10^{-3}$ and $3\cdot10^{-4}$. The [S+C finder]{} has been tested using clipping levels of 3$\sigma$ and 4$\sigma$. For each test, the raw reliability is given as a percentage in the legend of the figure. Here the completeness is the principal value to compare the source finders as the single value for raw reliability can be a misleading number. The number of possible settings or input parameters for each source finder is very large and we experimented with each source finder until we found a set of parameters that was representative for its performance. We emphasise that the scope of this paper is to compare the results of the different source finders, rather than to test them individually which has been done in other papers in this special issue. [Duchamp]{} performs very well on point sources, and the completeness is superior to the other source finders for all plotted parameters. The completeness starts at very low values, but rapidly increases to a completeness of about $\sim$50% at an integrated flux of $\sim$0.08 Jy km s$^{-1}$. There is a turnover in the plot reaching full completeness around $\sim$0.2 Jy km s$^{-1}$. The completeness does not stay at 100% as some of the bright sources become merged due to the wavelet reconstruction and multiple objects are counted as one. The other source finders show a very similar behaviour however the completeness levels are lower. There is a large variation in the reliability numbers, but apart from CNHI the reliabilities for all source finders have values above 70%. We have to stress here again that the raw reliability is an initial estimate of the quality of a source finder, but is likely to be improved upon in post-processing of the data. We will explain this in more detail in the discussion. The reliability will go down however with more realistic noise containing unpredictable features such as e.g. continuum sources and solar interference. In the top-right panel of Fig. \[point\_comp\_flux\] the completeness is plotted as a function of integrated signal-to-noise ratio $(F_{int}/{\sigma}_{int})$. The integrated noise is calculated as: $$\sigma_{int} = rms \cdot dV \cdot \sqrt{2.35 \cdot W_{50}/dV}.$$ where $dV$ is the spectral resolution of the cube and the coefficient is used to convert the $W_{50}$ value to the line width of a Gaussian. The general trend is very similar, here for the better performing source finders in terms of completeness, about 50% completeness can be achieved at an integrated signal-to-noise ratio of $\sim4-5$. The completeness increases very rapidly and for several source finders 100% completeness is achieved at an integrated signal-to-noise ratio around 10 while for the best [Duchamp]{} run this result is achieved at an integrated signal-to-noise ratio close to 6. ![image](model_compl_Fint-eps-converted-to.pdf){width="48.00000%"} ![image](model_compl_S2N-eps-converted-to.pdf){width="48.00000%"} ![image](model_compl_Fpeak-eps-converted-to.pdf){width="48.00000%"} ![image](model_compl_W50-eps-converted-to.pdf){width="48.00000%"} Model galaxies -------------- For the testing of the source finders on the cube with model galaxies, we analyse again two different runs for each of the source finders apart from the 2D-1D wavelet source finder. The tested parameters for [ Duchamp]{} are almost identical to two results as presented in [@westmeier2011], in table \[model\_duchamp\] we summarise the parameters that were used. The only difference between the two runs is that in the second run the objects are [“grown”]{} to a lower threshold once detected. When doing this, objects that are broken up into multiple detections can get merged. The [Gamma-finder]{} has been used with a 3$\sigma$ and a $5\sigma$ clipping level, while for the CNHI source finder we have used probability thresholds of $5\cdot10^{-4}$ and $5\cdot10^{-5}$ respectively. In the case of the [S+C finder]{} clipping levels of 3.5$\sigma$ and 4$\sigma$ have been used. Parameter Run 1 Run 2 Comment ----------------------- --------- --------- ------------------------ threshold 0.00186 0.00186 $1.0 \times$[rms]{} minPix 10 10 minChannels 3 3 flagAdjacent true true flagGrowth false true growthThreshold – 0.00093 $0.5 \times$[rms]{} flagRejectBeforeMerge false true flagATrous true true Wavelet reconstruction reconDim 3 3 in 3 dimensions snrRecon 2 2 scaleMin 3 3 In Fig. \[model\_completeness\] we plot again the completeness of the source finders as function of integrated flux, integrated signal-to-noise ratio, peak flux and velocity width ($W_{50}$). The integrated flux of the model galaxies is defined as: $$F_{int} [\textrm{Jy km s$^{-1}$}] = F_{peak} \cdot (2\pi)^{1.5} \cdot disp \cdot B_{maj} \cdot B_{min}$$ where $F_{peak}$ is the peak flux, $disp$ is the velocity dispersion and $B_{maj}$ and $B_{min}$ are the FWHM major and minor axis respectively of the 2-dimensional Gaussian describing the galaxy. The integrated noise is given by: $$\sigma_{int} = \sqrt{\frac{2.35 \cdot W_{50}}{dV}} \cdot \sqrt{\frac{1.13 \cdot B_{maj} \cdot B_{min}}{b_{min} \cdot b_{maj}}} \cdot rms \cdot dV \cdot 2.35$$ where $W_{50}$ is the velocity width FWHM given by the model catalogue, $dV$ is the channel separation, $b_{maj}$ and $b_{min}$ are the major and minor axis of the synthesised beam and [rms]{} is the noise in the cube. \ The general results are slightly different to the results as obtained from the cube with point sources. The performance of the different source finders is quite comparable, however in general both completeness and reliability levels are slightly lower than for the point sources. Sources that are extended in space or velocity can be almost hidden in the noise and hard to detect. For the better performing source finders, we reach 50% completeness around an integrated signal-to-noise ratio between 4 and 6 and 100% completeness for a signal-to-noise ratio between 10 and 15. These are very promising results given that the achieved completeness values are very close to the completeness of the point sources which should be much easier to detect. Compared to the point sources the S+C finder is performing much better and seems the best algorithm here in terms of completeness. This is due to the fact that with smoothing to different spatial or spectral scales the real shape of an object is matched as close as possible. In the case of point sources smoothing to a larger scale does not increase the signal to noise and hence the S+C finder does not benefit as much. The [Gamma-finder]{} performs much worse for model galaxies as this source finder is most sensitive to sudden changes in the spectrum, which are not as apparent in the case of extended sources. ![image](total_point.pdf){width="48.00000%"} ![image](total_model.pdf){width="48.00000%"} ![image](point_compl_duchamp1.pdf){width="48.00000%"} ![image](point_compl_duchamp2.pdf){width="48.00000%"} ![image](point_compl_comp_duchamp1.pdf){width="48.00000%"} ![image](point_compl_comp_duchamp2.pdf){width="48.00000%"} ![image](point_reli_duchamp1.pdf){width="48.00000%"} ![image](point_reli_duchamp2.pdf){width="48.00000%"} ![image](point_compl_CNHI1.pdf){width="48.00000%"} ![image](point_compl_CNHI2.pdf){width="48.00000%"} ![image](point_compl_comp_CNHI1.pdf){width="48.00000%"} ![image](point_compl_comp_CNHI2.pdf){width="48.00000%"} ![image](point_reli_CNHI1.pdf){width="48.00000%"} ![image](point_reli_CNHI2.pdf){width="48.00000%"} ![image](point_compl_smooth1.pdf){width="48.00000%"} ![image](point_compl_smooth2.pdf){width="48.00000%"} ![image](point_compl_comp_smooth1.pdf){width="48.00000%"} ![image](point_compl_comp_smooth2.pdf){width="48.00000%"} ![image](point_reli_smooth1.pdf){width="48.00000%"} ![image](point_reli_smooth2.pdf){width="48.00000%"} ![image](point_compl_wavelet.pdf){width="48.00000%"}\ ![image](point_compl_comp_wavelet.pdf){width="48.00000%"}\ ![image](point_reli_wavelet.pdf){width="48.00000%"}\ ![image](point_compl_gamma1.pdf){width="48.00000%"} ![image](point_compl_gamma2.pdf){width="48.00000%"} ![image](point_compl_comp_gamma1.pdf){width="48.00000%"} ![image](point_compl_comp_gamma2.pdf){width="48.00000%"} ![image](model_compl_duchamp1.pdf){width="48.00000%"} ![image](model_compl_duchamp2.pdf){width="48.00000%"} ![image](model_compl_comp_duchamp1.pdf){width="48.00000%"} ![image](model_compl_comp_duchamp2.pdf){width="48.00000%"} ![image](model_reli_duchamp1.pdf){width="48.00000%"} ![image](model_reli_duchamp2.pdf){width="48.00000%"} ![image](model_compl_CNHI1.pdf){width="48.00000%"} ![image](model_compl_CNHI2.pdf){width="48.00000%"} ![image](model_compl_comp_CNHI1.pdf){width="48.00000%"} ![image](model_compl_comp_CNHI2.pdf){width="48.00000%"} ![image](model_reli_CNHI1.pdf){width="48.00000%"} ![image](model_reli_CNHI2.pdf){width="48.00000%"} ![image](model_compl_smooth1.pdf){width="48.00000%"} ![image](model_compl_smooth2.pdf){width="48.00000%"} ![image](model_compl_comp_smooth1.pdf){width="48.00000%"} ![image](model_compl_comp_smooth2.pdf){width="48.00000%"} ![image](model_reli_smooth1.pdf){width="48.00000%"} ![image](model_reli_smooth2.pdf){width="48.00000%"} ![image](model_compl_wavelet.pdf){width="48.00000%"}\ ![image](model_compl_comp_wavelet.pdf){width="48.00000%"}\ ![image](model_reli_wavelet.pdf){width="48.00000%"}\ ![image](model_compl_gamma1.pdf){width="48.00000%"} ![image](model_compl_gamma2.pdf){width="48.00000%"} ![image](model_compl_comp_gamma1.pdf){width="48.00000%"} ![image](model_compl_comp_gamma2.pdf){width="48.00000%"} Discussion ========== A different way of demonstrating the performance of the source finders is by plotting the completeness of the source finders on a two dimensional plot as a function of integrated flux and velocity width. As a reference the total number of objects in both cubes is shown on this grid in Fig. \[total\_detections\_2D\]. Point sources ------------- In Fig. \[point\_2D\] we plot the completeness and the reliability results of the different source finders when applied to the point sources on a two-dimensional grid. For each result, completeness is plotted as a function of integrated flux and velocity width (represented by FWHM ($W_{50}$)) of the modelled point sources in the top panels. In the middle panels the ratio is shown between number of objects detected by the tested source finder and the number of sources detected by any source finder. Instead of showing the overall completeness this plot shows how a particular source finder performs compared to the other source finding results. Regions in the parameter space that appear blue in this plot are regions that can be improved upon, as other source finders do detect objects within this parameter space. Apart from showing how one source finder performs compared to the others, this plot also shows the parameter space that is covered by all the source finders combined. In the bottom panel of Fig. \[point\_2D\] reliability is plotted as function of measured integrated flux and velocity width $(W_{50})$. These panels are not included for the [Gamma-finder]{} as this source finder does not parameterise sources. The completeness plots in the top two panels all have the same scale as the parameters are based on the intrinsic parameters of the input catalogue. The scaling of the reliability panels is different in each plot as this is determined by the measured parameters of the different source finders. We have to emphasise here that the measured parameters are not by definition correct values as this depends on the capability to parameterise sources properly. Different parameterisation algorithms are used by the different source finders. We have not compared the parameters obtained from the source finders, but a possible difference has to be taken into account when comparing the plots. [Duchamp]{} is incomplete for small integrated fluxes, but is basically 100% complete for fluxes above 0.3 Jy km s$^{-1}$. It is expected that very low flux values are difficult to detect, however in quite a large area of the parameter space sources are detected which are not recovered by [Duchamp]{}. This indicates that although in Fig. \[point\_comp\_flux\] [Duchamp]{} appears to be the best performing source finder, another source finder is needed to detect the very low fluxes, or [Duchamp]{} has to be improved here. For both [Duchamp]{} tests the reliability is reasonable as most detections are true detections and the false detections are especially concentrated at very small fluxes. The [CNHI]{} source finder does not perform very well on the tested point sources, it misses almost all sources with a FWHM velocity width below 12 km s$^{-1}$. Apart from that this source finder also misses a very significant fraction of the bright sources. The number of false detections is relatively large and spread over the whole parameter range. Many of the false detections have low fluxes and very broad line widths, much broader than any of the real line widths. The [S+C finder]{} detects sources down to very low integrated fluxes, lower than most of the other source finders. As can be seen in the middle panel of the first [S+C finder]{} results, some of the sources with a low integrated flux are only recovered by this source finder. On the opposite side, the [S+C finder]{} is not 100% complete at either large fluxes or large line widths. False detections are quite difficult to distinguish when using this source finder, as the false detections are not clustered in a narrow range of the parameter space. For a large region in the plot the reliability fluctuates around 50%, indicating that the determined parameters of false detections are very similar to that of true detections. For the 2D-1D wavelet finder there seems to be a clear trend from 0% completeness at low fluxes to almost full completeness at high integrated flux values, very similar to the [Duchamp]{} results. In the parameter space covering the largest fluxes and line widths, the finder is not 100% complete. This could be caused by the fact that our model cube is very dense with many sources, and for the largest wavelet scales these sources start to merge. The wavelet finder can be improved here, as [Duchamp]{} also uses wavelet reconstructions, but appears to be less sensitive to this problem. The reliability of the 2D-1D wavelet finder is very good and 100% in most of the parameter space, although there are some false detections with a high integrated flux, we have no good explanation for why the reliability decreases here. The [Gamma-finder]{} seems to perform well on sources with a strong integrated flux and narrow line width. In fact it is the best finder for objects with a narrow line width below 5 km/s, although we have to question how realistic such sources are when observing real galaxies. As the [Gamma-finder]{} does not give a mask or parameters of the detected sources, we cannot make reliability plots for this source finder. Model galaxies -------------- In Fig. \[model\_2D\] we show very similar plots as in the previous figure, but now for the model galaxies. In the top panels the completeness of the different source finders is plotted, while the middle panels compare the completeness of the source finders with respect to each other. In the bottom panels the reliability of the source finders is plotted. These modelled galaxies have more complex structures compared to the point sources, and the completeness and reliability results are very different. Note the different scales in both integrated flux and velocity compared to the point sources in Fig. \[point\_2D\]. [Duchamp]{} is complete for objects with high flux in the first run, but in the second run misses a few sources that should be easy to detect due to their high flux. The only difference between the two [Duchamp]{} runs is the [growth]{} parameter, which has merged some of the extended sources. As can be seen in the plot, the missed sources have a large integrated flux but relatively narrow line width, which indicates that they are spatially extended. As the objects were all placed at a similar radial velocity in the cube, there is a high risk of merging. There is a clear transition phase between non-detected and detected objects and [Duchamp]{} misses objects with low integrated fluxes that are detected by at least one other source finder. The reliability of [Duchamp]{} looks very good, as almost all false detections are clustered in a limited area of the parameter space at small fluxes and narrow line widths. The CNHI finder also shows a transition phase from non-detected objects with a low flux to detected objects with high fluxes, however the transition is much broader than for [Duchamp]{}. The CNHI finder is less likely here to miss sources with a narrow line width as the velocity profiles of the model galaxies are much broader and more realistic than for the point sources. When compared to the other source finders, this finder detects a significant fraction of the objects at low integrated flux. The reliability is worse than for the other source finders, but a large fraction of the detected objects have very low fluxes, covering a large range in line width. For the [S+C finder]{} the results are very impressive as it even detects many of the sources with small flux and narrow line width. This finder also has a very small number of false detections that appear to be concentrated in a rather limited range of the parameter space. Although currently this appears to be the best source finder on the tested cube with model galaxies, it is not the best source finder on the full parameter range. In particular, objects with a small integrated flux and broad line width are missing, which in some cases are detected by the Gamma-finder. The 2D-1D wavelet source finder has a very narrow transition between detected and non-detected sources where almost all objects with a flux below 0.5 Jy km/s are missed, while almost all objects with an integrated flux above 1.5 Jy km/s are detected. The reliability of this source finder is very good and the completeness can probably be improved upon by decreasing the clipping threshold used on the reconstructed wavelet scales. The parameter space covered in the reliability plots is very different to the other source finders, the 2D-1D wavelet method seems to detect higher fluxes and smaller line-widths. The [Gamma-finder]{} has a relatively good performance in completeness as it detects the objects with high fluxes, but also a significant number of objects with low flux values. Interesting to see is that the first [Gamma-finder]{} results gives the best result for objects with a low flux and broad line width. Although not plotted in this figure, this good performance in completeness probably comes at the cost of reliability as the reliability of the first run is very low at 12%. ![image](point_scatter_duchamp1-eps-converted-to.pdf){width="48.00000%"} ![image](point_scatter_duchamp2-eps-converted-to.pdf){width="48.00000%"} ![image](point_scatter_CNHI1-eps-converted-to.pdf){width="48.00000%"} ![image](point_scatter_CNHI2-eps-converted-to.pdf){width="48.00000%"} ![image](point_scatter_smooth1-eps-converted-to.pdf){width="48.00000%"} ![image](point_scatter_smooth2-eps-converted-to.pdf){width="48.00000%"} ![image](point_scatter_wavelet-eps-converted-to.pdf){width="48.00000%"} Figure \[point\_scatter\]: Continued ![image](model_scatter_duchamp1-eps-converted-to.pdf){width="48.00000%"} ![image](model_scatter_duchamp2-eps-converted-to.pdf){width="48.00000%"} ![image](model_scatter_CNHI1-eps-converted-to.pdf){width="48.00000%"} ![image](model_scatter_CNHI2-eps-converted-to.pdf){width="48.00000%"} ![image](model_scatter_smooth1-eps-converted-to.pdf){width="48.00000%"} ![image](model_scatter_smooth2-eps-converted-to.pdf){width="48.00000%"} ![image](model_scatter_wavelet-eps-converted-to.pdf){width="48.00000%"}\ Figure \[model\_scatter\]: Continued Reliability of Source Finders ----------------------------- In Fig. \[point\_comp\_flux\] and \[model\_completeness\] the reliability of the different source finders is given by a single number, which can be misleading. This number could be completely dominated by a large number of false detections at a very low threshold, while the source finder is very reliable for high flux values. To better understand where the bulk of the false detections are, all detections are plotted in a scatter plot in Fig. \[point\_scatter\] for the point sources and Fig. \[model\_scatter\] for the model galaxies. The detections are again plotted as a function of velocity width and integrated flux, where true detections are plotted in green and false detections are plotted in red. In the [Duchamp]{} results for the point sources shown in the top panels of Fig. \[point\_scatter\] there are barely any false detections in the first run. In the second run, all false detections have low fluxes. A possibility that can improve the reliability is to apply a cut in integrated flux after the parametrisation of detections. In this example a cut a 0.05 Jy km/s would increase the reliability to $\sim$100% while the number of missed real detections is still limited. For the [CNHI]{} finder the difference between true and false detections is not so obvious. False detections are not clustered in a clearly defined parameter space, but rather mixed with real detections, making it more difficult to eliminate them after post-processing. There is however a very large bulge of detections with a low flux and broad line width. The [S+C finder]{} has a large number of false detections in the first run, however a very large fraction can be eliminated by applying a cut in integrated flux. In the second run the number of false detections is much lower, however they are very well mixed with true detections and difficult to eliminate. Although not shown in the plot, particular for this source finder is that it also reports negative fluxes. These are by default all considered false detections. Assuming that the noise is symmetric, the reliability of positive detections can be determined based on the properties of the negative detections. This method is further explored and explained in @serra2011b. The 2D-1D wavelet source finder is very reliable for point sources as shown before, with barely any false detections. The false detections are however difficult to eliminate as they are concentrated toward high fluxes and line widths. As mentioned this could be a consequence of the used test cube which has a very high source density. Especially in the case of strong sources the largest wavelet scales will merge sources, decreasing the number of detected objects and hence the completeness. \ A very similar set of plots is given in Fig. \[model\_scatter\], where true and false detections of the model galaxies are plotted for all the source finders apart from the [Gamma-finder]{}. The behaviour of the different algorithms is very similar to before, where the false detections of [Duchamp]{} tend to have a low integrated flux, although it is difficult to completely isolate them. The CNHI finder has a very large number of false detection with low flux and broad line-width, many of which can be rejected to refine the reliability. The performance of the [S+C finder]{} is very good when it comes to reliability as the number of false detections is relatively low. Also the reliability of the 2D-1D wavelet finder is very good, however the false detections are mixed with true detections. \ The reliability of the source finders can be dramatically improved upon through simple cuts in parameter space. To be able to do this, it is crucial to properly parameterise the detections which has not been done sufficiently at this stage. Nevertheless, to illustrate the concept, we applied a cut on the detections at different integrated flux levels. In Fig. \[cut\_results\] the results are shown, where completeness is plotted as function of reliability for the different source finders after applying cuts at different flux levels. For the point sources cuts have been applied at $F_{int}$ = 0.0, 0.01, 0.02, 0.03, 0.04 and 0.05 Jy km s$^{-1}$ while for the model galaxies at $F_{int}$ = 0.0, 0.1, 0.2, 0.3, 0.4 and 0.5 Jy km s$^{-1}$. The results move from high completeness and low reliabiliy when not applying a cut to low completeness and high realiability when applying the most extreme cut. Although the improvements in reliability vary amongst the different source finders, for each of them the raw reliability can be improved by tens of percent, while only losing a few percent in completeness. In the case of the second [Duchamp]{} test on the point sources the reliability increases from 72% to 96%, while the completeness drops by only 0.6% from 83% to 82% at the fourth data point. On the model galaxies the most impressive result is achieved with the [S+C finder]{} where in the second run the reliability increases to above 95%, while still maintaining a completeness of almost 70%. ![image](point_cut_results-eps-converted-to.pdf){width="48.00000%"} ![image](model_cut_results-eps-converted-to.pdf){width="48.00000%"} Conclusions =========== In this paper we have compared the performance of five potential ASKAP source finders. The tested source finders are 1) the [ Duchamp]{} source finder, 2) the [Gamma-finder]{}, 3) the [CNHI]{} source finder, 4) 2D-1D Wavelet reconstruction source finder and 5) the [S+C finder]{}, a source finder based on sigma clipping of smoothed versions of the original data cube. The source finders have been applied to two data cubes with model sources, the first containing point sources with a relatively narrow Gaussian line profile and the second containing extended galaxies with inclinations and rotation curves. We have to stress that apart from the [Gamma-finder]{} the tested source finders are not final products but are still under active development. In this paper we want to present the current status of the different source finders, however there is significant room for improvement as is also discussed in other papers in this issue describing some of the tested source finders individually. The testing of different source finding algorithms on different data cubes has proven that it is very difficult to find a good source finder which is reliable for many types of objects. Source finders perform very differently depending on the type of object that is detected. An important feature of a source finder is its reliability, which has not yet been fully explored. Although a number for the raw reliability can be given, in many cases the false detections are clustered within a certain range of flux and line width. We are confident that a large fraction of the false detections can be rejected through simple cuts in parameter space as has been demonstrated in the discussion, however to be able to do this properly all detections have to be parameterised accurately which has not been done yet. For the current source finders and datasets, we find that for point sources 50% completeness can be achieved at an integrated signal-to-noise ratio of $\sim$4-5 sigma, and 100% completeness can be achieved around an integrated signal-to-noise ratio of $\sim$10. For the extended sources the completeness estimates are very similar: for the best results 50% completeness is achieved at an integrated signal-to-noise ratio of $\sim$4-6 and 100% completeness is achieved at an integrated signal-to-noise ratio of $\sim$10. It is interesting to see that the different source finders achieve a different performance, depending on the type of object. Currently none of the source finders excels at being able to achieve the best result in the full parameter space when looking at integrated flux and line width. Nevertheless we have pointed out the strong and weak points of the different source finders, which provides input for future development and testing. For the tested parameters, currently [Duchamp]{} gives the best results on point sources, while the [S+C finder]{} gives the best result for extended objects when looking at the completeness. Due to the different smoothing levels that have been applied in the [S+C finder]{}, this algorithm is best capable of matching the true shape of an object. As the [S+C finder]{} concept is simple yet powerful, we recommend that the other source finders improve their performance by incorporating smoothing on multiple scales. Currently all the tested source finders perform reasonably well, however there is significant room for improvement to meet our goals. All of the source finders have a certain area in parameter space where they perform best and we will combine the algorithms of different source finders to optimise the result. @duffy2011 give predictions of the number of objects that will be detected with WALLABY and DINGO. They predict that at an angular resolution of $30''$, 14% of the WALLABY sources will be unresolved and the bulk of the remainder will be marginally resolved, while for DINGO 93.3% of the sources will be unresolved. This means that many of the unresolved sources in DINGO will have very different profiles to the ones tested in this paper. At an angular resolution of $10''$ these numbers change dramatically, as for WALLABY none of the sources will be point sources as all sources will be larger than one beam, and for DINGO 7.4% of the objects will be smaller than one beam. Although the two cubes that have been used for testing cover a large area in parameter space, they do not sample the full signal-to-noise ratio range properly. We have started efforts to test the source finders on models covering a large range of parameters, keeping integrated signal-to-noise values constant. These tests should give accurate estimates of how many sources can be detected by WALLABY and DINGO. We have a fairly good understanding of the different source finders on simulated objects as presented in this paper. The cubes that have been tested are ideal cubes in the sense that the noise is Gaussian and does not have any systematic artefacts caused by continuum sources, solar ripples, phase errors, radio frequency interference, etc. These contributions have not been taken into account but will have a very significant effect on the performance of source finders, especially in terms of reliability. The simulated model sources are perfectly symmetric sources without any weird or unexpected shapes or extended tails. To have a better understanding of the performance of the source finders, the next step will be to test the source finders on a cube containing data from real galaxies as they occur in the Universe. Boyce, P. 2003, GammaFinder: a Java application to find galaxies in astronomical spectral line datacubes, M.Sc. Dissertation, Cardiff University Deboer, D. R., et al. 2009, IEEE Proceedings, 97, 1507 Dewdney, P. E., Hall, P. J., Schilizzi, R. T., & Lazio, T. J. L. 2009, IEEE Proceedings, 97, 1482 Driver, S. P.et al. 1999, A&G, 50, 12 Duffy, A., R. et al. 2011, MNRAS in prep. Haynes, M. P. at al. 2011, AJ, 142, 170 Hibbard, J. E., van Gorkom, J. H., Rupen, M. P.& Schiminovich, D. 2001, ASPC, 240, 657 Johnston, S., et al. 2008, ExA 22, 151 Jonas, J. L. 2009, IEEE Proceedings, 97, 1522 Jones, A. J., Evans, D., Margetts, S. & Durrant, P. J., 2002,. The Gamma test. 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V., & van der Hulst, J. M. 2008, AIP Proceedings, 1035, 265 Westmeier, T., Popping, A. & Serra, P. 2011, PASA, special issue, arXiv1112.3093W Whiting, M. T. 2011, MNRAS, arXiv1201.2710 [^1]: Duchamp website: http://www.atnf.csiro.au/people/Matthew.Whiting/Duchamp/	ArXiv
--- abstract: 'dc and ac magnetic properties of two thin-walled superconducting Nb cylinders with a rectangular cross-section are reported. Magnetization curves and the ac response were studied on as-prepared and patterned samples in magnetic fields parallel to the cylinder axis. A row of micron-sized antidots (holes) was made in the film along the cylinder axis. Avalanche-like jumps of the magnetization are observed for both samples at low temperatures for magnetic fields not only above $H_{c1}$, but in fields lower than $H_{c1}$ in the vortex-free region. The positions of the jumps are not reproducible and they change from one experiment to another, resembling vortex lattice instabilities usually observed for magnetic fields larger than $H_{c1}$. At temperatures above $0.66T_c$ and $0.78T_c$ the magnetization curves become smooth for the patterned and the as-prepared samples, respectively. The magnetization curve of a reference planar Nb film in the parallel field geometry does not exhibit jumps in the entire range of accessible temperatures. The ac response was measured in constant and swept dc magnetic field modes. Experiment shows that ac losses at low magnetic fields in a swept field mode are smaller for the patterned sample. For both samples the shapes of the field dependences of losses and the amplitude of the third harmonic are the same in constant and swept field near $H_{c3}$. This similarity does not exist at low fields in a swept mode.' author: - 'M.I. Tsindlekht$^1$, V.M. Genkin$^1$, I. Felner$^1$, F. Zeides$^1$, N. Katz$^1$, $\check{\text{S}}$. Gazi$^2$, $\check{\text{S}}$. Chromik$^2$, O.V. Dobrovolskiy$^{3,4}$, R. Sachser$^3$, and M. Huth$^3$' title: 'dc and ac magnetic properties of thin-walled Nb cylinders with and without a row of antidots' --- Introduction ============ Penetration of magnetic flux into hollow superconducting cylinders is a long standing field of interest. The Little-Parks effect and the quantization of trapped flux were intensively studied during the last fifty years [@LITTLE; @DOU; @VEKHT]. Recent advances in nanotechnology have made it possible for studying experimentally superconducting properties of thin films with different arrays of antidots, see for example, [@Motta1] and references therein. In particular, for the observation of the aforementioned effects, cylinders or antidots of small diameter are required. At the same time, the study of hollow thin-walled cylinders with macroscopic sizes in magnetic fields parallel to its axis has been much less well studied. It was expected that quantum phenomena cannot be observed in such samples because of the fact that one flux quanta for cylinders with a cross section area of $\approx 1$ cm$^2$ corresponds to a magnetic field about $10^{-7}$ Oe. In this case magnetization will be a smooth function of the magnetic field. However, experimental results obtained recently for thin-walled macroscopic cylinders do not agree with this expectation. Namely, in such Nb cylinders we succeeded in monitoring the magnetic moment of the current circulating in the walls and observed dc magnetic moment jumps even in fields much lower than $H_{c1}$ of the film itself [@Katz1]. So far it is not clear what mechanism is responsible for such flux jumps. Under an axial magnetic field the cylinder walls screen weak external fields, provided that $L\equiv Rd/\lambda^{2} \gg 1$, where $R$ is the cylinder radius, $d$ is the wall thickness, and $\lambda$ is the London penetration depth [@DOU; @PG; @KITTEL]. Therefore, it is expected, that a dc magnetic field, $H_0$, will penetrate into the cylinder as soon as the current in the wall exceeds the critical current and no field penetration should be observed at lower fields. Only above $H_{c1}$, vortices created at the outer cylinder surface can move into the cylinder. For a magnetic field oriented perpendicular to the Nb film surface such vortex motion leads to flux jumps [@NOWAK; @STAM]. These flux jumps were interpreted as a thermomagnetic instability of the critical state. It was demonstrated that in a sample with an array of antidots flux jump propagates along the antidots row [@MOTTA2]. Nucleation of the superconducting phase in a thin surface sheath in decreasing magnetic fields parallel to the sample surface was predicted by Saint-James and de Gennes [@DSJ]. They showed that nucleation occurs in a magnetic field $H_0\leq H_{c3}\approx 1.695 H_{c2}$. Experimental confirmations of this prediction were obtained soon after their work appeared. The experimental methods for this confirmation were dc resistivity and ac susceptibility measurements [@ROLL]. It was found that low frequency losses in superconductors in surface superconducting states (SSS) can exceed losses in the normal state [@BURGER; @ROLL]. A swept dc magnetic field qualitatively changes the character of the ac response. Specifically, the penetration of the ac magnetic field into the sample takes place not only for $H_{c2}<H_0<H_{c3}$ but also for $H_{c1}<H_0<H_{c2}$, in sharp contrast to the case of constant dc fields [@STR2; @MAX; @GENKIN22]. The effect of a swept dc field can more suitably be investigated by using hollow thin-walled superconducting cylinders, rather than by bulk samples, because one can control the field transmission through their walls. Previously, we have shown [@Genkin1] that in a thin-walled cylinder in the mixed state, the effect of sweeping a dc field on the ac response is due to an enhancement of the vortex motion through the wall. Above $H_{c2}$, however, this picture is no longer appropriate and the experimental data were explained within the framework of a simple relaxation model [@Katz1]. The goal of this paper is to study how antidots affect the penetration of dc and ac magnetic fields into thin-walled superconducting Nb cylinders of macroscopic sizes, with a rectangular cross section. We show that at low enough temperatures for both, a flat and a patterned samples, even in the vortex-free regime at $H< H_{c1}$, the dc magnetic field penetrates through the cylinder walls in an “avalanche”-like fashion. Jumps of the dc magnetic moment also become apparent at fields above $H_{c1}$ at low temperatures. For both samples, the field values at which jumps occur vary from one measurement to another, indicating that one deals with transitions between metastable states. At temperatures above $0.66T_c$ and $0.78T_c$ the magnetization curves become smooth for the patterned and the as-prepared sample, respectively. The ac response of both cylinders was studied in the point-by-point and swept field modes. In these, the signals of the first, second and third harmonics were measured concurrently. The ac response of as-prepared and patterned samples is qualitatively different in a swept field mode. Experimental ============ The cylindrical samples were prepared by dc magnetron sputtering at room temperature on a rotated sapphire substrate. The sizes of the substrate with rounded corners (radius 0.2 mm) are $1.5\times3\times7.5$ mm$^3$. We fabricated, therefore, a thin-walled hollow superconducting cylinder with a rectangular cross section. The nominal film thickness of both samples was $d=100$ nm. A sketch of the sample geometry is presented in Fig. \[f1\]. The reference sample $A$ was kept as-grown, while the second one, sample $B$, was patterned with a row of antidots at the mid of the larger surface over the entire length of the sample. The row of antidots was milled by focused ion beam (FIB) in a scanning electron microscope (FEI, Nova Nanolab 600). The beam parameters were 30kV/0.5nA, while the defocus and blur were 560$\mu$m and 3$\mu$m, respectively. The pitch was equal to the antidot center-to-center distance of 1.8$\mu$m and the number of beam passes needed to mill 150nm-deep antidots was 2000. The antidots row with a length of $7.5$mm was milled by iteratively stitching the processing window with a long size of $400\,\mu$m. SEM images of the patterned surface of sample $B$ are shown in Fig. \[f2\]. The antidots have an average diameter of 1.5$\mu$m and an average edge-to-edge distance of 300nm. The dc magnetic properties were measured using a commercial superconducting quantum interference device (SQUID), Quantum Desing MPMS5, magnetometer. The ac response was measured by the pick-up coil method. The sample was inserted into one coil of a balanced pair of coils, and the unbalanced signal was measured by means of lock-in amplifier. The ac magnetic susceptibilities were measured in absolute units, see [@LEV2]. A “home-made” measurement cell of the experimental setup was adapted to the SQUID magnetometer. A block diagram of the experimental setup can be found elsewhere [@LEV2]. The ac response as a function of the dc field were carried out by two methods: (i) - point-by-point (PBP) mode, where the dc field was kept constant during the measurement, and (ii) - swept field (SF) mode, where the dc field was ramped with a rate of 20 Oe/s. Both external ac and dc fields were directed parallel to cylinder axis and hence, to the film surface. ![Sketch of the $B$ sample. Here $\text{L}_s = 7.5$ mm, $\text{W}_s=3$ mm, and $2\text{D}=1.4$ mm are the substrate length, width and thickness, respectively. Both dc and ac fields were parallel to $Z$-axis. Dimensions are not to scale.[]{data-label="f1"}](Fig1){width="0.6\linewidth"} ![SEM images of the surface of sample $B$. The antidots have an average diameter of 1.5 $\mu$m and an average edge-to-edge distance of 300 nm. An overview SEM image is presented in the bottom panel where the row of FIB-milled antidots is clearly seen. []{data-label="f2"}](Fig2a.eps "fig:"){width="0.94\linewidth"} ![SEM images of the surface of sample $B$. The antidots have an average diameter of 1.5 $\mu$m and an average edge-to-edge distance of 300 nm. An overview SEM image is presented in the bottom panel where the row of FIB-milled antidots is clearly seen. []{data-label="f2"}](Fig2b.eps "fig:"){width="0.94\linewidth"} Results ======= dc magnetization ---------------- The upper and lower panels of Fig. \[f3\] show the temperature dependences of the magnetic moments, $M_0$, in magnetic field $20\pm 2$ Oe, of samples $A$ and $B$, respectively. The critical temperatures, $T_c$, of both samples are almost the same, 8.3 K, the transition width for sample $A$ is 1.3 K but 2.7 K for sample $B$. Sample $B$ demonstrates a two-stage transition, see the inset to the lower panel of Fig. \[f3\]. At low temperatures, the magnetic moment of sample $A$ is a factor of two larger than that of sample $B$. Temperature and field dependences of the magnetic moment were measured after cooling the sample down to the desired temperatures in zero field (ZFC). ![(Color online). Temperature dependences of the magnetic moment of samples $A$ and $B$, upper and lower panels, respectively. Inset to lower panel shows temperature dependence of $M_0$ of $B$ sample near $T_c$. []{data-label="f3"}](Fig3a.eps "fig:"){width="0.98\linewidth"} ![(Color online). Temperature dependences of the magnetic moment of samples $A$ and $B$, upper and lower panels, respectively. Inset to lower panel shows temperature dependence of $M_0$ of $B$ sample near $T_c$. []{data-label="f3"}](Fig3b.eps "fig:"){width="0.98\linewidth"} ![(Color online). $M_0(H_0)$ of samples $A$ and $B$ after ZFC, upper panel. Expanded view of the magnetization curves in low magnetic fields for samples $A$ and $B$, lower panel. []{data-label="f5"}](Fig4a.eps "fig:"){width="0.98\linewidth"} ![(Color online). $M_0(H_0)$ of samples $A$ and $B$ after ZFC, upper panel. Expanded view of the magnetization curves in low magnetic fields for samples $A$ and $B$, lower panel. []{data-label="f5"}](Fig4b.eps "fig:"){width="0.98\linewidth"} The $M_0(H_0)$ dependences for samples $A$ and $B$ at 4.5 K are shown in the upper panel of Fig. \[f5\]. The magnetization curves in the ascending branch were measured in the hysteresis mode with 5 Oe step at low fields. Fig. \[f5\] shows that the $H_{c2}$ values are different. Determination of $H_{c2}$ for sample $B$ is less accurate than that of sample $A$, due to the magnetic moment relaxation, which at high fields is larger for sample $B$ [@MIT]. An expanded view of the magnetization curves at low fields is shown in the lower panel of Fig. \[f5\]. The fields of the first jumps, $H^$, are around 20 Oe and 10 Oe, while the number of jumps in magnetic fields up to 100 Oe are 5 and 7 for samples $A$ and $B$, respectively. Jumps of the magnetic moment were observed in a wide range of magnetic fields, including fields below $H_{c1}$ for both samples. This behavior is reminiscent of magnetic flux jumps in Nb thin films for $H_0$ perpendicular to the film surface [@NOWAK; @STAM]. The jumps observed in these papers were interpreted as a thermomagnetic instability of the Abrikosov vortex lattice [@NOWAK; @STAM]. However, existence of jumps in fields below than $H_{c1}$ and parallel to the surface have been reported in our recent work only [@Katz1]. $H_{c1}$ is $\approx 350$ Oe at 4.5 K in our samples. Direct determination of $H_{c1}$ for thin-walled cylindrical samples is impossible due to magnetic moment jumps at low fields. However, the estimation of $H_{c1}$ can be done using magnetization curves of the planar film as it shown in inset to Fig.\[f13\]. ac response ----------- The effective ac magnetic susceptibility of the sample in the external field $H(t)=H_0(t)+h_{ac}\sin (\omega t)$ is given by $$\label{Eq2} M(t)=Vh_{ac}\sum_n\{\chi_n^{'} \sin (n\omega t)-\chi_n^{''}\cos (n\omega t)\},$$ and it exhibits the appearance of the ac field penetration into the sample, i.e. $\chi_1^{'}\neq -1/4\pi$, ac losses $\chi_1^{''}>0$ and harmonics of the fundamental frequency, $\chi_n$. Here, $M(t)$ is the magnetic moment of the sample and $V$ is its volume. In what follows we consider the results of the ac measurements in both PBP and SF modes. ![(Color online). Field dependences of $\chi_1(H_0)$ of samples $A$ and $B$ in the PBP mode at 1465 and 293 Hz, upper and lower panels, respectively. Measurements were done at 4.5 K. Arrows on the lower panel show $H_{c3}$ for both samples.[]{data-label="f5aa"}](Fig5a.eps "fig:"){width="0.98\linewidth"} ![(Color online). Field dependences of $\chi_1(H_0)$ of samples $A$ and $B$ in the PBP mode at 1465 and 293 Hz, upper and lower panels, respectively. Measurements were done at 4.5 K. Arrows on the lower panel show $H_{c3}$ for both samples.[]{data-label="f5aa"}](Fig5b.eps "fig:"){width="0.98\linewidth"} The real and imaginary components of the ac susceptibility at 4.5 K for both samples measured in the PBP mode as a function of $H_0$ at two frequencies are shown in Fig. \[f5aa\]. Almost complete screening up to 12.5 kOe of the ac field by the superconducting walls is observed for both samples. This value is higher than $H_{c2} =11\pm 0.5$ kOe of sample $A$ (Fig. \[f5\], upper panel). Complete screening of ac fields by a type II superconductor at low frequencies ($\omega \ll \omega_p$, here $\omega_p$ is a depinning frequency) and amplitudes of excitation (ac current much lower than depinning current) in dc fields lower than $H_{c2}$ was observed years ago [@STR2]. The frequency dispersion of $\chi_1$ is weak for both samples. Third critical magnetic field was determined using ac data as follows. At low amplitude of excitation a loss peak located between $H_{c2}$ and $H_{c3}$. Losses disappear at $H_0>H_{c3}$ because in a normal state $\delta>>d$. Here $\delta$ is a skin depth in a normal state. Such determination of $H_{c3}$ was proposed years ago by Rollins and Silcox [@ROLL]. The lower panel of Fig.\[f5aa\] shows an example of determination of the third critical magnetic field. It was found that $H_{c3}\approx 17.5\pm 0.5$ and $16\pm 0.5$ kOe at 4.5 K for $A$ and $B$ samples, respectively. $H_{c3}/H_{c2} \approx 1.6$ for sample $A$. An accurate determination of $H_{c2}$ for sample $B$ is difficult, due to magnetic relaxation, as discussed above. The absorption line, $\chi_1^{''}(H_0)$, near $H_{c3}$ is different for samples $A$ and $B$. Thus, this line is nonuniform for sample $A$ and it is uniform but broadened for sample $B$. The ac response of superconductors even at very low amplitude of excitation, e.g., less than 1 Oe, is strongly nonlinear in the SSS [@ROLL; @Genkin1]. The second harmonic signal is absent in the PBP mode in the entire range of magnetic fields. At the same time, the third-harmonic signal exists in the vicinity of $H_{c3}$ only. The absence of the second harmonic in PBP mode is a common feature for the bulk samples as well [@CAMP]. Fig. \[f6a\] shows the field dependences of $\chi_3$, $\chi_{2,3}\equiv \sqrt{(\chi_{2,3}^{'})^{2}+(\chi_{2,3}^{''})^{2}}$, in PBP mode for samples $A$ and $B$, in the upper and lower panels, respectively. Perturbation theory with respect to the amplitude of excitation is not applicable for interpreting these experimental data. For example, according to perturbation theory, $\chi_3$ should be proportional $h_{ac}^{2}$ and this is not the case in our findings, Fig. \[f6a\]. It is known that perturbation theory cannot explain experimental data for bulk samples too [@ROLL; @GENKIN22]. We also note that there is a difference for the third harmonic signal between samples $A$ and $B$ in the PBP mode. ![(Color online). Field dependences of $\chi_3$ of samples $A$ and $B$ (upper and lower panels, respectively) in the PBP mode at 4.5 K.[]{data-label="f6a"}](Fig6a.eps "fig:"){width="0.9\linewidth"} ![(Color online). Field dependences of $\chi_3$ of samples $A$ and $B$ (upper and lower panels, respectively) in the PBP mode at 4.5 K.[]{data-label="f6a"}](Fig6b.eps "fig:"){width="0.9\linewidth"} A swept field affects the ac response more strongly at low frequencies or/and low excitation amplitudes for a given sweep rate. This was confirmed in experiments with bulk and thin-walled cylinders samples [@MAX; @GENKIN22] and [@Genkin1; @Katz1], respectively. Fig. \[f7c\] shows the field dependences $\chi_1$ for both samples $A$ and $B$ in the PBP and SF modes at 293 Hz and amplitude 0.04 Oe. The difference between the PBP and SF modes can easily be seen for both samples. The ac response at low magnetic fields in the SF mode are fluctuating due to magnetic flux jumps, Fig. \[f5\]. Near $H_{c3}$ the curves of $\chi_1$ coincide well in PBP and SF modes for both samples, Fig. \[f7c\]. The difference between the two samples in the SF mode is very pronounced in fields above 5 kOe. In particular, $\chi_1^{''}$ is a smooth function of the dc field for sample $A$, but for sample $B$ it shows step-like features in fields near 7 and 10 kOe. ![(Color online). Field dependences of $\chi_1$ for samples $A$ and $B$ (upper and lower panels, respectively) in the PBP and SF modes at 293 Hz and an excitation amplitude of 0.04 Oe. Measurements were done at 4.5 K.[]{data-label="f7c"}](Fig7a.eps "fig:"){width="0.98\linewidth"} ![(Color online). Field dependences of $\chi_1$ for samples $A$ and $B$ (upper and lower panels, respectively) in the PBP and SF modes at 293 Hz and an excitation amplitude of 0.04 Oe. Measurements were done at 4.5 K.[]{data-label="f7c"}](Fig7b.eps "fig:"){width="0.98\linewidth"} A nonlinearity can clearly be seen not only in the second and third harmonics, but in the first harmonic too. Fig. \[f11a\] shows the field dependences of $\chi_1$ of samples $A$ and $B$ at $h_{ac}=0.04$ and 0.2 Oe and T = 4.5 and 5.5 K in the SF mode. Panels $a$ and $b$ demonstrate: (i) that at low magnetic fields, losses in sample $A$ are significantly larger than losses in sample $B$ and: (ii) an increase of the excitation amplitude leads to a decrease of $\chi_1{''}$ for both samples. At $h_{ac}= 0.2$ Oe for $H_0> 5$ kOe there is a plateau and $\chi_1^{''}$ for both samples coincides with high precision. The plateau in the SF mode at high excitation amplitudes was observed at T = 4.5 K, Fig. \[f11a\]$c$ and also at 5.5 K for, Fig. \[f11a\]$b$. It appears that in this range of magnetic fields and at high enough amplitude, the first harmonic signal of the two samples is almost identical. However, a qualitative difference remains for the signals of the second and third harmonics, see Figs. \[f9a\] and \[f10a\]. ![(Color online). Field dependences of $\chi_1$ for samples $A$ and $B$ in the SF mode at 5.5 K (panels $a$ and $b$, respectively) and 4.5 K (panel $c$).[]{data-label="f11a"}](Fig8a.eps "fig:"){width="0.9\linewidth"} ![(Color online). Field dependences of $\chi_1$ for samples $A$ and $B$ in the SF mode at 5.5 K (panels $a$ and $b$, respectively) and 4.5 K (panel $c$).[]{data-label="f11a"}](Fig8b.eps "fig:"){width="0.9\linewidth"} ![(Color online). Field dependences of $\chi_1$ for samples $A$ and $B$ in the SF mode at 5.5 K (panels $a$ and $b$, respectively) and 4.5 K (panel $c$).[]{data-label="f11a"}](Fig8c.eps "fig:"){width="0.9\linewidth"} As for the second harmonic signal it is absent for both samples in the whole range of magnetic fields in the PBP mode, but becomes visible in the SF mode. Fig. \[f10a\] shows the field dependences of $\chi_2$ in the SF mode. Perturbation theory cannot explain the data for $\chi_2$ in the SF mode and $\chi_3$ in both modes. In accordance to this theory one could expect that $\chi_3\propto h_{ac}^2$ and $\chi_2\propto h_{ac}$. However, this is not the case in our experiment at any magnetic field. In our experiment, an increase of the excitation amplitude leads to a suppression of $\chi_2$. In the SF mode $\chi_2$ is larger than $\chi_3$ under the conditions of the experiment, see Figs. \[f9a\] and \[f10a\]. We note that the data for $\chi_1$, $\chi_2$ and $\chi_3$ fluctuate strongly at fields lower than 4 kOe at 4.5 K for sample $A$ due to magnetic flux jumps. ![(Color online). Field dependences of $\chi_3$ of $A$ and $B$ samples (upper and lower panels, respectively) in the SF mode at 4.5 K.[]{data-label="f9a"}](Fig9a.eps "fig:"){width="0.9\linewidth"} ![(Color online). Field dependences of $\chi_3$ of $A$ and $B$ samples (upper and lower panels, respectively) in the SF mode at 4.5 K.[]{data-label="f9a"}](Fig9b.eps "fig:"){width="0.9\linewidth"} ![(Color online). Field dependences of $\chi_2$ of $A$ and $B$ samples, upper and lower panels, respectively, in the SF mode at 4.5 K.[]{data-label="f10a"}](Fig10a.eps "fig:"){width="0.9\linewidth"} ![(Color online). Field dependences of $\chi_2$ of $A$ and $B$ samples, upper and lower panels, respectively, in the SF mode at 4.5 K.[]{data-label="f10a"}](Fig10b.eps "fig:"){width="0.9\linewidth"} It is interesting to note the following concerning the relation between field dependences of $\chi_1^{''}$ and $\chi_3$. Figs. \[f8c\] and \[f8b\] show field dependences of normalized $\chi_1^{''}$ and $\chi_3$ for samples $A$ and $B$. Upper panels in both figures correspond to the PBP mode and lower panels to SF mode. At low magnetic fields $\chi_1^{''}$ and $\chi_3$ are very small in the PBP mode for both samples. Both signals become measurable near $H_{c3}$ and the shape of these signals is identical with high precision. In the SF mode the shapes of $\chi_1^{''}$ and $\chi_3$ are again the same in the vicinity of $H_{c3}$. However, at low magnetic fields this similarity vanishes in the SF mode. Such similarity in the PBP mode can be proved in the frame of perturbation theory [@PAV], but it has not yet proven in the general case which we face in our experiment. ![(Color online). Field dependences of normalized $\chi_1^{''}$ and $\chi_3$ of sample $A$ in point-by-point and swept field modes (upper and lower panels, respectively). The shapes of $\chi_1^{"}$ and $\chi_3$ are with high accuracy identical in PBP and SF modes near $H_{c3}$. This similarity breaks in a SF mode at low magnetic fields.[]{data-label="f8c"}](Fig11a.eps "fig:"){width="0.9\linewidth"} ![(Color online). Field dependences of normalized $\chi_1^{''}$ and $\chi_3$ of sample $A$ in point-by-point and swept field modes (upper and lower panels, respectively). The shapes of $\chi_1^{"}$ and $\chi_3$ are with high accuracy identical in PBP and SF modes near $H_{c3}$. This similarity breaks in a SF mode at low magnetic fields.[]{data-label="f8c"}](Fig11b.eps "fig:"){width="0.9\linewidth"} ![(Color online). Field dependences of normalized of $\chi_1^{"}$ and $\chi_3$ of sample $B$ in point-by-point and swept field modes (upper and lower panels, respectively). The shapes of $\chi_1^{"}$ and $\chi_3$ are with high accuracy identical in PBP mode and in large fields in SF mode.[]{data-label="f8b"}](Fig12a.eps "fig:"){width="0.9\linewidth"} ![(Color online). Field dependences of normalized of $\chi_1^{"}$ and $\chi_3$ of sample $B$ in point-by-point and swept field modes (upper and lower panels, respectively). The shapes of $\chi_1^{"}$ and $\chi_3$ are with high accuracy identical in PBP mode and in large fields in SF mode.[]{data-label="f8b"}](Fig12b.eps "fig:"){width="0.9\linewidth"} Discussion ========== dc magnetization curves ----------------------- The physical reasons for the observed flux jumps at small magnetic fields are not clear. One can suggest that the alignment of the magnetic field with respect to the sample surface is not perfect. Indeed, the latter cannot be ruled out completely, and a small field component perpendicular to the surface, $H_{\bot}$, should create vortices which might be responsible for the flux jumps at small magnetic fields. Hence, one may expect that flux jumps could be present at small magnetic fields in a reference planar film as well. This assumption has been examined in an additional control experiment with a reference planar film. Figure \[f13\] displays ascending branches of the magnetization curves of the planar Nb film of 240nm thickness sputtered onto a silicon substrate, for the magnetic field inclination angles $\varphi =0^{\circ}$, $10^{\circ}$, and $45^{\circ}$. For $\varphi = 10^{\circ}$ and $45^{\circ}$ the component $H_{\bot}\approx0.17H_0$ and $H_{\bot}\approx0.71H_0$, respectively. Vortices created by this field component exist at small magnetic fields. This experiment demonstrates that in small fields the magnetic moment is a linear function of the magnetic field value and vortices created by $H_{\bot}$ do not induce any flux jumps* at small fields. The magnetic moment at small fields remains a linear function of the magnetic field for planar films of different thicknesses. Magnetic moment jumps first appear in the magnetization curve at inclination angles larger than $10^\circ$. Such a field inclination angle is at least a factor of 3 larger than the orientational misalignment of the sample orientation with respect to the field direction in our experiment. Therefore, the results obtained for planar films suggest that the vortices created by the small field component perpendicular to the surface are not the cause for magnetic moment jumps at small magnetic fields in the cylindrical samples. ![Ascending branches of magnetization curves of planar film in parallel and tilted magnetic fields. Inset shows determination of $H_{c1}$ of the planar film.[]{data-label="f13"}](Fig13.eps){width="0.9\linewidth"} The experimental data demonstrate the existence of magnetic instabilities in fields lower than $H_{c1}$. At 4.5 K, the flux starts to penetrate into the cylinders $A$ and $B$ at $H_0 = 20$ and 10 Oe, respectively, Fig. \[f5\] (lower panel). The field of the first jump, $H^$, is defined by the some critical current (not to be confused with a depairing current). If we assume that the critical current density in the isthmus between two antidots is the same as in the film, then the ratio $H_B^/ H_A^$ should be $\approx 0.16$. However, the experiment shows that this ratio is about 0.5, see Fig. \[f5\]. This means that the critical current density in the isthmuses is higher than in the as-prepared film. We note that the ratio of the magnetic moments in ZFC in field 20 Oe for samples $B$ and $A$ is 0.5, see Fig. \[f3\]. In accordance to the thermodynamic criterion [@Katz1] $H^ \propto\sqrt{d}$. Comparison $H^$ for $A$ sample and samples from [@Katz1] shows that the thermodynamics cannot describe these magnetization jumps in samples without antidots. It was demonstrated that at low temperature and at magnetic fields higher than some critical value, $H_{th}$, the magnetization curve becomes smooth and $H_{th}$ is sufficiently larger in the sample with an array of antidots [@Motta1]. The latter experiments were carried out with the field perpendicular to the film surface. In our case we deal with the row of antidots and the magnetic field parallel to the surface. We believe that this is the main reason why $H_{th}$ is lower for the sample with antidots, see upper panel of Fig. \[f5\]. We have to mention that the difference between perpendicular and parallel geometries is crucial. For example vortex velocity in the perpendicular geometry is a few orders magnitude larger than for the parallel one, see Ref. [@Genkin1]. ac response ----------- The field dependences of $\chi_1(H_0)$ in the PBP mode are different for $A$ and $B$ samples, Fig. \[f5aa\]. Losses appear and screening decreases in magnetic fields above $H_{c2}$. Near $H_{c3}$ there is a loss peak and the shape of this peak is different for samples $A$ and $B$. The shape of the loss peak for sample $A$ is nonuniform and for sample $B$ it is broadened. The third critical field of sample $A$ is larger than for sample $B$, Fig. \[f5aa\] lower panel. However, the determination of $H_{c3}$ for sample $B$ is questionable. The difference in the ac response of samples $A$ and $B$ becomes qualitative in the SF mode, Figs. \[f7c\], \[f9a\] and \[f10a\]. Whereas the field dependences of $\chi_1$, $\chi_2$ and $\chi_3$ are smooth for sample $A$, they have peculiarities in 7 and 10 kOe for sample $B$. As we have mentioned above, the data for $\chi_1$, $\chi_2$ and $\chi_3$ are noisy and fluctuating at fields lower than 4 kOe at 4.5 K and 2 kOe at 5.5 K due to magnetic flux jumps. The behavior of the ac response in the SF mode has some similar features for both samples. Thus, an increase of the excitation amplitude and frequency leads to a decrease of $\chi_1^{''}$ in fields down to $H_{c2}$ and $\chi_2$ in the whole field range. The reason for this behavior is the following. The main physical parameter defining the difference between the PBP and SF modes is $Q=\frac{\dot{H}_0}{\omega h_{ac}}$ [@MAX; @FINK2]. The PBP mode corresponds to $Q=0$. Parameter $Q$ decreases with the excitation amplitude and/or frequency tending to zero. This is why $\chi_1^{''}$ and $\chi_2$ decrease with $h_0$ and $\omega$ and in consequence of this in the SF mode perturbation theory is not applicable. In the limiting case of high frequencies, for example, in the GHz range, a swept field with sweep rate of few tens or hundred oersted per second does not affect the ac response [@VAT]. The ac response of sample $A$ in the SF mode is similar to that reported in our previous papers [@Genkin1; @Katz1]. In this sample we observe a smooth field dependence of $\chi_1^{''}$, $\chi_2$ and $\chi_3$. The models proposed in [@Genkin1; @Katz1] can explain the experimental data for sample $A$ in magnetic fields lower and higher than $H_{c2}$. The case with the sample $B$ is more complicated. It turned out that $\chi_1^{''}$ at magnetic fields of 4 kOe ($H_0< H_{c2}$) is lower for sample $B$ than for sample $A$, see Fig. \[f11a\]$c$. The following may be the reason for this. Vortex pinning and the current induced by ac and swept fields play an important role in ac response in a swept magnetic field [@Genkin1]. The area under the row of antidots is much smaller than the total film area. This is why vortex pinning by this row antidots cannot explain loss reduction. At the same time the total induced current is lower in sample $B$ than in sample $A$, Fig. \[f3\]. This reduces the forces dragging vortices into the substrate and leads to loss reduction [@Genkin1]. The jump at $H_0\approx 5$ kOe takes place only for sample $B$, see Figs. \[f9a\] and \[f10a\]. At fields higher than the jump field the losses for both samples at $h_{ac} =0.2$ Oe are equal, panels $b$ and $c$ of Fig. \[f11a\]. The weakening of pinning in high magnetic fields could be a cause for such behavior. The nature of the jump of $\chi_{3,2}$ in magnetic fields of 10 kOe (see panels b* of Figs.\[f9a\] and \[f10a\]) for sample $B$ in SF mode is not clear. ac amplitude is not smeared this jump completely in contrast with $\chi_1^{''}$, see panels a and c of Fig.\[f11a\]. This jump takes place in magnetic fields near $H_{c2}$ of sample $B$. Decreasing of ac losses and harmonics jump near $H_{c2}$ in a swept field was observed in single crystal Nb [@GENKIN22; @MT]. However, single crystal Nb has a well defined vortex structure and $H_{c2}$ but it is not the case with our sample. conclusion ========== We have studied the dc and ac magnetic properties of thin-walled cylinders of superconducting Nb with and without a row of antidots. Experiment showed that the critical current density is higher in the isthmus between antidots than in the film itself. The dc magnetization curves demonstrate an "avalanche”-like penetration of the magnetic flux into the cylinder for both samples. The effect was observed at a temperature of 4.5 K and completely disappeared at 7 and 5.5 K for samples $A$ and $B$, respectively. Such a behavior resembles a thermomagnetic instability of vortices but it was observed in fields below $H_{c1}$ of the films, i.e. in a vortex-free state. The effect of end faces, consisting in that the magnetic force lines is bending near the sample ends, could be another reason for flux jumps. The influence of the sample end faces on the flux jumps in such samples has to be studied using a local probe technique. The ac response of thin-walled cylinders with and without antidots is strongly nonlinear and perturbation theory cannot explain the experimental data. The ac response of $A$ and $B$ samples is similar in the point-by-point mode. However, in the swept field mode there is a qualitative difference between losses for samples $A$ and $B$. Thus, at low magnetic fields, losses in sample $B$ are lower than in sample $A$. There are jumps in $\chi_1$, $\chi_2$ and $\chi_3$ in high magnetic fields for sample $B$, but these quantities are smooth functions of the magnetic field in sample $A$. We demonstrate that field dependences of $\chi_1^{''}$ and $\chi_3$ have the same shapes in the point-by-point mode with high accuracy. In the swept field mode the shapes of $\chi_1^{''}$ and $\chi_3$ are the same in the vicinity of $H_{c3}$. This similarity has yet not been proved in the case of strong nonlinear response that we encounter in our experiment. The models developed in [@Genkin1; @Katz1] could describe the ac response of the as-prepared sample. However, these models are not applicable to the sample with a row of antidots. New models for samples with antidots have to be elaborated. As well as further experimental studies of samples with different lengths, wall thicknesses, sizes and geometry of antidots row or array have to be carried out. acknowledgments =============== We thank J. Kolacek, P. Lipavsky and V.A. Tulin for fruitful discussions. This work was done within the framework of the NanoSC-COST Action MP1201. Financial support of the grant agency VEGA in projects nos. 2/0173/13 and 2/0120/14 are kindly appreciated. Little W A and Parks R D 1962 Phys. Rev. Lett. [9]{} 9 Douglass D H, Jr.1963 Phys. Rev. [132]{}, 513 Aoyama K, Beaird R, Sheehy D E and Vekhter I 2013 Phys. Rev. Lett. [110]{}, 177004 Motta M, Colauto F, Otiz W A, Fritzsche J, Cuppens J, Gillijns J, Moshchalkov V V, Johansen T H, Sanchez A and Silhanek A V 2013 Appl. Phys. Lett., [102]{}, 212601 Tsindlekht M I, Genkin V M, Felner I, Zeides F, Katz N, Gazi S and Chromik S 2014 Phys. Rev. B [90]{} 014514 de Gennes P G 1966 [Superconductivity of metals and alloys]{} (W A Benjamin: INC New York) p 197 Kittel C, Fahy S, and Louie S G 1988 Phys. Rev. B [37]{} 642 Nowak E R, Taylor O W, Liu Li, Jager H M, and Selinder T I 1997 Phys. Rev. B [55]{} 11702 Esquinazi P, Setzer A, Fuchs D, Kopelevich Y, Zeldov E and Assmann C 1999 Phys. Rev. B [60]{} 12454; Stamopoulos D, Speliotis A and Niarchos D 2004 Supercond. Sci. Technol., [17]{}, 1261 Motta M, Colauto F, Zadorosny R, Johansen T H, Dinner R B, Blamire M G, Ataklti G W, Moshchalkov V V, Silhanek A V and Ortiz W A Phys. Rev. B 2011 [84]{} 214529 Saint-James D and Gennes P G 1963 Phys. Lett. [7]{} 306 Rollins R W and Silcox J 1967 Phys. Rev. [155]{} 404 Burger P, Deutscher G, Gueon E and Martinet A 1965 Phys. Rev., [137A]{} 853 Strongin M, Schweitzer D G, Paskin A, and Craig P P 1964 Phys. Rev. [136]{} A926 Maxwell E, Robbins W P 1966 Phys. Lett. [19]{} 629 Tsindlekht M I, Genkin V M, Leviev G I, Schlussel Y, Tulin V A, Berezin V A 2012 Physica C [473]{} 6 Tsindlekht M I, Genkin V M, Gazi S and Chromik S 2013 J. Phys.: Condens. Matter [25]{} 085701 Leviev G I, Genkin V M, Tsindlekht M I, Felner I, Paderno Yu B, Filippov V B 2005 Phys. Rev. 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--- abstract: 'The paper considers a coupled system of linear Boltzmann transport equations (BTE), and its Continuous Slowing Down Approximation (CSDA). This system can be used to model the relevant transport of particles used e.g. in dose calculation in radiation therapy. The evolution of charged particles (e.g. electrons and positrons) are in practice often modelled using the CSDA version of BTE because of the so-called forward peakedness of scattering events contributing to the particle fluencies (or particle densities), which causes severe problems in numerical methods. We shall verify, after the preliminary discussion, that CSDA-type modelling is actually necessary due to hyper-singularities in the differential cross-sections of certain interactions, that is, first-order partial derivatives with respect to energy must be included into the transport part of charged particles. The existence and uniqueness of (weak) solutions is shown, under sufficient criteria and in appropriate $L^2$-based spaces, for a single (particle) CSDA-equation by using three techniques, the Lions-Lax-Milgram Theorem (variational approach), the theory of $m$-dissipative operators and the theory evolution operators (semigroup approach). The necessary a priori estimates are derived. In addition, we prove the corresponding results and estimates for the system of coupled transport equations. The related results are given for the adjoint problem as well. We also give some computational points (e.g. certain explicit formulas), and we outline a related inverse problem at the end of the paper.' address: - '$^1$University of Eastern Finland, Department of Applied Physics, Kuopio, Finland' - '$^2$Varian Medical Systems Finland Oy, Helsinki, Finland' - '$^3$RWTH Aachen University, MATHCCES, Schinkelstrasse 2, 52062, Germany' - '$^4$RWTH Aachen University, IGPM, Templergraben 5, 52062, Germany' author: - 'J. Tervo$^1$' - 'P. Kokkonen$^2$' - 'M. Frank$^3$' - 'M. Herty$^4$' title: 'On existence of $L^2$-solutions of Coupled Boltzmann Continuous Slowing Down transport equation system' --- Introduction {#intro} ============ The Boltzmann transport equation (BTE) models changes of the number density of particles in phase space (position, velocity direction, energy). In this paper the species of particles include photons, electrons and positrons and the explored analysis of transport equations is mainly intended for dose calculation in radiation treatment planning. However, various other kinds of transport phenomena can be modelled by equations of similar type including in e.g. transport of particles in optical tomography ([@anikonov02], [@arridge]), in cosmic radiation ([@wilson]) and in solid state physics ([@madelung78]). For general theory of linear BTE with relevant boundary conditions we refer to [@dautraylionsv6] and [@agoshkov]. See also [@case], [@cercignani], [@duderstadt], [@pomraning] where the subject is considered from a physical point of view. For some recent issues (including certain inverse problems) related to linear BTE can be found in [@mokhtarkharroubi], and general non-linear aspects e.g. in [@ukai], [@bellamo]. A thorough mathematical survey (mathematical and physical foundations, results, problems) of non-linear collision theory of particle transport is given in [@villani]. This survey is mainly intended to collision processes in dilute gases and plasmas but analogous results and problems arise in other fields of particle physics. Finally, for topics related to Monte–Carlo methods in the context of BTE, both from theoretical and practical point of view, we refer to [@lapeyre], [@seco] and [@spanier08]. Dose calculation is of crucial importance in radiation therapy. Relevant dose calculation models require (approximate) solution of a coupled system of (linear) transport equations for fluencies (number densities in the phase space) for all considered particles. This is a difficult problem, at least from computational point of view, due to the different particle species and their dependence on a high–dimensional phase space. For that reason traditional dose calculation algorithms have applied some closed-form formulas which have their origins in analytical solutions, or Monte–Carlo derived solutions, of simplified problems. The latter however contain often empirically derived corrections to take more accurately into account the underlying particle physics ([@mayles07], [@seco]). Certain “factors” which account for e.g. the spatial inhomogeneities must be included to improve the accuracy of the final solution. These approaches lead to methods that are fast enough but have typically a limited accuracy. Commonly used models are based on the so-called pencil beams, or point kernels, see [@asadzadeh], [@borgers], [@larsen], [@mayles07], [@tillikainen08], [@ulmer] for more details. A notable exception to these approximate (deterministic) methods is the Acuros code [@vassiliev], which is based on a discretization of the BTE. In radiation therapy BTE describes the evolution of radiative particles due to scattering and absorption in tissue. The dose delivery methods can be roughly divided into two categories. In external therapy the sources (below denoted by $g$) of high energy particles (usually photons, electrons or protons) are on the patches of patient’s surface. In internal therapy, on the other hand, the sources (below denoted by $f$) are inside the patient close to the cancerous tissue. In the energy range, say up to 25 MeV, relevant for photon and electron therapy, the three species of particles whose simultaneous evolution should be taken into account in a realistic transport model, are photons, electrons and positrons. In this setting, the potential creation of (or contamination by) other heavy particles will not be taken into account since their contribution to the dose is negligible (see [@seco]). The transport of relevant particles in tissue (in an appropriate energy range) can be modelled by the following linear coupled system of three BTEs \[intro1\] \_x\_j(x,,E)+\_j(x,,E)\_j(x,,E)-(K\_j)(x,,E)=f\_j(x,,E), for $j=1,2,3$, combined with an inflow boundary condition (for the definition of $\Gamma_-$, see section \[fs\]) \[intro2\] [\_j]{}\_[\|\_-]{}=g\_j,j=1,2,3, where for $j=1,2,3$, \[intro3\] (K\_j)(x,,E)=\_[k=1]{}\^3\_[SI]{}\_[kj]{}(x,’,,E’,E)\_k(x,’,E’) [[d]{}]{}’ [[d]{}E]{}’. For a derivation of linear BTE, see e.g. [@agoshkov], [@allaire12], [@duderstadt], [@stacey01]. The first term on the left in (\[intro1\]) is called a convection (or advection) operator, the second term is a (total) scattering operator and the third one is a collision operator. Notice that the (total) scattering operator $$\Sigma=\Sigma_{\rm t}=\Sigma_{\rm a}+\Sigma_{\rm s}$$ (we drop the index $j$ here to simplify the notation) contains contribution from both the absorption (term $\Sigma_{\rm a}$) and the scattering (term $\Sigma_{\rm s}$), see [@stacey01 Sec. 9.1]. On the right in (\[intro1\]), the functions $f_j$ represent (internal) sources and $g_j$ in (\[intro2\]) are (inflow) boundary sources. The system is coupled through the integral operators $K_j$ (unless, of course, $\sigma_{kj}=0$ for $j\not= k$). The solution $\psi=(\psi_1,\psi_2,\psi_3)$ of the problem (\[intro1\])-(\[intro2\]) is a vector-valued function whose components describe the radiation fluxes of photons, electrons and positrons, respectively. Roughly speaking, the flux $\psi(x,\omega,E)$ is the flux of energy through a surface located at $x$ and normal to the direction $\omega$. The particle number density $N$, which is another usual unknown in kinetic theory, is related to $\psi$ by $\psi = {\left\Vert v\right\Vert} N$, where ${\left\Vert v\right\Vert}$ is the particle speed ([@stacey01]), which is often relativistic. The equation (\[intro1\]) is a steady state counterpart of the dynamical equation \[intro4\] [1]{}[t]{}+\_x\_j+\_j\_j-K\_j=f\_j,j=1,2,3, where $v_j$ is the velocity of the $j$-th particle type. In radiation therapy related applications, it is sufficient to consider the steady state equations because the flux $\psi$ reaches the steady state nearly instantly ([@borgers99]). The existence of solutions for the problem (\[intro1\]), (\[intro2\]), as well as for the time-dependent problem (\[intro4\]), (\[intro2\]) (with an appropriate initial condition) in $L^1$-based spaces has been studied in [@tervo14] (the results of which remaining valid, after slight modifications, for any $1\leq p<\infty$). In [@tervo14] it is assumed that the collision operator $K$ satisfies a so-called Schur criterion (for boundedness) which is not valid for all species of particle interactions. The differential cross-sections may have singularities, or even hyper-singularities, and in these cases the integral $\int_{S\times I}$ appearing in the collision term must be interpreted as the Hadamard finite part integral. In section \[coll\] we shall present some details of real, physical collision operators. In particular, we find that certain differential cross sections may contain hyper-singularities like ${1\over{(E'-E)^2}}dE'$. These kind of singularities lead to extra partial differential (or pseudo-differential) terms in the transport equation ([@hsiao Sec. 7.1, pp. 353–394]). The analysis reveals the exact form of transport operators, and serves as a basis for better approximations and error analysis. These considerations, however, would require further knowledge about the regularity properties of solutions, which, to our understanding, remains an open question. Likewise the existence analysis of (non-CSDA) exact (coupled) system of transport equations is open. In this work, we demonstrate how the, here considered, CSDA-approximation follows from “hyper-singular analysis” and how the CSDA equations can be derived. When the collision terms containing hyper-singularities are removed, the remaining operators are, however, in Schur form (at least approximately). Furthermore, we notice that in the expression of $K_j$ the integration is performed only with respect to $\omega'$ and $E'$, while $x$-variable is kept fixed. Therefore, $K_j$ is only a so-called partial (hyper-singular) integral operator (cf. [@appell]). These facts imply that the familiar properties of (singular) integral operators (compactness, for example) are not valid even for restricted collision operators. Due to the above mentioned hyper-singularities, the transport of new primary electrons (and positrons) is forward-peaked. This implies that the grid in numerical computations needs to be very tight in order to achieve reliable results. Traditionally, to overcome the computational complexity, one has applied to the evolution of electrons and positrons the so-called continuous slowing down approximation (CSDA). The CSDA equations are vastly applied e.g. in various cosmic radiation problems ([@rockell], [@wilson]). The analysis given in section \[coll\] show that the exact transport model inherently contains first order partial differential (pseudo-differential) terms, and so the traditional use of CSDA-approximation can be justified. The CSDA-method replaces Eqs. (\[intro1\]) for $\psi_j$, $j=2,3$ (electrons and positrons) by the following equations ([@wilson], [@larsen], [@frank10]) \[intro5\] -[E]{}+\_x\_j+\_[j,r]{}\_j-K\_[j,r]{}=f\_j,j=2,3, where (K\_[2,r]{})(x,,E)=&\_[SI]{}\_[12]{}(x,’,,E’,E)\_1(x,’,E’) [[d]{}]{}’ [[d]{}E]{}’\ & +\_[SI]{}\_[22,r]{}(x,’,,E’,E)\_2(x,’,E’) [[d]{}]{}’ [[d]{}E]{}’\ & +\_[SI]{}\_[32]{}(x,’,,E’,E)\_3(x,’,E’) [[d]{}]{}’ [[d]{}E]{}’and (K\_[3,r]{})(x,,E)=&\_[SI]{}\_[13]{}(x,’,,E’,E)\_1(x,’,E’) [[d]{}]{}’ [[d]{}E]{}’\ & +\_[SI]{}\_[23]{}(x,’,,E’,E)\_2(x,’,E’) [[d]{}]{}’ [[d]{}E]{}’\ & +\_[SI]{}\_[33,r]{}(x,’,,E’,E)\_3(x,’,E’) [[d]{}]{}’ [[d]{}E]{}’.Here, for $j=2,3$, the functions $\Sigma_{j,r}=\Sigma_{j,r}(x,\omega,E)$ are the restricted total cross-sections, $S_{j,r}=S_{j,r}(x,E)$ are the restricted stopping powers, and $\sigma_{jj,r}(x,\omega',\omega,E',E)$ are the restricted differential cross-sections, which do not include soft inelastic interactions ([@boman]). Besides the inflow boundary condition (\[intro2\]), one needs to impose on the solutions $\psi_j$, $j=2,3$, of an energy-boundary condition. We make the reasonable assumption that at some (high enough, but finite) cut-off energy $E_{\rm m}>0$ the fluxes $\psi_2$, $\psi_3$ vanish, i.e. we take the energy-boundary condition to be $$\psi_2(x,\omega, E_{\rm m})=\psi_3(x,\omega, E_{\rm m})=0,$$ (below, we often call this an initial condition as well). One could also demand the cut-off energy $E_{\rm m}$ to be infinite, in which case the corresponding energy boundary condition would become, $$\lim_{E\to\infty}\psi_2(x,\omega, E)=\lim_{E\to\infty}\psi_3(x,\omega, E)=0.$$ However, we shall restrict our discussion to the case where $E_{\rm m}$ is finite. The requirement that such energy initial condition be satisfied by $\psi_2,\psi_3$, makes the overall problem mathematically well-posed, that is, under relevant (physical) assumptions the problem has a unique (weak) solution. We give here a short heuristic derivation of CSDA based on the presentation given in [@wilson pp. 14-17], while a more transparent justification of it will be given in section \[coll\]. Firstly, we decompose for $k=2,3$ the differential cross-section into two parts $$\sigma_{kk}(x,\omega',\omega,E',E)= \sigma_{kk}^{\rm at}(x,\omega',\omega,E',E)+\sigma_{kk}^{\rm nu}(x,\omega',\omega,E',E), \ k=2,3$$ where “${\rm at}$” refers to interactions with atomic electrons, and “${\rm nu}$” refers to nuclear interactions. The term $\sigma_{kk}^{\rm at}(x,\omega',\omega,E',E)$ contains the “problematic” features that are to be approximated by the continuous slowing down model. Secondly, we write formally $$\sigma_{kk}^{\rm at}(x,\omega',\omega,E',E) = \sum_{n=0}^N\sigma^{\rm at}_{k,n}(x,E+\epsilon_n)\tilde\delta_{\omega}(\omega')\delta_{E+\epsilon_n}(E'),$$ where $\tilde{\delta}_{\omega}$ is the Dirac measure on the 2-dimensional unit sphere $S=S_2$ concentrated at $\omega$, and $\delta_{E+\epsilon_n}$ is the Dirac measure on ${\mathbb{R}}$ concentrated at $E+\epsilon_n$. This approximation assumes small changes in energy but not changes in angular direction. Applying these concepts and Taylor’s formula up to first–order, we find that &\_[SI]{}\_[kk]{}\^[at]{}(x,’,,E’,E)\_k(x,’,E’) [[d]{}]{}’ [[d]{}E]{}’ = \_[n=0]{}\^N\^[at]{}\_[k,n]{}(x,E+\_n)\_k(x,,E+\_n)\ & \_[n=0]{}\^N\^[at]{}\_[k,n]{}(x,E)\_k(x,,E) +\_[n=0]{}\^N[E]{}(x,,E)\_n. \[f6\] The restricted total atomic cross-sections and the restricted stopping powers are $$\Sigma^{\rm at}_{k,r}(x,E):=\sum_{n=0}^N\sigma^{\rm at}_{k,n}(x,E), \quad S_{k,r}(x,E):=\sum_{n=0}^N\sigma^{\rm at}_{k,n}(x,E)\epsilon_n.$$ Let $\sigma_{kk,r}:=\sigma_{kk}^{\rm nu}$. Then by (\[f6\]) we obtain for $k=2,3$, \[intro9\] & \_[SI]{}\_[kk]{}(x,’,,E’,E)\_k(x,’,E’) [[d]{}]{}’ [[d]{}E]{}’\ =& \_[SI]{}\_[kk]{}\^[at]{}(x,’,,E’,E)\_k(x,’,E’) [[d]{}]{}’ [[d]{}E]{}’ + \_[SI]{}\_[kk,r]{}(x,’,,E’,E)\_k(x,’,E’) [[d]{}]{}’ [[d]{}E]{}’\ & \_[k,r]{}\^[at]{}(x,E)\_k(x,,E) +[E]{}(x,,E) + \_[SI]{}\_[kk,r]{}(x,’,,E’,E)\_k(x,’,E’) [[d]{}]{}’ [[d]{}E]{}’ . Finally, writing $\Sigma_{j,r}:=\Sigma_{j}-\Sigma_{j,r}^{\rm at}$ and substituting (\[intro9\]) into (\[intro1\]), we obtain approximately the equations (\[intro5\]) for $j=2,3$. The so-called CSDA-Fokker-Plank model includes (cf. [@adams97]) additionally (roughly speaking) the second order partial derivatives with respect angle in the above Taylor expansion (for $\psi_2$, $\psi_3$). It is a further approximation that in addition to small energy changes assumes small angle changes for the scattering. The hierarchy of the Boltzmann equation, its CSDA approximation and the Fokker-Planck approximation is detailed in Appendix A of [@Larsen97]. Typically, the Fokker-Planck approximation is considered not valid for electrons in tissues [@Larsen97]. In addition, in [@pom92] a formal asymptotic analysis is performed that sheds light onto the validity of these approximations. For example, it was shown that the Fokker-Planck approximation is not valid for the often used Henyey-Greenstein kernel. The dose $D(x)=(D\psi)(x)$ is calculated from the solution of the problem $$\begin{gathered} \omega\cdot\nabla_x\psi_1+\Sigma_1\psi_1-K_1\psi=f_1, \label{intro10a}\\ -{{\frac{\partial (S_{j,r}\psi_j)}{\partial E}}}+\omega\cdot\nabla_x\psi_j+\Sigma_{j,r}\psi_j-K_{j,r}\psi=f_j,\quad j=2,3, \label{intro10} \\ {\psi_j}_{\|\Gamma_-}=g_j,\quad j=1,2,3, \label{intro11} \\[2mm] \psi_j(\cdot,\cdot,E_{\rm m})=0,\quad j=2,3, \label{intro12}\end{gathered}$$ by \[intro13\] D(x)=\_[j=1]{}\^3\_[SI]{}\_j(x,E)\_j(x,,E) [[d]{}]{}[[d]{}E]{}, where $\varsigma_j(x,E)$ are [stopping powers]{}, which in general can be different from the restricted stopping powers $S_{j,r}$. The dose calculation is a [forward problem]{}. The determination of the external particle flux $g=(g_1,g_2,g_3)$ and/or the distribution of internal source $f=(f_1,f_2,f_3)$ is called inverse radiation treatment planning problem (IRTP) which is an inverse problem. It always requires a dose calculation model. We refer to [@schepard], [@tervo14], [@webb] and references therein for some details concerning the IRTP-problem. In [@frank10] the IRTP-problem has been studied in the context of CSDA-equation for a single particle (when the stopping power is independent of $x$). See also [@boman] where related spatially $3$-dimensional numerical simulations (real case simulations applying finite element methods, FEM) have been explored. This paper contains several novel contributions to the study of particle transport in tissues. In the beginning we discuss the preliminaries including details (many of which are reproductions of known results) of the so-called escape-time mapping and inflow trace theory. These tools are essential in the treated analysis. After that we consider the existence and uniqueness of solutions for a single (particle) CSDA equation $$\begin{gathered} -{{\frac{\partial (S_{0}\psi)}{\partial E}}}+\omega\cdot\nabla_x\psi+\Sigma\psi-K\psi=f, \label{intro14} \\ \psi_{\|\Gamma_-}=g, \label{intro15} \\[2mm] \psi(\cdot, \cdot, E_{\rm m})=0, \label{intro16}\end{gathered}$$ in $L^2(G\times S\times I)$-based spaces. Here $G\subset{\mathbb{R}}^3$ is the spatial domain, $S=S_2\subset {\mathbb{R}}^3$ is the unit sphere and $I=[0,E_{\rm m}]$ is the energy interval. The set $\Gamma_-$ is “the inflow boundary part of $\partial G$”. We extend the results of [@frank10], where a spatially homogeneous stopping power was assumed. In addition, we notice that the single CSDA equation is (symmetric) hyperbolic in nature, and so the hyperbolic theory (e.g. [@rauch12]) is applicable at least in the case where $K=0$ and $G={\mathbb{R}}^3$. For $G\not={\mathbb{R}}^3$ the existence and regularity of solutions is more subtle because of the inflow boundary condition. We also remark that the coupled system (\[intro10a\]), (\[intro10\]) is not hyperbolic. In section \[esols\] we give a variational formulation of the problem (\[intro14\]), (\[intro15\]), (\[intro16\]) in appropriate spaces. We give sufficient conditions under which the corresponding bilinear and linear forms obey the assumptions of the so-called [Lions-Lax-Milgram Theorem]{}, which then provides an existence result for solution of the equation (\[intro14\]). Under a certain additional assumption on traces, this solution is unique and, additionally, it satisfies the boundary and initial conditions (\[intro15\]), (\[intro16\]). In Section \[m-d\] one applies the classical theory of (formally) dissipative operators (e.g. by [@friedrich58], [@lax], [@rauch85]), and it is via this approach that we achieve the most satisfactory existence and uniqueness result for the problem considered. In section \[evcsd\] we show the existence and uniqueness results for the solution of the problem (\[intro14\]), (\[intro15\]), (\[intro16\]) by using a yet another technique, based on the theory of evolution equations and ($m$-)dissipative operators, but for a restricted class of collision operators $K$. In section \[cosyst\] we extend the corresponding existence and uniqueness results for the coupled transport system (\[intro10a\])-(\[intro12\]), which has not been studied in the literature, in $L^2(G\times S\times I)^3$-based spaces. We also show certain a priori estimates (needed e.g. in section \[irtpre\]) which in particular show (under specific assumptions) that the solution depends continuously on the data. Analogous results for the [adjoint problem]{} are formulated in section \[adjoint\]. Section \[comp\] considers certain computational aspects. The emphasis is on how to calculate numerical solution (for the forward problem), in principle, without inversion of (huge) matrices. Finally, in the last section \[irtpre\] we outline a related IRTP-problem but its thorough study remains open for future work. Preliminaries {#pre} ============= Notations, Assumptions and Introduction of Relevant Function Spaces {#fs} ------------------------------------------------------------------- We assume that $G$ is an open bounded connected set in ${\mathbb{R}}^3$ such that $\ol{G}$ is a $C^1$-manifold with boundary (as a submanifold of ${\mathbb{R}}^3$; cf. [@lee]). In particular, it follows from this definition that $G$ lies on one side of its boundary. The unit outward (with respect to $G$) pointing normal on $\partial G$ is denoted by $\nu$, and the surface measure (induced by the Lebesgue measure) on $\partial G$ is written as $\sigma$. We let $S=S_2$ be the unit sphere in ${\mathbb{R}}^3$ equipped with the usual rotationally invariant surface measure $\mu_{S}$. Here and in what follows we typically refer to the measures of interest to us, namely Lebesgue measure ${\mathcal{L}}^3$ in ${\mathbb{R}}^3$, and the above (surface) measures $\sigma$ and $\mu_{S}$ simply by $dx$, ${{d}\sigma}$ and ${{d}\omega}$ in the sense that for all sets $A$ measurable with respect to relevant one of them, $${\mathcal{L}}^3(A)=\int_A {d}x, \quad \sigma(A)=\int_A {{d}\sigma}, \quad \mu_{S}(A)=\int_A {{d}\omega}.$$ Furthermore, let $I=[0,E_{\rm m}]$ where $0<E_{\rm m}<\infty$. We could replace $I$ by $I=[E_0,E_{\rm m}]$ or $I=[E_0,\infty[$ where $E_0\geq 0$ but we omit this generalization here. We shall denote by $I^\circ$ the interior $]0,E_m[$ of $I$. The interval $I$ in equipped with the 1-dimensional Lebesgue measure ${\mathcal{L}}^1$, which we typically write as $dE$ in the sense that ${\mathcal{L}}^1(A)=\int_A dE$. For $(x,\omega)\in G\times S$ the escape time (in the direction $\omega$) $t(x,\omega)=t_-(x,\omega)$ is defined by t(x,)&={s>0 \| x-sG}\ &={T>0 \| x-sG [for all]{} 0<s<T}, for $(x,\omega)\in G\times S$. The escape time function $t(\cdot,\cdot)$ is known to be lower semicontinuous in general, and continuous if $G$ is convex, see e.g. [@tervo14]. We define $$\Gamma:={}&(\partial G)\times S\times I,$$ and $$\Gamma_0 :={}&\{(y,\omega,E)\in \Gamma\ \|\ \omega\cdot\nu(y)=0\} \\[2mm] \Gamma_{-}:={}&\{(y,\omega,E)\in \Gamma\ \|\ \omega\cdot\nu(y)<0\} \\[2mm] ={}&\{(y,\omega)\in \partial G\times S\ \|\ \omega\cdot\nu(y)<0\}\times I \\[2mm] \Gamma_{+}:={}&\{(y,\omega,E)\in \Gamma\ \|\ \omega\cdot\nu(y)>0\} \\[2mm] ={}&\{(y,\omega)\in \partial G\times S\ \|\ \omega\cdot\nu(y)>0\}\times I.$$ Let $\mu_\Gamma = \sigma\otimes \mu_S\otimes {\mathcal{L}}^1$, written typically as ${d}\sigma{d}\omega{d}E$ in the same sense as discussed above. It follows that $\mu_{\Gamma}(\Gamma_0)=0$ and $$\Gamma=\Gamma_0\cup \Gamma_-\cup \Gamma_+.$$ Let \[N0\] N\_0:={(x,,E)GSI \| (x-t(x,))=0}, and \[D\] D:=(GSI)N\_0. Recall that $N_{0}$ has a measure zero in $ G\times S\times I$ ([@tervo14 Theorem 3.8]). Define escape-time mappings $\tau_{\pm}(y,\omega)$ from boundary to boundary in the direction $\omega$ as follows $$\begin{aligned} {3} & \tau_-(y,\omega):= \inf\{s>0\ \|\ y+s\omega\not\in G\},\quad && (y,\omega)\in \partial G\times S, \\ & \tau_+(y,\omega):= \inf\{s>0\ \|\ y-s\omega\not\in G\},\quad && (y,\omega)\in \partial G\times S.\end{aligned}$$ Note that for $(y,\omega,E)\in\Gamma_-$ the vector $(y_+,\omega,E)\in\Gamma_+$ where $y_+:=y+ \tau_-(y,\omega)\omega\in\Gamma_+$ and $$\tau_-(y,\omega)=\tau_+(y_+,\omega).$$ \[re:S\_Radon\] For most purposes of this paper, we could have chosen, instead of the particular measure $\mu_S$, any positive Radon measure $\rho$ on (Borel sets of) the unit sphere $S$, as long as relevant additional assumptions are supposed, for example that $\Gamma_0$ has measure zero with respect to $\sigma\otimes \rho\otimes {\mathcal{L}}^1$, i.e. $\int_{\Gamma} \chi_{\Gamma_0}(y,\omega,E)d\sigma(y)d\rho(\omega)dE=0$. Note that it follows from this assumption that $N_0$ has measure zero as well. Especially, the main existence and uniqueness results for the solutions of the CSDA Boltzmann transport problem, as given in Sections \[single-eq\] and \[cosyst\] (Theorems \[csdath3\], \[coth2-d\], \[evoth1\], \[cosystth2\], \[m-d-j-co1\], \[coupthev\], along with their corollaries), as well as the non-negativity result of Subsection \[possol\], remain essentially true even with this more general choice of measure on $S$. Below the maps $t:G\times S\to [0,\infty]$ and $\tau_{\pm}:\partial G\times S\to [0,\infty]$, will often be interpret as maps $t:G\times S\times I\to [0,\infty]$ and $\tau_{\pm}:\Gamma\to [0,\infty]$, by dropping off the energy variable, e.g. $t(x,\omega,E)=t(x,\omega)$. \[le:esccont\] Let $(y_0,\omega_0)\in \partial G\times S$ be such that there exists a bounded open cone $C:=\{x\in{\mathbb{R}}^3\ \|\ \Big\|{x\over{{\left\Vert x\right\Vert}}}-{a\over{{\left\Vert a\right\Vert}}}\Big\|<r, 0<{\left\Vert x\right\Vert}<r\}\subset{\mathbb{R}}^3,\ a\not=0$ containing $-\omega_0$ such that for some $\lambda_0>0$ one has $y_0+\lambda_0 C\subset {\mathbb{R}}^3\backslash G$. Then we have the following: \(i) The map $t(\cdot,\cdot)$ is continuous at any point $(x_0,\omega_0)\in G\times S$ such that $y_0=x_0-t(x_0,\omega_0)\omega_0$. \(ii) Letting $y_+ = y_0 + \tau_-(y_0,\omega_0)\omega_0$, one has the following limits $$& \lim_{(x,\omega)\to (y_0,\omega_0)} t(x,\omega)=0, \\ & \lim_{(x,\omega)\to (y_+,\omega_0)} t(x,\omega)=\tau_-(y_0,\omega_0),$$ where $(x,\omega)\in G\times S$ when taking the limits. By using charts of a $C^1$-manifold with boundary and the assumption, there exists a bounded open cone $C'$ containing $-\omega_0$, a $\lambda_0'>0$ and a neighbourhood $U$ of $y_0$ in ${\mathbb{R}}^3$ such that $$y+\lambda_0' C'\subset {\mathbb{R}}^3\backslash G,\quad \forall y\in U\cap\partial G.$$ \(i) First, let us show that $t(\cdot,\cdot)$ is lower-semicontinous on $G\times S$. Indeed, if $(x,\omega)\in G\times S$ and $0<s<t(x,\omega)$, we have $x-[0,s]\omega:=\{x-t\omega\ \|\ t\in [0,s]\}\subset G$. Thus if $(x_n,\omega_n)$ is a sequence in $G\times S$ converging to $(x,\omega)$, one has $x_n-[0,s]\omega_n\subset G$ for all large enough $n$, which implies $t(x_n,\omega_n)\geq s$, and thus $\liminf_{n\to\infty} t(x_n,\omega_n)\geq s$. Letting $s\to t(x,\omega)$ from the left, we conclude that $\liminf_{n\to\infty} t(x_n,\omega_n)\geq t(x,\omega)$, which gives the lower semi-continuity we were to show. It remains to show that $t(\cdot,\cdot)$ is upper semi-continuous at $(x_0,\omega_0)$ which then implies that $t(\cdot,\cdot)$ is continuous at $(x_0,\omega_0)$ since we already known that $t(\cdot,\cdot)$ is lower-semicontinous at $(x_0,\omega_0)$. Indeed, let $(x_n,\omega_n)\to (x_0,\omega_0)$ when $n\to\infty$. Fix $\lambda\in ]0,\lambda_0]$. Then $$y_0-\lambda \omega_0\in y_0+\lambda C'\subset y_0+\lambda_0 C'\subset {\mathbb{R}}^3\backslash G,$$ from which it follows that for all large enough $n$ (which we assume from now on), $$x_n-(t(x_0,\omega_0)+\lambda)\omega_n\in y_0+\lambda_0 C'\subset{\mathbb{R}}^3\backslash G.$$ Therefore, $t(x_n,\omega_n)\leq t(x_0,\omega_0)+\lambda$, and hence $$\limsup_{n\to\infty} t(x_n,\omega_n)\leq t(x_0,\omega_0)+\lambda,$$ for all $\lambda>0$, which gives us the desired upper semi-continuity at $(x_0,\omega_0)$, $$\limsup_{n\to\infty} t(x_n,\omega_n)\leq t(x_0,\omega_0).$$ \(ii) Arguments are analogous to the ones employed in case (i). Consider first the case $(x_n,\omega_n)\to (y_0,\omega_0)$ where $(x_n,\omega_n)\in G\times S$ . Then $y_0-\lambda_0'\omega_0 \in y_0+\lambda_0' C'\subset{\mathbb{R}}^3\backslash G$ implies that for all $n$ large enough, one has $x_n-\lambda \omega_n\in y_0+\lambda_0' C'\subset{\mathbb{R}}^3\backslash G$ for all $0<\lambda\leq \lambda_0'$, hence $t(x_n,\omega_n)\leq \lambda$, from which $\limsup_{n\to\infty} t(x_n,\omega_n)\leq \lambda$, and finally $\limsup_{n\to\infty} t(x_n,\omega_n)=0$. We then suppose that $(x_n,\omega_n)\to (y_+,\omega_0)$. Since $y_0=y_+ - \tau_-(y_0,\omega_0)\omega_0$ and $y_0-\lambda_0'\omega_0\in y_0+\lambda_0'C'\subset{\mathbb{R}}^3\backslash G$, one has for all $n$ large enough and all $0<\lambda\leq \lambda_0'$, that $$x_n- (\tau_-(y_0,\omega_0)+\lambda)\omega_n \in y_0+\lambda_0'C'\subset{\mathbb{R}}^3\backslash G,$$ whence $t(x_n,\omega_n)\leq \tau_-(y_0,\omega_0)+\lambda$, which gives the upper limit $$\limsup_{n\to\infty} t(x_n,\omega_n)\leq \tau_-(y_0,\omega_0).$$ In order to obtain the corresponding lower limit, which then shows the existence and the correct value of the limit we were set out to show, notice that if $0<\sigma<\tau_-(y_0,\omega_0)$, then $y_0+\sigma\omega_0\in G$, hence for all $n$ large enough $x_n-(\tau_-(y_0,\omega_0)-\sigma)\omega_n \in G$ (since $x_n-(\tau_-(y_0,\omega_0)-\sigma)\omega_n\to x_0-(\tau_-(y_0,\omega_0)-\sigma)\omega_0=y_0+\sigma\omega_0$) , which implies that $t(x_n,\omega_n)\geq \tau_+(y_+,\omega_0)-\sigma$, and thus $\liminf_{n\to\infty} t(x_n,\omega_n)\geq \tau_-(y_0,\omega_0)-\sigma$. Letting $\sigma\to 0+$ allows us to conclude that $$\liminf_{n\to\infty} t(x_n,\omega_n)\geq \tau_-(y_0,\omega_0).$$ \[le:esccont:1\] Let $(x_0,\omega_0,E_0)\in (G\times S\times I)\cup \Gamma_+\cup \Gamma_-$. Then $$\lim_{(x,\omega,E)\to (x_0,\omega_0,E_0)} t(x,\omega) =\begin{cases} t(x_0,\omega_0),\ & \textrm{if}\ (x_0,\omega_0,E_0)\in G\times S\times I \\ & \textrm{and}\ \big(x_0-t(x_0,\omega_0)\omega_0,\omega_0,E_0\big)\in \Gamma_-, \\ \tau_+(x_0,\omega_0),\ & \textrm{if}\ (x_0,\omega_0,E_0)\in \Gamma_+ \\ & \textrm{and}\ (x_0-\tau_+(x_0,\omega_0)\omega_0,\omega_0,E_0)\in \Gamma_-,\\ 0,\ & \textrm{if}\ (x_0,\omega_0,E_0)\in \Gamma_-. \end{cases}$$ where $(x,\omega,E)\in G\times S\times I$ when taking the limits. Define $$z_0=\begin{cases} x_0-t(x_0,\omega_0)\omega_0,\ & \textrm{if}\ (x_0,\omega_0,E_0)\in G\times S\times I \\ x_0-\tau_+(x_0,\omega_0)\omega_0,\ & \textrm{if}\ (x_0,\omega_0,E_0)\in \Gamma_+ \\ x_0,\ & \textrm{if}\ (x_0,\omega_0,E_0)\in \Gamma_-. \end{cases}$$ In each of the limits concerned, we have thus assumed that $(z_0,\omega_0,E_0)\in\Gamma_-$, which implies, using the fact that $\ol{G}$ is $C^1$-manifold with boundary and $\omega_0\cdot\nu(z_0)<0$, that there is an open finite cone $C\subset{\mathbb{R}}^3$ and $\lambda_0>0$ such that $z_0+\lambda_0 C\subset{\mathbb{R}}^3\backslash G$ and $-\omega_0\in C$. The claim then follows from Lemma \[le:esccont\]. Note in particular that when considering the limit $\lim t(x,\omega)=\tau_+(x_0,\omega_0)=\tau_-(y_0,\omega_0)$ one takes in the lemma $y_0=x_0-\tau_+(x_0,\omega_0)\omega_0$ and (hence) $y_+=x_0$. \[prop-ex\] Define $$(\Gamma_+)_c:=\{(y,\omega,E)\in \Gamma_+\ \|\ (y-\tau_+(y,\omega)\omega,\omega,E)\in \Gamma_-\}.$$ Then $\Gamma_+\backslash (\Gamma_+)_c$ has zero-measure in $\Gamma$ and there is a continuous extension $\ol{t}$ of $t:D\to {\mathbb{R}}$ onto ${\mathcal{D}}_-:=D\cup \Gamma_-\cup (\Gamma_+)_c$ given by $$\ol{t}(x,\omega)=\begin{cases} t(x,\omega), & \textrm{if}\ (x,\omega,E)\in D, \\ \tau_+(x,\omega), & \textrm{if}\ (x,\omega,E)\in (\Gamma_+)_c, \\ 0, & \textrm{if}\ (x,\omega,E)\in \Gamma_-. \end{cases}$$ The result is a straightforward consequence of the above Lemmas. Note that in the case where $G$ is convex and its boundary is $C^1$-regular the above extension $\ol{t}$ is continuous on $\ol G\times S$. For example, for the ball $G=B(0,r)$ we have $\tau_+(x,\omega)=2\|{\left\langle}x,\omega{\right\rangle}\|$ and $t(x,\omega)={\left\langle}x,\omega{\right\rangle}+\sqrt{{\left\langle}x,\omega{\right\rangle}^2+r^2-{\left\Vert x\right\Vert}^2}$ ([@tervo14 Example 3.1]). We find that for $(y,\omega)\in \Gamma'=\partial G\times S$ \[et\] \_[(x,)(y,),(x,)GS]{}t(x,)=y,+\|y,\|. For $(y,\omega)\in \Gamma'_+$ the projection ${\left\langle}y,\omega{\right\rangle}$ is non-negative (and then the limit (\[et\]) is $\tau_+(y,\omega)$) and for $(y,\omega)\in \Gamma'_-$ it is non-positive and then the limit (\[et\]) is $0$). Hence $\ol t:\ol G\times S\to{\mathbb{R}}$ is continuous. Due to the Proposition \[prop-ex\] we can (almost everywhere) uniquely set $t(y,\omega)=0$ for $(y,\omega,E)\in\Gamma_{-}$. For $(y,\omega,E)\in \Gamma_+$ we set $t(y,\omega):=\tau_+(y,\omega)$. For further (unexplained) notations we refer to [@tervo14]. In the sequel we denote for $k\in{\mathbb{N}}_0$ $$C^k(\ol G\times S\times I):=\{\psi\in C^k(G\times S\times I^\circ)\ \|\ \psi\ {\rm has\ continuous\ partial\ derivatives\ on}\ \ol G\times S\times I\}$$ and $$D^k(\ol G\times S\times I):=\{\psi\in C^k(G\times S\times I^\circ)\ \|\ \psi=f_{\|G\times S\times I^\circ},\ f\in C_0^k({\mathbb{R}}^n\times S\times{\mathbb{R}})\}.$$ It is well known that these spaces are equal, i.e. $C^k(\ol G\times S\times I)=D^k(\ol G\times S\times I)$, for a given $k$, if the boundary $\partial G$ of $G$ is of class $C^k$ (see [@friedman Part 1, Lemma 5.2]). Thus, in particular, since our standing assumption in this paper is that $\ol{G}$ is (at least) of class $C^1$, we have $$C^1(\ol G\times S\times I)=D^1(\ol G\times S\times I).$$ Define the (Sobolev) space $W^2(G\times S\times I)$ by \[fseq1\] W\^2(GSI) ={L\^2(GSI) \| \_x L\^2(GSI) } and its subspace $W^2_1(G\times S\times I)$ by \[fseq2\] W\^2\_1(GSI) ={W\^2(GSI) \| L\^2(GSI)}. Here $\omega\cdot\nabla_x\psi$ and ${{\frac{\partial \psi}{\partial E}}}$ are understood in the distributional sense. In what follows, $\omega\cdot\nabla_x\psi$ will stand for (the distribution) $\Omega\cdot\nabla_x \psi$, where $\Omega:G\times S\times I\to{\mathbb{R}}^3$; $\Omega(x,\omega,E)=\omega$. The spaces $W^2(G\times S\times I)$, $W^2_1(G\times S\times I)$ are equipped with the inner products, respectively \[fs4\] ,v\_[W\^2(GSI)]{}=,v\_[L\^2(GSI)]{}+ \_x,\_x v\_[L\^2(GSI)]{} and \[fs5\] ,v\_[W\^2\_1(GSI)]{}=,v\_[L\^2(GSI)]{}+ \_x,\_x v\_[L\^2(GSI)]{} +,\_[L\^2(GSI)]{}. These spaces are Hilbert spaces. Let C\^1(GSI):={\_[\| GSI]{} \| C\_0\^1(\^3S)}. We have (cf. [@friedrichs], [@bardos]. The proof can also be shown by the similar considerations as in [@friedman pp. 11-19]) \[denseth\] The space $C^1(\ol G\times S\times I)$ is a dense subspace of $W^2(G\times S\times I)$ and of $W^2_1(G\times S\times I)$. For $\Gamma_-=\{(y,\omega,E)\in (\partial G)\times S\times I)\\|\ \omega\cdot\nu(y)<0\}$ we define the space of $L^2$-functions with respect to the measure $\|\omega\cdot\nu\|\ d\sigma d\omega dE$ which is denoted by $T^2(\Gamma_-)$ that is, $T^2(\Gamma_-)=L^2(\Gamma_-,\|\omega\cdot\nu\|\ d\sigma d\omega dE)$. $T^2(\Gamma_-)$ is a Hilbert space and its inner product is (in this paper all functions are real-valued) \[fs8\] [h\_1]{},h\_2\_[T\^2(\_-)]{}=\_[\_-]{}h\_1(y,,E)h\_2(y,,E) \|\| dddE. The space $T^2(\Gamma_+)$ (and its inner product) of $L^2$-functions on $\Gamma_+=\{(y,\omega,E)\in (\partial G)\times S\times I)\|\ \omega\cdot\nu(y)>0\}$ with respect to the measure $\|\omega\cdot\nu\|\ d\sigma d\omega dE$ is similarly defined. We denote by $T^2(\Gamma)$ the space of $L^2$-functions with respect to the measure $\|\omega\cdot\nu\|\ d\sigma d\omega dE $ that is, $ T^2(\Gamma)=L^2(\Gamma,\|\omega\cdot\nu\|\ d\sigma d\omega dE). $ The inner product in $T^2(\Gamma)$ is \[fs10\] [h\_1]{},h\_2\_[T\^2()]{}= \_h\_1(y,,E)h\_2(y,,E)\|\| dddE. One can show ([@dautraylionsv6 pp. 230-231], [@tervo14]) that for any compact set $K\subset \Gamma_-$ \[ttha\] \_K\|(y,,E)\|\^2\|\| dddEC\_K\_[W\^2(GSI)]{}\^2 C\^1(GSI). Hence any element $\psi\in W^2(G\times S\times I)$ has well defined trace $\psi_{\|\Gamma_-}$ in $L^2_{\rm loc}(\Gamma_-,\|\omega\cdot\nu\|\ d\sigma d\omega dE)$ defined by \[tthb\] \_[\|K]{}:=\_[j]{} [\_j]{}\_[\|K]{} [for any compact subset ]{} K\_-, where $\{\psi_j\}\subset C^1(\ol G\times S\times I)$ is a sequence such that $\lim_{j\to\infty}{\left\Vert \psi_j-\psi\right\Vert}_{W^2(G\times S\times I)} =0$. In addition the trace mapping $\gamma_-:W^2(G\times S\times I)\to \ L^2_{\rm loc}(\Gamma_-,\|\omega\cdot\nu\|\ d\sigma d\omega dE)$ such that $\gamma_-(\psi)=\psi_{\|\Gamma_-}$ is continuous. Similarly one has a continuous trace mapping $\gamma_+:W^2(G\times S\times I)\to \ L^2_{\rm loc}(\Gamma_+,\|\omega\cdot\nu\|\ d\sigma d\omega dE)$ and so we can define (a.e. unique) the trace $\gamma(\psi)$ on $\Gamma$ for $\psi\in W^2(G\times S\times I)$. By the Sobolev Embedding Theorem for $\psi\in C^1(\ol G\times S\times I)$ (cf. [@friedman p. 22], or [@treves p. 220]; or see in the proof of Theorem \[tth\] below) \[sobim\] [(,,E)]{}\_[L\^2(GS)]{}C(\_[L\^2(GSI)]{}+\_[L\^2(GSI)]{}),EI, and then the traces $\psi(\cdot,\cdot,0),\ \psi(\cdot,\cdot,E_{\rm m})\ \in L^2(G\times S)$ are well-defined for any $\psi\in W_1^2(G\times S\times I)$. The trace $\gamma(\psi),\ \psi\in W^2(G\times S\times I)$ is not necessarily in the space $T^2(\Gamma)$. Hence we define the spaces \[fs12\] W\^2(GSI)={W\^2(GSI) \| ()T\^2() }. which is equipped with the inner product \[fs14\] ,v\_[W\^2(GSI)]{}=,v\_[W\^2(GSI)]{}+ [()]{},(v)\_[T\^2()]{}. The space $\tilde W^2(G\times S\times I)$ is a Hilbert space (cf. [@tervo14]). Moreover, we define subspaces $\tilde{W}_{\pm,0}^2(G\times S\times I)$ of it by \_[,0]{}\^2(GSI)={W\^2(GSI) \| \_()=0}. For $v\in \tilde W^2( G\times S\times I)$ and $\psi\in\tilde W^2(G\times S\times I)$ it holds the Green’s formula $$\begin{aligned} \label{green} \int_{G\times S\times I}(\omega\cdot \nabla_x \psi)v\ dxd\omega dE +\int_{G\times S\times I}(\omega\cdot \nabla_x v)\psi\ dxd\omega dE= \int_{\partial G\times S\times I}(\omega\cdot \nu) v\ \psi\ d\sigma d\omega dE,\end{aligned}$$ which is obtained by Stokes Theorem for $v,\psi\in C^1(\ol G\times S\times I)$ and then by the limiting considerations for general $v\in \tilde W^2(G\times S\times I)$ and $\psi\in\tilde W^2(G\times S\times I)$. \[re:W2\_0\_trace\] By Green’s formula $${\left\Vert \gamma_{\pm}(\psi)\right\Vert}_{T^2(\Gamma_{\pm})}\leq {\left\Vert \psi\right\Vert}_{\tilde{W}^2_{\mp}(G\times S\times I)}$$ and thus $\gamma_{\pm} :\tilde{W}^2_{\mp,0}(G\times S\times I)\to T^2(\Gamma_{\pm})$ is bounded. Occasionally we work in the (energy independent) spaces $L^2(G\times S)$. The corresponding Hilbert spaces $W^2(G\times S)$, $T^2(\Gamma'_{\pm})$, $T^2(\Gamma')$ and $\tilde W^2(G\times S)$ are similarly defined, where $$\Gamma':={}&(\partial G)\times S, \\[2mm] \Gamma'_{0}:={}&\{(y,\omega)\in (\partial G)\times S\ \|\ \omega\cdot\nu(y)=0\}, \\[2mm] \Gamma'_{-}:={}&\{(y,\omega)\in (\partial G)\times S\ \|\ \omega\cdot\nu(y)<0\}, \\[2mm] \Gamma'_{+}:={}&\{(y,\omega)\in (\partial G)\times S\ \|\ \omega\cdot\nu(y)>0\}.$$ In addition, the trace $\gamma'(\psi):=\psi_{\|\Gamma'}$ for $\psi\in \tilde{W}^2(G\times S)$ is defined as $\gamma(\psi)$ above. In the context of CSDA-equations we need the following additional Hilbert spaces. Let $H$ be the completion of $C^1(\ol G\times S\times I)$ with respect to the inner product ,v\_H:=,v\_[L\^2(GSI)]{}+ (),(v)\_[T\^2()]{} The elements of $H$ are of the form $\tilde\psi=(\psi,q)\in L^2(G\times S\times I)\times T^2(\Gamma)$. Actually, they are exactly elements of the closure of the graph of trace operator $\gamma:C^1(\ol G\times S\times I)\to C^1(\partial G\times S\times I)$ in $L^2(G\times S\times I)\times T^2(\Gamma)$. The inner product in $H$ is $$\begin{aligned} \label{spaceH} {\left\langle}\tilde\psi,\tilde \psi'{\right\rangle}_{H}={\left\langle}\psi, \psi'{\right\rangle}_{L^2(G\times S\times I)} +{\left\langle}q,q'{\right\rangle}_{T^2(\Gamma)},\end{aligned}$$ for $\tilde\psi=(\psi,q)$, $\tilde\psi'=(\psi',q')\in H$. Furthermore, let $H_1$ be the completion of $C^1(\ol G\times S\times I)$ with respect to the inner product $$\begin{aligned} \label{spaceH1} {\left\langle}\psi,v{\right\rangle}_{H_1}:={}&{\left\langle}\psi,v{\right\rangle}_{L^2(G\times S\times I)}+ {\left\langle}\gamma(\psi),\gamma(v){\right\rangle}_{T^2(\Gamma)}\nonumber \\ & + {\left\langle}\psi(\cdot,\cdot,0),v(\cdot,\cdot,0){\right\rangle}_{L^2(G\times S)} + {\left\langle}\psi(\cdot,\cdot,E_m),v(\cdot,\cdot,E_m){\right\rangle}_{L^2(G\times S)}.\end{aligned}$$ The elements of $H_1$ are of the form $\tilde\psi=(\psi,q,p_0,p_m)\in L^2(G\times S\times I)\times T^2(\Gamma)\times L^2(G\times S)^2$. More precisely, the elements of $H_1$ are the elements of the closure (in $L^2(G\times S\times I)\times T^2(\Gamma)\times L^2(G\times S)^2$) of the graph of the (trace) operator $C^1(\ol G\times S\times I)\to C^1(\partial G\times S\times I)\times C^1(\ol G\times S)^2$ defined by $\psi\mapsto (\gamma(\psi),\psi(\cdot,\cdot,0),\psi(\cdot,\cdot,E_m))$. The inner product in $H_1$ is $${\left\langle}\tilde\psi,\tilde \psi'{\right\rangle}_{H_1}={}&{\left\langle}\psi, \psi'{\right\rangle}_{L^2(G\times S\times I)} +{\left\langle}q,q'{\right\rangle}_{T^2(\Gamma)}\nonumber\\ & + {\left\langle}p_0,p_0'{\right\rangle}_{L^2(G\times S)}+{\left\langle}p_{\rm m},p_{\rm m}'{\right\rangle}_{L^2(G\times S)},$$ for $\tilde\psi=(\psi,q,p_0,p_{\rm m})$, $\tilde\psi=(\psi',q',p_0',p_{\rm m}')\in H_1$. Finally, let $H_2$ be the completion of $C^1(\ol G\times S\times I)$ with respect to the inner product \[inph2\] ,v\_[H\_2]{} :=&,v\_[W\^2(GSI)]{}+,[E]{} \_[L\^2(GSI)]{}\ =&,v\_[L\^2(GSI)]{}+ \_x ,\_x v\_[L\^2(GSI)]{}\ &+ (),(v)\_[T\^2()]{} + ,[E]{} \_[L\^2(GSI)]{}. Obviously $H_2\subset \tilde W^2(G\times S\times I)\cap W_1^2(G\times S\times I)$ and the inner product in $H_2$ is given by (\[inph2\]). Some Details on (Inflow) Trace Theory {#trath} ------------------------------------- We still bring up a refinement of the above explained trace theory. Let us define $$T^2_{\tau_{\pm}}(\Gamma_{\pm}):=L^2(\Gamma_{\pm},\tau_{\pm}(\cdot,\cdot)\|\omega\cdot\nu\| d\sigma d\omega dE),$$ equipped with the inner product \[fs8a\] [h\_1]{},h\_2\_[T\^2\_[\_]{}(\_)]{}=\_[\_]{}h\_1(y,,E)h\_2(y,,E)\_(y,) \|\| dddE. Suppose that $g\in C(\Gamma_-)$ such that ${{\frac{\partial g}{\partial \tilde y_i}}}\in C(\Gamma_-), i=1,2$, where ${\partial\over{\partial \tilde y_i}}$ denotes any local basis of the tangent space of $\partial G$ (and ${{\frac{\partial g}{\partial \tilde y_i}}}\in C(\Gamma_-)$ is to be understood in a local sense), and $\Sigma\in C(\ol G\times S\times I)$ such that ${{\frac{\partial \Sigma}{\partial x_j}}}\in C(\ol G\times S\times I),\ j=1,2,3$. Then the unique (classical) solution of the homogeneous convection-scattering equation (recall the definition of $D$ in ) \[trath1\] \_x+=0 D, satisfying the inhomogeneous inflow boundary condition \[trath2\] (y,,E)=g(y,,E)(y,,E)\_-, is given by ([@tervo14 Theorem 3.13]) \[trath3\] (x,,E)=e\^[-\_0\^[t(x,)]{}(x-s,,E)ds]{} g(x-t(x,),,E). We denote by $L_-g$ ($=\psi$) the solution of the problem (\[trath1\])-(\[trath2\]) that is, $L_-g$ is given by the right hand side of (\[trath3\]). Note that $L_-g$ is not generally even continuous (for non-convex $G$) even if $g$ happens to be smooth. We show, however, that the formula (\[trath3\]) gives a weak solution of the problem (\[trath1\])–(\[trath2\]) that is, $$\begin{aligned} {3} & {\left\langle}\psi,-\omega\cdot \nabla_xv+\Sigma v{\right\rangle}_{L^2(G\times S\times I)}=0 \quad && {\rm for\ all}\ v\in C_0^\infty(G\times S\times I^\circ),\nonumber\\[2mm] & \psi(y,\omega,E)=g(y,\omega,E)\quad && {\rm for\ a.e.} \ (y,\omega,E)\in\Gamma_-,\end{aligned}$$ for any $g\in T_{\tau_-}^2(\Gamma_{-})$, and that $L_-g\in W^2(G\times S\times I)$. We record the following Lemma for later use. \[le:lift\] Assume that $\Sigma\in L^\infty(G\times S\times I)$ and that $\Sigma\geq 0$. Then for any $g\in T_{\tau_-}^2(\Gamma_{-})$ the function $L_-g$ defined by (\[trath3\]) (is measurable and) belongs to $L^2(G\times S\times I)$, and \[trpr9\] [L\_-g]{}\_[L\^2(GSI)]{}[g]{}\_[T\^2\_[\_-]{}(\_-)]{}. Moreover, equality holds here if $\Sigma=0$. Write $L_{\Sigma,-}$ for the lift-operator defined in for a given $\Sigma\geq 0$, i.e. $\psi=L_{\Sigma,-} g$. Then $${\left\Vert L_{0,-} g\right\Vert}_{L^2(G\times S\times I)}^2 ={}&\int_{G\times S\times I} g(x-t(x,\omega)\omega,\omega,E)^2 dx d\omega dE \\ ={}&\int_{\Gamma_-} g(y,\omega,E)^2 \tau_-(y,\omega)\|\omega\cdot\nu(y)\|d\sigma(y) d\omega dE \\ ={}&{\left\Vert g\right\Vert}_{T^2_{\tau_-}(\Gamma_-)}^2,$$ where in the second step we applied the change of variables in integration explained in the proof of Theorem \[tth\] below (see Remark \[changevar\]), and noticed that $t(y+s\omega,\omega)=s$ whenever $(y,\omega,E)\in \Gamma_-$. Therefore, $${\left\Vert L_{\Sigma,-} g\right\Vert}_{L^2(G\times S\times I)}^2 =&{}\int_{G\times S\times I} \big(e^{-\int_0^{t(x,\omega)}\Sigma(x-s\omega,\omega,E)ds}g(x-t(x,\omega)\omega,\omega,E)\big)^2 dx d\omega dE \\ \leq &{}\int_{G\times S\times I} g(x-t(x,\omega)\omega,\omega,E)^2 dx d\omega dE \\ =&{}{\left\Vert L_{0,-} g\right\Vert}_{L^2(G\times S\times I)}^2 ={\left\Vert g\right\Vert}_{T^2_{\tau_-}(\Gamma_-)}^2.$$ Since $t(y,\omega)=0$ a.e. in $\Gamma_-$ we see that $\gamma_-(L_-g)=g.$ When we verify (Lemma \[trathle1\]) that $\omega\cdot\nabla_x(L_-g)+\Sigma(L_-g)=0$ (weakly) in $G\times S\times I$ we can conclude that $L_-g\in W^2(G\times S\times I)$. \[trathle1\] Assume that $\Sigma\in L^\infty(G\times S\times I)$ and that $\Sigma\geq 0$. Let $L_{-}:T^2_{\tau_{-}}(\Gamma_{-})\to L^2(G\times S\times I)$ be defined by $$(L_{-} g)(x,\omega,E)= e^{-\int_0^{t(x,\omega)}\Sigma(x-s\omega,\omega,E)ds} g(x- t(x,\omega)\omega,\omega,E).$$ Then in the weak sense on $G\times S\times I$, $$\omega\cdot \nabla_x (L_- g)+\Sigma (L_- g)=0.$$ Given $g\in T^2_{\tau_-}(\Gamma_-)$, choosing a sequence $g_n$ in $C^1_0(\Gamma_-)$ that converges to $g$ in $T^2_{\tau_-}(\Gamma_-)$ (the proof of the existence of this kind sequence is quite standard and is omitted), we have by the continuity of $L_-$ (see (\[trpr9\])), that $L_-g_n\to L_- g$ in $L^2(G\times S\times I)$, hence in ${\mathcal{D}}'(G\times S\times I^\circ)$, where $I^\circ:=]0,E_{\rm m}[$, from which we deduce that $\omega\cdot\nabla_x (L_-g_n)+\Sigma(L_-g_n)\to \omega\cdot\nabla_x (L_- g)+\Sigma (L_-g)$. in ${\mathcal{D}}'(G\times S\times I^\circ)$. This shows that we may assume $g\in C^1_0(\Gamma_-)$. Let $\varphi\in C_0^\infty (G\times S\times I^\circ)$. Then by Fubini’s Theorem $$&-(\omega\cdot \nabla_x (L_- g))(\varphi) =(L_- g)(\omega\cdot \nabla_x\varphi) \\ ={}&\int_{G\times S\times I} e^{-\int_0^{t(x,\omega)}\Sigma(x-s\omega,\omega,E)ds} g(x-t(x,\omega)\omega,\omega,E)(\omega\cdot \nabla_x \varphi)(x,\omega,E){d}x {d}\omega {d}E \\ ={}&\int_{S\times I}\int_{G_{\omega}}\int_{J_{y,\omega}} e^{-\int_0^{t(y+\tau\omega,\omega)}\Sigma(y+(\tau-s)\omega,\omega,E)ds} g(y+\tau\omega-t(y+\tau\omega,\omega)\omega,\omega,E) \\ & \cdot{\frac{{d}}{{d}\tau}}\varphi(y+\tau\omega,\omega,E){d}s{d}y{d}\omega{d}E \\ ={}&\int_{S\times I}\int_{G_{\omega}}\int_{J_{y,\omega}} e^{-\int_{\tau-t(y+\tau\omega,\omega)}^\tau\Sigma(y+s\omega,\omega,E)ds} g(y+\tau\omega-t(y+\tau\omega,\omega)\omega,\omega,E) \\ & \cdot{\frac{{d}}{{d}\tau}}\varphi(y+\tau\omega,\omega,E){d}\tau{d}y{d}\omega{d}E$$ Here $G_\omega$ is the orthogonal projection of $G$ along $\omega$, $$G_\omega=\{x-(x\cdot\omega)\omega\ \|\ x\in G\},$$ which is an $(n-1)$-dimensional open submanifold of ${\mathbb{R}}^n$. Moreover, $J_{y,\omega}$ is the intersection of $G$ and the straight line $y+{\mathbb{R}}\omega$, $$J_{y,\omega}=\{\tau\in{\mathbb{R}}\ \|\ y+\tau\omega\in G\}.$$ Notice that $J_{y,\omega}$ is open subset of ${\mathbb{R}}$, hence disjoint union of countably many open intervals $J_{y,\omega}^i=]a_{y,\omega}^i,b_{y,\omega}^i[$, $i\in{\mathbb{N}}$ (we take $J_{y,\omega}^i=\emptyset$ for all big enough $i$ if there is only finitely many non-empty ones) $$J_{y,\omega}=\coprod_{i\in{\mathbb{N}}} J^i_{y,\omega}.$$ Noticing also that for all $\tau\in J^i_{y,\omega}$, $$t(y+\tau\omega,\omega)=\tau-a^i_{y,\omega}$$ we have $$&-(\omega\cdot \nabla_x (L_- g))(\varphi) \\ ={}&\int_{S\times I}\int_{G_{\omega}}\sum_i \int_{J^i_{y,\omega}} e^{-\int_{a^i_{y,\omega}}^\tau\Sigma(y+s\omega,\omega,E)ds}g(y+a^i_{y,\omega}\omega,\omega,E) {\frac{{d}}{{d}\tau}}\varphi(y+\tau\omega,\omega,E){d}\tau{d}y{d}\omega{d}E \\ ={}&\int_{S\times I}\int_{G_{\omega}}\sum_i g(y+a^i_{y,\omega}\omega,\omega,E)\Big( e^{-\int_{a^i_{y,\omega}}^\tau\Sigma(y+s\omega,\omega,E)ds}\varphi(y+\tau\omega,\omega,E) \Big\|_{\tau=a^i_{y,\omega}}^{\tau=b^i_{y,\omega}} \Big){d}y{d}\omega{d}E \\ &+\int_{S\times I}\int_{G_{\omega}}\sum_i \int_{J^i_{y,\omega}} \Sigma(y+\tau\omega,\omega,E)e^{-\int_{a^i_{y,\omega}}^\tau\Sigma(y+s\omega,\omega,E)ds} \\ &\cdot g(y+a^i_{y,\omega}\omega,\omega,E)\varphi(y+\tau\omega,\omega,E){d}y{d}\omega{d}E \\ ={}&0 + \int_{G\times S\times I} \Sigma(x,\omega,E)e^{-\int_0^{t(x,\omega)}\Sigma(x-s\omega,\omega,E)ds} \\ &\cdot g(x-t(x,\omega)\omega,\omega,E)\varphi(x,\omega,E){d}x {d}\omega {d}E \\ ={}&\big(\Sigma (L_-g)\big)(\varphi),$$ where at the 3rd equality we used the fact that $\varphi$ vanishes on the boundary of $G$ and $y+a^i_{y,\omega}\omega\in\partial G$ and $y+b^i_{y,\omega}\omega\in\partial G$. This completes the proof. Note that if we assume that there exists $c>0$ such that $\Sigma\geq c$ on $G\times S\times I$, then in the previous lemma one can take $G$ to be unbounded as well. Analogously to Lemma \[trathle1\] for any $g\in T^2_{\tau_+}(\Gamma_+)$ the weak solution of the problem $$\begin{aligned} \omega\cdot \nabla_x\psi+\Sigma\psi&=0,\\ \psi(y,\omega,E)&=g(y,\omega,E)\quad {\rm for}\ (y,\omega,E)\in \Gamma_+\ \end{aligned}$$ is given by (note that $(y,\omega)\in\Gamma_-$ if and only if $(y,-\omega)\in\Gamma_+$) $$(L_+g)(x,\omega,E):=\psi(x,\omega,E)=e^{-\int_0^{t(x,-\omega)}\Sigma(x-s\omega,\omega,E)ds} g(x+t(x,-\omega)\omega,\omega,E).$$ For later use (section \[comp\]) we also treat the inhomogeneous convection-scattering equation with the homogeneous boundary data. Suppose that $f\in C(\ol G\times S\times I)$ such that ${{\frac{\partial f}{\partial x_j}}}\in C(\ol G\times S\times I)$, and let $\Sigma\in C(\ol G\times S\times I)$ such that ${{\frac{\partial \Sigma}{\partial x_j}}}\in C(\ol G\times S\times I),\ j=1,2,3$ . Then the unique (classical) solution of the equation \[trath7\] \_x+=f D, satisfying the homogeneous inflow boundary condition \[trath8\] (y,,E)=0 (y,,E)\_-, is given by (cf. [@dautraylionsv6], or [@tervo14 Section 3.3]) \[trath9\] (x,,E)=\_0\^[t(x,)]{}e\^[-\_0\^t(x-s,,E)ds]{} f(x-t,,E) dt. We need the next lemma. Assume that $G$ is bounded, $d:={\rm diag}(G)$, $\Sigma\in L^\infty(G\times S\times I)$ and that $\Sigma\geq 0$. Then for any $f\in L^2(G\times S\times I)$ the formula (\[trath9\]) defines $L^2(G\times S\times I)$-function $\psi=S_{\Sigma} f$, and \[trath12\] [S\_ f]{}\_[L\^2(GSI)]{}d[f]{}\_[L\^2(GSI)]{}. We have $$&{\left\Vert S_{\Sigma} f\right\Vert}_{L^2(G\times S\times I)}^2 =\int_{G\times S\times I} \Big(\int_0^{t(x,\omega)}e^{-\int_0^t\Sigma(x-s\omega,\omega,E)ds} f(x-t\omega,\omega,E) dt\Big)^2 dx d\omega dE \\ ={}&\int_{\Gamma_-} \int_0^{\tau_-(y,\omega)} \Big(\int_0^{t(y+r\omega,\omega)}e^{-\int_0^t\Sigma(y+(r-s)\omega,\omega,E)ds} f(y+(r-t)\omega,\omega,E) dt\Big)^2 dr \|\omega\cdot\nu(y)\|d\sigma(y) d\omega dE \\ ={}&\int_{\Gamma_-} \int_0^{\tau_-(y,\omega)} \Big(\int_0^{r}e^{-\int_0^t\Sigma(y+(r-s)\omega,\omega,E)ds} f(y+(r-t)\omega,\omega,E) dt\Big)^2 dr \|\omega\cdot\nu(y)\|d\sigma(y) d\omega dE \\ \leq &\int_{\Gamma_-} \int_0^{\tau_-(y,\omega)} r\int_0^{r}\|f(y+(r-t)\omega,\omega,E)\|^2 dt dr \|\omega\cdot\nu(y)\|d\sigma(y) d\omega dE \\ ={}&\int_{\Gamma_-} \int_0^{\tau_-(y,\omega)} r\int_0^{r}\|f(y+t\omega,\omega,E)\|^2 dt dr \|\omega\cdot\nu(y)\|d\sigma(y) d\omega dE \\ \leq &\int_{\Gamma_-} \tau_-(y,\omega)^2\int_0^{\tau_-(y,\omega)}\|f(y+t\omega,\omega,E)\|^2 dt \|\omega\cdot\nu(y)\|d\sigma(y) d\omega dE,$$ where we again in the second step applied the change of variables in integration explained in the proof of Theorem \[tth\] below and in the fourth step applied the Cauchy-Schwartz inequality. Since $\tau_-(y,\omega)\leq d$, this gives $${\left\Vert S_{\Sigma} f\right\Vert}_{L^2(G\times S\times I)}^2 \leq {}& d^2\int_{\Gamma_-} \int_0^{\tau_-(y,\omega)}\|f(y+t\omega,\omega,E)\|^2 dt\|\omega\cdot\nu(y)\|d\sigma(y) d\omega dE \\ ={}&d^2{\left\Vert f\right\Vert}_{L^2(G\times S\times I)}^2.$$ More generally, we have the following. \[trathle2\] Assume that $\Sigma\in L^\infty(G\times S\times I)$ and that $\Sigma\geq 0$. Then $\psi$ defined by satisfies weakly in $G\times S\times I$, \[trath11\] \_x +=f, and the inflow boundary condition (\[trath8\]) is valid. Due to (\[trath12\]) it suffices to show (\[trath11\]) only for $f\in C^1(\ol G\times S\times I)$. Using the notations from the proof of Lemma \[trathle1\], for $\varphi\in C_0^\infty(G\times S\times I^\circ)$ we get by the Fubin’s Theorem $$&-(\omega\cdot \nabla_x \psi)(\varphi) =\psi(\omega\cdot\nabla_x\varphi) \\ ={}&\int_{G\times S\times I} \psi(x,\omega,E) (\omega\cdot \nabla_x \varphi)(x,\omega,E){d}x {d}\omega {d}E \\ ={}&\int_{S\times I}\int_{G_{\omega}}\sum_i \int_{J^i_{y,\omega}} \psi(y+\tau\omega,\omega,E) {\frac{{d}}{{d}\tau}}\varphi(y+\tau\omega,\omega,E){d}\tau{d}y{d}\omega{d}E \\ ={}&\int_{S\times I}\int_{G_{\omega}}\sum_i \int_{J^i_{y,\omega}} \int_0^{\tau-a^i_{y,\omega}}e^{-\int_0^t\Sigma(y+(\tau-s)\omega,\omega,E)ds} f(y+(\tau-t)\omega,\omega,E)\\ &\cdot{\frac{{d}}{{d}\tau}}\varphi(y+\tau\omega,\omega,E) dt{d}\tau{d}y{d}\omega{d}E \\ ={}&\int_{S\times I}\int_{G_{\omega}}\sum_i \int_{J^i_{y,\omega}} \int_{a^i_{y,\omega}}^\tau e^{-\int_0^{\tau-t}\Sigma(y+(\tau-s)\omega,\omega,E)ds} f(y+t\omega,\omega,E)\\ &\cdot{\frac{{d}}{{d}\tau}}\varphi(y+\tau\omega,\omega,E) dt{d}\tau{d}y{d}\omega{d}E \\ ={}&\int_{S\times I}\int_{G_{\omega}}\sum_i \int_{J^i_{y,\omega}} f(y+t\omega,\omega,E) \int_t^{b^i_{y,\omega}} e^{-\int_{t}^\tau\Sigma(y+s\omega,\omega,E)ds} \\ &\cdot{\frac{{d}}{{d}\tau}}\varphi(y+\tau\omega,\omega,E) {d}\tau{d}t{d}y{d}\omega{d}E.$$ In the last step, we changed the order of integration $dtd\tau\to d\tau dt$, in which the domain of integration $$\{(\tau,t)\ \|\ \tau\in J^i_{y,\omega}=]a^i_{y,\omega},b^i_{y,\omega}[,\ t\in ]a^i_{y,\omega},\tau[\}$$ changes into $$\{(t,\tau)\ \|\ t\in J^i_{y,\omega}=]a^i_{y,\omega},b^i_{y,\omega}[,\ \tau\in ]t,b^i_{y,\omega}[\}$$ as usual. Observing that $$&\int_t^{b^i_{y,\omega}} e^{-\int_{t}^\tau\Sigma(y+s\omega,\omega,E){d}s}\cdot{\frac{{d}}{{d}\tau}}\varphi(y+\tau\omega,\omega,E) {d}\tau \\ ={}&\Big(e^{-\int_{t}^\tau\Sigma(y+s\omega,\omega,E){d}s} \varphi(y+\tau\omega,\omega,E) \Big\|_{\tau=t}^{\tau=b^i_{y,\omega}} \Big) \\ &+\int_t^{b^i_{y,\omega}} \Sigma(y+\tau\omega,\omega,E)e^{-\int_{t}^\tau\Sigma(y+s\omega,\omega,E){d}s} \varphi(y+\tau\omega,\omega,E) {d}\tau \\ ={}&-\varphi(y+t\omega,\omega,E)+\int_t^{b^i_{y,\omega}} \Sigma(y+\tau\omega,\omega,E)e^{-\int_{t}^\tau\Sigma(y+s\omega,\omega,E){d}s} \varphi(y+\tau\omega,\omega,E) {d}\tau ,$$ we obtain $$&-(\omega\cdot \nabla_x \psi)(\varphi) \\ ={}&\int_{S\times I}\int_{G_{\omega}}\sum_i \int_{J^i_{y,\omega}} f(y+t\omega,\omega,E)\Big(-\varphi(y+t\omega,\omega,E) \\ &+ \int_{t}^{b^i_{y,\omega}} \Sigma(y+\tau\omega,\omega,E)e^{-\int_{t}^\tau\Sigma(y+s\omega,\omega,E){d}s} \varphi(y+\tau\omega,\omega,E) {d}\tau\Big){d}t{d}y{d}\omega{d}E \\ ={}& -\int_{G\times S\times I}f(x,\omega,E)\varphi(x,\omega,E){d}x{d}\omega{d}E \\ &+\int_{G\times S\times I} \Sigma(x,\omega,E)\int_0^{t(x,\omega)} e^{-\int_0^t\Sigma(x-s\omega,\omega,E){d}s}f(x-t\omega,\omega,E){d}t \varphi(x,\omega,E){d}x{d}\omega{d}E \\ ={}&\int_{G\times S\times I} \big(-f(x,\omega,E)+\Sigma(x,\omega,E)\psi(x,\omega,E)\big)\varphi(x,\omega,E){d}x{d}\omega{d}E$$ which is what we set out to prove. Choosing especially $\Sigma=0, \ f=1$ in Lemma \[trathle2\] we find that in the weak sense $\omega\cdot\nabla_x t=1$ in $G\times S\times I$. We are now ready to prove the inflow trace theorem. Since ${\left\Vert \omega\right\Vert}=1$ and since the domain $G$ is bounded we have $\tau_{\pm}(y,\omega)\leq d$. Hence from [@dautraylionsv6 p. 252], [@cessenat85] or [@choulli] (where the result is considered for a more general $G$) we obtain the following theorem. For completeness we give its detailed proof. \[tth\] The trace mappings $$\gamma_{\pm}:W^2(G\times S\times I)\to T^2_{\tau_{\pm}}(\Gamma_{\pm})$$ are (well-defined) bounded surjective operators with bounded right inverses $L_{\pm}:T^2_{\tau_{\pm}}(\Gamma_{\pm})\to W^2(G\times S\times I)$ that is, $\gamma_{\pm}\circ L_{\pm}=I$ (the identity). The operators $L_{\pm}$ are called [lifts]{}. Below we use abbreviation $L:=L_-$ if no confusion is possible. A. For the first instance we recall an elementary estimate for (smooth) functions defined on an interval $[0,T]\subset{\mathbb{R}}$. Let $f\in C^1([0,T])$. Then for any $s\in [0,T]$ $$f(0)=-\int_0^sf'(t)dt +f(s)$$ which implies (by the Cauchy-Schwartz’s inequality) that \[trpr1\] \|f(0)\|\^2 2(s\_0\^s\|f’(t)\|\^2dt+\|f(s)\|\^2) 2(T[f’]{}\^2\_[L\^2(\[0,T\])]{}+\|f(s)\|\^2). Hence integrating over $[0,T]$ we obtain the estimate \[trpr2\] T\|f(0)\|\^2 2(T\^2[f’]{}\^2\_[L\^2(\[0,T\])]{}+[f]{}\_[L\^2(\[0,T\])]{}\^2). B. We shall only deal with the trace $\gamma_-$ because the treatment of the trace $\gamma_+$ is analogous. At first we show the boundedness of $\gamma_-$. Since $C^1(\ol G\times S\times I)$ is dense in $W^2(G\times S\times I)$ (by Theorem \[denseth\]) it suffices to prove that there exists a constant $C>0$ such that \[trpr3\] [\_-()]{}\_[T\^2\_[\_-]{}(\_-)]{}C\_[W\^2(GSI)]{}C\^1(GSI). We apply the change of variables given in [@tervo14 proof of Lemma 5.8] (see also [@choulli Prop. 2.1]). Assume for simplicity that $\partial G$ has a $C^1$-parametrization (which is almost global), say $h:V\to\partial G\setminus \Gamma_1 $ where $\Gamma_1$ has zero surface measure. Generally we have a finite number of parametrized patches that cover $\partial G$. Applying [for each fixed]{} $\omega$ the change of variables (in $x$-variable) $x=h(v)+t\omega=:H(v,t)$, we find that the Jacobian $J_H$ of $H$ is $$J_H(v,t)=\omega\cdot (\partial_1h\times\partial_2h)(v)= \omega\cdot\nu(h(v)) {\left\Vert (\partial_1h\times\partial_2h)(v)\right\Vert},$$ since $\nu(h(v))={{(\partial_1h\times\partial_2h)(v)}\over{{\left\Vert (\partial_1h\times\partial_2h)(v)\right\Vert}}}$. We notice that $J_H(v,t)$ depends only on $v$ (and $\omega$), but not on $t$, and hence we write it as $J_H(v)$. Moreover, almost everywhere $H(W)=G$, where $W:=\{(v,t)\ \|\ v\in V_-,\ 0<t<\tau_-(h(v),\omega)\}$ and $V_-:=\{v\in V\ \|\ \omega\cdot\nu(h(v))< 0\}$ (which depend on $\omega$). Hence for any $\psi\in C^1(\ol G\times S\times I)$ \[trpr5\] & \_[GSI]{}\|(x,,E)\|\^2 dx ddE = \_[SI]{}(\_G\|(x,,E)\|\^2 dx )ddE\ =& \_[SI]{}\_W\|(H(v,t),,E)\|\^2 \|J\_H(v,t)\|dv dt ddE\ =& \_[SI]{}\_[V\_-]{}\_0\^[\_-(h(v),)]{}\|(h(v)+t,,E)\|\^2 \|J\_H(v,t)\|dtdv ddE. For a fixed $(v,\omega)\in V_-$ and $E\in I$ we apply (\[trpr2\]) to the $C^1$-mapping $f:[0,\tau_-(h(v),\omega)]\to{\mathbb{R}}$ defined by $$f(t):=\psi(h(v)+\omega t,\omega,E).$$ Noting that $f'(t)=(\omega\cdot\nabla_x\psi)(h(v)+\omega t,\omega,E)$ we obtain the estimate \[trpr6\] \_-(h(v),)\|(h(v),,E)\|\^2 & 2(\_-(h(v),)\^2 \_0\^[\_-(h(v),)]{}\|(\_x)(h(v)+t,,E)\|\^2 dt\ & + \_0\^[\_-(h(v),)]{}\|(h(v)+t,,E)\|\^2 dt ). Utilizing these preliminaries and the fact that $\tau_-(h(v),\omega)\leq d$ we get \[trpr4\] & [\_-()]{}\_[ T\^2\_[\_-]{}(\_-)]{}\^2 = \_[\_-]{}\|(y,,E)\|\^2\_-(y,)\|(y)\| dddE\ =& \_[SI]{}\_[V\_-]{}\|(h(v),,E)\|\^2 \_-(h(v),)\|(h(v))\| [(\_1h\_2h)(v)]{} dv ddE\ =& \_[SI]{}\_[V\_-]{}\|(h(v),,E)\|\^2 \_-(h(v),)\|J\_H(v)\| dv ddE\ & \_[SI]{}\_[V\_-]{} 2(d\^2 \_0\^[\_-(h(v),)]{}\|(\_x)(h(v)+t,,E)\|\^2 \|J\_H(v)\|dt\ & +\_0\^[\_-(h(v),)]{}\|(h(v)+t,,E)\|\^2\|J\_H(v)\| dt )\ =& 2(d\^2[\_x]{}\^2\_[L\^2(GSI)]{} + \^2\_[L\^2(GSI)]{}) where we in the last step applied (\[trpr5\]) to $\omega\cdot\nabla_x\psi$ and to $\psi$. This completes the boundedness claim of $\gamma_-$. C. Next we prove the existence of the right inverses. Again we consider only the case of $\gamma_-$. We choose $\Sigma=0$ in Lemma \[trathle1\] and define the right inverse $L_-:T_{\tau_-}^2(\Gamma_-)\to W^2(G\times S\times I)$ by \[liftb\] L\_-g:=g(x-t(x,),,E). Then by (\[trpr9\]) $L_-: T^2_{\tau_{-}}(\Gamma_-)\to W^2(G\times S\times I)$ is a well-defined bounded linear operator, which is even isometric embedding (which follows since $\Sigma=0$). This completes the proof. \[changevar\] From the proof of the previous lemma we get the following useful formulas (when the integrals exist) \_[\_-]{}g(y,,E)dddE = \_[SI]{}\_[V\_-]{}g(h(v),,E) [(\_1h\_2h)(v)]{}dv ddE, where $V_-=\{v\in V\ \|\ \omega\cdot\nu(h(v))< 0\}$ (which depends on $\omega$), and (see (\[trpr5\])) \[trpr5-1\] & \_[GSI]{}\|(x,,E)\|\^2 dx ddE\ =& \_[SI]{}\_[V\_-]{}\_0\^[\_-(h(v),)]{}\|(h(v)+t,,E)\|\^2 \|(h(v))\|[(\_1h\_2h)(v)]{}dt dv ddE\ =& \_[\_-]{}\_0\^[\_-(y,)]{}\|(y+t,,E)\|\^2 \|(y)\| dt d(y) ddE. \[wl\] A. For any compact set $K\subset\Gamma_-$ we have $\tau_-(y,\omega)\geq c_K>0$ for all $(y,\omega,E)\in K$ and so the estimate (\[ttha\]) follows from Theorem \[tth\]. B. The formula (\[liftb\]) gives the lift $L_-g$ explicitly which is useful e.g. in numerical computations. Note that the lift is not unique (for example, we are able to define $L_-=L_{\Sigma,-}$ for any $\Sigma$ given above). Analogously to $L_-$ the (isometric) lift $L_+:T_{\tau_+}^2(\Gamma_+)\to W^2(G\times S\times I)$ can be chosen to be $$(L_+g)(x,\omega,E):= g(x+t(x,-\omega)\omega,\omega,E).$$ C. We have for any $w\in L^2(G\times S\times I)$ and $g\in T^2(\Gamma_-)$ (as in (\[trpr5\]) & L\_-g,w\_[L\^2(GSI)]{} = \_[GSI]{}g(x-t(x,),,E) w(x,,E) dx ddE\ & = \_[SI]{}\_[V\_-]{}\_0\^[\_-(h(v),)]{}g(h(v)+t-t(h(v)+t),,E) w(h(v)+t,,E)\|J\_H(v,t)\| dv ddE dt\ & = \_[SI]{}\_[V\_-]{}g(h(v),,E) (L\_-\^\w)(h(v),,E) \_-(h(v),)\|(h(v))\|[\_1h\_2h]{} dv ddE\ & =g,L\_-\^\w\_[T\_[\_-]{}\^2(\_-)]{} where we used the fact $t(h(v)+t\omega)=t$ and defined \[liftad\] (L\_-\^\w)(y,,E):=[1]{}\_0\^[\_-(y,)]{}w(y+t,,E)dt. Hence $L_-^$ is the adjoint of the operator $L_-:T_{\tau_-}^2(\Gamma_-)\to L^2(G\times S\times I)$. Applying the preceding computations we find that $${\left\Vert L_-^w\right\Vert}_{T_{\tau_-}^2(\Gamma_-)}\leq {\left\Vert w\right\Vert}_{L^2(G\times S\times I)}.$$ D. Interpreting $\gamma_-$ as a densely defined linear operator $L^2(G\times S\times I)\to T^2_{\tau_-}(\Gamma_-)$ we recall that the adjoint $\gamma_-^:T^2_{\tau_-}(\Gamma_-)\to L^2(G\times S\times I)$ exists and it is defined by $${\left\langle}\gamma_-(\psi),g{\right\rangle}_{T_{\tau_-}^2(\Gamma_-)} ={\left\langle}\psi,\gamma_-^(g){\right\rangle}_{L^2(G\times S\times I)} \ {\rm for}\ \psi\in W^2(G\times S\times I),\ g\in D(\gamma_-^).$$ The knowledge of adjoints is significant in the theory of existence of solutions. For example, the Fredholm alternative theorem is essential in the cases where the unique solution does not exist. To demonstrate that, consider (for simplicity) the problem (\[intro1\]), (\[intro2\]). Let $T=(T_1,T_2,T_3)$ be the transport operator related to the problem that is, $T_j\psi:=\omega\cdot\nabla_x\psi_j+\Sigma_j\psi_j-K_j\psi_j$. Then the transport problem reads A:= ( T\_- ) = ( fg ) where $A$ is interpreted as a densely defined closed operator $L^2(G\times S\times I)^3\to L^2(G\times S\times I)^3\times T_{\tau_-}^2(\Gamma_-)^3$ with $D(A):=W^2(G\times S\times I)^3$. We find that the adjoint $A^$ of $A$ is a densely defined operator $L^2(G\times S\times I)^3\times T_{\tau_-}^2(\Gamma_-)^3\to L^2(G\times S\times I)^3$ given by A\^\ ( uv ) := ( T\^\& \_-\^\ ) ( uv ) [for]{} ( uv ) D(A\^\) where $T^=(T_1^,T_2^,T_3^)$ with $T_j^u:=-\omega\cdot\nabla_x u_j+\Sigma_j u_j-K_j^u$ (the operator $K_j^$ is given in section \[adjoint\]). Supposing that $A$ is a Fredholm operator we have $R(A)=N(A^)^\perp$. Hence a necessary and sufficient condition for the existence of solutions of $A\psi={ \left( \begin{matrix} f\cr g\cr \end{matrix} \right) }$ is the orthogonality criterion (f,g),(u,v)\_[ L\^2(GSI)\^3 T\_[\_-]{}\^2(\_-)\^3]{}=0 for all $(u,v)\in N(A^)$ that is, for all $(u,v)\in D(A^)\subset L^2(G\times S\times I)^3\times T_{\tau_-}^2(\Gamma_-)^3 $ which obey $$T^u+\gamma_-^v=0.$$ Under the due assumptions the transport operator $T$ is (after an appropriate change of variables) a pseudo-differential operator (actually only the term $K$ needs careful treatment). Hence the application of the pseudo-differential boundary value operator calculus is possible. We omit here the details of such approaches. E. For $(y,\omega,E)\in\Gamma_+$ it is reasonable to set (cf. Proposition \[prop-ex\]) $$(L_-g)(y,\omega,E):=g(y-\tau_+(y,\omega)\omega,\omega,E),\quad g\in T^2_{\tau_-}(\Gamma_-).$$ From [@dautraylionsv6 p. 253] (or [@cessenat85]) it follows that for $g\in T^2_{\tau}(\Gamma)$, where $\tau\|_{\Gamma_-}=\tau_-$, $\tau\|_{\Gamma_+}=\tau_+$ and $\tau\|_{\Gamma_0}=0$, there exists an element $\tilde\psi\in \tilde W^2(G\times S\times I)$ such that $$\gamma_-(\tilde\psi)=g_{\|\Gamma_-}=:g_-\quad {\rm and}\quad \gamma_+(\tilde\psi)=g_{\|\Gamma_+}=:g_+$$ if and only if $$(g-L_-(g_-))_{\|\Gamma_+}\in L^2(\Gamma_+,\tau_+^{-1}(y,\omega)\|\omega\cdot\nu(y)\|d\sigma d\omega dE).$$ G. It can also be shown that $\gamma_-:\tilde W_{-,0}^2(G\times S\times I)\to L^2(\Gamma_-,\tau_{-}^{-1}(y,\omega)\|\omega\cdot\nu\| d\sigma dE)$ is bounded (note again that for bounded $G$ we have $\tau_-(x,\omega)\leq d$) and that it has a bounded right inverse $L_{-,0}: L^2(\Gamma_-,\tau_-^{-1}(y,\omega)\|\omega\cdot\nu\| d\sigma dE)\to \tilde W_{-,0}^2(G\times S\times I)$ ([@cessenat84] or [@dautraylionsv6 p.252]). Similar result hold for $\gamma_+$. We still consider the following special case of the trace theory for an exterior of a [convex]{} bounded domain $G$. Let $G_e$ be the complement (the exterior of $G$) $G_e:={\mathbb{R}}^3\setminus \ol G$. Then $\partial G_e=\partial G$. Denote (as above for $G$) $$\Gamma_{e,+}:={}&\{(y,\omega,E)\in \partial G_e\times S\times I)=\partial G\times S\times I)\|\ \omega\cdot\nu_e(y)>0\}, \\[2mm] \Gamma_{e,-}:={}&\{(y,\omega,E)\in \partial G_e\times S\times I)=\partial G\times S\times I)\|\ \omega\cdot\nu_e(y)<0\}, \\[2mm] \Gamma_e:={}&\Gamma_{e,+}\cup\Gamma_{e,-}$$ where $\nu_e$ is the unit outward pointing normal vector on $\partial G_e$. We find that $\nu_e=-\nu$ and then $\Gamma_{e,\pm}=\Gamma_{\mp}$ and $\gamma_{e,\pm}(\psi):=\psi_{\|\Gamma_{e,\pm}}=\gamma_{\mp}(\psi)$. Furthermore, let $t_e(x,\omega)$ be the escape time mapping for the domain $G_e$ and let for an element $(y,\omega,E)\in \Gamma_{e,-}$ (as above) $\tau_{e,-}(y,\omega)=\inf\{s>0\ \|\ y+s\omega\not\in G_e \}.$ We observe that for the convex set $G$ actually $\tau_{e,-}(y,\omega)=\infty$ for all $(y,\omega,E)\in \Gamma_{e,-}$. \[tthcon\] The trace mappings $$\gamma_{e,\pm}:W^2(G_e\times S\times I)\to T^2(\Gamma_{e,\pm})$$ are (well-defined) bounded surjective operators with bounded right inverses (lifts) $L_{e,\pm}: T^2(\Gamma_{e,\pm})\to W^2(G_e\times S\times I)$. Again it needs only to consider the trace operator $\gamma_-$. The proof runs similarly to the proof of Theorem \[tth\] with following changes. Instead of estimate (\[trpr2\]) we utilize the inequality $$\begin{gathered} \|f(0)\|^2=\Big\|\int_0^\infty {d\over{dt}}(f(t)^2)dt\Big\| =\Big\|\int_0^\infty 2f'(t)f(t)dt\Big\|\\ \leq 2\Big(\int_0^\infty\|f'(t)\|^2dt\Big)^{1/2} \Big(\int_0^\infty\|f(t)\|^2dt\Big)^{1/2} \leq \int_0^\infty\|f'(t)\|^2dt +\int_0^\infty\|f(t)\|^2dt, \label{trpr13}\end{gathered}$$ which is valid for all $f\in C_0^1([0,\infty[)$ (note that $f(t)=0$ for sufficiently large $t$). In addition, one applies the change of variables $H$ we have $H(W)=G_e$ where $W:=\{(v,t)\ \|\ v\in V_-,\ 0<t<\infty\}$ and $V_-:=\{v\in V\ \|\ \omega\cdot\nu_e(h(v))<0\}$. Let $\lambda>0$. For any $g_{e}\in C(\Gamma_{e,-})$ such that ${{\frac{\partial g_{e}}{\partial \tilde y_i}}}\in C(\Gamma_{e,-})$ the (classical) solution $\Psi$ of the problem \[exle1\] \_x+=&0 D\_e,\ \_[\|\_[e,-]{}]{}=&g\_[e]{}, where $D_e$ is the set (\[D\]) corresponding to $G_e\times S\times I$, is given explicitly by (cf. (\[trath3\])) \[exle2\] (x,,E)= e\^[-t\_e(x,)]{}g\_[e]{}(x-t\_e(x,),,E), &[when]{} t\_e(x,) [is finite]{}\ 0, &[otherwise]{} . Similarly to the proof of Lemma \[trathle1\] we find that (\[exle1\]) holds weakly in $G_e\times S\times I$. For $g_{e}\in T^2(\Gamma_{e,-})$ we define the lift explicitly by \[trpr14\] L\_[e,-]{}g\_[e]{}:=. Then as in the proof of Lemma 5.8 given in [@tervo14] we get that $L_{e,-}g_{e}\in \tilde W^2(G_e\times S\times I)$ and (note that ${\left\Vert \omega\cdot\nabla_x\Psi\right\Vert}_{L^2(G_e\times S\times I)}= \lambda{\left\Vert \Psi\right\Vert}_{L^2(G_e\times S\times I)}$) \[trpr15\] [L\_[e,-]{}g\_[e]{}]{}\_[W\^2(G\_eSI)]{} =&\_[W\^2(G\_eSI)]{}= \_[L\^2(G\_eSI)]{}\ =& [g\_[e]{}]{}\_[T\^2(\_[e,-]{})]{}, since (cf. Remark \[changevar\]) $${\left\Vert \Psi\right\Vert}^2_{L^2(G_e\times S\times I)} ={}&\int_{\Gamma_{e,-}} \int_0^{\tau_{e,-}(x,\omega)} \big(e^{-\lambda s}g_{e}(y,\omega,E)\big)^2 \|\omega\cdot\nu(y)\| ds d\sigma(y)d\omega dE \\ ={}&\frac{1}{2\lambda}\int_{\Gamma_{e,-}} g_{e}(y,\omega,E)^2 \|\omega\cdot\nu(y)\| d\sigma(y)d\omega dE \\ ={}&\frac{1}{2\lambda}{\left\Vert g_e\right\Vert}_{T^2(\Gamma_{e,-})}^2,$$ where $\tau_{e,-}$ is $\tau_-$ for the domain $G_e\times S\times I$. We omit further details. \[wla\] A. By (\[exle2\]) one can show that $$L_{e,-}(g_{e,-})_{\|\Gamma_{e,+}}=0.$$ B. The lift $L_{e,+}:T^2(\Gamma_{e,+})\to W^2(G_e\times S\times I)$ is given by $$(L_{e,+})(x,\omega,E)=\begin{cases}e^{-\lambda t_e(x,-\omega)}g_{e}(x+t_e(x,-\omega)\omega,\omega,E),\ &{\rm when}\ t_e(x,-\omega)\ {\rm is\ finite},\\ 0,\ &{\rm otherwise}.\end{cases}$$ We find that $$L_{e,+}(g_{e,+})_{\|\Gamma_{e,-}} =0.$$ C. Let $g_e\in T^2(\Gamma_e)$ and let $g_{e,\pm}:={g_e}_{\|\Gamma_{e,\pm}}$. Then for $$\tilde{\Psi}:=L_{e,-}(g_{e,-})+L_{e,+}(g_{e,+})\in W^2(G_e\times S\times I),$$ we find that $\tilde{\Psi}\in \tilde{W}^2(G_e\times S\times I)$ and $$\tilde\Psi_{\|\Gamma_{e,\pm}}=g_{e,\pm}.$$ (See Remark \[wl\], Part E.) As a corollary we show the following extension result. In the proof we explicitly construct the [extension of]{} $\psi$ (cf. [@dautraylionsv6 p. 415, proof of Lemma 2]). \[exle\] Suppose that $G\subset{\mathbb{R}}^3$ is as above and that it is convex. Then for any $\psi\in \tilde W^2(G\times S\times I)$ there exists an extension ${{{\mathcal{}}}E}\psi\in W^2({\mathbb{R}}^3\times S\times I)$ of $\psi$ that is, ${{{\mathcal{}}}E}\psi_{\|G\times S\times I}=\psi$. In addition, the linear operator ${{{\mathcal{}}}E}:\tilde W^2(G\times S\times I)\to W^2({\mathbb{R}}^3\times S\times I)$ is bounded. Suppose that $\psi\in\tilde W^2(G\times S\times I)$. Denote $g:=\psi_{\|\Gamma}$, which belongs to $T^2(\Gamma)=T^2(\Gamma_{e})$. Let $\tilde\Psi\in \tilde W^2(G_e\times S\times I)$ given in Remark \[wla\], Part C. Define ${{{\mathcal{}}}E}\psi$ by \[cg5\] [E]{}:= & [on]{} GSI\ \_()=\_[e,]{}() & [on]{} \_\ & [on]{} G\_eSI Then ${{{\mathcal{}}}E}\psi$ is in $W^2({\mathbb{R}}^3\times S\times I)$. This follows from the Green’s formula (\[green\]) since for all $v\in C_0^\infty({\mathbb{R}}^3\times S\times I^\circ)$ &\_[\^3SI]{}([E]{}) (\_x v) dx ddE\ =& \_[GSI]{}([E]{}) (\_x v) dx ddE+ \_[G\_eSI]{}([E]{}) (\_x v) dx ddE\ =& -\_[GSI]{} (\_x ) v dx ddE + \_[GSI]{}() (v) ()dddE\ & - \_[G\_eSI]{} (\_x ) v dx ddE + \_[G\_eSI]{}\_e() \_e(v) (\_e)dddE\ =& -\_[GSI]{} (\_x ) v dx ddE -\_[G\_eSI]{} (\_x ) v dx ddE where we used the facts that $\partial G=\partial G_e$ and $\nu_e=-\nu$ and so $\gamma_{\pm}(\psi)=:g_{\pm}=g_{e,\mp}:=\gamma_{e,\mp}(\tilde\Psi)$. Hence $\omega\cdot\nabla_x ({{{\mathcal{}}}E}\psi)\in L^2({\mathbb{R}}^3\times S\times I)$, as desired. Finally, we find that by (\[trpr15\]) (recall Remark \[wla\], Part C.) &\_[W\^2(\^3SI)]{} =\_[W\^2(GSI)]{}+ \_[W\^2(G\_eSI)]{}\ & \_[W\^2(GSI)]{}+[L\_[e,-]{}(g\_[e,-]{})]{}\_[W\^2(G\_eSI)]{}+ [L\_[e,+]{}(g\_[e,+]{})]{}\_[W\^2(G\_eSI)]{}\ =& \_[W\^2(GSI)]{}+ ([\_+()]{}\_[T\^2(\_[e,-]{})]{}+ [\_-()]{}\_[T\^2(\_[e,+]{})]{}) which implies the boundedness of ${{{\mathcal{}}}E}$. This completes the proof. Let $\tilde {\bf W}^2(G\times S\times I)$ be the completion of $C^1(\ol G\times S\times I)$ with respect to ${\left\Vert \cdot\right\Vert}_{\tilde { W}^2(G\times S\times I)}$-norm. For a convex set $G\subset{\mathbb{R}}^3$ have the following density result. \[convg\] Suppose that $G\subset{\mathbb{R}}^3$ is as above and that it is convex. Then \[cg1\] \^2(GSI)=\^2(GSI), and \[cg2\] H\_2= \^2(GSI)W\_1\^2(GSI). (Definitions of the spaces $W_1^2$, $\tilde{W}^2$ and $H_2$ were given in , and , respectively.) At first, we deal with the claim (\[cg1\]). The inclusion $"\subset"$ is clear and then it suffices to prove only the opposite inclusion. Because $G$ is convex we have by Theorem \[tthcon\] \[trth\] [\_[e]{}()]{}\_[T\^2(\_[e]{})]{}C\_[W\^2(G\_eSI)]{} W\^2(G\_eSI). Let $\psi\in\tilde W^2(G\times S\times I)$ and let ${{{\mathcal{}}}E} \psi\in W^2({\mathbb{R}}^3\times S\times I) $ be its extension provided by Lemma \[exle\]. Since $C^1_0({\mathbb{R}}^3\times S\times {\mathbb{R}})$ is dense in $W^2({\mathbb{R}}^3\times S\times I)$ there exists a sequence $\{\Psi_n\}\subset C^1_0({\mathbb{R}}^3\times S\times {\mathbb{R}})$ such that ${\left\Vert \Psi_n-{{{\mathcal{}}}E}\psi\right\Vert}_{W^2({\mathbb{R}}^3\times S\times I)}\to 0$ for $n\to\infty$. Let $\psi_n:={\Psi_n}_{\|G\times S\times I}$. We have $${\left\Vert \gamma(\psi_n-\psi)\right\Vert}_{T^2(\Gamma)}= {\left\Vert \gamma_e(\Psi_n-{{{\mathcal{}}}E}\psi)\right\Vert}_{T^2(\Gamma_e)}.$$ Hence we get by (\[trth\]) \[cg6\] [\_n-]{}\_[W\^2(GSI)]{}\^2 =& [\_n-]{}\_[ W\^2(GSI)]{}\^2+ [(\_n-)]{}\_[T\^2()]{}\^2\ & [\_n-[E]{}]{}\_[W\^2(GSI)]{}\^2 + C\^2[\_n-[E]{}]{}\_[W\^2(G\_eSI)]{}\^2\ & [\_n-[E]{}]{}\_[W\^2(\^3SI)]{}\^2 + C\^2[\_n-[E]{}]{}\_[W\^2(\^3SI)]{}\^2, which implies that $\psi\in \tilde {\bf W}^2(G\times S\times I)$. This completes the proof of (\[cg1\]). Since $C^1_0({\mathbb{R}}^3\times S\times I)$ is dense in $W_1^2({\mathbb{R}}^3\times S\times I)$ the proof of (\[cg2\]) is quite similar and so the proof is complete. Finally, we notice that the following continuous inclusions are valid $$\begin{gathered} \tilde W^2(G\times S\times I)\subset H; \\ \psi\mapsto (\psi,\gamma(\psi)),\end{gathered}$$ and $$\begin{gathered} \tilde W^2(G\times S\times I)\cap W_1^2(G\times S\times I)\subset H_1;\\ \psi\mapsto (\psi,\gamma(\psi),\psi(\cdot,\cdot,0),\psi(\cdot,\cdot,E_m)).\end{gathered}$$ \[fre1\] Let $\rho_1,\ \rho_2:I\to {\mathbb{R}}$ be positive (weight) functions in $L^\infty(I)$. We can define a linear space (more generally instead of $W^2(G\times S\times I)$) by \[f16\] & W\^2\_[\_1,\_2]{}(GSI) ={L\^2(GSI) \|& & \_1 \_x L\^2(GSI), &\ & & & \_2[E]{}L\^2(GSI)}, which can be equipped with the inner product \[f17\] ,v\_[W\^2\_[\_1,\_2]{}(GSI)]{} =&,v\_[L\^2(GSI)]{}+ \_1\_x,\_1\_x v\_[L\^2(GSI)]{}\ &+\_2[E]{},\_2\_[L\^2(GSI)]{}, rendering $W^2_{\rho_1,\rho_2}(G\times S\times I)$ to a Hilbert space. Similar weighted spaces can be defined generalizing other spaces above. These spaces are needed e.g. in the context of time-dependent transport equations (where $\rho_1=\rho_2=\sqrt{E}$). On Collision Operators {#coll} ---------------------- The differential cross-sections may have singularities, or even hyper-singularities, which would lead to extra (partial differential and) pseudo-differential terms in the transport equation ([@hsiao Sec. 7.1, pp. 353-394]). Instead of explaining systematically the underlying theory, the following slightly informal description suffices for the purposes of this work. First of all, in the case where $\sigma(x,\omega',\omega,E',E)$ has hyper-singularities (like Møller differential cross section given in the below example) the integral $\int_S\int_I$ occurring in the collision operator must be understood in the sense of [Cauchy principal value]{} ${\rm p.v.}\int_S\int_I$ or more generally in the sense of [Hadamard finite part integral]{} ${\mathrm{p.f.}}\int_S\int_I$ ([@hsiao Sec. 3.2], [@chan], [@martin-rizzo], [@schwartz pp. 104-105]). We remark that one encounters this kind of hyper-singularities frequently in physical models. In addition, we must assume that $E_0>0$ in the energy interval $I=[E_0,E_{\rm m}]$, because otherwise $K\psi$, for $\psi\in C_0^\infty(G\times S\times I^\circ)$, might turn out to be (strictly) a distribution, which would increase the complexity of what is presented here. In [@lorence p. 7], it is reported that the differential cross sections are not necessarily valid for very small energies which supports this assumption. Consider the following partial hyper-singular integral operator; for clarity we denote by $S'$ and $I'$ the set $S$ and $I$ when its variable is $\omega'$ and $E'$, respectively, \[coll-1\] (K\_0)(x,,E)=\_[I’]{}\_[S’]{}(x,’,,E’,E)(x,’,E’)d’ dE’. The simplest case is where $\sigma=\sigma_0(x,\omega',\omega,E',E)$ is a measurable non-negative function $G\times S'\times S\times (I'\times I\setminus D)\to{\mathbb{R}}$, where $D=\{(E,E)\ \|\ E\in I=I'\}$ is the diagonal of $I'\times I$, obeying for $E\neq E'$ the estimates $$\begin{aligned} {\mathop{\mathrm{ess\hspace{1mm}sup}}}_{(x,\omega)}\int_{S'}\sigma_0(x,\omega',\omega,E',E)d\omega'\leq {}& {C\over{\|E-E'\|^\kappa}}, \label{coll-3} \\ {\mathop{\mathrm{ess\hspace{1mm}sup}}}_{(x,\omega')}\int_{S}\sigma_0(x,\omega',\omega,E',E)d\omega\leq {}& {C\over{\|E-E'\|^\kappa}}, \label{coll-3a}\end{aligned}$$ where $\kappa<1$, meaning that $\sigma_0(x,\omega',\omega,E',E)$ may have a so-called weak singularity* with respect to energy. We see that $$\begin{gathered} {\mathop{\mathrm{ess\hspace{1mm}sup}}}_{(x,\omega,E)\in G\times S\times I}\int_{I'}\int_{S'}\sigma_0(x,\omega'\omega,E',E)d\omega'dE' \leq \sup_{E} C\int_{I'}{1\over{\|E-E'\|^\kappa}}dE' \\ ={} \sup_{E} C{1\over{1-\kappa}}[(E_{\rm m}-E)^{1-\kappa}+(E-E_0)^{1-\kappa}]\leq {{2CE_{\rm m}^{1-\kappa}}\over{1-\kappa}}, \label{coll-4a}\end{gathered}$$ and similarly for $\int_{I}\int_{S}\sigma_0(x,\omega'\omega,E',E)d\omega dE$. Hence we see that $\sigma_0(x,\omega',\omega,E',E)$ satisfies conditions below, and the corresponding collision operator (K\_0)(x,,E) = \_[I’]{}\_[S’]{}\_0(x,’,,E’,E)(x,’,E’)d’ dE’, is the usual partial Schur (singular) integral operator. It is bounded $L^2(G\times S\times I)\to L^2(G\times S\times I)$. Nevertheless, the collision operator $K$ is not generally of the above form $K_0$. $(E',E)$-dependence in differential cross section $\sigma(x,\omega'\omega,E',E)$ may contain hyper-singularities of higher order, ${1\over{(E'-E)}^m}$, for $m=1,2$ for example; see Example \[moller\] below. Moreover, the $(\omega',\omega)$-dependence in differential cross-sections typically contain Dirac’s $\delta$-distributions (on ${\mathbb{R}}$). More precisely, in $\sigma(x,\omega'\omega,E',E)$ there may occur terms like $\delta(\omega\cdot\omega'-\mu(E',E))$ which require special treatment. We remark, however that $\delta$-distribution can be approximated by smooth functions $\eta_{\epsilon}\in C_0^\infty({\mathbb{R}})$ in the sense that \|()-\_,\_[L\^2()]{}\| [-\_]{}\_[H\^[-1]{}()]{}\_[H\^1()]{},H\^1(), where ${\left\Vert \delta-\eta_\epsilon\right\Vert}_{H^{-1}({\mathbb{R}})}\to 0$ when $\epsilon\to 0^+$. Typically $\eta_\epsilon$ is chosen to be the convolution $\eta_\epsilon:=\delta\star\theta_\epsilon$, where $\theta\in C_0^\infty({\mathbb{R}})$ such that $\int_{{\mathbb{R}}}\theta(x) dx=1$, and $\theta_\epsilon(x):=\epsilon\theta(\epsilon x)$. Hence we are able to replace $\delta(\omega\cdot\omega'-\mu(E',E))$, with $\eta_\epsilon(\omega\cdot\omega'-\mu(E',E))$ which is a well-behaved function. The standard method is first to solve the regularized problem (where $\delta$ is replaced with $\eta_\epsilon$), and then retrieve existence results for the original problem using a limiting process. We shall see that the cross section $\sigma$ may be the form (e.g. in [Møller]{} electron-electron cross-section) $$\begin{gathered} \sigma(x,\omega',\omega,E',E) = \chi(E',E)\Big( {1\over{(E'-E)^2}}\sigma_2(x,\omega',\omega,E',E)\\ +{1\over{E'-E}}\sigma_1(x,\omega',\omega,E',E)+\sigma_0(x,\omega',\omega,E',E)\Big) \label{coll-2}\end{gathered}$$ where $\chi(E',E)$ is a certain product of characteristic functions. Here each of $\sigma_j(x,\omega',\omega,E',E)$, $j=0,1,2$ may contain the above explained $\delta$-distributions, and hence they are not necessarily measurable functions. Denote for $j=0,1,2$, $$(\ol {{{\mathcal{}}}K}_j\psi)(x,\omega,E',E):={}&\int_{S'}\sigma_j(x,\omega',\omega,E',E)\psi(x,\omega',E') d\omega', \\[2mm] (\hat {{{\mathcal{}}}K}_j\psi)(x,\omega,E',E):={}&\chi(E',E)(\ol {{{\mathcal{}}}K}_j\psi)(x,\omega,E',E).$$ In the sequel the integral $\int_{S'}$ is interpreted as a distribution when needed. We shall see, according to the examples below, that at worst $K$ can be of the form (this is corresponding the [Møller]{} scattering for electrons) $$\begin{gathered} ({K}\psi)(x,\omega,E) = {{{\mathcal{}}}H}_2\big((\ol {{{\mathcal{}}}K}_2\psi)(x,\omega,\cdot,E)\big)(E) \\ + {{{\mathcal{}}}H}_1\big((\ol {{{\mathcal{}}}K}_1\psi)(x,\omega,\cdot,E)\big)(E) +\int_{I'}(\hat {{{\mathcal{}}}K}_0\psi)(x,\omega,E',E) dE', \label{co-bb}\end{gathered}$$ where ${{{\mathcal{}}}H}_m$, $m=1,2$, are the [Hadamard finite part operators]{} with respect to $E'$-variable defined by $$({{{\mathcal{}}}H}_m u)(E):={\mathrm{p.f.}}\int_{E}^{E_m}{1\over{(E'-E)^m}}u(E')dE'.$$ The expression is the hyper-singular integral form of $K$. We shall verify that can be equivalently given in the “pseudo-differential form” by \[co-cc\] ([K]{})(x,,E) =& ( [H]{}\_1((\_2)(x,,,E))(E)) - [H]{}\_1(([E]{}(x,,,E))(E)\ & +( (\_2)(x,,E’,E))\_[\|E’=E]{}\ & + [H]{}\_1((\_1)(x,,,E))(E) +\_[I’]{}(\_0)(x,,E’,E) dE’ where only ${{{\mathcal{}}}H}_1$ appears. This formulation reveals the nature of charged particles’ collisions and it enables among others to retrieve existence results of solutions. Note that ${{{\mathcal{}}}H}_1$ is well-defined (at least) for all $u\in C^\alpha(I),\ \alpha>0$ and (cf. [@chan]) \[c-0-a\] ([H]{}\_1u)(E)=\_[E]{}\^[E\_m]{}[[u(E’)-u(E)]{}]{}dE’+ u(E)([E\_m-E]{}) . Moreover, it can perhaps be shown that ${{{\mathcal{}}}H}_1$ is a zeroth-order pseudo-differential operator (cf. [@hsiao Chapter 7]). As a conclusion we find that some interactions produce the first-order partial derivatives with respect to energy $E$ combined with the “zeroth-order” Hadamard part operator. A closer analysis of the operators $\ol{{{\mathcal{}}}K}_j$ reveals that in addition, partial derivatives with respect to $\omega$ may appear. We shall demonstrate that below for $n=2$. The problematic interactions are the primary electron-electron (considered below), primary positron-positron collisions and bremsstrahlung. The operator (\[co-cc\]) (or equivalently (\[co-bb\])) contains two features that require further study: i) The analysis of operators $\hat {{{\mathcal{}}}K}_j,\ j=0,1,2$, and ii) the analysis of the Hadamard finite part operator ${{{\mathcal{}}}H}_1$. The analysis of the existence of the solutions for the transport problem, in the case where these operators are included in the transport operator remains to our understanding open. For the analysis of the examples below we need some preliminaries. Recall the definition of Hadamard finite part integral for discontinuous functions $f:[a,b]\to{\mathbb{R}}$ by [@martin-rizzo], pp. 5 and 32, formulas (14) and (32) therein or [@schwartz p. 104] Applying these definitions (for a fixed $x$) to the function $F_x(t):=\chi_{[x,b]}(t)f(t)$, where $\chi_{[x,b]}(t)$ is the characteristic of the interval $[x,b]$, we have \_a\^b[[F\_x(t)]{}]{}dt =\_x\^[b]{}[[f(t)]{}]{}dt =\_[0]{}(\_[x+]{}\^[b]{}[[f(t)]{}]{}dt +f(x\^+)()) and \_a\^b[[F\_x(t)]{}]{}dt =&\_x\^[b]{}[[f(t)]{}]{}dt\ =&\_[0]{}(\_[x+]{}\^[b]{}[[f(t)]{}]{}dt +f’(x\^+)()-[1]{}f(x\^+)). These formulas give \[hada1\] \_x\^[b]{}[1]{}dt =(b-x), \[hada2\] \_x\^[b]{}[1]{}dt =-[1]{}. Recall the Taylor’s formula (of order $r\in{\mathbb{N}}_0$) for sufficiently smooth functions $f:U\to{\mathbb{R}}$ for open $U\subset {\mathbb{R}}^N$, \[taylor\] f(x)=\_[\|\|r]{}[1]{}[[\^f]{}]{}(x\_0)(x-x\_0)\^+ \_[\|\|=r+1]{} R\_(x)(x-x\_0)\^where the residual (in one of its variant forms) is $$R_\alpha(x):={{\|\alpha\|}\over{\alpha !}}\int_0^1(1-t)^{\|\alpha\|-1} {{\partial^\alpha f}\over{\partial x^\alpha}}(x_0+t(x-x_0))dt.$$ \[hadale\] Suppose that $u\in C^2([a,b])$. Then for $x\in [a,b]$ \[ch-id\] [d]{}(\_x\^[b]{}[[f(t)]{}]{}dt) =\_x\^[b]{}[[f(t)]{}]{}dt-f’(x). In virtue of the Taylor’s formula and (\[hada1\]) \[ch-id1\] I(x):=&\_x\^[b]{}[[f(t)]{}]{}dt\ =& \_x\^[b]{}[[f(x)+(\_0\^1f’(x+s(t-x))ds)(t-x)]{}]{}dt\ =&f(x)(b-x)+\_x\^[b]{}\_0\^1f’(x+s(t-x))dsdt. Hence \[ch-id2\] I’(x)=&f’(x)(b-x)-f(x)[1]{}-f’(x) +\_x\^[b]{}\_0\^1f”(x+s(t-x))(1-s)dsdt On the other hand by the Taylor’s formula \[ch-id3\] &\_x\^[b]{}[[f(t)]{}]{}dt\ =& \_x\^[b]{}[[f(x)+f’(x)(t-x)+(\_0\^1(1-s)f”(x+s(t-x))ds)(t-x)\^2]{}]{}dt\ =&-f(x)[1]{}+f’(x)(b-x)+\_x\^[b]{}\_0\^1(1-s)f”(x+s(t-x))dsdt\ =&I’(x)+f’(x) which implies the claim. We remark that the operators of the form \[r-1\] (Pu)(x,E):=& [p.f.]{}\_[E\_0]{}\^[E\_m]{}[[\_0(x,E,E’)]{}]{}u(x,E,E’)dE’,uC\_0\^(GI\^I\^), can be treated as in [@hsiao Chapter 7]. Note that in (\[r-1\]) the integration is over the whole interval $[E_0,E_m]$. Under relevant assumptions on $\sigma_0$, the operators (\[r-1\]) can (probably) be shown to be pseudo-differential operators. In particular we recall that the [partial Hilbert transform]{} $$(Hu)(x,E):={}& {\rm p.f.}\int_{E_0}^{E_m}{{u(x,E,E')}\over{E-E'}}dE'$$ is a pseudo-differential operator with symbol $-{\rm i}\ {\rm sign}(\xi)$. Recall that the operator ${{{\mathcal{}}}H}_1$ introduced above can be presented in the form \[r-1-a\] ([H]{}\_1u)(x,E):=& [p.f.]{}\_[E]{}\^[E\_m]{}[[\_0(x,E,E’)]{}]{}u(x,E,E’)dE’,uC\_0\^(GI\^I\^). The problematic feature in the expression of $({{{\mathcal{}}}H}_1u)(x,E)$ is that the integration is over $[E,E_m]$. Similar observations concern the operator ${{{\mathcal{}}}H}_2$. In consistence with [@hsiao], formal computations suggest that the expected symbols of ${{{\mathcal{}}}H}_j$, $j=1,2$, are p\_1(x,E,)= \_E\^[[\_0(x,E,E’)]{}]{} e\^[(E’-E)]{}dE’ = \_0\^[[\_0(x,E,E+z)]{}]{} e\^[z]{}dz, and p\_2(x,E,)= \_0\^[[\_0(x,E,E+z)]{}]{} e\^[z]{}dz, respectively. Careful analysis of these operators (whether they are correspondingly zeroth-order and first-order pseudo-differential operators, for example) remains to our knowledge open. The next examples of the two real collision operators, relevant in e.g. radiation therapy, illustrate the above observations. \[compton\] [Photon-electron scattering - Compton-Klein-Nishina]{}. This scattering process describes the collision of a photon with (free) electron, and the corresponding photon$\to$ photon (i.e. $1\to 1$) scattering cross-section is given by (for its derivation from QED, see [@weinberg Section 8.7]; see also [@lorence Section VII]) \_[11]{}(x,’,,E’,E) =& \_[11]{}(x,E’,E)\_[11]{}(E,E’) (’-\_[11]{}(E’,E)), where $$\hat{\sigma}_{11}(x,E',E):={}&\sigma_0(x)\Big({{1}\over{E'}}\Big)^2\Big({{E'}\over{E}}+{{E}\over{E'}}-1+\mu_{11}(E',E)^2\Big) \nonumber\\ \chi_{11}(E',E):={}&\chi_{{\mathbb{R}}_+}(E-E_0)\chi_{{\mathbb{R}}_+}\big(E-\frac{E'}{1+2E'}\big)\chi_{{\mathbb{R}}_+}(E'-E) \\ \mu_{11}(E',E):={}&1+{{1}\over{E'}}-{{1}\over{E}}.$$ Here $(\omega', E')$ and $(\omega, E)$ are, respectively, the (direction, energy) of the incident and the scattered (outgoing) photons. If the scattering angle is written as $\theta_{11}$, then $\omega\cdot\omega'=\cos(\theta_{11})=\mu_{11}(E',E)$, and this condition is enforced by the delta-distribution term in $\sigma_{11}$. We point out that if one defines (for a presentation more or less in this way, see [@hensel Appendix A.1.]), $$\ol{\sigma}_{11}(x,\omega',\omega,E',E) =&\sigma_0(x)\ol{\sigma}_{11}'(E',E)\delta(E-E'\hspace{0.5mm}\ol{P}(\omega,\omega',E')),$$ where $$& P(E',E):=\frac{1}{1+E'(1-\mu_{11}(E',E))}=\frac{E}{E'}, \\ & \ol{P}(\omega,\omega',E'):=\frac{1}{1+E'(1-\omega'\cdot\omega)}, \\ & \ol{\sigma}_{11}'(E',E) :=P(E',E)^2\Big(P(E',E)+\frac{1}{P(E',E)}-1+\mu_{11}(E',E)^2\Big),$$ Then $\sigma_0(x)\ol{\sigma}_{11}'(E',E)=E^2\hat{\sigma}_{11}(x,E',E)$, using which one can further show that the collision operator, say $\ol{K}_{11}$, corresponding to $\ol{\sigma}_{11}$ is equal to the collision operator $K_{11}$ defined by $\sigma_{11}$, i.e. $K_{11}=\ol{K}_{11}$. We find that the operator $\hat {{{\mathcal{}}}K}_{11}$ is given by &(\_[11]{})(x,,E’,E)\ =& \_[11]{}(x,E’,E)\_[11]{}(E’,E)\_[S’]{} (’-\_[11]{}(E’,E))(x,’,E’)d’\ =&\_[11]{}(x,E’,E)\_[11]{}(E’,E)\_[0]{}\^[2]{}(x,(s),E’)ds, where $\gamma=\gamma_{11}(E',E,\omega):[0,2\pi]\to S$ is a parametrization with (constant) speed $${\left\Vert \gamma'(s)\right\Vert}=\sqrt{1-\mu_{11}(E',E)^2},\quad s\in [0,2\pi],$$ of the curve $\Gamma(E',E,\omega)$ which is the intersection of $S$ and the plane which is orthogonal to $\omega$ and whose distance from origin is $\mu_{11}(E',E)$, that is $$\Gamma(E',E,\omega)=\{\omega'\in S\ \|\ \omega'\cdot\omega-\mu_{11}(E',E)=0\}.$$ For example, we can choose (s)=R()((s),(s),\_[11]{}),swhere $\mu_{11}=\mu_{11}(E',E)$, and $R(\omega)$ is any rotation matrix which maps the vector $(0,0,1)$ into $\omega$. Indeed, let $\eta_\epsilon\subset C_0^\infty({\mathbb{R}})$ be such that $\lim_{\epsilon\to 0} \eta_\epsilon\to\delta$ in $H^{-1}({\mathbb{R}})$. Then for $\psi\in C_0^\infty(G\times S\times I^\circ)$, we have (by definition) $$\begin{gathered} \label{path-1} \int_{S'} \delta(\omega'\cdot\omega-\mu_{11}(E',E))\psi(x,\omega',E')d\omega' =\lim_{\epsilon\to 0} \int_{S'} \eta_\epsilon(\omega'\cdot\omega-\mu_{11}(E',E))\psi(x,\omega',E')d\omega'.\end{gathered}$$ For each $t\in [-1,1]$, let $$\gamma_t(s):=R(\omega)(\sqrt{1-t^2}\cos(s),\sqrt{1-t^2}\sin(s),t),\quad s\in [0,2\pi],$$ and let $\Gamma_t$ be the curve (Jordan loop) corresponding to $\gamma_t$, i.e. $\Gamma_t=\gamma_t([0,2\pi])$. Then the (differential) surface measure $d\mu_S(\omega)=d\omega$ on $S$ can be disintegrated into the family $\big(\frac{1}{\sqrt{1-t^2}}d\ell_t\big)\otimes dt$, with $t\in ]-1,1[$, where $d\ell_t (s)=\gamma_t'(s)ds$ is the (differential) path length measure (on $\Gamma_t$) along $\gamma_t$. Note that the differential path length of the circle $S_1(0,t)$ is $\frac{1}{\sqrt{1-t^2}} dt$. In virtue of the Fubini’s Theorem (in disintegration sense), \[path-2\] & \_[S’]{} \_(’-\_[11]{}(E’,E))(x,’,E’)d’\ &= \_[-1]{}\^1\_[\_t]{} \_(-\_[11]{}(E’,E))(x,,E’)d\_t() dt\ &= \_[-1]{}\^1\_[0]{}\^[2]{} \_(t-\_[11]{}(E’,E))(x,\_t(s),E’)[\_t’(s)]{} dsdt\ &= \_[-1]{}\^1\_[0]{}\^[2]{} \_(t-\_[11]{}(E’,E))(x,\_t(s),E’) dsdt\ & \_[0]{} \_[-1]{}\^1\_[0]{}\^[2]{} (t-\_[11]{}(E’,E))(x,\_t(s),E’) dsdt\ &= \_[0]{}\^[2]{} (x,\_[\_[11]{}(E’,E)]{}(s),E’)ds. Hence combining (\[path-1\]) and (\[path-2\]) we get $$\int_{S'} \delta(\omega'\cdot\omega-\mu_{11}(E',E))\psi(x,\omega',E')d\omega' = \int_{0}^{2\pi} \psi(x,\gamma_{\mu_{11}(E',E)}(s),E')ds,$$ as desired, since $\gamma_{\mu_{11}(E',E)}(s)=\gamma(s)$ ($=\gamma_{11}(E',E,\omega)(s)$). It thus follows that (K\_[11]{})(x,,E) =&\_[I’]{} (\_[11]{} )(x,,E’,E)dE\ =& \_[I’]{}\_[11]{}(E’,E)\_[11]{}(x,E’,E) \_[0]{}\^[2]{}(x,(s),E’)ds. Let us approximate $\delta$-distribution with a smooth function, say $\eta_\epsilon$ as above. Then $$(\hat {{{\mathcal{}}}K}_{11}\psi)(x,\omega,E',E) \approx \hat{\sigma}_{11}(x,E',E)\chi_{11}(E',E)\int_{S'} \eta_\epsilon(\omega'\cdot\omega-\mu_{11}(E',E))\psi(x,\omega',E')d\omega'$$ and so &(K\_[11]{})(x,,E’,E) =\_I (\_[11]{} )(x,,E’,E)dE\ & \_[I’]{}\_[S’]{}\_[11]{}(x,E’,E)\_[11]{}(E’,E)\_(’-\_[11]{}(E’,E))(x,’,E’)d’ dE’\ =:& \_[I’]{}\_[S’]{}\_[11]{}(x,’,,E’,E)(x,’,E’)d’ dE’\ =:& (K\_[11,]{})(x,,E), where $K_{11,\epsilon}$ is a partial Schur integral operator satisfying the below assumptions (\[ass2\]). In the case where $\psi$ has the first order derivatives (in the weak sense) with respect to $\omega$ that is, $\psi$ belongs to the mixed-norm Sobolev-Slobodevskij space $H^{2,(0,1,0)}(G\times S\times I^\circ)$. We conjecture that [K\_[11,]{}-K\_[11]{}]{}\_[L\^2(GSI)]{}C[\_-]{}\_[H\^[-1]{}()]{}\_[H\^[2,(0,1,0)]{}(GSI\^)]{} and so the approximation of $K_{11}$ with $K_{11,\epsilon}$ would be under control. \[re:S\_Radon:example\] We make the following remark regarding the fact, as explained in Remark \[re:S\_Radon\], that the unit sphere $S$ could have been equipped with more general Radon measure $\rho$, instead of its typical (Lebesgue-measure induced) measure $\mu_S$. Letting $\rho=H^1$ be the 1-dimensional Hausdorff measure on $S$ (see [@falconer86 pp. 7–10]), and writing $$\underline{\sigma}_{11}(x,\omega',\omega,E',E) =\chi_{11}(E',E)\frac{\sigma_{11}(x,E',E)}{\sqrt{1-\mu_{11}(E',E)^2}}\chi_{{\mathcal{M}}}(\omega',\omega,E',E),$$ where $\chi_{{\mathcal{M}}}$ is the characteristic function of the set $${\mathcal{M}}:=\{(\omega',\omega,E',E)\in S^2\times I^2\ \|\ \omega'\cdot\omega-\mu_{11}(E',E)=0\},$$ then we would have (assuming that pertinent functions are Borel integrable) \[k-11\] (K\_[11]{})(x,,E)=\_[S’I’]{} \_[11]{}(x,’,,E’,E)(x,’,E’)d(’)dE’. This expression has the pleasant feature that the differential cross section $\underline{\sigma}_{11}(x,\omega',\omega,E',E)$ becomes a measurable function on space $G\times S^2\times I^2$. Indeed, the key observation here is that in , the last line can be written as an integral, $$\begin{gathered} \int_{0}^{2\pi}\psi(x,\gamma_{11}(E',E,\omega)(s),E')ds =\frac{1}{\sqrt{1-\mu_{11}(E',E)^2}}\int_{\Gamma(E',E,\omega)} \psi(x,\cdot,E')d\ell, \\ = \frac{1}{\sqrt{1-\mu_{11}(E',E)^2}}\int_{S'} \chi_{{\mathcal{M}}}(\omega',\omega,E',E)\psi(x,\omega',E')d\rho(\omega')\end{gathered}$$ where $\int_{\Gamma(E',E,\omega)} (\cdots)d\ell$ is the path integral along the curve $\Gamma(E',E,\omega)$. Hence we get the expression (\[k-11\]). Finally, for the sake of completeness, let us mention that the Compton-Klein-Nishina photon$\to$ electron (i.e. $1\to 2$) cross section, with corresponding operator $K_{12}$, is given by the following formulas $$\sigma_{12}(x,\omega',\omega,E',E) ={}& \hat\sigma_{12}(x,E',E)\chi_{12}(E,E') \delta(\omega'\cdot\omega-\mu_{12}(E',E)), \\ \hat{\sigma}_{12}(x,E',E):={}& \hat{\sigma}_{11}(x,E',E'-E)\frac{(1+E')^2(1-\mu_{11}(E',E'-E))^2}{\mu_{12}(E',E)^3} \\ \mu_{12}(E',E):={}&\Big(1+\frac{1}{E'}\Big)\sqrt{\frac{E}{E+2}}, \\ \chi_{12}(E',E):={}&\chi_{{\mathbb{R}}_+}(E-E_0)\chi_{{\mathbb{R}}_+}\big(\frac{2{E'}^2}{1+2E'}-E\big)\chi_{{\mathbb{R}}_+}(E'-E),$$ where $(\omega', E')$ is the (direction, energy) of the incident photon, while $(\omega,E)$ is the (direction, kinetic-energy) of the (outgoing) recoil electron. If the scattering angle is written as $\theta_{12}$, then $\omega\cdot\omega'=\cos(\theta_{12})=\mu_{12}(E',E)$. \[moller\] [Electron-electron scattering - Møller]{}. We denote the corresponding differential cross section by $\sigma_{22}(x,\omega',\omega,E',E)$. It has a decomposition ([@duclous], [@lorence], [@boman], [@hensel]) \[i-e-e1\] \_[22]{}(x,’,,E’,E) =\^p\_[22]{}(x,’,,E’,E)+\^s\_[22]{}(x,’,,E’,E). where $\sigma^p_{22}(x,\omega',\omega,E',E)$ is corresponding to the (new) primary electrons and $\sigma^s_{22}(x,\omega',\omega,E',E)$ is corresponding to the secondary electrons. In this scattering process the spins have been averaged out, and the two electrons completely lose their identity. Therefore, categorizing the electrons as “primary” and “secondary” is simply done by assigning the electron exiting the scattering event with the highest energy to be the primary one. The scattering cross section for primary electron $\sigma^p_{22}(x,\omega',\omega,E',E)$ has an expression $$\begin{gathered} \sigma^p_{22}(x,\omega',\omega,E',E) = \sigma_{0}(x){{(E'+1)^2}\over{E'(E'+2)}}\Big({1\over{E^2}} +{1\over{(E'-E)^2}}+{1\over{(E'+1)^2}} \\ -{{2E'+1}\over{(E'+1)^2E(E'-E)}}\Big)\chi_{22,p}(E',E)\delta(\omega'\cdot\omega-\mu_{22,p}(E',E)),\end{gathered}$$ where $\sigma_{0}(x)$ depends on the background material, and $$& \mu_{22,p}(E',E):=\sqrt{{E(E'+2)}\over{E'(E+2)}} \\ & \chi_{22,p}(E',E):=\chi_{{\mathbb{R}}_+}(E-E_0)\chi_{{\mathbb{R}}_+}(E-{E'\over 2})\chi_{{\mathbb{R}}_+}(E'-E),$$ while the cross section for the secondary electron $\sigma^s_{22}(x,\omega',\omega,E',E)$ is $$\begin{gathered} \sigma^s_{22}(x,\omega',\omega,E',E) = \sigma_0(x){{(E'+1)^2}\over{E'(E'+2)}}\Big({1\over{E^2}} +{1\over{(E'-E)^2}}+{1\over{(E'+1)^2}}\\ -{{2E'+1}\over{(E'+1)^2E(E'-E)}}\Big)\chi_{22,s}(E',E)\delta(\omega'\cdot\omega-\mu_{22,s}(E',E)),\end{gathered}$$ where $$& \mu_{22,s}(E',E):=\mu_{22,p}(E',E-E), \\ & \chi_{22,s}(E',E):=\chi_{{\mathbb{R}}_+}({{E'}\over 2}-E)\chi_{{\mathbb{R}}_+}(E-E_0).$$ Since $\sigma^s_{22}(x,\omega',\omega,E',E)=0$ for $E'\leq 2E$ the singularities at $E'=E$ do not cause any problems for the secondary electrons. Write $$\chi_{22}(E',E) :={}& \chi_{22,p}(E',E)+\chi_{22,s}(E',E), \\[2mm] \mu_{22}(E',E) :={}& \begin{cases} \mu_{22,p}(E',E),& E'\leq 2E \\[2mm] \mu_{22,s}(E',E),& E'\geq 2E \end{cases}, \\[2mm] \hat\sigma_{22,0}(x,E',E) :={}& \sigma_{0}(x){{(E'+1)^2}\over{E'(E'+2)}}\Big({1\over{E^2}}+{1\over{(E'+1)^2}}\Big), \\[2mm] \hat\sigma_{22,1}(x,E',E) :={}& -\sigma_{0}(x){{2E'+1}\over{E'(E'+2)E}}, \\[2mm] \hat\sigma_{22,2}(x,E',E) :={}& \sigma_{0}(x){{(E'+1)^2}\over{E'(E'+2)}}.$$ Then we find that $$\begin{gathered} \label{i-e-e2} \sigma_{22}(x,\omega',\omega,E',E) = \chi_{22}(E',E)\Big({1\over{(E'-E)^2}}\hat\sigma_{22,2}(x,E',E)\delta(\omega'\cdot\omega-\mu_{22}(E',E))\\ + {1\over{E'-E}}\hat\sigma_{22,1}(x,E',E)\delta(\omega'\cdot\omega-\mu_{22}(E',E))\\ + \hat\sigma_{22,0}(x,E',E)\delta(\omega'\cdot\omega-\mu_{22}(E',E))\Big).\end{gathered}$$ The operators $\ol {{{\mathcal{}}}K}_{22,j}$ are for any $j=0,1,2$, (\_[22,j]{})(x,,E’,E) = & \_[22,j]{}(x,E’,E) \_[S’]{} (’- \_[22]{}(E,E’))(x,’,E’)d’\ = & \_[22,j]{}(x,E’,E)\_0\^[2]{}(x,(s),E’)ds, where $\gamma=\gamma_{22}(E',E,\omega):[0,2\pi]\to S$ is a parametrization with (constant) speed $${\left\Vert \gamma'(s)\right\Vert}=\sqrt{1-\mu_{22}(E',E)^2},\quad s\in [0,2\pi],$$ of the curve $$\Gamma(E',E,\omega)=\{\omega'\in S\ \|\ \omega'\cdot\omega-\mu_{22}(E',E)=0\}.$$ Writing for $j=0,1,2$, $$(\hat {{{\mathcal{}}}K}_{22,j}\psi)(x,\omega,E',E)=\chi_{22}(E',E) (\ol {{{\mathcal{}}}K}_{22,j}\psi)(x,\omega,E',E),$$ the collision operator $K_{22}$ decomposes into $$K_{22}=K_{22,2}+K_{22,1}+K_{22,0}$$ where \[k220\] (K\_[22,0]{})(x,,E) = \_[I’]{}(\_[22,0]{})(x,,E’,E), dE’ \[k221\] (K\_[22,1]{})(x,,E) =& \_[I’]{}[[(\_[22,1]{})(x,,E’,E)]{}]{} dE’\ =& \_[E]{}\^[E\_m]{}[[(\_[22,1]{})(x,,E’,E)]{}]{} dE’\ =& [H]{}\_1((\_[22,1]{})(x,,,E))(E), and \[k222\] (K\_[22,2]{})(x,,E) =& \_[I’]{}[[(\_[22,2]{})(x,,E’,E)]{}]{}dE’\ =& \_[E]{}\^[E\_m]{}[[(\_[22,2]{})(x,,E’,E)]{}]{}dE’\ =& [H]{}\_2((\_[22,2]{})(x,,,E))(E). Hence by , , , \[k22\] (K\_[22]{})(x,,E) =& [H]{}\_2((\_[22,2]{})(x,,,E))(E) + [H]{}\_1((\_[22,1]{})(x,,,E))(E)\ & +\_[I’]{}(\_[22,0]{})(x,,E’,E) dE’ which is the hyper-singular integral form of $K_{22}$. Applying Lemma \[hadale\] we see that \[k22-a\] & [H]{}\_2((\_[22,2]{})(x,,,E))(E)\ =& ( \_[E]{}\^[E\_m]{}[[(\_[22,2]{})(x,,E’,E)]{}]{}dE’) - \_[E]{}\^[E\_m]{}[1]{}[E]{}(x,,E’,E)dE’\ & +( (\_[22,2]{})(x,,E’,E))\_[\|E’=E]{}\ =& ( [H]{}\_1((\_[22,2]{})(x,,,E))(E)) - [H]{}\_1([E]{}(x,,,E))(E)\ & +( (\_[22,2]{})(x,,E’,E))\_[\|E’=E]{}, Finally, we remark that the approximative $\hat {{{\mathcal{}}}K}_{22,j}$ for $j=0,1,2$ are $$(\hat {{{\mathcal{}}}K}_{22,j}\psi)(x,\omega,E) \approx {}& \int_{S'}\hat\sigma_j(x,E',E)\chi_{22}(E',E) \eta_\epsilon(\omega'\cdot\omega-\mu_{22}(E,E'))\psi(x,\omega',E')d\omega' \\ =: & \int_{S'}\tilde\sigma_j(x,\omega',\omega,E',E)\psi(x,\omega',E')d\omega' =:(\tilde { K}_{22,j}\psi)(x,\omega,E',E).$$ Note that $\tilde K_{22,0}$ is the usual partial Schur integral operator. The approximations $\tilde {{{\mathcal{}}}K}_{22,j}$ may be useful from theoretical and practical point of view. A. When the spatial dimension $n=2$, $$\begin{gathered} (\hat {{{\mathcal{}}}K}_{11}\psi)(x,\omega,E',E) \\ = \hat{\sigma}_{11}(x,E',E)\chi_{11}(E',E)\big(\psi(x,\mu_{11}(E',E)\omega+\sqrt{1-\mu_{11}(E',E)^2}\omega^\perp,E') \\ + \psi(x,\mu_{11}(E',E)\omega-\sqrt{1-\mu_{11}(E',E)^2}\omega^\perp,E')\big),\end{gathered}$$ where $\omega^\perp:=(-\omega_2,\omega_1)$ (the tangent vector of the unit circle $S=S_1$ at $\omega$). B. We compute further some of the above terms. For simplicity we restrict ourselves to the case $n=2$. In this case $$\begin{gathered} \label{n2k-1} (\ol {{{\mathcal{}}}K}_{22,2}\psi)(x,\omega,E',E) = \hat\sigma_2(x,E',E)\big(\psi(x,\mu_{22}(E',E)\omega+\sqrt{1-\mu_{22}(E',E)^2}\omega^\perp,E') \\ + \psi(x,\mu_{22}(E',E)\omega-\sqrt{1-\mu_{22}(E',E)^2}\omega^\perp,E')\big).\end{gathered}$$ Denote $$\xi_{\pm}(E',E,\omega):=\mu_{22}(E',E)\omega\pm\sqrt{1-\mu_{22}(E',E)^2}\omega^\perp.$$ Then \[n2k-2\] ((\_[22,2]{})(x,,E’,E)) =& (\_2(x,E’,E)((x,\_+(E’,E,),E’))\ &+ (\_2(x,E’,E)((x,\_-(E’,E,),E’)). Furthermore, \[n2k-3\] & (\_2(x,E’,E)((x,\_+(E’,E,),E’))\ = & (x,E’,E)(x,\_+(E’,E,),E’)\ &+ \_2(x,E’,E)(\_)(x,\_+(E’,E,),E’) (E’,E,)\ &+ \_2(x,E’,E)(x,\_+(E’,E,),E’) and similarly for the last term in (\[n2k-2\]). Since $\mu_{22}(E,E)=\mu_{22,p}(E,E)=1$ we have \[n2k-4\] & ((\_[22,2]{})(x,,E’,E))\_[\|E’=E]{}\ =& (x,E,E)(x,,E)\ &+ \_2(x,E,E)(\_)(x,,E)(E,E,) + \_2(x,E,E)(x,,E)\ &+ (x,E,E)(x,,E)\ &+ \_2(x,E,E)(\_)(x,,E)(E,E,) + \_2(x,E,E)(x,,E)\ =& 2\_2(x,E,E)(x,,E)+ \_2(x,E,E)((E,E,)+(E,E,)) (\_)(x,,E)\ &+ 2(x,E,E)(x,,E). Finally, \[n2k-5\] & [E]{}(x,,E’,E)\ =& (\_2(x,E’,E)(x,\_+(E’,E,),E’) +\_2(x,E’,E)(x,\_-(E’,E,),E’) )\ =& (x,E’,E)(x,\_+(E’,E,),E’)\ & +\_2(x,E’,E)(\_)(x,\_+(E’,E,),E’)(E’,E)\ & +(x,E’,E)(x,\_-(E’,E,),E’)\ & +\_2(x,E’,E)(\_)(x,\_-(E’,E,),E’) (E’,E). As a conclusion we see that for $n=2$ $$\begin{gathered} (K_{22,2}\psi)(x,\omega,E) = {\partial\over{\partial E}}\Big( {{{\mathcal{}}}H}_1((\ol{{{\mathcal{}}}K}_{22,2}\psi)(x,\omega,\cdot,E))(E)\Big) \\ - {{{\mathcal{}}}H}_1\big(({{\frac{\partial (\ol{{{\mathcal{}}}K}_{22,2}\psi)}{\partial E}}}(x,\omega,\cdot,E)\big)(E) \\ +2\hat\sigma_2(x,E,E){{\frac{\partial \psi}{\partial E}}}(x,\omega,E) +2{{\frac{\partial \hat\sigma_2}{\partial E'}}}(x,E,E)\psi(x,\omega,E) \\ +\hat\sigma_2(x,E,E) \big({{\frac{\partial \xi_+}{\partial E'}}}(E,E,\omega)+{{\frac{\partial \xi_-}{\partial E'}}}(E,E,\omega)\big) \cdot (\nabla_\omega\psi)(x,\omega,E), \label{n2k-6}\end{gathered}$$ where ${{\frac{\partial (\ol{{{\mathcal{}}}K}_{22,2}\psi)}{\partial E}}}$ is computed by (\[n2k-5\]). The corresponding collision operator can be analogously computed in the general dimension $n$ (which we omit here). The [Møller]{} collision term produces first order partial differential terms, along with a Hadamard finite part operator. The exact form of [Møller]{} collision operator allows for accessing relevant approximation schemes for which the error analysis can be carried out. We find that the CSDA-approximation does not take into account the change of angle for the (new) primary electron during transport. Hence the angular derivative ($\nabla_\omega$) is missing from it. On the other hand, CSDA-Focker-Plank approximation contains also second order partial derivatives (with respect to angle) which do not show up in (\[n2k-6\]). As an application of the above analysis we derive a CSDA-type approximation for the [Møller]{} scattering. We apply the hyper-singular integral form of the collision operator $K_{22}$ that is (recall \[k22\])) $$\begin{gathered} \label{csda1} (K_{22}\psi)(x,\omega,E)= {{{\mathcal{}}}H}_2((\ol{{{\mathcal{}}}K}_{22,2}\psi)(x,\omega,\cdot,E))(E) \\ + {{{\mathcal{}}}H}_1((\ol{{{\mathcal{}}}K}_{22,1}\psi)(x,\omega,\cdot,E))(E) + \int_{I'}(\hat{{{\mathcal{}}}K}_{22,0}\psi)(x,\omega,E',E) dE'.\end{gathered}$$ Recall also that the operators $\ol {{{\mathcal{}}}K}_j$, $j=0,1,2$ are $$(\ol {{{\mathcal{}}}K}_{22,j}\psi)(x,\omega,E',E)= \hat\sigma_j(x,E',E) \int_{S'} \delta(\omega'\cdot\omega-\mu_{22}(E,E'))\psi(x,\omega',E')d\omega'$$ and $$(\hat {{{\mathcal{}}}K}_{22,j}\psi)(x,\omega,E',E) =\chi_{22}(E',E)(\ol {{{\mathcal{}}}K}_{22,j}\psi)(x,\omega,E',E).$$ In the case where $E\approx E'$ (as in the case of forward peaked primary electrons) we have \[csda2-a\] \_[22,p]{}(E’,E)1 and then for $E\leq E'\leq 2E$ \[csda3-aa\] (\_[22,j]{})(x,,E’,E) & \_j(x,E’,E) \_[S’]{} (’- 1)(x,’,E’)d’\ =& \_j(x,E’,E) (x,,E’). Assuming (\[csda2-a\]), we have an approximation $$\begin{gathered} \label{csda4} (K_{22,0}\psi)(x,\omega,E)\approx (\tilde K_{22,0}\psi)(x,\omega,E) \\ :=\int_{E}^{2E}\hat\sigma_0(x,E',E) \psi(x,\omega,E')dE'+\int_{2E}^{E_m}(\ol {{{\mathcal{}}}K}_{22,0}\psi)(x,\omega,E',E).\end{gathered}$$ Eq. is the [first CSDA-type approximation]{}. When we apply the approximation \[d-approx\] (’-\_[22]{}(E,E’))\_(’-\_[22]{}(E,E’)) we immediately see that $\tilde K_{22,0}$ is the usual partial Schur integral operator. Consider now the term ${{{\mathcal{}}}H}_1\big((\ol{{{\mathcal{}}}K}_{22,1}\psi)(x,\omega,\cdot,E)\big)(E)$. We have by , \[csda5\] & [H]{}\_1((\_[22,1]{})(x,,,E))(E) = \_E\^[E\_m]{}[1]{}(\_[22,1]{})(x,,E’,E)dE’\ =& \_E\^[2E]{}[1]{}(\_[22,1]{})(x,,E’,E)dE’+ \_[2E]{}\^[E\_m]{}[1]{}(\_[22,1]{})(x,,E’,E)dE’\ & \_E\^[2E]{}[1]{}\_1(x,E’,E)(x,,E’)dE’+ \_[2E]{}\^[E\_m]{}[1]{}(\_[22,1]{})(x,,E’,E)dE’. Due to the Taylor’s formula, & \_E\^[2E]{}[1]{}\_1(x,E’,E)(x,,E’)dE’\ & \_E\^[2E]{}[1]{}(\_1(x,E,E)(x,,E)+ R\_[1,1]{}(E’)(E’-E))dE’ where \[csda5-b\] R\_[1,1]{}(E’)=\_0\^1 (\_1(x,,E)(x,,))(E+t(E’-E))dt. The [second CSDA-type approximation]{} (which is valid if $E\approx E'$) is that \[csda5-c\] R\_[1,1]{}(E’)(E’-E)=0, which gives by (\[hada1\]) that approximately \[csda5-a\] & [H]{}\_1((\_[22,1]{})(x,,,E))(E)\ & \_E\^[2E]{}[1]{}\_1(x,E,E)(x,,E)dE’\ & +\_[2E]{}\^[E\_m]{}[1]{}(\_[22,1]{})(x,,E’,E)dE’\ =& \_1(x,E,E)(x,,E)(E) +(K\_[22,1,2]{})(x,,E), where $$(K_{22,1,2}\psi)(x,\omega,E):= \int_{2E}^{E_m}{1\over{E'-E}}(\ol {{{\mathcal{}}}K}_{22,1}\psi)(x,\omega,E',E)dE'.$$ Applying the approximation (\[d-approx\]), $K_{22,1,2}$ is approximately a partial Schur integral operator, say $\tilde K_{22,1,2}$. Next, consider the term ${{{\mathcal{}}}H}_2((\ol{{{\mathcal{}}}K}_{22,2}\psi)(x,\omega,\cdot,E))(E)$. In virtue of the Taylor’s formula, $$\begin{gathered} \label{csda6} \hat\sigma_2(x,E',E)\psi(x,\omega,E') = \hat\sigma_2(x,E,E)\psi(x,\omega,E) \\ + {\partial\over{\partial E'}}\big(\hat\sigma_2(x,\cdot,E)\psi(x,\omega,\cdot)\big)(E)(E'-E)+R_{2,2}(E')(E'-E)^2, \end{gathered}$$ where $$R_{2,2}(E')={2\over{2!}} \int_0^1(1-t){\partial^2\over{\partial E'^2}}\big(\hat\sigma_2(x,\cdot,E)\psi(x,\omega,\cdot)\big)(E+t(E'-E))dt$$ In the case where $E\approx E'$ we can omit the residual term that is, \[csda6-a\] R\_[2,2]{}(E’)(E’-E)\^2=0 which is the [third CSDA-type approximation]{}. Then we get by (\[hada1\]), (\[hada2\]) approximately \[csda8\] & [H]{}\_2((\_[22,2]{})(x,,,E))(E)\ & \_E\^[2E]{}[1]{}( \_2(x,E,E)(x,,E) + (\_2(x,,E)(x,,))(E))dE’\ & +\_[2E]{}\^[E\_m]{}[1]{}(\_[22,2]{})(x,,E’,E)dE’\ = & -\_2(x,E,E)(x,,E)[1]{} + (\_2(x,,E)(x,,))(E)(E)\ & +\_[2E]{}\^[E\_m]{}[1]{}(\_[22,2]{})(x,,E’,E)dE’\ = & \_2(x,,E)(E)(x,,E) +(K\_[22,2,2]{})(x,,E)\ & +( -\_2(x,E,E)[1]{} +[E]{}(x,E,E)(E))(x,,E), where $$(K_{22,2,2}\psi)(x,\omega,E):= \int_{2E}^{E_m}{1\over{(E'-E)^2}}(\ol{{{\mathcal{}}}K}_{22,2}\psi)(x,\omega,E',E)dE'.$$ Applying the approximation (\[d-approx\]), $K_{22,2,2}$ is approximately a partial Schur integral operator, say $\tilde K_{22,2,2}$. Combining (\[csda8\]), (\[csda5-a\]), (\[csda4\]) we obtain an approximation of the exact [Møller]{} collision operator $$\begin{gathered} \label{csda-app} (\tilde K_{22}\psi)(x,\omega,E)= \hat\sigma_2(x,\cdot,E)\ln(E){{\frac{\partial \psi}{\partial E}}}(x,\omega,E) \\ +\Big( -\hat\sigma_2(x,E,E){1\over{E}} +{{\frac{\partial \hat\sigma_2}{\partial E}}}(x,E,E)\ln(E) +\hat\sigma_1(x,E,E)\ln(E) \Big)\psi(x,\omega,E) \\ + (\tilde K_{22,2,2}\psi)(x,\omega,E)+(\tilde K_{22,2,1}\psi)(x,\omega,E)+ (\tilde K_{22,0}\psi)(x,\omega,E).\end{gathered}$$ Here $\tilde K_{22,2,2}+\tilde K_{22,2,1}+\tilde K_{22,0}$ is a partial Schur integral operator. The above approximations are reasonable only if $\psi\in H^{(0,0,2)}(G\times S\times I^\circ)\cap H^{(0,1,0)}(G\times S\times I^\circ)$. We end this section by bringing up some open issues. The existence and uniqueness analysis for the exact transport equation $T\psi=f$ (with the due inflow boundary and initial conditions). Here $T$ is of the form \[T\] T=& -( [H]{}\_1((\_2)(x,,,E))(E)) + [H]{}\_1(([E]{}(x,,,E))(E)\ & -( (\_2)(x,,E’,E))\_[\|E’=E]{}+ \_x+F\_+\ & - [H]{}\_1((\_1)(x,,,E))(E)-K\_0, where \[K\] K\_0= \_[I’]{}(\_0)(x,,E’,E) dE’ =\_[I’]{}\_0(x,E’,E)\_0\^[2]{}(x,(s),E’)ds dE’, and where $\gamma$ is defined similarly as $\gamma_{11}(E',E,\omega)(s)$ and $\gamma_{22}(E',E,\omega)(s)$ above. In addition, the real, physical model is a coupled system $T=(T_1,T_2,T_3)$ of the operators like (\[T\]) and some terms may be missing in $T_j$ (cf. the system considered in this paper). The existence and uniqueness properties for the exact transport equation $T\psi=f$ with the given inflow boundary and initial conditions remains to be analysed. Potential methods are (as in this paper) Lions-Lax-Milgram Theorem, theory of maximally dissipative operators and the theory of evolution operators. It might be useful to first approximate operator $\ol {{{\mathcal{}}}K}$ by $\ol {{{\mathcal{}}}K}_\epsilon$ as we have outlined above. Then the corresponding transport operator, say $T_\epsilon$, might be a pseudo-differential operator, and hence a rich calculus of pseudo-differential operators could be applied. In this context, one would seek solutions for the approximative problem $T_\epsilon\psi_\epsilon=f$, and then one could try to show that $\psi_\epsilon$ convergences to a solution of $T\psi=f$. It is also important to understand regularity of solutions of BTE in the mixed-norm (anisotropic) Sobolev-Slobodevskij spaces. This is needed e.g. in approximation analysis and, in particular, in numerical analysis (e.g. FEM). Uniqueness and regularity analysis the above derived pseudo-differential form-like expressions of collision operators might also be useful. \[tuned FEM\] At least in existence and uniqueness analysis of solutions, it is more fruitful to use the partial differential (or pseudo-differential) form of the exact transport equation. Nevertheless, the numerical methods might apply directly the hyper-singular partial integral equation (\[co-bb\]). For instance, the Galerkin (discontinuous) finite element methods (FEM) are able to consider hyper-singular partial integral terms. These techniques are well-known e.g. in field of boundary element methods (BEM) where the hyper-singular integral kernels are emerging from single and double layer potentials. We remark that carefully chosen (special) numerical integration schemes, and the choice of basis functions are needed in computing element matrices for hyper-singular integral operators. The applicability of these methods for the problem considered remains an open question. Single Continuous Slowing Down Equation {#single-eq} ======================================= Preliminaries {#presingle-eq} ------------- At first we consider a [single CSDA transport equation]{} given by \[se1\] -[E]{}+\_x+- K= f GSI, where the solution satisfies inflow boundary and initial value conditions $$\begin{aligned} {3} \psi_{\|\Gamma_-}&=g\quad && \textrm{on}\ \Gamma_-, \label{se2} \\[2mm] \psi(\cdot,\cdot,E_{\rm m})&=0\quad && \textrm{on}\ G\times S. \label{se3}\end{aligned}$$ We assume that \[ass1\] L\^(GSI),0 a.e. on $G\times S\times I$. Furthermore, the collision operator is given for $\psi\in L^2(G\times S\times I)$ by \[colla\] (K)(x,,E)=\_[SI]{}(x,’,,E’,E)(x,’,E’)d’ dE’, where $\sigma:G\times S^2\times I^2\to{\mathbb{R}}$ is a non-negative measurable function such that \[ass2\] &\_[SI]{}(x,’,,E’,E)d’ dE’M\_1,\ &\_[SI]{}(x,,’,E,E’)d’ dE’M\_2, for a.e. $(x,\omega,E)\in G\times S\times I$, and we assume that there exists $c\geq 0$ such that \[ass3\] &(x,,E)-\_[SI]{}(x,,’,E,E’)e\^[C(E’-E)]{} d’ dE’ c,\ &(x,,E)-\_[SI]{}(x,’,,E’,E)e\^[C(E-E’)]{} d’ dE’ c, for a.e. $(x,\omega,E)\in G\times S\times I$, and where the constant $c\geq 0$ is specified below (see ). The criterion (\[ass2\]) is called the [Schur criterion]{} (emerged from measure theory) and the corresponding collision operators are [Schur operators]{}. Note that in some cases we will assume $c$ to be strictly positive, $c>0$. This assumption has been relaxed in [@egger2014] (see also [@dautraylionsv6 Remark 15, pp. 241-242]). We have by Cauchy-Schwarz inequality and , \[k-norma\] [K]{}& [\_[SI]{}(,’,,E’,)d’ dE’]{}\_[L\^]{}\^[1/2]{} [\_[SI]{}(,,’,,E’)d’ dE’]{}\_[L\^]{}\^[1/2]{}\ & [M\_1]{}\^[1/2]{}[M\_2]{}\^[1/2]{}, where $L^\infty = L^\infty(G\times S\times I)$, ${\left\Vert K\right\Vert}$ is the norm of $K$ as an operator in $L^2(G\times S\times I)$, and $$\begin{aligned} \label{eq:sigma_norms_Linfty_L1} &{\left\Vert \int_{S\times I}\sigma(\cdot,\omega',\cdot,E',\cdot)d\omega' dE'\right\Vert}_{L^\infty(G\times S\times I)}={\mathop{\mathrm{ess\hspace{1mm}sup}}}_{(x,\omega,E)\in G\times S\times I}\int_{S\times I}\sigma(x,\omega',\omega,E',E)d\omega' dE', \nonumber\\ & {\left\Vert \int_{S\times I}\sigma(\cdot,\cdot,\omega',\cdot,E')d\omega' dE'\right\Vert}_{L^\infty(G\times S\times I)}={\mathop{\mathrm{ess\hspace{1mm}sup}}}_{(x,\omega,E)\in G\times S\times I}\int_{S\times I}\sigma(x,\omega,\omega',E,E')d\omega' dE'.\end{aligned}$$ In what follows, we assume that the stopping power $S_0:\ol G\times I\to{\mathbb{R}}$ satisfies the following assumptions: & S\_0L\^(GI), \[csda9\]\ & L\^(GI), \[csda9aa\]\ & :=\_[(x,E)GI]{}S\_0(x,E)>0,\[csda9a\]\ & \_x S\_0L\^(GI). \[csda9b\] We remark that the assumption (\[csda9b\]) will be needed only in the context of the theory of evolution operators in section \[evcsd\]. \[rese1a\] Assume that $\tilde{\sigma}:G\times S^2\times I\to{\mathbb{R}}$ is a measurable non-negative function, and that the cross-section $\sigma$ is (formally) of the form $$\sigma(x,\omega',\omega,E',E)=\tilde\sigma(x,\omega',\omega,E)\delta(E'-E),$$ in the sense that the collision operator $K$ is given by \[collb\] (K)(x,,E)=\_S(x,’,,E)(x,’,E)d’,L\^2(GSI). In this case the assumptions (\[ass2\]) and (\[ass3\]) for any $C$ mean that \[ass2a\] &\_S(x,’,,E)d’ M\_1,\ &\_S(x,,’,E)d’M\_2, for a.e. $(x,\omega)\in G\times S\times I$, and \[ass2b\] &(x,,E)-\_S(x,,’,E) d’ c,\ &(x,,E)-\_S(x,’,,E) d’ c, for a.e. $(x,\omega,E)\in G\times S\times I$. All results proved in this paper are valid for these (simplified) collision operators. The estimate (\[k-norma\]) in this case is \[k-normb\] & [K]{} [\_S(,’,,)d’]{}\_[L\^(GSI)]{}\^[1/2]{} [\_S(,,’,)d’]{}\_[L\^(GSI)]{}\^[1/2]{} \^[1/2]{}[M\_2]{}\^[1/2]{}, where ${\left\Vert K\right\Vert}$ is the norm of $K$ as an operator in $L^2(G\times S\times I)$, and $$&{\left\Vert \int_S\tilde\sigma(\cdot,\omega',\cdot,\cdot)d\omega'\right\Vert}_{L^\infty(G\times S\times I)}={\mathop{\mathrm{ess\hspace{1mm}sup}}}_{(x,\omega,E)\in G\times S\times I} \int_S\tilde\sigma(x,\omega',\omega,E)d\omega' \\ &{\left\Vert \int_S\tilde\sigma(\cdot,\cdot,\omega',\cdot)d\omega'\right\Vert}_{L^\infty(G\times S\times I)}={\mathop{\mathrm{ess\hspace{1mm}sup}}}_{(x,\omega,E)\in G\times S\times I} \int_S\tilde\sigma(x,\omega,\omega',E)d\omega'$$ We begin with a lemma. \[csdale0\] For all $\psi\in C^1(\ol G\times S\times I)$, $$\begin{gathered} \label{se4} {\left\langle}{{\frac{\partial (S_0\psi)}{\partial E}}},\psi{\right\rangle}_{L^2(G\times S\times I)}\leq q{\left\Vert \psi\right\Vert}^2_{L^2(G\times S\times I)} \\ +\frac{1}{2}\int_{G\times S} \big(S_0(x,E_m)\psi^2(x,\omega,E_m)- S_0(x,0)\psi^2(x,\omega,0)\big)dx d\omega,\end{gathered}$$ where \[q\] q:=[12]{}\_[(x,E)GI]{}[E]{}(x,E). Integrating by parts, we have \[se5\] &,\_[L\^2(GSI)]{} = ,\_[L\^2(GSI)]{} +,S\_0\_[L\^2(GSI)]{}\ =& ,\_[L\^2(GSI)]{} -, [E]{}\_[L\^2(GSI)]{}\ &+\_[GS]{} (S\_0(x,E\_m)(x,,E\_m)\^2 - S\_0(x,0)(x,,0)\^2)dx d, and therefore &2,\_[L\^2(GSI)]{}\ =& ,\_[L\^2(GSI)]{} +\_[GS]{} (S\_0(x,E\_m)(x,,E\_m)\^2 - S\_0(x,0)(x,,0)\^2) dx d\ & 2q\^2\_[L\^2(GSI)]{} +\_[GS]{} (S\_0(x,E\_m)(x,,E\_m)\^2-S\_0(x,0)(x,,0)\^2) dx d. This finishes the proof. Note that if $E\mapsto S_0(x,E)$ is decreasing for every $x\in G$ then $q\leq 0$ (and therefore $C$ below vanishes). Let $$\begin{aligned} \label{eq:def_C} C:=\frac{\max\{q,0\}}{\kappa}.\end{aligned}$$ We make the following change of the unknown function. We replace $\psi$ by $$\begin{aligned} \label{eq:exp_trick} \phi(x,\omega,E):=e^{CE}\psi(x,\omega,E).\end{aligned}$$ This substitution changes the equation (\[se1\]) to (here and below by writing $e^{CE}$ we mean a function $(x,\omega,E)\mapsto e^{CE}$) \[csda3A\] -[E]{}+\_x+C S\_0+-K\_C=e\^[CE]{}f, where $K_C$ is given by \[collc\] (K\_C)(x,,E)=\_[SI]{}(x,’,,E’,E) e\^[C(E-E’)]{}(x,’,E’)d’ dE’ . The inflow boundary and the initial conditions are \_[\|\_-]{}=&e\^[CE]{}g, \[finalbc\]\ (x,,E\_m)=&0, \[finalic\] the latter (initial) condition holding for a.e. $(x,\omega)\in G\times S$. \[csdale1a\] Assume that the conditions (\[ass1\]), (\[ass2\]) and (\[ass3\]) are valid. Then $$\Sigma-K_C:L^2(G\times S\times I)\to L^2(G\times S\times I)$$ is a bounded operator and it satisfies the following accretivity condition \[se7\] (-K\_C),\_[L\^2(GSI)]{}c\^2\_[L\^2(GSI)]{}, L\^2(GSI). Since $I=[0,E_{\rm m}]$, where $E_{\rm m}<\infty$, we find that the conditions (\[ass2\]) are valid also for $\sigma_C(x,\omega',\omega,E',E):=\sigma(x,\omega',\omega,E',E)e^{C(E-E')}$, possibly with different $M_1,\ M_2$. This observation combined with assumptions (\[ass1\]), (\[ass3\]), implies the result by a fairly simple application of Cauchy-Schwarz inequality, see [@dautraylionsv6 Theorem 4, p. 241]. In the study below, the standing assumption for the (inflow) boundary condition is $g\in T^2(\Gamma_-)$ (single equation), or $g\in T^2(\Gamma_-)^3$ (coupled equation), however we point out that in some parts, one could do with the more general assumption $g\in T^2_{\tau_-}(\Gamma_-)$, or $g\in T^2_{\tau_-}(\Gamma_-)^3$. In this paper we omit these generalizations. Existence of Solutions for a Single Continuous Slowing Down Equation by a Variational Formulation {#esols} ------------------------------------------------------------------------------------------------- We shall consider the existence of solutions for the problem (\[csda3A\]), (\[finalbc\]), (\[finalic\]) by applying the so-called Lions-Lax-Milgram Theorem (or generalized Lax-Milgram Theorem) which is based on the variational formulation. Hence we begin with some computations which lead one to find the related bilinear and linear forms. In the following we denote \[eq:b\_fg\] [f]{}=e\^[CE]{}f, =e\^[CE]{}g . Let $f_-$ ($f_+$) be the negative (positive) part of a function. Recall that f=f\_+-f\_-\|f\|=f\_++f\_-. Applying the Green’s formula (\[green\]) and integrating by parts, we have for and $\phi\in C^1(\ol G\times S\times I)$ that satisfies the equation (\[csda3A\]) and for any $v\in C^1(\ol G\times S\times I)$, \[csda25\] &-,v\_[L\^2(GSI)]{}+\_x,v\_[L\^2(GSI)]{}+CS\_0,v\_[L\^2(GSI)]{}\ &+,v\_[L\^2(GSI)]{}-K\_C,v\_[L\^2(GSI)]{}\ =& ,S\_0[E]{}\_[L\^2(GSI)]{}- \_[GS]{} S\_0v\|\_[E=0]{}\^[E=E\_[m]{}]{} dx d\ &-,\_x v\_[L\^2(GSI)]{} +\_[GSI]{}()v dddE\ &+,CS\_0v\_[L\^2(GSI)]{} +,\^\* v\_[L\^2(GSI)]{}-,K\_C\^\v\_[L\^2(GSI)]{}\ =&[f]{},v\_[L\^2(GSI)]{} where \^\=and (K\_C\^\v)(x,,E)= \_[SI]{}(x,,’,E,E’) e\^[C(E’-E)]{}v(x,’,E’)d’ dE’. Assuming that the inflow boundary condition $\phi_{\|\Gamma_-}={\bf g}$ and the initial condition $\phi(\cdot,\cdot,E_{\rm m})=0$ are valid, the equation (\[csda25\]) is equivalent to \[csda26\] & ,S\_0[E]{}\_[L\^2(GSI)]{}+ (,,0),S\_0(,,0) v(,,0)\_[L\^2(GS)]{} -,\_x v\_[L\^2(GSI)]{}\ &+\_[GSI]{}()\_+v dddE +,CS\_0v\_[L\^2(GSI)]{}\ &+,\^\ v\_[L\^2(GSI)]{}-,K\_C\^\v\_[L\^2(GSI)]{}\ =&[f]{},v\_[L\^2(GSI)]{}+\_[GSI]{}()\_-[g]{} v dddE. Clearly \_[GSI]{}()\_-[g]{} v dddE= [g]{},\_-(v)\_[T\^2(\_-)]{} and \_[GSI]{}()\_+v dddE= \_+(),\_+(v)\_[T\^2(\_+)]{}. One thus deduces that the relevant bilinear from $B$ and linear form $F$ are \[csda27\] B(,v)=& ,S\_0[E]{}\_[L\^2(GSI)]{} -,\_x v\_[L\^2(GSI)]{}\ &+C,S\_0v\_[L\^2(GSI)]{}+,(\^\-K\_C\^\) v\_[L\^2(GSI)]{}\ &+\_+(),\_+(v)\_[T\^2(\_+)]{} +(,,0),S\_0(,0) v(,,0)\_[L\^2(GS)]{}, and $$F(v)={\left\langle}{\bf f},v{\right\rangle}_{L^2(G\times S\times I)}+{\left\langle}{\bf g},\gamma_-(v){\right\rangle}_{T^2(\Gamma_-)}.$$ The variational equation corresponding to the problem (\[csda3A\]), (\[finalbc\]), (\[finalic\]) (in the classical sense) is $$B(\phi,v)=F(v)\quad \forall v\in C^1(\ol G\times S\times I).$$ We show that the bilinear form $B:C^1(\ol G\times S\times I)\times C^1(\ol G\times S\times I)\to{\mathbb{R}}$ obeys the following [boundedness and coercivity]{} conditions: \[csdath1\] Suppose that the assumptions (\[ass1\]), (\[ass2\]), (\[ass3\]) (with $C=\frac{\max\{q,0\}}{\kappa}$ and $c>0$) and (\[csda9\]), (\[csda9aa\]), (\[csda9a\]) are valid. Then there exists a constant $M>0$ such that \[csda29\] \|B(,v)\|M\_[H\_1]{}[v]{}\_[ H\_2]{}, vC\^1(GSI) and \[csda30\] B(,)c’\_[H\_1]{}\^2C\^1(GSI) where we assume that \[cprime\] c’:={[12]{},[2]{},c} is strictly positive. Note that $\kappa$ is defined in and $c$ in . (Recall that the spaces $H_1$ and $H_2$ were defined in equations and , respectively.) A. At first we show the boundedness of $B(\cdot,\cdot)$. The assumptions , and the fact that the (energy) interval $I$ is bounded imply by the Sobolev Embedding theorem (see ) that $S_0(\cdot,0)\in L^\infty(G)$. Hence we find that \[csda31\] \|B(,v)\|& \_[L\^2(GSI)]{}[S\_0]{}\_[L\^(GI)]{}\_[L\^2(GSI)]{}\ &+\_[L\^2(GSI)]{} [\_x v]{}\_[L\^2(GSI)]{}\ &+C[S\_0]{}\_[L\^(GI)]{}\_[L\^2(GSI)]{}[v]{}\_[L\^2(GSI)]{}\ &+\_[L\^2(GSI)]{}[(-K\_C)\^\]{}[v]{}\_[L\^2(GSI)]{}\ &+[()]{}\_[T\^2()]{} [(v)]{}\_[T\^2()]{}\ &+[(,,0)]{}\_[L\^2(GS)]{}[S\_0(,0)]{}\_[L\^(G)]{}[v(,,0)]{}\_[L\^2(GS)]{}. This implies the assertion by the definition of the spaces $H_1$, $H_2$, and after observing that Sobolev Embedding theorem (see ) implies the existence of a constant $C'\geq 0$ such that $${\left\Vert w(\cdot,\cdot,0)\right\Vert}_{L^2(G\times S)}\leq C'{\left\Vert w\right\Vert}_{H_2},\quad \forall w\in H_2.$$ B. We verify the coercitivity (\[csda30\]). Integrating by parts we have $$\begin{gathered} \label{csda32} {\left\langle}\phi,S_0{{\frac{\partial \phi}{\partial E}}}{\right\rangle}_{L^2(G\times S\times I)} = -{\left\langle}\phi,{{\frac{\partial (S_0\phi)}{\partial E}}}{\right\rangle}_{L^2(G\times S\times I)} \\ +{\left\langle}\phi(\cdot,\cdot,E_m),S_0(\cdot,E_m)\phi(\cdot,\cdot,E_m){\right\rangle}_{L^2(G\times S)}- {\left\langle}\phi(\cdot,\cdot,0),S_0(\cdot,0)\phi(\cdot,\cdot,0){\right\rangle}_{L^2(G\times S)}\end{gathered}$$ Using the Green’s formula (\[green\]) we have \[csda33\] -, \_x \_[L\^2(GSI)]{} =\_x,\_[L\^2(GSI)]{} -\_[GSI]{}()\^2 dddE\ which implies \[csda33a\] \_x,\_[L\^2(GSI)]{} =&\_[GSI]{}()\^2 dddE\ =& \_[GSI]{}(()\_+-()\_-)\^2 dddE\ =&([\_+()]{}\_[T\^2(\_+)]{}\^2-[\_-()]{}\_[T\^2(\_-)]{}\^2). By the definition of $C$ and assumption , we have $$C=\frac{\max\{q,0\}}{\kappa}\geq \frac{\max\{q,0\}}{S_0}\geq \frac{q}{S_0}$$ a.e. and hence $$C{\left\langle}\phi,S_0\phi{\right\rangle}_{L^2(G\times S\times I)}\geq q{\left\Vert \phi\right\Vert}_{L^2(G\times S\times I)}^2.$$ Taking Lemmas \[csdale0\] and \[csdale1a\] into account, one can thus estimate $$B(\phi,\phi)={}& -{\left\langle}\phi,{{\frac{\partial (S_0\phi)}{\partial E}}}{\right\rangle}_{L^2(G\times S\times I)} +{\left\langle}\phi(\cdot,\cdot,E_m),S_0(\cdot,E_m)\phi(\cdot,\cdot,E_m){\right\rangle}_{L^2(G\times S)} \nonumber\\ &-\frac{1}{2}\big({\left\Vert \gamma_+(\phi)\right\Vert}_{T^2(\Gamma_+)}^2-{\left\Vert \gamma_-(\phi)\right\Vert}_{T^2(\Gamma_-)}^2\big) +{\left\Vert \gamma_+(\phi)\right\Vert}_{T^2(\Gamma_+)}^2 \nonumber\\ &+C{\left\langle}\phi,S_0\phi{\right\rangle}_{L^2(G\times S\times I)}+{\left\langle}(\Sigma-K_C)\phi,\phi{\right\rangle}_{L^2(G\times S\times I)} \nonumber\\ \geq {}& -q{\left\Vert \phi\right\Vert}^2_{L^2(G\times S\times I)} -\frac{1}{2}{\left\langle}\phi(\cdot,\cdot,E_m),S_0(\cdot,E_m)\phi(\cdot,\cdot,E_m){\right\rangle}_{L^2(G\times S)} \\ &+\frac{1}{2}{\left\langle}\phi(\cdot,\cdot,0),S_0(\cdot,0)\phi(\cdot,\cdot,0){\right\rangle}_{L^2(G\times S)} +{\left\langle}\phi(\cdot,\cdot,E_m),S_0(\cdot,E_m)\phi(\cdot,\cdot,E_m){\right\rangle}_{L^2(G\times S)}\nonumber\\ & +\frac{1}{2}{\left\Vert \gamma(\phi)\right\Vert}_{T^2(\Gamma)}^2 +q{\left\Vert \phi\right\Vert}_{L^2(G\times S\times I)}^2+c{\left\Vert \phi\right\Vert}^2_{L^2(G\times S\times I)} \\ \geq {}& \frac{\kappa}{2}\big({\left\Vert \phi(\cdot,\cdot,0)\right\Vert}_{L^2(G\times S)}^2 +{\left\Vert \phi(\cdot,\cdot,E_m)\right\Vert}_{L^2(G\times S)}^2\big) +\frac{1}{2}{\left\Vert \gamma(\phi)\right\Vert}_{T^2(\Gamma)}^2 +c{\left\Vert \phi\right\Vert}^2_{L^2(G\times S\times I)}.$$ Because $C^1(\ol G\times S\times I)\times C^1(\ol G\times S\times I)$ is dense in $H_1\times H_2$ and since holds, the bilinear form $B(\cdot,\cdot): C^1(\ol G\times S\times I)\times C^1(\ol G\times S\times I)\to{\mathbb{R}}$ has an unique extension $\tilde B(\cdot,\cdot):H_1\times H_2\to{\mathbb{R}}$ which satisfies \[csda36\] \|B(,v)\|M\_[H\_1]{}[v]{}\_[H\_2]{}H\_1, vH\_2 and \[csda37\] B(v,v)c’[v]{}\_[H\_1]{}\^2vH\_2. We see that actually $$\begin{aligned} \label{coex} \tilde B(\tilde\phi,v) ={}&{\left\langle}\phi,S_0{{\frac{\partial v}{\partial E}}}{\right\rangle}_{L^2(G\times S\times I)} -{\left\langle}\phi,\omega\cdot\nabla_x v{\right\rangle}_{L^2(G\times S\times I)} \nonumber\\ &+C{\left\langle}\phi,S_0v{\right\rangle}_{L^2(G\times S\times I)}+{\left\langle}\phi,(\Sigma^-K_C^) v{\right\rangle}_{L^2(G\times S\times I)} \nonumber\\ &+{\left\langle}q_{\|\Gamma_+},\gamma_+(v){\right\rangle}_{T^2(\Gamma_+)}+{\left\langle}p_0,S_0(\cdot,0) v(\cdot,\cdot,0){\right\rangle}_{L^2(G\times S)},\end{aligned}$$ when $\tilde\phi=(\phi,q,p_0,p_m)\in H_1$ and $v\in H_2$. In addition, since for $v\in C^1(\ol G\times S\times I)$ we have ${\left\Vert \gamma_-(v)\right\Vert}_{T^2(\Gamma_-)}\leq {\left\Vert \gamma(v)\right\Vert}_{T^2(\Gamma)}$, it follows that \[csda39\] \|F(v)\| & \|[f]{},v\_[L\^2(GSI)]{}\|+\|[g]{},\_-(v)\_[T\^2(\_-)]{}\|\ & \_[L\^2(GSI)]{}[v]{}\_[L\^2(GSI)]{}+\_[T\^2(\_-)]{}[(v)]{}\_[T\^2()]{}, and therefore, since $C^1(\ol{G}\times S\times I)$ is dense in $H_1$, the linear form $F:C^1(\ol{G}\times S\times I)\to{\mathbb{R}}$ has a unique bounded extension, which we still denote by $F$, $$\begin{aligned} \label{Fex} F:H_1\to{\mathbb{R}};\quad F(\tilde{\phi})={\left\langle}{\bf f},\phi{\right\rangle}_{L^2(G\times S\times I)}+{\left\langle}{\bf g}, q{\right\rangle}_{T^2(\Gamma_-)},\end{aligned}$$ when $\tilde\phi=(\phi,q,p_0,p_m)\in H_1$. Recall also that the embedding $H_2\subset H_1$ is continuous. We need the following result, so called [Lions-Lax-Milgram Theorem]{} (generalized Lax-Milgram Theorem). \[glm\] Let $X$ and $Y$ be Hilbert spaces, with $Y$ continuously embedded into $X$. Assume that $B(\cdot,\cdot):X\times Y\to{\mathbb{R}}$ is a bilinear form satisfying the following properties with $M\geq 0,\ c>0$, \[csda38\] \|B(u,v)\|M[u]{}\_[X]{}[v]{}\_[Y]{}uX, vY ([boundedness]{}) and \[csda37a\] B(v,v)c[v]{}\_[X]{}\^2vY ([coercivity]{}). Suppose that $F:X\to{\mathbb{R}}$ is a bounded linear form. Then there exists $u\in X$ (possibly non-unique) such that \[csda38-a\] B(u,v)=F(v)vY. See e.g. [@treves p. 403] or [@grisvard p. 234]. Let $$P(x,\omega,E,D)\phi:= -{{\frac{\partial (S_0\phi)}{\partial E}}}+\omega\cdot\nabla_x\phi$$ The space $$\begin{gathered} {{{\mathcal{}}}H}_P(G\times S\times I^\circ):=\{\phi\in L^2(G\times S\times I)\ \| \\ P(x,\omega,E,D)\phi\in L^2(G\times S\times I)\ {\rm in\ the \ weak\ sense}\}\end{gathered}$$ is a Hilbert space when equipped with the inner product (cf. section \[m-d\]) $${\left\langle}\phi,v{\right\rangle}_{{{{\mathcal{}}}H}_P(G\times S\times I^\circ)}={\left\langle}\phi,v{\right\rangle}_{L^2(G\times S\times I)}+{\left\langle}P(x,\omega,E,D)\phi,P(x,\omega,E,D)v{\right\rangle}_{L^2(G\times S\times I)}.$$ With this notation, the equations and can be written as $$\begin{aligned} P(x,\omega,E,D)\psi+\Sigma\psi - K\psi = f,\end{aligned}$$ and $$\begin{aligned} P(x,\omega,E,D)\phi+C S_0\phi+\Sigma\phi-K_C\phi=e^{CE}f,\end{aligned}$$ respectively. In the context if Lions-Lax-Milgram Theorem we shall make us of the following assumption which we call as ${\bf TC}$: \[as:TC\] Let $\gamma_{\pm}(\phi)=\phi_{\|\Gamma_{\pm}}$ and $\gamma_{\rm m}(\phi):=\phi(\cdot,\cdot,E_{\rm m}),\ \gamma_{0}(\phi):=\phi(\cdot,\cdot,0)$. The assumption is that the linear maps $$\begin{aligned} \gamma_{\pm}:{{{\mathcal{}}}H}_P(G\times S\times I^\circ) & \to L_{\rm loc}^2(\Gamma_{\pm},\|\omega\cdot\nu\| d\sigma d\omega dE), \\ \gamma_{\rm m}:{{{\mathcal{}}}H}_P(G\times S\times I^\circ) & \to L_{\rm loc}^2(G\times S), \\ \gamma_{0}:{{{\mathcal{}}}H}_P(G\times S\times I^\circ) & \to L_{\rm loc}^2(G\times S),\end{aligned}$$ are well-defined and continuous. In the case where $S=S(E)$ is independent of $x$ and $S\in C(I)$ one can show that the assumption ${\bf TC}$ holds. The proof can be based on the techniques of the proof of Theorem \[tth\] and Example \[desolex1\]. If $S=S(x,E)$ depends also on $x$, we conjecture that it suffices only to assume that $S$ is regular enough. Let $$P'(x,\omega,E,D)v= S_0{{\frac{\partial v}{\partial E}}}-\omega\cdot\nabla_x v$$ be the formal transpose of $P(x,\omega,E,D)$. Making the assumption ${\bf TC}$ the (extended) Green formula \[green-ex\] & \_[GSI]{}(P(x,,E,D) )v dxddE -\_[GSI]{}(P’(x,,E,D) v) dxddE\ =& \_[GSI]{}() v dddE\ & + \_[GS]{}(S\_0(,0)(,,0)v(,,0)-S\_0(,E\_[m]{})(,,E\_[m]{})v(,,E\_[m]{}))dx dis valid for all $\phi,\ v\in {{{\mathcal{}}}H}_P(G\times S\times I^\circ)$ for which $({\rm supp}(v))\cap \partial (G\times S\times I)$ is a compact subset of $\Gamma_-\cup \Gamma_+\cup (G\times S\times\{E_m\})\cup(G\times S\times\{0\})$. Moreover, (\[green-ex\]) holds for $\phi,\ v\in {{{\mathcal{}}}H}_P(G\times S\times I^\circ)$ when $\gamma_{\pm}(\phi)\in T^2(\Gamma_{\pm})$ and $\gamma_{\rm m}(\phi),\ \gamma_0(\psi)\in L^2(G\times S)$. We omit the proof of both these claims. We are now in position to formulate and prove the following theorem. \[csdath3\] Suppose that the assumptions (\[ass1\]), (\[ass2\]), (\[ass3\]) (with $C=\frac{\max\{q,0\}}{\kappa}$ and $c>0$) and (\[csda9\]), (\[csda9aa\]), (\[csda9a\]) are valid. Let ${\bf f}\in L^2(G\times S\times I)$ and ${\bf g}\in T^2(\Gamma_-)$. Then the following assertions hold. \(i) The variational equation (see , ) \[csda40a\] (,v)=F(v)vH\_2, has a solution $\tilde\phi=(\phi,q,p_0,p_{\rm m})\in H_1$. Furthermore, $\phi \in {{{\mathcal{}}}H}_P(G\times S\times I^\circ)$ and it is a weak (distributional) solution of the equation (\[csda3A\]). \(ii) Suppose that additionally the assumption ${\bf TC}$ holds. Then a solution $\phi$ of the equation obtained in part (i) is a solution of the problem , , . In addition, we have $q_{\|\Gamma_+}=\gamma_+(\phi)$ and $p_0=\phi(\cdot,\cdot,0)$, when $\tilde{\phi}=(\phi,q,p_0,p_m)$ is a solution in $H_1$ obtained in part (i). \(iii) Under the assumptions imposed in part (ii), any solution $\phi\in {\mathcal{H}}_P(G\times S\times I^\circ)$ of the problem , , that further satisfies \[asscl\] \_[\|\_+]{}T\^2(\_+)(,,0)L\^2(GS), is unique and obeys the estimate \[csda40aa\] \_[[H\_1]{}]{} (\_[L\^2(GSI)]{}+\_[T\^2(\_-)]{}), where $c'$ is given in . The proof is based on “variations” and it is quite standard. \(i) We apply Theorem \[glm\] with $X=H_{1}$, $Y=H_{2}$, and with $B(\cdot,\cdot)=\tilde B(\cdot,\cdot)$ and $F$ given by and , respectively. As mentioned above $\tilde B(\cdot,\cdot)$ satisfies (\[csda38\]) and (\[csda37a\]), while $F$ is a bounded linear functional, hence Theorem \[glm\] guarantees the existence of a solution $\tilde\phi=(\phi,q,p_0,p_m)\in H_1$ such that (\[csda40a\]) holds. We verify that $\phi\in L^2(G\times S\times I)$ is a weak solution of the equation (\[csda3A\]). Let $I^\circ:=]0,E_m[$. From (\[csda40a\]) it follows that \[csda41\] B(,v)=F(v),vC\_0\^(GSI\^). Since for $v\in C_0^\infty(G\times S\times I^\circ)$ we have $v(\cdot,\cdot,0)=v(\cdot,\cdot,E_m)=0$ and $v_{\|\Gamma}=0$, we see from (\[coex\]) that \[csda42\] (,v) =&,S\_0[E]{}\_[L\^2(GSI)]{} -, \_x v\_[L\^2(GSI)]{}\ &+CS\_0,v\_[L\^2(GSI)]{}+(-K\_C),v\_[L\^2(GSI)]{}\ =&F(v) =[f]{},v\_[L\^2(GSI)]{}, for all $v\in C_0^\infty(G\times S\times I^\circ)$, which means that holds in the weak sense. Since $\phi\in L^2(G\times S\times I)$, and by the above $${\left\langle}\phi,P'(x,\omega,E,D)v{\right\rangle}_{L^2(G\times S\times I)}={\left\langle}-(CS_0+(\Sigma-K_C))\phi + {\bf f},v{\right\rangle}_{L^2(G\times S\times I)},$$ we see that $\phi\in {{{\mathcal{}}}H}_P(G\times S\times I^\circ)$. \(ii) Suppose that the assumption ${\bf TC}$ holds and that $\tilde\phi=(\phi,q,p_0,p_m)\in H_1$ satisfies (\[csda40a\]). Then for all $v\in H_2$ \[coexpr\] &,S\_0[E]{}\_[L\^2(GSI)]{} -,\_x v\_[L\^2(GSI)]{}\ &+C,S\_0v\_[L\^2(GSI)]{}+,(\^\-K\_C\^\) v\_[L\^2(GSI)]{}\ &+q,\_+(v)\_[T\^2(\_+)]{}+p\_0,S\_0(,0) v(,,0)\_[L\^2(GS)]{}\ & = B(,v)= [f]{},v\_[L\^2(GSI)]{}+[g]{}, \_-(v)\_[T\^2(\_-)]{}. Recall that $\Gamma'=\partial G\times S$ and $\Gamma'_{-}=\{(y,\omega)\in \partial G\times S\ \|\ \omega\cdot\nu(y)<0\}$. Choose any $\eta\in C^1_0(I^\circ)$ and $\theta\in C^1(\ol G\times S)$ such that ${\rm supp}(\theta)\cap \Gamma'$ is a compact subset of $\Gamma_-'$. Then $w(x,\omega,E):=\theta(x,\omega)\eta(E)\in C^1(\ol{G}\times S\times I)$ and $w(\cdot,\cdot,0)=w(\cdot,\cdot,E_{\rm m})=0$, $w_{\|\Gamma_+}=0$. Hence ${\left\langle}p_0,S_0(\cdot,0) w(\cdot,\cdot,0){\right\rangle}_{L^2(G\times S)}=0$ and ${\left\langle}q,\gamma_+(w){\right\rangle}_{T^2(\Gamma_+)}=0$, and so by (\[coexpr\]) for these $w$, \[coexpra\] &,S\_0[E]{}\_[L\^2(GSI)]{} -,\_x w\_[L\^2(GSI)]{}\ &+, CS\_0w\_[L\^2(GSI)]{}+, (\^\-K\_C\^\) w\_[L\^2(GSI)]{}\ &=[f]{},w\_[L\^2(GSI)]{}+[g]{}, \_-(w)\_[T\^2(\_-)]{}. Since the solution $\phi$ obtained in part (i) belongs to ${\mathcal{H}}_P(G\times S\times I^\circ)$, we have by virtue of the Green’s formula and that \[coexpra-a\] & [f]{},w\_[L\^2(GSI)]{}+[g]{}, \_-(w)\_[T\^2(\_-)]{}\ =& [-]{}+\_x+CS\_0+(-K\_C),w\_[L\^2(GSI)]{} +\_-(),\_-(w) \_[T\^2(\_-)]{}\ =& [f]{},w\_[L\^2(GSI)]{} +\_-(),\_-(w) \_[T\^2(\_-)]{} and hence \[csda42d\] \_-(),\_-(w) \_[T\^2(\_-)]{} = [g]{},\_-(w) \_[T\^2(\_-)]{}, for any $w$ of the form as chosen above. This clearly implies that $\gamma_-(\phi)={\bf g}\in T^2(\Gamma_-)$. Next, choose $\tilde{\eta}\in C^1(I)$ such that $\tilde{\eta}(0)=0$ and choose $\tilde{\theta}\in C_0^1(G)$. Let $\tilde{w}:=\tilde{\eta} \tilde{\theta}$. Then by similar calculation as above, we see that the Green’s formula (\[green-ex\]) and (\[coexpr\]) imply $${\left\langle}\phi(\cdot,\cdot,E_m),S_0(\cdot,E_{\rm m}) \tilde w(\cdot,\cdot,E_m){\right\rangle}_{L^2(G\times S)}=0$$ for these $\tilde{w}$. Consequently, $\phi(\cdot,\cdot,E_m)=0$ a.e. in $G\times S$ (since $S_0\geq\kappa>0$), as desired. Finally, if $v\in C^1(\ol{G}\times S\times I)$, we have $$& {\left\langle}{\bf f},v{\right\rangle}_{L^2(G\times S\times I)}+{\left\langle}{\bf g},\gamma_-(v){\right\rangle}_{T^2(\Gamma_-)} =F(v)=\tilde{B}(\tilde{\phi},v) \\[2mm] ={}& {\left\langle}\phi, P'(x,\omega,E,D)v{\right\rangle}_{L^2(G\times S\times I)}+{\left\langle}CS_0\phi,v{\right\rangle}_{L^2(G\times S\times I)}+{\left\langle}(\Sigma-K_C)\phi,v{\right\rangle}_{L^2(G\times S\times I)} \\ {}&+{\left\langle}q_{\|\Gamma_+},\gamma_+(v){\right\rangle}_{T^2(\Gamma_+)}+{\left\langle}p_0,S_0(\cdot,0)v(\cdot,\cdot,0){\right\rangle}_{L^2(G\times S)} \\[2mm] ={}&{\left\langle}T_C\phi,v{\right\rangle}_{L^2(G\times S\times I)} +{\left\langle}q_{\|\Gamma_+}-\gamma_+(\phi),\gamma_+(v){\right\rangle}_{T^2(\Gamma_+)} +{\left\langle}\gamma_-(\phi),\gamma_-(v){\right\rangle}_{T^2(\Gamma_-)} \\ {}& +{\left\langle}p_0-\phi(\cdot,\cdot,0),S_0(\cdot,0)v(\cdot,\cdot,0){\right\rangle}_{L^2(G\times S)}+{\left\langle}\phi(\cdot,\cdot,E_m),S_0(\cdot,E_m)v(\cdot,\cdot,E_m){\right\rangle}_{L^2(G\times S)} \\[2mm] ={}&{\left\langle}{\bf f},v{\right\rangle}_{L^2(G\times S\times I)}+{\left\langle}q_{\|\Gamma_+}-\gamma_+(\phi),\gamma_+(v){\right\rangle}_{T^2(\Gamma_+)}+{\left\langle}{\bf g},\gamma_-(v){\right\rangle}_{T^2(\Gamma_-)} \\ {}&+{\left\langle}p_0-\phi(\cdot,\cdot,0),S_0(\cdot,0)v(\cdot,\cdot,0){\right\rangle}_{L^2(G\times S)}$$ where on the second to last phase we wrote $T_C\phi=(P(x,\omega,E,D)+CS_0+\Sigma-K_C)\phi$ and used Green’s formula , and on the last phase we made use of the already proven facts: $T_C\phi={\bf f}$, $\gamma_-(\phi)={\bf g}$, and $\phi(\cdot,\cdot,E_m)=0$. Thus it holds $${\left\langle}q_{\|\Gamma_+}-\gamma_+(\phi),\gamma_+(v){\right\rangle}_{T^2(\Gamma_+)}+{\left\langle}p_0-\phi(\cdot,\cdot,0),S_0(\cdot,0)v(\cdot,\cdot,0){\right\rangle}_{L^2(G\times S)}=0,$$ for all $v\in C^1(\ol{G}\times S\times I)$, which clearly implies that $q_{\|\Gamma_+}=\gamma_+(\phi)$ and $p_0=\phi(\cdot,\cdot,0)$. \(iii) By assumption, $\phi_{\|\Gamma_{\pm}}\in T^2(\Gamma_{\pm})$ and $\phi(\cdot,\cdot,0)$, $\phi(\cdot,\cdot,E_{\rm m})\in L^2(G\times S)$, and moreover $\phi\in {\mathcal{H}}_P(G\times S\times I^\circ)$. These properties allow us to apply the Green’s formula (\[green-ex\]), which in combination with the fact that $$P'(x,\omega,E,D)\phi=-P(x,\omega,E,D)\phi-{\frac{\partial S_0}{\partial E}}\phi,$$ leads us to $${\left\langle}P(x,\omega,E,D)\phi,\phi{\right\rangle}_{L^2(G\times S\times I)} ={}& -{\left\langle}P(x,\omega,E,D)\phi,\phi{\right\rangle}_{L^2(G\times S\times I)} -{\left\langle}{\frac{\partial S_0}{\partial E}}\phi,\phi{\right\rangle}_{L^2(G\times S\times I)} \\ {}& +{\left\Vert \gamma_+(\phi)\right\Vert}_{T^2(\Gamma_+)}^2-{\left\Vert \gamma_-(\phi)\right\Vert}_{T^2(\Gamma_-)}^2 \\ {}& +{\left\langle}S_0(\cdot,0)\gamma_0(\phi),\gamma_0(\phi){\right\rangle}_{L^2(G\times S)} -{\left\langle}S_0(\cdot,E_m)\gamma_m(\phi),\gamma_m(\phi){\right\rangle}_{L^2(G\times S)}.$$ Using this equation, and performing estimations as in the proof of Theorem \[csdath1\], allows us to deduce the inequality $$\begin{aligned} {\left\langle}{\bf f},\phi{\right\rangle}_{L^2(G\times S\times I)}+{\left\langle}{\bf g}, \gamma_-(\phi){\right\rangle}_{T^2(\Gamma_-)} \geq c'{\left\Vert \phi\right\Vert}_{H_1}^2,\end{aligned}$$ from which the desired estimate , and therefore uniqueness of solutions, follow. \[ttc\] Suppose that the assumption ${\bf TC}$ is valid and that $\phi\in {{{\mathcal{}}}H}_P$ such that \[assttc\] \_[\|\_-]{}T\^2(\_-) (,,E\_[m]{})L\^2(GS). Then at least in some cases one is able to show that (cf. [@cessenat85]) \[asscl-a\] \_[\|\_+]{}T\^2(\_+) (,,0)L\^2(GS). This would make the assumption of part (iii) of Theorem \[csdath3\] superfluous. We omit further considerations of this issue here. Similarly as above we see that the variational equation corresponding to the original problem (\[se1\]), (\[se2\]), (\[se3\]) is $$\tilde{B}_0(\tilde{\psi},v)=F_0(v)\quad \forall v\in H_2,$$ where $\tilde\psi \in H_1$ and $\tilde{B}_0(\cdot,\cdot)$ is the continuous extension onto $H_1\times H_2$ of the bilinear form $B_0(\cdot,\cdot):C^1(\ol G\times S\times I)\times C^1(\ol G\times S\times I)\to{\mathbb{R}}$ defined by (that is, the bilinear form (\[csda27\]) with $C=0$) \[csda27a\] B\_0(,v)=& ,S\_0[E]{}\_[L\^2(GSI)]{} -,\_x v\_[L\^2(GSI)]{} +,(\^\* -K\^\)v\_[L\^2(GSI)]{}\ &+\_+(), \_+(v)\_[T\^2(\_+)]{} +(,,0),S\_0(,0) v(,,0)\_[L\^2(GS)]{}. The linear form $F_0:C^1(\ol G\times S\times I)\to{\mathbb{R}}$ is given by $$F_0(v)= {\left\langle}{f},v{\right\rangle}_{L^2(G\times S\times I)}+ {\left\langle}g, \gamma_-(v){\right\rangle}_{T^2(\Gamma_-)},$$ and it admits a unique extension to a bounded linear form $F_0:H_1\to{\mathbb{R}}$. Note that the bilinear form (\[csda27a\]) is not necessarily coercive that is (\[csda30\]) does not necessarily hold, which justifies the need for the change of unknown $\phi=e^{CE}\psi$ performed above. We have the following immediate corollary for the existence of solutions of the original CSDA-problem. \[csdaco1\] Suppose that the assumptions (\[ass1\]), (\[ass2\]), (\[ass3\]) (with $c>0$), (\[csda9\]), (\[csda9aa\]) and (\[csda9a\]) are valid. Let ${ f}\in L^2(G\times S\times I)$ and ${ g}\in T^2(\Gamma_-)$. Then the following assertions hold. \(i) The variational equation \[vareq1\] B\_0(,v)=F\_0(v)vH\_2 has a solution $\tilde\psi=(\psi,q,p_0,p_m)\in H_1$. Furthermore, $\psi\in{\mathcal{H}}_P(G\times S\times I)$ and it is a weak (distributional) solution of the equation (\[se1\]). \(ii) Suppose that additionally the assumption ${\bf TC}$ holds. Then a solution $\psi$ of the equation obtained in part (i) is a solution of the problem , , . In addition, we have $q_{\|\Gamma_+}=\gamma_+(\psi)$ and $p_0=\psi(\cdot,\cdot,0)$, when $\tilde{\psi}=(\psi,q,p_0,p_m)$ is a solution in $H_1$ obtained in part (i). \(iii) Under the assumptions imposed in part (ii) any solution $\psi\in {\mathcal{H}}_P(G\times S\times I^\circ)$ of the problem , , that further satisfies \[asscl-aa\] \_[\|\_+]{}T\^2(\_+)(,,0)L\^2(GS), is unique and obeys the estimate \[csda40aaa\] \_[[H\_1]{}]{} (\_[L\^2(GSI)]{}+\_[T\^2(\_-)]{}). (Recall that $C$ is defined in , $c'$ in and that $E_m$ is the cutoff energy.) Let ${ f}\in L^2(G\times S\times I)$ and ${ g}\in T^2(\Gamma_-)$. Since $I$ is finite interval we see that ${\bf f}:=e^{CE}f\in L^2(G\times S\times I)$ and ${\bf g}:=e^{CE}g\in T^2(\Gamma_-)$, where $C=\frac{\max\{q,0\}}{\kappa}$ (see , ). By Theorem \[csdath3\] the variational problem (\[csda40a\]) has a solution $\tilde\phi=(\phi,q',p_0',p_m')\in H_1$, from which one deduces without difficulty that $\tilde\psi=(\psi,q,p_0,p_m):=e^{-CE}\tilde\phi\in H_1$ is a solution of the variational problem (\[vareq1\]). Similarly, the fact that $\phi\in {\mathcal{H}}_P(G\times S\times I^\circ)$ implies that $\psi\in{\mathcal{H}}_P(G\times S\times I^\circ)$. This can be seen by substituting $e^{-CE}v\in H_2$ instead of $v$ into (\[csda40a\]). Besides the estimate , all the claims are consequences of the corresponding items in Theorem \[csdath3\]. Recalling that ${\bf f}=e^{CE}f$, ${\bf g}=e^{CE}g$ the estimate (\[csda40aaa\]) is obtained as follows, \_[H\_1]{} =&[e\^[-CE]{}]{}\_[H\_1]{} \_[H\_1]{} (\_[L\^2(GSI)]{}+\_[T\^2(\_-)]{})\ & [[e\^[CE\_m]{}]{}]{} (\_[L\^2(GSI)]{}+\_[T\^2(\_-)]{}). This completes the proof. A. Let $$T\psi:=-{{\frac{\partial (S_0\psi)}{\partial E}}}+\omega\cdot\nabla_x\psi+\Sigma\psi-K\psi$$ and let $$T^v=S_0{{\frac{\partial v}{\partial E}}}-\omega\cdot\nabla_x v+\Sigma^v-K^v$$ be the formal transpose of $T$. Furthermore, let the assumptions (\[ass1\]), (\[ass2\]), (\[ass3\]), (\[csda9\]), (\[csda9aa\]) and (\[csda9a\]) of Theorem \[csdath3\] be valid, and let $\tilde\psi=(\psi,q,p_0,p_m)\in H_1$ be a solution of (\[vareq1\]) (guaranteed by Corollary \[csdaco1\]). By (\[csda27a\]), we find that for any $v\in C^1(\ol G\times S\times I)$ for which the [adjoint boundary conditions]{} $v(\cdot,\cdot,0)=0$ and $\gamma_+(v)=0$ hold, we have ,T\^\v= B\_0(,v)=f,v\_[L\^2(GSI)]{}+g,\_-(v)\_[T\^2(\_-)]{}, that is ,T\^\v-T,v\_[L\^2(GSI)]{}=g,\_-(v)\_[T\^2(\_-)]{}. This means that $\tilde\psi\in H_1$ is a [weak solution of the boundary (initial) value problem]{} (\[se1\]), (\[se2\]), (\[se3\]), a terminology which goes back to [@lax], [@sarason]. Also the terminology that boundary initial values are weakly valid is used. The validity of the trace theorems and the Green formula (\[green-ex\]) with $q=\gamma(\psi)$, $p_0=\psi(\cdot,\cdot,0)$, $ p_m=\psi(\cdot,\cdot,E_m)$ are keys for obtaining well-defined solutions (that is, solutions for which the boundary (initial) values really hold). Trace theorems always demand geometrical treatments where the smoothness of the boundary $\partial G$ is essential. B. We also remark that the following generalized Green formula (see e.g. [@rauch85 Theorems 1 and 7]) \[green-gene\] T , v\_[L\^2(GSI]{} -T\^\* v,\_[L\^2(GSI]{}=([A]{}\_)(v), \_P(GSI), for $v\in C^1(\ol G\times S\times I)$, is valid where ${{{\mathcal{}}}A}_\nu\psi$ is interpreted as an element of the dual $H^{1/2}(\partial (G\times S\times I))^$. The value ${{{\mathcal{}}}A}_\nu\psi$ is obtained by (uniquely) extending the bilinear form (cf. (\[green-ex\])) $$\begin{gathered} ({{{\mathcal{}}}A}_\nu\psi)(v) := \int_{\partial G\times S\times I}(\omega\cdot \nu) v\ \psi\ d\sigma d\omega dE\nonumber\\ + \int_{G\times S}\big(S_0(\cdot,0)\psi(\cdot,\cdot,0)v(\cdot,\cdot,0)-S_0(\cdot,E_{\rm m})\psi(\cdot,\cdot,E_{\rm m})v(\cdot,\cdot,E_{\rm m})\big)dx d\omega,\end{gathered}$$ where $\psi,\ v\in C^1(\ol G\times S\times I)$. \[reg-re\] Define the transport operator $$T_C\phi:= -{{\frac{\partial (S_0\phi)}{\partial E}}}+\omega\cdot\nabla_x\phi+CS_0\phi+(\Sigma-K_C)\phi.$$ Then from the estimate (\[csda30\]) it follows that for all $\phi\in C^1(\ol G\times S\times I)$, $$c'{\left\Vert \phi\right\Vert}_{H_1} \leq {\left\Vert T_C\phi\right\Vert}_{L^2(G\times S\times I)}+{\left\Vert \gamma_-(\phi)\right\Vert}_{T^2(\Gamma_-)} +{\left\Vert S_0\right\Vert}_{L^\infty(G\times I)}{\left\Vert \phi(\cdot,\cdot,E_{\rm m})\right\Vert}_{L^2(G\times S)}.$$ This a priori* estimate might be (as is standard) an appropriate starting point for developing [regularity results of solutions]{} at least in the case where $G={\mathbb{R}}^3$. For these kind of transport equations more regularity must be sought in mixed-norm (anisotropic) Sobolev-Slobodevskij spaces $H^{2,(s_1,s_2,s_3)}(G\times S\times I^\circ)$ where $s_j\geq 0$. Regularity results are needed e.g. in considerations of various approximation errors and in convergence analysis of numerical schemes. The regularity analysis remains open. The variational methods used above apply (after relevant modifications of the assumptions) also to more general problems of the form $$\begin{gathered} a{{\frac{\partial \psi}{\partial E}}}+ F\cdot\nabla_{\tilde\omega}\psi+\omega\cdot\nabla_x\psi+(\Sigma-K)\psi=f \nonumber\\ \psi_{\|\Gamma_-}=g,\quad \psi(\cdot,\cdot,E_{\rm m})=0, \label{add1}\end{gathered}$$ where $a=a(x,\omega,E)$, $F=(F_1(x,\omega,E),F_2(x,\omega,E),F_3(x,\omega,E))$ and $\nabla_{\tilde{\omega}}$ is the gradient operator with respect to the Riemannian metric on the sphere $S$ induced by the Euclidean metric on ${\mathbb{R}}^3$. Notice that since $\nabla_{\tilde{\omega}}\psi$ is tangent to $S$, we may assume that $\omega\cdot F(x,\omega,E)=0$ for all $(x,\omega,E)$. More explicitly, if $\phi\in C^1(S)$, and if $\tilde{\phi}:{\mathbb{R}}^3\backslash\{0\}\to{\mathbb{R}}$ is given by $\tilde{\phi}(y)=\phi(\frac{y}{{\left\Vert y\right\Vert}})$, then $$\nabla_{\tilde{\omega}}\phi =\nabla_y\tilde{\phi}-(\omega\cdot\nabla_y\tilde\phi)\omega$$ where $\nabla_{\tilde{\omega}} \phi=((\nabla_{\tilde{\omega}} \phi)_1, (\nabla_{\tilde{\omega}} \phi)_2, (\nabla_{\tilde{\omega}} \phi)_3)$ and $\omega=(\omega_1,\omega_2,\omega_3)$ as elements of ${\mathbb{R}}^3$. In this context the space $H_2$ must be replaced with the space ${{{\mathcal{}}}H}_2$ defined as follows. The space ${{{\mathcal{}}}H}_2$ is the completion of $C^1(G\times S\times I)$ with respect to the inner product$${\left\langle}\psi,v{\right\rangle}_{H_2}+{\left\langle}F\cdot\nabla_{\tilde\omega}\psi,F\cdot\nabla_{\tilde\omega}v{\right\rangle}_{L^2(G\times S\times I)}.$$ This observation is based on the fact that $$2{\left\langle}F\cdot\nabla_{\tilde\omega}\psi,\psi{\right\rangle}_{L^2(G\times S\times I)}={\left\langle}d\psi,\psi{\right\rangle}_{L^2(G\times S\times I)},$$ where $$d:=-\mathrm{div}_{\tilde{\omega}}(F),$$ and $\mathrm{div}_{\tilde{\omega}}$ is the divergence operator on $S$ with respect to its Riemannian metric. The equation (\[add1\]) in velocity coordinates $(x,v)\in{\mathbb{R}}^3\times{\mathbb{R}}^3$ can be written as \[add2\] F\_v+v\_x+(-K)=f, which is known as the (linear) Vlasov-Boltzmann equation. Similar observations concern the methods used in the next sections. Nevertheless, the $m$-dissipativity of the operator $\psi\mapsto {1\over a}( F\cdot\nabla_{\tilde\omega}\psi+\omega\cdot\nabla_x\psi+C\psi)$, for $C$ large enough (in relevant modified spaces), requires extra analysis. Finally notice that no boundary condition is needed in with respect to $\omega$-variable because the unit sphere $S$ is a compact manifold without boundary. Existence Results Based on $m$-dissipativity {#m-d} -------------------------------------------- In this section we apply an alternative method based on the results of dissipative first-order partial differential operators. Let $$P(x,\omega,E,D)\phi := -{{\frac{\partial (S_0\phi)}{\partial E}}}+\omega\cdot\nabla_x\phi +CS_0\phi$$ where $S_0\in C^1(\ol G\times S\times I)$ and $C$ is constant. Recall that the [formal transpose (adjoint) ]{} of $P(x,\omega,E,D)$ is P’(x,,E,D)v := S\_0-\_x v+CS\_0v. It should be pointed out that here the notation for $P(x,\omega,E,D)$ and $P'(x,\omega,E,D)$ differs from that used in section \[esols\] in that here the operators $P(x,\omega,E,D)$ and $P'(x,\omega,E,D)$ also include the term $CS_0$. Define linear operators $P,P':L^2(G\times S\times I)\to L^2(G\times S\times I)$ with domains of definition $D(P)$, $D(P')$ by setting D(P):=D\^1(GSI),P:=P(x,,E,D), and $$D(P'):=C_0^\infty(G\times S\times I^\circ),\quad P'v:=P'(x,\omega,E,D)v.$$ Clearly both $P$ and $P'$ are densely defined. Let $P'^:L^2(G\times S\times I)\to L^2(G\times S\times I)$ be the adjoint operator of $P'$. Then $\phi\in L^2(G\times S\times I)$ is said to be a weak solution* of \[d1\] P=f,fL\^2(GSI) if and only if $$\phi\in D(P'^)\quad \mathrm{and}\quad P'^\phi=f.$$ Since the adjoint $P'^$ is a closed operator, the space $$\begin{gathered} {{{\mathcal{}}}H}_{P}(G\times S\times I^\circ):=\{\phi\in L^2(G\times S\times I)\ \| \\ P(x,\omega,E,D)\phi\in L^2(G\times S\times I)\ {\rm in\ the \ weak\ sense}\}\end{gathered}$$ is a Hilbert space when equipped with the inner product $${\left\langle}\phi,v{\right\rangle}_{{{{\mathcal{}}}H}_{P}(G\times S\times I^\circ)}:={\left\langle}\phi,v{\right\rangle}_{L^2(G\times S\times I)}+{\left\langle}P(x,\omega,E,D)\phi,P(x,\omega,E,D)v{\right\rangle}_{L^2(G\times S\times I)}.$$ Notice that $$\begin{aligned} \label{eq:H_P_is_D-P-dot-star} {{{\mathcal{}}}H}_{P}(G\times S\times I^\circ)=D(P'^),\end{aligned}$$ when $D(P'^)$ is equipped with the graph norm of $P'^$. We say that $\phi\in L^2(G\times S\times I)$ is a [strong solution of (\[d1\]) (without boundary conditions)]{} if there exists a sequence $\{\phi_n\}\subset D^1(\ol G\times S\times I)$ ($=D(P)$) such that $${\left\Vert \phi-\phi_n\right\Vert}_{L^2(G\times S\times I)}+{\left\Vert P\phi_n-f\right\Vert}_{L^2(G\times S\times I)}{\mathop{\longrightarrow}}0\quad {\rm when}\ n\to\infty.$$ Let $\tilde P:L^2(G\times S\times I)\to L^2(G\times S\times I)$ be the smallest closed extension (closure) of $P$. Then one sees that $\phi$ is a strong solution (without boundary conditions) if and only if $\phi\in D(\tilde P)$ and $\tilde P\phi=f$. \[re:P\_closure\] To see that the closure $\tilde{P}$ exists, notice that if $\phi_n\in D(P)$, $f\in L^2(G\times S\times I)$ and $\phi_n\to 0$, $P\phi_n\to f$ in $L^2(G\times S\times I)$, then for every $v\in C_0^\infty(G\times S\times I^\circ)$ we have $${\left\langle}f,v{\right\rangle}_{L^2(G\times S\times I)}={}&\lim_{n\to\infty} {\left\langle}P(x,\omega,E,D)\phi_n,v{\right\rangle}_{L^2(G\times S\times I)} \\ ={}&\lim_{n\to\infty} {\left\langle}\phi_n, P'(x,\omega,E,D)v{\right\rangle}_{L^2(G\times S\times I)}=0,$$ which implies $f=0$. One immediately sees that every strong solution of (\[d1\]) is its weak solution. When the boundary $\partial G$ is smooth enough the converse is also true, the result which goes back to [Friedrich ]{} [@friedrich44] (the main theorem on page 135 together with Theorem 4.2, p. 144); see also [@rauch85]. \[density\] Let $G\subset{\mathbb{R}}^n$ be an open bounded subset, lying on one side of its boundary, and with boundary of class $C^1$. Then every weak solution of the equation (\[d1\]) is its strong solution (without boundary conditions). In other words, $D^1(\ol G\times S\times I)$ is dense in ${{{\mathcal{}}}H}_{P}( G\times S\times I^\circ)$. See [@friedrich44], or [@rauch85 Proposition 1]. The claim of the above theorem can also be equivalently stated as $$\tilde{P}=P'^.$$ We still formulate an existence result of strong solutions concerning homogeneous inflow boundary (initial) value problems. To this end, the following modified definition of the strong solution is needed. One says that $\phi\in L^2(G\times S\times I)$ is a [strong solution of (\[d1\]) with homogeneous inflow boundary conditions]{} if there exists a sequence $\{\phi_n\}\subset \tilde W^2(G\times S\times I)\cap H^1(I,L^2(G\times S))$ such that $${\left\Vert \phi-\phi_n\right\Vert}_{L^2(G\times S\times I)}+{\left\Vert P(x,\omega,E,D)\phi_n-f\right\Vert}_{L^2(G\times S\times I)}{\mathop{\longrightarrow}}0,\quad {\rm when}\ n\to\infty,$$ and \[inf1\] [\_n]{}\_[\|\_[-]{}]{}=0,\_n(,,E\_[m]{})=0. Define a linear operator $P_{0}:L^2(G\times S\times I)\to L^2(G\times S\times I)$ by D(P\_[0]{}):=&{W\^2(GSI)H\^1(I,L\^2(GS)) \| \_[\|\_[-]{}]{}=0,(,,E\_[m]{})=0}\ P\_[0]{}:=&P(x,,E,D). Let $\tilde P_{0}:L^2(G\times S\times I)\to L^2(G\times S\times I)$ be the smallest closed extension (closure) of $P_{0}$ (the existence of which can be seen using the argument of Remark \[re:P\_closure\]). Then one sees that $\phi$ is a strong solution with the homogeneous inflow boundary conditions if and only if $\phi\in D(\tilde P_{0})$ and $\tilde P_{0}\phi=f$. Moreover, one sees that every strong solution with homogeneous inflow boundary conditions is a weak solution of $P(x,\omega,E,D)\phi=f$, that is $$\begin{aligned} \label{eq:P_0_in_P-dot-star} \tilde P_{0}\subset P'^.\end{aligned}$$ Since $\tilde P_{0}$ is a closed operator, the space \[eq:H\_tilde-P\_0\_is\_D-tilde-P\_0\] [H]{}\_[P\_[0]{}]{}(GSI\^):= D(P\_[0]{}) is a Hilbert space when equipped with the inner product $${\left\langle}\phi,v{\right\rangle}_{{{{\mathcal{}}}H}_{P_0}(G\times S\times I^\circ)}:={\left\langle}\phi,v{\right\rangle}_{L^2(G\times S\times I)}+{\left\langle}\tilde{P}_0\phi,\tilde{P}_0v{\right\rangle}_{L^2(G\times S\times I)}.$$ When $\phi\in {{{\mathcal{}}}H}_{P_{0}}(G\times S\times I^\circ)$ we say that the (homogeneous) initial and boundary conditions $$\phi_{\|\Gamma_-}=0,\quad \phi(\cdot,\cdot,E_{\rm m})=0$$ are valid in the strong sense. We show the $m$-dissipativity of $\tilde{P}_0$ using the theory of evolution operators presented in section \[evcsd\] below. \[md-evoth\] Suppose that $$\begin{aligned} {}& S_0\in C^2(I,L^\infty(G)),\label{evo16} \\[2mm] {}& \kappa:=\inf_{(x,E)\in \ol{G}\times I}S_0(x,E)>0, \label{evo8-a} \\[2mm] {}& \nabla_xS_0\in L^\infty(G\times I), \label{evo9-a} \\[2mm] {}& -{{\frac{\partial S_0}{\partial E}}}+2CS_0\geq 0. \label{flp1-ab}\end{aligned}$$ Then \[inf4-a\] R(I+P\_0)=L\^2(GSI) and \[inf5-a\] P\_0,\_[L\_2(GSI)]{}0,D(P\_0). We apply Theorem \[evoth1\] (see below) with $K=0,\ \Sigma=CS_0+1,\ g=0$. Let $f\in C_0^\infty(G\times S\times I^\circ)$. Then it follows that the problem -[E]{}+\_x+(CS\_0+1)= (I + P(x,,E,D))=f,(,,E\_[m]{})=0, has a unique solution $\phi\in C(I,\tilde{W}^2_{-,0}(G\times S))\cap C^1(I,L^2(G\times S))$. We find that $$\big\{\psi\in C(I,\tilde W^2_{-,0}(G\times S))\cap C^1(I,L^2(G\times S))\ \big\|\ \psi(\cdot,\cdot,E_m)=0\big\}\subset D(P_0),$$ and so for any $f\in C_0^\infty(G\times S\times I)$ the equation $(I+ P_0)\phi=f$ has a solution. Since $C_0^\infty(G\times S\times I^\circ)$ is dense in $L^2(G\times S\times I)$ we find that the range $R(I+ P_0)$ is dense that is, \[evo17-a\] =L\^2(GSI). As in the proof of Lemma \[csdale0\] (note that here the assumptions are somewhat weaker), we have for all $\phi\in D(P_0)$, (we write $L^2=L^2(G\times S\times I)$) \[ineq\] & P\_0,\_[L\^2]{} =-[E]{},\_[L\^2]{} +\_x,\_[L\^2]{}+ CS\_0,\_[L\^2]{}\ =& (-+CS\_0),\_[L\^2]{} + ,\_[T\^2(\_+)]{}\^2 +[12]{}\_[GS]{}S\_0(,0)(,,0)\^2 dx d, which in combination with the assumptions , implies $${\left\langle}P_0\phi,\phi{\right\rangle}_{L^2(G\times S\times I)}\geq 0,\quad \forall \phi\in D(P_0).$$ If $\phi\in D(\tilde{P}_0)$, choose a sequence $\phi_n\in D(P_0)$ such that $\phi_n\to \phi$ and $P_0\phi_n\to \tilde{P}_0\phi$ in $L^2(G\times S\times I)$ when $n\to\infty$. By the above inequality, we have $${\left\langle}\tilde{P}_0\phi,\phi{\right\rangle}_{L^2(G\times S\times I)} =\lim_n {\left\langle}P_0\phi_n,\phi_n{\right\rangle}_{L^2(G\times S\times I)}\geq 0,$$ which gives . Finally, from it follows that \[evo18-ab\] [(I+P\_0)]{}\_[L\^2(GSI)]{}\_[L\^2(GSI)]{},D(P\_0). and therefore $R(I+\tilde P_0)$ is closed in $L^2(G\times S\times I)$. This result, the observation that $R(I+P_0)\subset R(I+\tilde{P}_0)$, and show that $R(I+\tilde P_0)=L^2(G\times S\times I)$. The proof is complete. Theorem \[md-evoth\] says that: \[d-cor\] The operator $-\tilde P_0:L^2(G\times S\times I)\to L^2(G\times S\times I)$ is $m$-dissipative, or equivalently $\tilde P_0$ is $m$-accretive. See e.g. [@dautraylionsv5 p. 340]. \[F-L-P-R\] The general theory of initial boundary value problems of symmetric formally dissipative first order partial differential operators can alternatively be applied to show the $m$-dissipativity of $-\tilde P_0$. The (classical) results for positive symmetric initial boundary value problems can be found in [@lax], Theorem 3.2 and discussion in section 4 therein; [@friedrich58], together with discussion in section 17 therein; [@sarason] and [@rauch85]. The spatial domain there is replaced with $G\times S$ (here the additional smooth compact manifold $S$ without boundary does not affect to the conclusions; for the construction of the Friedrich’s mollifier on $S$ we refer to [@fukuoka]). The above references are, however, valid only for problems in which the dimension of the kernel ${\rm Ker}(A_\nu)$, given below, is constant on $\Gamma$ (i.e. $A_\nu$ has constant multiplicity), and thus we are not able to directly apply them. Some results for variable multiplicity can be found e.g. in [@rauch94], [@nishitani96], [@nishitani98] and [@takayama02] together with their references. The latter researches require the an additional assumption which concerns the “transition with a non-zero derivative” on the smooth $(6-2)$-dimensional manifold $\Gamma_0$. For simplicity we deal only with the case where $S_0=S_0(x)$ is independent of $E$ and use formulations of [@nishitani96]. More general formulations can be based on [@nishitani98] and related references but we omit them here. We make the change of variables and of the unknown function \[evo2\] (x,,E):=(x,,E\_m-E) and denote $$\tilde f(x,\omega,E)=f(x,\omega,E_{\rm m}-E).$$ Making the changes, we find that the problem $(I+\tilde P_0)\phi=f$ is equivalent to $\phi$ satisfying the equation \[evo3\] [E]{}+[1]{}\_x+[1]{}(C S\_0+1)=[1]{}f=:[f]{}, on $G\times S\times I$, along with satisfying the following inflow boundary and initial value conditions \[evo4\] \_[\|\_-]{}=0, (,,0)=0. Define $$Q(x,\omega,E,D)\Phi:={1\over{ S_0}}\omega\cdot\nabla_x\Phi +{1\over{ S_0}}(C S_0+1)\Phi$$ and the boundary matrix $$A_{\nu}(z)={1\over{ S_0}}\omega\cdot\nu(y),\quad z=(y,\omega,E)\in\Gamma.$$ We assume that $\partial G=\{x\in{\mathbb{R}}^3\ \|\ f(x)=0\}$ is a level set of class $C^\infty$, i.e. $f\in C^\infty({\mathbb{R}}^3)$ and $\nabla_x f(x)\neq 0$ whenever $f(x)=0$, and we also assume that $S_0\in C^\infty(G)$. The outward unit normal vector field $\nu(y)$ on $\partial G$ is given by $$\nu(y)={{(\nabla_x f)(y)}\over{{\left\Vert (\nabla_x f)(y)\right\Vert}}}.$$ The additional assumption is as follows. Suppose that there exist $C^\infty$-functions $h$ and $A:V_{y_0}\times S\to {\mathbb{R}}$ such that $$\begin{aligned} & A_f(x,\omega):={1\over{S_0(x)}}\omega\cdot \nabla_x f(x)=-h(x,\omega)A(x,\omega) \label{ass-on-G-1} \\[2mm] & (V_{y_0}\times S)\cap\Gamma_+'=\{(y,\omega)\in\partial G\times S\ \|\ h(y,\omega)>0\}, \label{ass-on-G-2} \\[2mm] & (V_{y_0}\times S)\cap\Gamma_-'=\{(y,\omega)\in\partial G\times S\ \|\ h(y,\omega)<0\}, \label{ass-on-G-3} \\[2mm] & (V_{y_0}\times S)\cap\Gamma_0'=\{(y,\omega)\in\partial G\times S\ \|\ h(y,\omega)=0\}. \label{ass-on-G-4}\end{aligned}$$ Let $$A_h(x,\omega):={1\over{S_0(x)}}(\omega\cdot \nabla_x h)(x,\omega).$$ The additional assumption is \[add-ass\] A(y,) [and]{} A\_h(y,) [are positive definite on]{} (V\_[y\_0]{}S)\_0’. For example, in the case of the ball $G=B(0,1)\subset{\mathbb{R}}^3$ we can choose $f(x)=1-{\left\Vert x\right\Vert}^2$. Then $A_f(y,\omega)=-2{1\over{S_0(y)}}(y\cdot \omega)$ and we choose $h(y,\omega)=2(y\cdot\omega)$, $A(y,\omega)={1\over{S_0(y)}}>0$. Moreover, we find that $A_h(y,\omega)=2{1\over{S_0(y)}}>0$. Hence the stated assumptions can be met for the case of the ball $G=B(0,1)$. The following result holds in this context. Suppose that $\partial G$ is in the class $C^1$ and that the assumption (\[add-ass\]) holds. Furthermore, suppose that $S_0\in C^1(\ol G\times I)$ such that $$\begin{aligned} S_0>0\quad {\rm on}\ \ol G\times I. \label{flp2}\end{aligned}$$ Then \[inf4\] C\_0\^(GSI\^)R(I+P\_0) and \[inf5\] P\_0,\_[L\_2(GSI)]{}0,D(P\_0). Since our equation is scalar valued it is symmetric in the sense of [@nishitani96]. We choose the linear subspace of [@nishitani96] by $$M(y,\omega)=\begin{cases} {\mathbb{R}},\quad & (y,\omega)\in \Gamma'_+\cup\Gamma'_0\\ \{0\},\quad & (y,\omega)\in\Gamma'_- \end{cases}.$$ Then we find that $M(y,\omega)$ is maximal positive in the sense of [@nishitani96]. Due to the Theorem 5.5 of [@nishitani96], for any $f\in C_0^\infty(G\times S\times I^\circ)$ the equation ${{\frac{\partial \Phi}{\partial E}}}+Q(x,\omega,E,D)\Phi={\bf f}$ has a unique strong solution $\Phi$ satisfying the initial and boundary values. Then $\phi(x,\omega,E):=\Phi(x,\omega,E_{\rm m}-E)$ is the solution of $(I+P_0)\phi=f$. This completes the proof of (\[inf4\]). The inequality (\[inf5\]) can be shown similarly as above and so the conclusion follows. \[md-gene\] The previous theorem has the following generalization which can be applied for more general transport problems. Let $$P(x,\omega,E,D)\phi= -S_0{{\frac{\partial \phi}{\partial E}}}+F_1\cdot\nabla_x\phi +F_2\cdot\nabla_{\tilde\omega}\phi+a\phi$$ be the first order partial differential operator with coefficients $S_0,\ F_1,\ F_2,\ a\in C^1(\ol G\times S\times I)$. Assume that $\partial G$ is in the class $C^2$ and that $\Gamma=\Gamma_{+}\cup\Gamma_{-}\cup\Gamma_{0}$ where $\Gamma_{0}$ has the zero surface measure. In addition, we assume that $\Gamma_{\pm}$ are open in $\partial G\times S\times I^\circ$. Finally, suppose that $$\begin{aligned} & {{\frac{\partial S_0}{\partial E}}}-{\rm div}_x(F_1)-{\rm div}_{\omega}(F_2)+2a\geq 0, \label{flp1-aa} \\[2mm] & \inf_{(x,\omega,E)\in G\times S\times I}S_0(x,\omega,E)>0, \label{flp2-aa}\end{aligned}$$ and that \[flp3-aa\] F\_10 [on]{} \_[+]{}. Then \[inf4-aa\] R( I+P\_0)=L\^2(GSI) and \[inf5-aa\] P\_0,\_[L\_2(GSI)]{}0,D(P\_0). \[dis-re\] In certain cases, the $m$-dissipativity of $-\tilde P_0$ -like operator can even be proved by using explicit formulas for the solution. We only sketch the idea here. Suppose that $S_0=S_0(E)$ is independent of $x$ and that $a=a(x,\omega)\in C^1(\ol G\times S)$ is independent of $E$. Let $$P(x,\omega,E,D)\phi=-{{\frac{\partial (S_0\phi)}{\partial E}}}+\omega\cdot\nabla_x \phi+a\phi.$$ Assume that ${ f}\in C_0^\infty(G\times S\times I^\circ)$. Then by the formula (\[comp17:1\]) below, the solution of the problem \[S0const\] P(x,,E,D)=[f]{},\_[\|\_-]{}=0,(,,E\_[m]{})=0 is given by \[comp17:1-a\] (x,,E) = [1]{}( \_0\^[r(x,,E)]{} e\^[-\_0\^s a(x-,)d]{}(x-s,,R(E)+s) ds), where $$r(x,\omega,E):={}&\min\{R(E_m)-R(E),t(x,\omega)\}, \\[2mm] R(E):={}&\int_0^E{1\over{S_0(\tau)}}d\tau, \\[2mm] \tilde f(x,\omega,\eta):={}& S_0(R^{-1}(\eta))f(x,\omega,R^{-1}(\eta)).$$ We find that there exists a constant $C_1>0$ such that \[inv\] \_[L\^2(GSI)]{}C\_1[f]{}\_[L\^2(GSI)]{}. Let ${ f}\in L^2(G\times S\times I)$ and let $\{f_n\}\subset C_0^\infty(G\times S\times I^\circ)$ be a sequence such that ${\left\Vert f_n-f\right\Vert}_{L^2(G\times S\times I)}\to 0$ when $n\to\infty$. Define $$\phi_n:= {1\over{S_0(E)}}\Big( \int_0^{r(x,\omega,E)} e^{-\int_0^sa(x-\tau s\omega,\omega)d\tau}\tilde{f_n}(x-s\omega,\omega,R(E)+s) ds\Big).$$ We find that $$\phi_n\in C^0(\ol G\times S\times I)\cap H^1(G\times S\times I^\circ),\quad {\phi_n}_{\|\Gamma_-}=0,\quad \phi_n(\cdot,\cdot,E_{\rm m})=0,$$ and then $\phi_n\in D( P_{0})$. In showing that $\phi_n\in H^1(G\times S\times I^\circ)$ notice that $$\tilde{f}_n(x-t(x,\omega)\omega,\omega,R(E)+t(x,\omega))=0,\quad \textrm{if}\ R(E_m)-R(E)>t(x,\omega),$$ and so ${{\frac{\partial t}{\partial x_j}}}$, ${{\frac{\partial t}{\partial \tilde\omega_j}}}$ do not appear in ${{\frac{\partial \phi_n}{\partial x_j}}}$, ${{\frac{\partial \phi_n}{\partial \tilde\omega_j}}}$. By (\[inv\]) there exists $\phi\in L^2(G\times S\times I)$ such that ${\left\Vert \phi_n-\phi\right\Vert}_{L^2(G\times S\times I)}\to 0$. In addition, ${\left\Vert P_0\phi_n-f\right\Vert}_{L^2(G\times S\times I)}={\left\Vert f_n-f\right\Vert}_{L^2(G\times S\times I)}\to 0,\ n\to\infty$, which shows that $\phi\in D(\tilde P_0)$ and $\tilde P_0\phi=f$. Hence $R(\tilde P_0)=L^2(G\times S\times I)$. Consequently, in this special case these methods (based only on explicit solution formulas) give an alternative proof for the surjectivity of $\tilde P_{0}$ (and hence together with (\[inf5\]) the $m$-dissipativity of $-\tilde P_{0}$; cf. the treatise of $A_0$ in [@tervo14 proof of Theorem 4.7]). We return to the existence and uniqueness of solutions for the following problem. Given ${\bf f}\in L^2(G\times S\times I)$, find $\phi\in L^2(G\times S\times I)$ such that $$\begin{gathered} -{{\frac{\partial (S_0\phi)}{\partial E}}}+\omega\cdot\nabla_x\phi+CS_0\phi +\Sigma\phi -K_C\phi ={\bf f}, \nonumber\\ {\phi}_{\|\Gamma_-}=0,\quad\ \phi(\cdot,\cdot,E_{\rm m})=0. \label{co3-d}\end{gathered}$$ Let (for clarity, we have included here the subscript $C$ into $P$) P\_C(x,,E,D):=& -[E]{}+\_x+CS\_0\ =& -S\_0[E]{}+\_x-[E]{}+CS\_0. We shall seek a strong solution of with the homogeneous inflow boundary conditions. Using the above notations, the problem is equivalent to $$(\tilde{P}_{C,0}+\Sigma-K_C)\phi={\bf f},$$ where $\phi\in D(\tilde{P}_{C,0})$. The following arguments are analogous to those used in [@tervo14 Section 5.3]. \[coth2-d\] Suppose that the assumptions (\[ass1\]), (\[ass2\]), (\[ass3\]), (\[evo16\]), (\[evo8-a\]) and (\[evo9-a\]) are valid with $C={{\max\{q,0\}}\over{\kappa}}$ and $c>0$. Then for every ${\bf f}\in L^2(G\times S\times I)$ the problem has a unique strong solution $\phi\in D(\tilde P_{C,0})={{{\mathcal{}}}H}_{P_{C,0}}(G\times S\times I^\circ)$ with homogeneous inflow boundary conditions. Recall that $q={1\over 2}\sup_{(x,E)\in G\times I}{{\frac{\partial S_0}{\partial E}}}(x,E)$ (see ). For $C={{\max\{q,0\}}\over{\kappa}}$ we have $${{\frac{\partial S_0}{\partial E}}}\leq 2q=2\kappa{q\over\kappa}\leq 2S_0{{\max\{q,0\}}\over\kappa}=2S_0C,$$ that is \[flp1-d\] -[E]{}+2CS\_00, and hence by Corollary \[d-cor\], the operator $-{\tilde P}_{C,0}:L^2(G\times S\times I)\to L^2(G\times S\times I)$ is $m$-dissipative. On the other hand, due to Lemma \[csdale1a\] the bounded operator $-(\Sigma-K_C)+c I:L^2(G\times S\times I)\to L^2(G\times S\times I)$ is dissipative. These facts allow us to conclude that $-{\tilde P}_{C,0}-(\Sigma-K_C)+cI :L^2(G\times S\times I)\to L^2(G\times S\times I)$ is $m$-dissipative ([@pazy83 Theorem 4.3 and Corollary 3.3], or [@tervo14 Theorem 4.4]). This implies, as $c>0$, that $R\big(cI-(-{\tilde P}_{C,0}-(\Sigma-K_C)+cI)\big)=R({\tilde P}_{C,0}+\Sigma-K_C)=L^2(G\times S\times I)$, and so the existence of solutions follows. Because $c>0$ and because $-{\tilde P}_{C,0}-(\Sigma-K_C)+cI$ is dissipative, we have \[eq:uniquesol-d\] [([P]{}\_[C,0]{}+-K\_C)]{}\_[L\^2(GSI)]{}c\_[L\^2(GSI)]{},D(P\_[C,0]{}), which implies the uniqueness of the solution. This completes the proof. We note that the inequality implies that for all ${\bf f}\in L^2(G\times S\times I)$ \[bestim-d\] [(-[P]{}\_[C,0]{}+-K)\^[-1]{}[f]{}]{}\_[ L\^2(GSI)]{}\_[ L\^2(GSI)]{}, or in other words, the solution of the problem (\[co3-d\]) satisfies \[bestim1-d\] \_[L\^2(GSI)]{} \_[L\^2(GSI)]{}. The next results addresses the case with inhomogeneous inflow boundary data. We begin with a lemma (see Lemmas 5.8 and 5.11 in [@tervo14]). \[le:H1\_lift\] Let $d:=\mathrm{diam}(G)<\infty$ (diameter of $G$). Then for any $g\in H^1(I,T^2(\Gamma_-'))$ (recall $\Gamma_-'$ from section \[pre\]) we have $Lg\in H^1(I, L^2(G\times S))$, where $$(Lg)(x,\omega,E):=g(x-t(x,\omega)\omega,\omega,E),$$ is the lift of $g$ (see Lemma \[le:lift\] and Eq. ), and $${\left\Vert Lg\right\Vert}_{H^1(I, L^2(G\times S))}\leq \sqrt{d}{\left\Vert g\right\Vert}_{H^1(I,T^2(\Gamma_-'))}.$$ Taking into account that $\|\tau_-\|\leq d$ on $\Gamma_-$, we have by Lemma \[le:lift\] (with $\Sigma=0$), $${\left\Vert Lg\right\Vert}_{L^2(G\times S\times I)}\leq \sqrt{d}{\left\Vert g\right\Vert}_{T^2(\Gamma_-)} = \sqrt{d}{\left\Vert g\right\Vert}_{L^2(I,T^2(\Gamma'_-))}.$$ On the other hand, ${\frac{\partial (Lg)}{\partial E}}=L{\frac{\partial g}{\partial E}}$, and therefore one has, again by Lemma \[le:lift\] (with $\Sigma=0$), $${\left\Vert {\frac{\partial (Lg)}{\partial E}}\right\Vert}_{L^2(G\times S\times I)}\leq \sqrt{d}{\left\Vert {\frac{\partial g}{\partial E}}\right\Vert}_{L^2(I,T^2(\Gamma'_-))}.$$ This completes the proof. \[coth3-dd\] Suppose that the assumptions (\[ass1\]), (\[ass2\]), (\[ass3\]), (\[evo16\]), (\[evo8-a\]) and (\[evo9-a\]) are valid with $C={{\max\{q,0\}}\over{\kappa}}$ and $c>0$. Furthermore, suppose that ${\bf f}\in L^2(G\times S\times I)$, and ${\bf g}\in H^1(I,T^2(\Gamma_-'))$ is such that the compatibility condition \[comp-dd\] [g]{}(,,E\_[m]{})=0 holds. Then the problem $$\begin{gathered} -{{\frac{\partial (S_0\phi)}{\partial E}}}+\omega\cdot\nabla_x\phi+CS_0\phi+\Sigma\phi -K_C\phi={\bf f},\ \nonumber\\ {\phi}_{\|\Gamma_-}={\bf g},\quad \phi(\cdot,\cdot,E_{\rm m})=0, \label{co3aa-dd}\end{gathered}$$ has a unique solution \[solin\] \_[P\_[C,0]{}]{}(GSI\^)+ L(H\^1(I,T\^2(\_-’)))\_[P\_C]{}(GSI\^), Substitute $\phi$ in the problem (\[co3aa-dd\]) by $u:=\phi-L{\bf g}$ to obtain $$& -{{\frac{\partial (S_0u)}{\partial E}}}+ \omega\cdot\nabla_x u+CS_0u+\Sigma u-K_C u, \nonumber\\ ={}& {\bf f}+{{\frac{\partial (S_0L{\bf g})}{\partial E}}} -CS_0(L{\bf g})-\Sigma(L{\bf g}) +K_C(L {\bf g})=:\tilde{\bf f}(x,\omega,E),\nonumber$$ where we have used the fact that $\omega\cdot\nabla_x(L{\bf g})=0$ (see Lemma \[trathle1\] with $\Sigma=0$). On the other hand, $$u_{\|\Gamma_-}={\phi}_{\|\Gamma_-}-(L{\bf g})_{\|\Gamma_-}={\bf g}-{\bf g}=0,$$ and, using the compatibility condition (\[comp-dd\]), $$u(\cdot,\cdot,E_{\rm m})=\phi(\cdot,\cdot,E_{\rm m})-{\bf g}(x-t(x,\omega)\omega,\omega,E_{\rm m})=0.$$ We find that the assumptions guarantee that $\tilde {\bf f}\in L^2(G\times S\times I)$ (for more details, see the proof of Corollary \[cdd\] below), and hence by Theorem \[coth2-d\] the problem (for $u$) $$\begin{gathered} -{{\frac{\partial (S_0u)}{\partial E}}}+ \omega\cdot\nabla_x u+CS_0u+\Sigma u -K_C u=\tilde{\bf f}, \\ u_{\|\Gamma_-}=0,\quad u(\cdot,\cdot,E_{\rm m})=0,\end{gathered}$$ has a unique solution $u\in {{{\mathcal{}}}H}_{P_{C,0}}(G\times S\times I^\circ)$. It follows that then $$\phi:=u+L{\bf g}\in {{{\mathcal{}}}H}_{P_{C,0}}(G\times S\times I^\circ) + L\big(H^1(I,T^2(\Gamma_-'))\big),$$ is the wanted unique solution of (\[co3aa-dd\]). Finally, the last inclusion in is justified by the inclusion (see , , ) $${{{\mathcal{}}}H}_{P_{C,0}}(G\times S\times I^\circ)\subset {{{\mathcal{}}}H}_{P_{C}}(G\times S\times I^\circ),$$ and by the fact that (see the proof of Corollary \[cdd\] below) $$P_C(x,\omega,E,D)L\tilde{{\bf g}} = -{{\frac{\partial (S_0L\tilde{{\bf g}})}{\partial E}}}+CS_0(L\tilde{{\bf g}})\in L^2(G\times S\times I),$$ where the equality $\omega\cdot\nabla_x(L \tilde{{\bf g}})=0$ has been used again. We additionally obtain the following a priori estimate. \[cdd\] Under the assumptions of Theorem \[coth3-dd\] the solution $\phi$ of the problem (\[co3aa-dd\]) satisfies, with a constant $C_1\geq 0$, the estimate \[ess1-d\] \_[L\^2(GSI)]{}C\_1(\_[L\^2(GSI)]{} +\_[H\^1(I,T\^2(\_-’))]{}). By estimate (\[bestim-d\]), we have \[ess1-da\] \_[L\^2(GSI)]{} =[u+L[g]{}]{}\_[L\^2(GSI)]{} \_[L\^2(GSI)]{} +[L[g]{}]{}\_[L\^2(GSI)]{}. By Lemma \[le:H1\_lift\], $${\left\Vert L{\bf g}\right\Vert}_{L^2(G\times S\times I)}={\left\Vert L{\bf g}\right\Vert}_{L^2(I,L^2(G\times S))} \leq {\left\Vert L{\bf g}\right\Vert}_{H^1(I, L^2(G\times S))}\leq \sqrt{d}{\left\Vert {\bf g}\right\Vert}_{H^1(I,T^2(\Gamma'_-))},$$ where $d=\mathrm{diam}(G)<\infty$, and similarly, $${\left\Vert {\frac{\partial (L{\bf g})}{\partial E}}\right\Vert}_{L^2(G\times S\times I)} ={\left\Vert {\frac{\partial (L{\bf g})}{\partial E}}\right\Vert}_{L^2(I,L^2(G\times S))} \leq {\left\Vert L{\bf g}\right\Vert}_{H^1(I,L^2(G\times S))} \leq \sqrt{d}{\left\Vert {\bf g}\right\Vert}_{H^1(I,T^2(\Gamma'_-))}.$$ Finally, due to these estimates, Lemma \[csdale1a\], and the fact that $\omega\cdot\nabla_x(L{\bf g})=0$, one has $${\left\Vert \tilde{{\bf f}}\right\Vert}_{L^2} \leq {}& {\left\Vert {\bf f}\right\Vert}_{L^2}+{\left\Vert S_0{\frac{\partial (L{\bf g})}{\partial E}}+\Big({\frac{\partial S_0}{\partial E}} -CS_0\Big)L{\bf g}-(\Sigma-K_C)L{\bf g}\right\Vert}_{L^2} \\ \leq {}& {\left\Vert {\bf f}\right\Vert}_{L^2} +\sqrt{d}\Big({\left\Vert {\frac{\partial S_0}{\partial E}}\right\Vert}_{L^\infty}+(C+1){\left\Vert S_0\right\Vert}_{L^\infty}+{\left\Vert \Sigma-K_C\right\Vert}\Big){\left\Vert {\bf g}\right\Vert}_{H^1(I,T^2(\Gamma'_-))},$$ where we wrote, unambiguously, $L^2=L^2(G\times S\times I)$ and $L^\infty=L^\infty(G\times I)$ in order to compress the formulas. This proves the estimate as claimed. For the original problem we get \[m-d-co1\] Suppose that the assumptions (\[ass1\]), (\[ass2\]), (\[ass3\]), (\[evo16\]), (\[evo8-a\]) and (\[evo9-a\]) are valid with $C=\frac{\max\{q,0\}}{\kappa}$ and $c>0$. Furthermore, suppose that ${f}\in L^2(G\times S\times I)$, and $g\in H^1(I,T^2(\Gamma_-'))$ is such that \[comp-d-d\] g(,,E\_[m]{})=0. Then the problem $$\begin{gathered} -{{\frac{\partial (S_0\psi)}{\partial E}}}+\omega\cdot\nabla_x\psi+\Sigma\psi -K\psi=f,\ \nonumber\\ {\psi}_{\|\Gamma_-}=g,\quad \psi(\cdot,\cdot,E_{\rm m})=0, \label{co3aa-ddd}\end{gathered}$$ has a unique solution $\psi\in {{{\mathcal{}}}H}_P(G\times S\times I^\circ)$. In addition, there exists a constant $C_1'>0$ such that a priori estimate \[csda40aaa-dd\] \_[L\^2(GSI)]{}C\_1’( \_[L\^2(GSI)]{}+\_[H\^1(I,T\^2(\_-’))]{}), holds. Recalling (see , ) that $\psi$ is a solution of the problem with data $(f,g)$ if and only if $\phi=e^{CE}\psi$ is a solution of the problem with data $({\bf f}, {\bf g})=(e^{CE}f, e^{CE}g)$, we deduce the claims from Theorem \[coth3-dd\] and Corollary \[cdd\]. On the Non-negativity of Solutions {#possol} ---------------------------------- We will outline an argument to show that the solution $\psi$ obtained by Corollary \[m-d-co1\], when the assumptions of it are valid, is non-negative if the data $f$ and $g$ are non-negative. Using a change of unknown $\phi=e^{CE}\psi$, with $C$ given in , we return the problem to the one considered in Theorem \[coth3-dd\], that is $$\begin{aligned} {3} -{{\frac{\partial (S_0\phi)}{\partial E}}}+\omega\cdot\nabla_x\phi+CS_0\phi+\Sigma\phi-K_C\phi={}&{\bf f} &&\quad \textrm{on}\ G\times S\times I, \label{p-s1-b} \\ \phi_{\|\Gamma_-}={}&{\bf g} &&\quad \textrm{on}\ \Gamma_-,\label{p-s2-b} \\[2mm] \phi(\cdot,\cdot,E_{\rm m})={}&0 &&\quad \textrm{on}\ G\times S. \label{p-s3-b}\end{aligned}$$ Clearly, the positivity condition on data, $f\geq 0$ and $g\geq 0$, is equivalent to having ${\bf f}\geq 0$ and ${\bf g}\geq 0$. Recall that $$P_C(x,\omega,E,D)\phi:=-{{\frac{\partial (S_0\phi)}{\partial E}}}+\omega\cdot\nabla_x\phi+CS_0\phi.$$ A. In the first instance, we assume that $g=0$. Let $\tilde P_{C,0}$ be the smallest closed extension defined above. Using the these notations, the problem (\[p-s1-b\])-(\[p-s3-b\]) is equivalent to $$\begin{aligned} \label{eq:problem_x} (\tilde P_{C,0}+\Sigma-K_C)\phi={\bf f} \end{aligned}$$ where $\phi\in D(\tilde P_{C,0})$. Let $$T_{C,0}:=\tilde P_{C,0}+\Sigma-K_C.$$ Since the operator $-T_{C,0}+cI$ is $m$-dissipative (see the proof of Theorem \[coth2-d\], and recall that $c>0$ is a part of the assumption ), it generates a contraction $C^0$-semigroup $G_c(t)$, (cf. [@engelnagel], [@dautraylionsv5]), and hence $-T_{C,0}$ generates a (contraction) $C^0$-semigroup $G(t)=G_c(t)e^{-ct}$, $t\geq 0$, such that ${\left\Vert G(t)\right\Vert}\leq e^{-ct}$ for all $t\geq 0$. In addition, the solution $\phi$ of is given by (see [@engelnagel Chapter II, Theorem 1.10]) \[p-s8\] =T\_[C,0]{}\^[-1]{}[f]{} =(0-(-T\_[C,0]{}))\^[-1]{}[f]{} =\_0\^G(t)[f]{} dt. We decompose the operator $-T_{C,0}$ as follows \[p-s9\] -T\_[C,0]{}= B\_0+A\_0-(+CS\_0 I)+K\_C, where linear operators $B_0$ and $A_0$ are defined by $$\begin{gathered} D(A_0):=\tilde W^2_{-,0}(G\times S\times I), \quad A_0\phi:=-\omega\cdot\nabla_x\phi\nonumber\\ D(B_0):=\{\phi\in W_1^2(G\times S\times I)\ \|\ \phi(\cdot,\cdot,E_{\rm m})=0\}, \quad B_0\phi:={{\frac{\partial (S_0\phi)}{\partial E}}}.\end{gathered}$$ The semi-groups generated by the last three components are given by (below $H$ is the Heaviside function) \[p-s10\] & T\_[K\_C]{}(t)=e\^[tK\_C]{},\ & T\_[-(+CS\_0 I)]{}(t)=e\^[-t(+CS\_0I)]{},\ & (T\_[A\_0]{}(t)f)(x,,E)=H(t(x,)-t)f(x-t,,E), and they are clearly all of positive type. Also, the semi-group $T_{B_0}(t)$ generated by $B_0$ is of positive type. Indeed, letting $(U(t))(x,\omega,E):=(T_{B_0}(t)f)(x,\omega,E)$ for a given $f\in D(B_0)$, then $U$ satisfies the Cauchy problem $${{\frac{\partial U}{\partial t}}}-B_0U=0,\quad U(0)=f,$$ or equivalently, \[p-s11\] [U[t]{}]{}-[E]{}=0,U(0)=f. The solution of can be written in the form (cf. Example \[desolex1\] below) $$\begin{gathered} U(x,\omega,E,t) = (U(t))(x,\omega,E) \\ = H(R_x(E_m)-R_x(E)-t) {{S_0(x,R_x^{-1}(R_x(E)+t))}\over{S_0(x,E)}} f(x,\omega,R_x^{-1}(R_x(E)+t)),\end{gathered}$$ where $H$ is the Heaviside function and $$R_x(E):=\int_0^E{1\over{S_0(x,\tau)}}d\tau, \quad E\in [0,E_m].$$ In other words, $$\begin{gathered} (T_{B_0}(t)f)(x,\omega,E) \\ = H(R_x(E_m)-R_x(E)-t) {{S_0(x,R_x^{-1}(R_x(E)+t))}\over{S_0(x,E)}} f(x,\omega,R_x^{-1}(R_x(E)+t)),\end{gathered}$$ and therefore $T_{B_0}(t)$ is evidently of positive type. Due to and there is $M>0$ such that $0<\kappa\leq S_0\leq M$ a.e. on $\ol{G}\times I$. For fixed $E\in I$, letting $s_E(t):=R_x^{-1}(R_x(E)+t)-E$, we have $s_E(0)=0$, and $s_E'(t)=S_0(x,s_E(t)+E)$, and hence the estimates $$\kappa t\leq s_E(t)\leq Mt,\quad \forall t\geq 0$$ hold, and thus in particular, $$E\leq \kappa t+E\leq R_x^{-1}(R_x(E)+t).$$ Assumption , written in the form $\frac{1}{S_0}{\frac{\partial S_0}{\partial E}}\leq 2C$ yields after integration from $E$ to $E'$, where $E\leq E'$, $$\frac{S_0(x,E')}{S_0(x,E)}\leq e^{2C(E'-E)},$$ and hence by the above, $$\frac{S_0(x,R_x^{-1}(R_x(E)+t))}{S_0(x,E)}\leq e^{2Cs_E(t)}\leq e^{2CMt},$$ On the other hand, $$J_{x,t}(E):={{{\partial}\over{\partial E}}} R_x^{-1}(R_x(E)+t)=\frac{R_x'(E)}{R_x'\big(R_x^{-1}(R_x(E)+t)\big)}=\frac{S_0(x,R_x^{-1}(R_x(E)+t))}{S_0(x,E)},$$ we therefore we obtain the following estimate for $T_{B_0}(t)$, $t\geq 0$, $${\left\Vert T_{B_0}(t)f\right\Vert}_{L^2(G\times S\times I)}^2 \leq{}& e^{2CMt}\int_{G\times S\times I} \|\ol f(x,\omega,R_x^{-1}(R_x(E)+t))\|^2 J_{x,t}(E) dxd\omega dE \\ \leq {}&e^{2CMt}\int_{G\times S\times I} \|f(x,\omega,E')\|^2 dxd\omega dE' =e^{2CMt}{\left\Vert f\right\Vert}_{L^2(G\times S\times I)}^2$$ where $\ol f$ is the extension by zero of $f$ onto $G\times S\times [0,\infty[$. The above computations show that for any $n\in{\mathbb{N}}$, we have $${\left\Vert \Big[T_{B_0}(t/n)T_{A_0}(t/n)T_{-(\Sigma+CS_0 I)}(t/n)T_{K_C}(t/n)\Big]^n\right\Vert} \leq e^{t(CM+{\left\Vert \Sigma-CS_0I\right\Vert}+{\left\Vert K_C\right\Vert})}.$$ Therefore, by Trotter’s product formula ([@engelnagel Corollary 5.8, p. 227]) we have for ${\bf f}\geq 0$, and for all $t\geq 0$ that G(t)[f]{}=\_[n]{}\^n[f]{}0, and thus $\phi\geq 0$ by . This implies that $\psi=e^{-CE}\phi\geq 0$. (See [@dautraylionsv6 Section XXI-§2, Proposition 2, pp. 226-227], and [@tervo14 Theorem 5.16]). B. Suppose that $g\geq 0$ is more general. Moreover, in accordance with Theorem \[coth3-dd\] (or Corollary \[m-d-co1\]), we assume that $g\in H^1(I, T^2(\Gamma'_-))$ for which $g(E_{\rm m})=0$. We decompose the solution as follows. Let $u$ be the solution of the problem $$\begin{gathered} -{{\frac{\partial (S_0u)}{\partial E}}}+\omega\cdot\nabla_x u+CS_0 u+\Sigma u=0,\nonumber\\ u_{\|\Gamma_-}={\bf g}, \quad u(\cdot,\cdot,E_{\rm m})=0, \label{eq:problem_simple_u}\end{gathered}$$ and let $w$ be the solution of the problem $$\begin{gathered} -{{\frac{\partial (S_0w)}{\partial E}}}+\omega\cdot\nabla_x w+CS_0 w+\Sigma w-K_Cw={\bf f}+K_Cu, \nonumber \\ w_{\|\Gamma_-}=0, \quad w(\cdot,\cdot,E_{\rm m})=0. \label{eq:problem_w_u}\end{gathered}$$ Then $\phi:=w+u$ is the solution of -. Since ${\bf g}\geq 0$, one can show that the solution $u$ of the problem is positive, i.e. $u\geq 0$ (see Remark \[re:general\_positivity\_of\_u\] below). For example, in the special case of Example \[desolex1\] given below, where $S(x,E)=S(E)$ does not depend on $x$, and $\Sigma(x,\omega,E)=\Sigma(x,E)$ does not depend on $\omega$, the solution $u$ can be expressed explicitly in the form u=[1]{}H(R(E\_[m]{})-Q(x,,E))e\^[-\_0\^[t(x,)]{}(x-s,)ds]{}(x-t(x,),,Q(x,,E)), \[eq:explicit\_u\_in\_simplified\_case\] where $$R(E):={}& \int_0^E{1\over{S_0(\tau)}}d\tau, \\[2mm] Q(x,\omega,E):={}&R(E)+t(x,\omega), \\[2mm] \tilde{\bf g}(y,\omega,\eta):={}& S_0(R^{-1}(\eta)){\bf g}(y,\omega,R^{-1}(\eta)),$$ From it is clear that $u\geq 0$ whenever $g\geq 0$. Finally, because ${\bf f}\geq 0$, and $K_Cu\geq 0$ as $u\geq 0$, it follows from the part A above that $w\geq 0$, where $w$ is the solution of the problem . This allows us to conclude that $\phi=w+u\geq 0$, and so $\psi\geq 0$ as desired. The same argument is valid for the coupled system considered below. \[re:general\_positivity\_of\_u\] We sketch here a more general proof that, under the standing assumptions of this section, the solution $u$ of the problem is positive, when ${\bf g}\geq 0$. We allow $S_0(x,E)$ to depend on both $x$ and $E$, but, in order to make the technicalities easier, we replace the assumption by a slightly stronger one, namely $$\begin{aligned} \label{eq:strong_S_0_ass} S_0\in C^2(I,C_b(G)),\end{aligned}$$ where $C_b(G)$ is the set of bounded continuous functions on $G$, equipped with the norm ${\left\Vert \cdot\right\Vert}_{L^\infty(G)}$, making it a closed subspace of $L^\infty(G)$. Clearly, assumption implies . In fact, for the argument below to work, it suffices to assume $S_0\in C^1(I,C_b(G))$. Since ${\frac{\partial S_0}{\partial E}}\in C(I,C_b(G))$, there is a constant $M>0$ such that ${\left\Vert {\frac{\partial S_0}{\partial E}}(E)\right\Vert}_{L^\infty(G)}\leq M$, and hence for every $E,E'\in I$, it holds $${\left\Vert S_0(E')-S_0(E)\right\Vert}_{L^\infty(G)}\leq M\|E-E'\|.$$ For a given $(x,\omega)\in G\times S$, letting $$P_{x,\omega}:[0,t(x,\omega)]\times I\to{\mathbb{R}}; \quad P_{x,\omega}(t,E):=S_0(x-t\omega,E),$$ we see that $P$ is continuous, and $$\|P_{x,\omega}(t,E')-P_{x,\omega}(t,E)\|\leq M\|E'-E\|,$$ i.e. the map $(t,E)\mapsto P_{x,\omega}(t,E)$ satisfies Lipschitz condition on $I$ uniformly with respect to $[0,t(x,\omega)]$. Therefore, Cauchy-Lipschitz (or Picard-Lindelöf) theorem (cf. [@lang95 Chapter IV, Proposition 1.1]) implies that for every $(x,\omega,E)\in G\times S\times I$ the problem $$\dot{\gamma}(t)={}&S_0(x-t\omega,\gamma(t)), \quad t\in [0,t(x,\omega)],\\ \gamma(0)={}&E,$$ has a unique solution $\gamma\in C^1([0,\ol{\tau}])$ defined on the maximal interval $[0,\ol{\tau}]$, such that $\gamma(t)\in I$ for all $t\in [0,\ol{\tau}]$. We write this solution as $\gamma(x,\omega,E,t)$, and write $\ol{\tau}(x,\omega,E)$ for the end-point of its maximal interval of existence. Because $S_0\geq \kappa>0$ on $G\times I$ (by ), we have $$\begin{aligned} \label{eq:inequality_gamma} \gamma(x,\omega,E,t')\geq \gamma(x,\omega,E,t)+\kappa(t'-t),\end{aligned}$$ whenever $t\leq t'$, and therefore one can see that (i) or (ii) (or both) below holds: $$\begin{aligned} \label{eq:alpha_endpoint} \textrm{(i)}\ \gamma\big(x,\omega,E,\ol{\tau}(x,\omega,E)\big)=E_m, \quad\textrm{or}\quad \textrm{(ii)}\ \ol{\tau}(x,\omega,E)=t(x,\omega).\end{aligned}$$ Denote for $(x,\omega,E,t)\in G\times S\times I\times [0,\ol{\tau}(x,\omega,E)]$, $$\Gamma(x,\omega,E,t):=(x-t\omega, \omega, \gamma(x,\omega,E,t)),$$ and notice that $\Gamma(x,\omega,E,0)=(x,\omega,E)$. It follows from that $$\begin{aligned} \label{eq:T_boundary} \beta(x,\omega,E):=\Gamma(x,\omega,E,\ol{\tau}(x,\omega,E))\in \Gamma_-\cup (G\times S\times \{E_m\}),\end{aligned}$$ for all $(x,\omega,E)\in G\times S\times I$. Below we shall understand that $S_0(\Gamma(x,\omega,E,t))$ means $S_0(x-t\omega, \gamma(x,\omega,E,t))$, and similarly for ${\frac{\partial S_0}{\partial E}}$, since $S_0(x,E)$ is assumed not to depend on $\omega$. Finally, if $\varphi:G\times S\times I\to{\mathbb{R}}$ is smooth enough (say $C^1$), then $${{{\partial}\over{\partial t}}} \Big(\varphi\big(\Gamma(x,\omega,E,t)\big)\Big) = S_0\big(\Gamma(x,\omega,E,t)\big){\frac{\partial \varphi}{\partial E}}\big(\Gamma(x,\omega,E,t)\big)-(\omega\cdot\nabla_x\varphi)\big(\Gamma(x,\omega,E,t)\big),$$ from which one can deduce that the solution $u$ of the problem is given by $$\begin{aligned} \label{eq:general_explicit_u} u(x,\omega,E)=h(x,\omega,E)e^{-\int_0^{\ol{\tau}(x,\omega,E)}W(\Gamma(x,\omega,E,t))dt},\end{aligned}$$ where $$W(x,\omega,E):=-{\frac{\partial S_0}{\partial E}}(x,E)+CS_0(x,E)+\Sigma(x,\omega,E),$$ and (recall ) $$h(x,\omega,E) :=\begin{cases} 0, & {\rm if}\ \beta(x,\omega,E)\in G\times S\times \{E_m\}, \\ {\bf g}(\beta(x,\omega,E)), & {\rm if}\ \beta(x,\omega,E)\in \Gamma_-. \\ \end{cases}$$ Since $h\geq 0$ by the assumption ${\bf g}\geq 0$, the explicit formula for the solution $u$ clearly implies that $u\geq 0$, which is what we were to show. An Existence Result Based on the Theory of Evolution Equations {#evcsd} -------------------------------------------------------------- Due to the existence result (part (i)) of Corollary \[csdaco1\] one has, $$-{{\frac{\partial (S_0\psi)}{\partial E}}}+\omega\cdot\nabla_x\psi\in L^2(G\times S\times I),$$ but we do not know a priori if ${{\frac{\partial \psi}{\partial E}}}\in L^2(G\times S\times I)$, or if $\omega\cdot\nabla_x\psi\in L^2(G\times S\times I)$. Hence it is not known if $\psi\in W^2_1(G\times S\times I)$, and so the regularity assumption in parts (ii)-(iii) of Corollary \[csdaco1\] is not in general guaranteed. Here we give an alternative approach for a less general problem (\[se1\]), (\[se2\]), (\[se3\]), which gives more regularity for its solution $\psi$, and in particular provides the regularity criterion $\psi\in W_1^2(G\times S\times I)$ used above. We assume that the collision operator is of the form \[ecsd1\] (K)(x,,E)=\_S(x,’,,E)(x,’,E)d’. This situation corresponds to the case where the differential cross-section is of the form $\sigma(x,\omega',\omega,E',E)=\tilde\sigma(x,\omega',\omega,E)\delta(E'-E)$ (cf. Remark \[rese1a\]). Then $K$ can be written as $(K\psi)(E)=K(E)\psi(E)$, where $\psi(E)(x,\omega):=\psi(x,\omega,E)$ and where for any fixed $E\in I$, the linear operator $K(E)$ is defined by $$(K(E)\phi)(x,\omega)=\int_S \tilde{\sigma}(x,\omega',\omega,E)\phi(x,\omega') d\omega', \quad \phi\in L^2(G\times S).$$ For a condition related to the property $(K\psi)(E)=K(E)\psi(E)$, see [@engelnagel Chapter VI, Prop. 9.13]. By an argument similar to the one leading to the estimate , we have for all $E\in I$, $$\begin{aligned} \label{co(b)} {\left\Vert K(E)\right\Vert}\leq {\left\Vert \int_S \tilde{\sigma}(\cdot,\omega',\cdot,E) d\omega'\right\Vert}_{L^\infty(G\times S)}^{1/2} {\left\Vert \int_S \tilde{\sigma}(\cdot,\cdot,\omega',E) d\omega'\right\Vert}_{L^\infty(G\times S)}^{1/2},\end{aligned}$$ where ${\left\Vert K(E)\right\Vert}$ is the norm of $K(E)$ as an operator in $L^2(G\times S)$. Hence under the assumption (\[ass2a\]) the operator $K(E):L^2(G\times S)\to L^2(G\times S)$ is bounded and $${\left\Vert K(E)\right\Vert}\leq M_1^{1/2} M_2^{1/2},$$ uniformly for $E\in I$. Consider the problem , , , where $K$ is of the particular form given in . We shall assume that the restricted cross-sections $\Sigma$, $\tilde\sigma$ and the stopping power $S_0$ satisfy somewhat different assumptions than in the previous section. We make the following change of variables and of the unknown function \[ecsd2\] (x,,E):=&(x,,E\_m-E),\ :=&e\^[-CE]{}, and denote S(x,E)=&S\_0(x,E\_[m]{}-E), \ (x,,E)=&(x,,E\_[m]{}-E)\ (x,,’,E)=&(x,,’,E\_[m]{}-E)\ f(x,,E)=&f(x,,E\_[m]{}-E)\ g(y,,E)=&g(y,,E\_[m]{}-E)\ (K)(x,,E)=&\_S(x,’,,E)(x,’,E)d’, with $\phi\in L^2(G\times S\times I)$ in the definition of $\tilde{K}$. Making the changes in , we find that the problem , , is equivalent to $\phi$ satisfying the equation \[se1a\] &[E]{}+[1]{}\_x+C+[1]{}[E]{}+[1]{}-[1]{}K=[1]{}e\^[-CE]{}f, on $G\times S\times I$, along with satisfying the following inflow boundary and initial value conditions, $$\begin{aligned} \phi_{\|\Gamma_-}={}&e^{-CE}\tilde g, \label{se2a} \\ \phi(\cdot,\cdot,0)={}&0. \hspace{1.5cm} \textrm{(on $G\times S$)}. \label{se3a}\end{aligned}$$ Furthermore, define for any fixed $E\in I$ and $C\geq 0$ the linear operator $A_C(E):L^2(G\times S)\to L^2(G\times S)$ with domain $D(A_C(E))$ by (here $\tilde S(E)=\tilde S(\cdot,E)$ and $\tilde \Sigma(E)=\tilde \Sigma(\cdot,\cdot,E)$), \[ecsd4\] &D(A\_C(E))=W\^2\_[-,0]{}(GS):={W\^2(GS) \| \_-’()=0},\ &A\_C(E)=-([1]{}\_x+C +[1]{}(E)+[1]{}[E]{}(E)-[1]{} K(E)), and a function ${\bf f}(E):G\times S\to{\mathbb{R}}$ such that $${\bf f}(E)(x,\omega)={1\over{\tilde S(x,E)}}e^{-CE}\tilde f(x,\omega,E),$$ where $$(\tilde K(E)\phi)(x,\omega)=\int_S\ol\sigma(x,\omega',\omega,E)\phi(x,\omega')d\omega',\quad \phi\in L^2(G\times S),$$ and where $\Gamma'_{-}=\{(y,\omega)\in \partial G\times S\ \|\ \omega\cdot\nu(y)<0\}$, while $\gamma'_-:\tilde{W}^2(G\times S)\to \Gamma'_{-}$; $\gamma'_-(\psi)=\psi\|_{\Gamma'_{-}}$ is the trace mapping (see section \[fs\]). We interpret $\phi$ as a mapping $I\to L^2(G\times S)$ by defining $\phi(E)(x,\omega):=\phi(x,\omega,E)$. Assuming that $\phi(E)\in D(A_C(E))$ for any $E\in I$ (which takes care of the inflow boundary condition) the problem (\[se1a\]), (\[se2a\]), (\[se3a\]) [for $g=0$]{} can be put into the abstract form \[ecsd6\] [E]{}-A\_C(E)=[f]{}(E),(0)=0. We recall the following result from the theory of evolution equations. \[evoth\] Suppose that $X$ is a Banach space and that for any fixed $t\in [0,T]$ the operator $A(t):X\to X$ is linear and closed, with domain $D(A(t))\subset X$. In addition, we assume that the following conditions hold: \(i) The domain $D:=D(A(t))$ is independent of $t$ and is a dense subspace of $X$. \(ii) The operator $A(t)$ is $m$-dissipative for any fixed $t\in [0,T]$ \(iii) For every $u\in D$, the mapping $f_u:[0,T]\to X$ defined by $f_u(t):=A(t)u$ is in $C^1([0,T],X)$. \(iv) $f\in C^1([0,T],X)$ and $u_0\in D$. Then the (evolution) equation \[ecsd5\] [t]{}-A(t)u=f,u(0)=u\_0, has a unique solution $u\in C([0,T],D)\cap C^1([0,T],X)$. In addition, the solution is given by \[solev\] u(t)=U(t,0)u\_0+\_0\^t U(t,s)f(s) ds where $U(t,s):X\to X$, $0\leq t\leq s\leq T$, is a family of bounded operators, strongly continuous in $(t,s)$, called the (two-parameter) evolution system of operators of $A(t)$, $t\in [0,T]$, and $U(\cdot,s)u_0$ solves (for a fixed $s$) for every $u_0\in D$ the Cauchy problem \[solevb\] (U(t,s)u\_0)-A(t)U(t,s)u\_0=0,U(s,s)u\_0=u\_0. See [@tanabe Theorem 4.5.3, pp. 89-106], [@pazy83 pp. 126-182], [@engelnagel pp. 477-496]. \[re:evoth\_norm\] We make a brief remark concerning the meaning of the claim in Theorem \[evoth\] that $u\in C([0,T],D)$, since here the topology of $D$ needs to be specified in order to speak of continuity of maps into $D$. In fact, in [@tanabe Corollary to Theorem 4.4.2, pp. 102-103] one equips $D$ with the graph norm of $A(0)$. Recall that for $t\in [0,T]$, the graph norm of $A(t)$ on $D(A(t))$ ($=D$) is defined as ${\left\Vert v\right\Vert}_{A(t)}:={\left\Vert v\right\Vert}_X+{\left\Vert A(t)v\right\Vert}_X$, for $v\in D$. In Theorem \[evoth\] the definition of the concept of a solution of comprises that $u(t)\in D=D(A(t))$ for any $t\in [0,T]$. Define a closed (see [@tervo14]) densely defined linear operator $A_0(E):L^2(G\times S)\to L^2(G\times S)$ by &D(A\_0(E))=W\^2\_[-,0]{}(GS)=:D([independent of]{} E)\ &A\_0(E)=-[1]{}\_x. We begin with a lemma on dissipativity properties of $A_0(E)$. \[csdale1\] Suppose that the assumptions (\[csda9\]), (\[csda9a\]) and (\[csda9b\]) are valid, and let $$\begin{aligned} \label{eq:C0} C_0:={1\over 2}\kappa^{-2}{\left\Vert \nabla_x S\right\Vert}_{L^\infty(G\times I)}.\end{aligned}$$ Then for any $C\geq C_0$ the operator $A_0(E)-CI:L^2(G\times S)\to L^2(G\times S)$ is $m$-dissipative for all $E\in I$. A. Dissipativity. We have for $\phi\in D$, \[csda12a\] \_x()=[1]{}\_x+\_x([1]{})and $\nabla_x\Big({1\over{\tilde{S}(E)^{1/2}}}\Big)=-{1\over 2}\tilde{S}(E)^{-3/2}\nabla_x \tilde{S}(E)$. By the assumptions (\[csda9a\]) and (\[csda9b\]) we get that ${\phi\over{\tilde{S}(E)^{1/2}}}\in \tilde W^2_{-,0}(G\times S)=D$. From the Green’s formula (\[green\]) we obtain (since $\omega\cdot\nu >0$ on $\Gamma_+'$ and since $\phi_{\|\Gamma_-'}=0$) that for $\phi\in D$, \[csda13a\] \_x(),\_[L\^2(GS)]{} =&[12]{}\_[GS]{}()[[\^2]{}]{} dd\ =& [12]{}\_[\_+’]{}()[[\^2]{}]{} dd 0. That is why by (\[csda12a\]) we have \[csda17\] &-A\_0(E),\_[L\^2(GS)]{} =\_[GS]{}[1]{}(\_x)ddx\ =&\_[GS]{}[1]{}(\_x) [1]{}ddx\ =& \_[GS]{}\_x( ) ( ) ddx -\_[GS]{}\_x( [1]{})[[\^2]{}]{} ddx\ & -\_[GS]{}\_x( [1]{})[[\^2]{}]{} ddx\ & -[\_x( [1]{})[1]{}]{}\_[L\^(GI)]{} \_[L\^2(GS)]{}\^2\ & -[12]{}\^[-2]{}[\_xS]{}\_[L\^(GI)]{} \_[L\^2(GS)]{}\^2\ =&-C\_0\_[L\^2(GS)]{}\^2, and hence $${\left\langle}(A_0(E)-CI)\phi,\phi{\right\rangle}_{L^2(G\times S)} ={}&{\left\langle}A_0(E)\phi,\phi{\right\rangle}_{L^2(G\times S)}-C{\left\Vert \phi\right\Vert}^2_{L^2(G\times S)} \\ \leq{}&(C_0-C){\left\Vert \phi\right\Vert}^2_{L^2(G\times S)}.$$ Choosing $C\geq C_0$, one finds that $A_0(E)-CI$ is dissipative for any $E\in I$. B. We still have to show that $R(\lambda I-(A_0(E)-CI))=L^2(G\times S)$ for (any) $\lambda>0$. The equation \[csda18op\] (I-(A\_0(E)-CI))=f, means that $\phi\in \tilde W_{-,0}^2(G\times S)$ and that \[csda18\] [1]{}\_x+(+C)=f, which is equivalent to \[csda19\] \_x+(+C)S(E)=S(E)f, since by (\[csda9\]) and (\[csda9a\]) $f\in L^2(G\times S)$ if and only if $\tilde{S}(E)f\in L^2(G\times S)$. Let $B_0:L^2(G\times S)\to L^2(G\times S)$ be a linear operator with domain $D(B_0)$ defined by $$& D(B_0)=\tilde W^2_{-,0}(G\times S),\\ & B_0\phi=-\omega\cdot\nabla_x\phi.$$ Then $B_0$ is $m$-dissipative ([@tervo14], [@dautraylionsv6]). Let $\lambda':=\kappa (\lambda+C)$. The equation (\[csda19\]) is equivalent to \[csda20\] (’I -(B\_0+B\_1))=S(E)f where $B_1$ is defined by $$B_1\phi=-((\lambda+C)\tilde S(E)-\lambda')\phi.$$ It is clear that the operator $B_1:L^2(G\times S)\to L^2(G\times S)$ is bounded, and since $(\lambda+C)\tilde{S}(E)-\lambda'\geq 0$ by the assumption and the definition of $\lambda'$, it follows that $B_1$ is dissipative. These observations imply that $B_0+B_1$ is $m$-dissipative (cf. [@engelnagel Chapter III], or [@tervo14 Theorem 4.2]). Since is equivalent to as explained above, this shows that $R(\lambda I-(A_0(E)-CI))=L^2(G\times S)$, and thus completes the proof. For $E\in I$, let $A_1(E):L^2(G\times S)\to L^2(G\times S)$ be the linear operator $$A_1(E)\phi:=-{1\over {\tilde S(E)}}\tilde\Sigma(E)\phi-{1\over {\tilde S(E)}}{{\frac{\partial \tilde S}{\partial E}}}(E)\phi+{1\over {\tilde S(E)}}\tilde K(E)\phi.$$ We have the following uniform bound for the family of operators $\{A_1(E)\ \|\ E\in I\}$. \[csdale2\] Under the assumptions , and the operator $A_1(E)$ is bounded for any fixed $E\in I$, and collectively they obey a uniform bound, $$\begin{aligned} \label{eq:A1E_unif_bound} \sup_{E\in I} {\left\Vert A_1(E)\right\Vert}\leq \kappa^{-1}\Big({\left\Vert \tilde \Sigma\right\Vert}_{L^\infty(G\times S\times I)}+ {\left\Vert {{\frac{\partial \tilde S}{\partial E}}}\right\Vert}_{L^\infty(G\times I)}+{M_1}^{1/2}{M_2}^{1/2}\Big)=:C_0'<\infty,\end{aligned}$$ where $M_j\geq 0$, $j=1,2$ are as in . The (uniform) estimate follows immediately from the assumptions and the estimate . Using the above notations we have for $C:=C_0+C_0'$ the following decomposition \[eq:ACE\_decomp\] A\_C(E)=A\_0(E)-C\_0I-C\_0’I+A\_1(E). Recall that $C_0$ and $C_0'$ were defined in and . Since by Lemma \[csdale2\], $$&{\left\langle}(-C_0'I+A_1(E))\phi,\phi{\right\rangle}_{L^2(G\times S\times I)}=-C_0'{\left\Vert \phi\right\Vert}^2_{L^2(G\times S\times I)}+{\left\langle}A_1(E))\phi,\phi{\right\rangle}_{L^2(G\times S\times I)}\nonumber\\ \leq & -C_0'{\left\Vert \phi\right\Vert}^2_{L^2(G\times S\times I)}+C_0'{\left\Vert \phi\right\Vert}^2_{L^2(G\times S\times I)}=0,$$ we see that $-C_0'I+A_1(E)$ is bounded and dissipative. On the other hand, according to Lemma \[csdale1\], $A_0(E)-C_0I$ is $m$-dissipative. Hence $A_C(E)$ is $m$-dissipative for any $E\in I$ (cf. [@engelnagel Chapter III, Theorem 2.7], or [@tervo14 Theorem 4.2]). We record this observation into the next lemma. \[le:ACE\_m\_diss\] For $C=C_0+C_0'$ and for every fixed $E\in I$, the operator $A_C(E)$ is $m$-dissipative. We shall assume that $$\tilde{\sigma}\in C(I,L^\infty(G\times S,L^1(S')))\cap C(I,L^\infty(G\times S',L^1(S))),$$ where $\tilde{\sigma}$ interpreted as an element of $C(I,L^\infty(G\times S,L^1(S')))$ is $$\tilde\sigma(E)(x,\omega)(\omega')=\tilde\sigma(x,\omega',\omega,E),$$ and when interpreted as an element of $C(I,L^\infty(G\times S',L^1(S)))$ it is $$\tilde{\sigma}(E)(x,\omega')(\omega) = \tilde\sigma(x,\omega',\omega,E).$$ Furthermore, in order to avoid ambiguity, we have denoted by $S'$ (resp. $S$) the unit sphere in ${\mathbb{R}}^3$ for the variable $\omega'$ (resp. $\omega$). We find that the conditions are then satisfied valid for $\ol\sigma$ with $$M_1:=&\max_{E\in I}{\left\Vert \ol\sigma(E)\right\Vert}_{L^\infty(G\times S),L^1(S'))} = {\left\Vert \ol{\sigma}\right\Vert}_{C(I, L^\infty(G\times S),L^1(S')))}, \\ M_2:=&\max_{E\in I}{\left\Vert \ol\sigma(E)\right\Vert}_{L^\infty(G\times S'),L^1(S))} = {\left\Vert \ol{\sigma}\right\Vert}_{C(I, L^\infty(G\times S'),L^1(S)))}.$$ Moreover, for all $E_1,\ E_2\in I$, \[ccc-a\] & \_[(x,)GS]{}\_[S’]{}\| (x,’,,E\_1)-(x,’,,E\_2)\| d’ =[(E\_1)-(E\_2)]{}\_[L\^(GS,L\^1(S’))]{}\ & \_[(x,’)GS’]{}\_S\| (x,’,,E\_1)-(x,’,,E\_2)\| d=[(E\_1)-(E\_2)]{}\_[L\^(GS’,L\^1(S))]{}. Supposing additionally that $$\begin{aligned} \label{eq:ass_sigma_C1} \tilde\sigma\in C^1(I,L^\infty(G\times S,L^1(S')))\cap C^1(I,L^\infty(G\times S',L^1(S))),\end{aligned}$$ then for any fixed $E$ the operator ${{\frac{\partial K}{\partial E}}}(E):L^2(G\times S)\to L^2(G\times S)$ defined by $$\begin{aligned} \label{eq:dKE} \Big({{\frac{\partial \tilde K}{\partial E}}}(E)\phi\Big)(x,\omega):=\int_S{{\frac{\partial \ol\sigma}{\partial E}}}(x,\omega',\omega,E)\phi(x,\omega') d\omega'\end{aligned}$$ is a bounded operator and \[ccc-b\] & \_[(x,)GS]{}\_[S’]{}\| [E]{}(x,’,,E\_1)-[E]{}(x,’,,E\_2)\| d’ =[(E\_1)-(E\_2)]{}\_[L\^(GS,L\^1(S’))]{},\ & \_[(x,’)GS’]{}\_S\| [E]{}(x,’,,E\_1)-[E]{}(x,’,,E\_2)\| d=[(E\_1)-(E\_2)]{}\_[L\^(GS’,L\^1(S))]{}. The following lemma justifies the notation given in . \[le:K\_C1\] Under the assumption , for any fixed $\phi\in L^2(G\times S)$ the map $$k_\phi:I\to L^2(G\times I);\quad k_\phi(E)=\tilde{K}(E)\phi$$ is in $C^1(I, L^2(G\times I))$ and ${\frac{\partial k_\phi}{\partial E}}(E)={\frac{\partial \tilde{K}}{\partial E}}(E)\phi$. If $\beta:G\times S'\times S\times I\to {\mathbb{R}}$ is measurable, and one defines for any $\phi\in L^2(G\times S)$ such that the integral converges, $$L_{\beta}(E)\phi:=\int_{G\times S} \beta(x,\omega',\omega,E)\phi(x,\omega')d\omega dx,$$ then for every $E\in I$ one has (compare to ), $$\begin{gathered} \label{eq:LE_estim} {\left\Vert L_{\beta}(E)\phi\right\Vert}^2_{L^2(G\times S)} \leq {\left\Vert \beta(E)\right\Vert}_{L^\infty(G\times S, L^1(S'))} {\left\Vert \beta(E)\right\Vert}_{L^\infty(G\times S', L^1(S))} {\left\Vert \phi\right\Vert}_{L^2(G\times S)}^2,\end{gathered}$$ with the conventions $\beta(E)(x,\omega)(\omega')=\beta(E)(x,\omega')(\omega)=\beta(x,\omega',\omega,E)$ as above. For a fixed $E_0\in I$, taking $\beta(x,\omega',\omega,E)=\ol{\sigma}(x,\omega',\omega,E)-\ol{\sigma}(x,\omega',\omega,E_0)$, we have $k_\phi(E)-k_\phi(E_0)=L_{\beta}(E)\phi$, and the above estimate shows that $k_\phi$ is continuous at $E_0$, by assumption . Similarly, one sees that $E\mapsto {\frac{\partial \tilde{K}}{\partial E}}(E)\phi$, is continuous at $E_0$, by choosing $\beta(x,\omega',\omega,E)={\frac{\partial \ol{\sigma}}{\partial E}}(x,\omega',\omega,E)-{\frac{\partial \ol{\sigma}}{\partial E}}(x,\omega',\omega,E_0)$. Finally, the differentiability of $k_\phi$ at $E_0$ is obtained by taking in the above estimate, $$\beta(x,\omega',\omega,E)=\frac{\ol{\sigma}(x,\omega',\omega,E)-\ol{\sigma}(x,\omega',\omega,E_0)}{E-E_0}-{\frac{\partial \ol{\sigma}}{\partial E}}(x,\omega',\omega,E_0),$$ whence $\frac{k_\phi(E)-k_\phi(E_0)}{E-E_0}-{\frac{\partial \tilde{K}}{\partial E}}(E_0)\phi=L_{\beta}(E)\phi$, and we have $k_\phi'(E_0)={\frac{\partial \tilde{K}}{\partial E}}(E_0)\phi$. This completes the proof of the lemma. In fact, the proof of the previous lemma shows that $\tilde{K}$ as a map $I\to {\mathcal{L}}(L^2(G\times S))$ is $C^1$, where for a Banach space $Z$ the set ${\mathcal{L}}(Z)$ is the space of bounded linear operators $Z\to Z$ equipped with the uniform operator norm topology. With the above notations and results at hand, we are ready to state the central result of this section. \[evoth1\] Suppose that the assumptions , , and are valid, that $\tilde{\sigma}\geq 0$, and that & C\^1(I,L\^(GS)),\[ecsd6a\]\ & S\_0C\^2(I,L\^(G)),\[ecsd6a-a\]\ & C\^1(I,L\^(GS,L\^1(S’)))C\^1(I,L\^(GS’,L\^1(S))).\[ecsd6a-b\] Let $f\in C^1(I,L^2(G\times S))$ and let $g\in C^2(I,T^2(\Gamma_-))$ which satisfies the compatibility condition \[cc\] g(E\_m)=0. Then the problem (\[se1\]), (\[se2\]), (\[se3\]) has a unique solution $\psi\in C(I,\tilde W^2(G\times S))\cap C^1(I,L^2(G\times S))$. If in addition the assumptions (\[ass2b\]) (with $c>0$) are also valid, the estimate \[evoest\] \_[[H\_1]{}]{} (\_[L\^2(GSI)]{}+\_[T\^2(\_-)]{}). holds. (The constants $\kappa$, $q$ and $c'$ were defined in , and , respectively.) A. Assume at first that $g=0$. We make the change of variables and the change of unknown function as above by setting $\tilde\psi(x,\omega,E)=\psi(x,\omega,E_m-E)$ and $\phi=e^{-CE}\tilde\psi$. Choose $C=C_0+C_0'$ (see , ). Then, as observed above, the problem (\[se1\]), (\[se2\]), (\[se3\]) can be cast into an equivalent form (see ) \[ecsd7\] [E]{}-A\_C(E)=[f]{}(E),(0)=0, where the domain $D(A_C(E))=\tilde{W}^2_{-,0}(G\times S)=:D$ of definition of $A_C(E)$ is independent of $E$. We have, moreover, demonstrated (see Lemma \[le:ACE\_m\_diss\]) that the (densely defined) operator $A_C(E):L^2(G\times S)\to L^2(G\times S)$ is $m$-dissipative for any fixed $E\in I$. The assumptions , , imply that for any fixed $\phi\in D$ the mapping $$h_{\phi}:I\to L^2(G\times S);\quad h_\phi(E):=A_C(E)\phi,$$ is differentiable and $$h_\phi'(E)=&-{\partial\over{{\partial E}}}\Big({1\over{\tilde S(E)}}\Big)\omega\cdot \nabla_x\phi -{\partial\over{{\partial E}}}\Big({1\over{\tilde S(E)}}\tilde\Sigma(E)\Big)\phi -{\partial\over{{\partial E}}}\Big({1\over{\tilde S(E)}}{{\frac{\partial \tilde S}{\partial E}}}\Big)\phi \\ &+{\partial\over{{\partial E}}}\Big({1\over{\tilde S(E)}}\Big)\tilde K(E)\phi +{1\over{\tilde S(E)}}{{\frac{\partial \tilde K}{\partial E}}}(E)\phi,$$ where ${{\frac{\partial \tilde K}{\partial E}}}(E)\phi$ is defined in , and the derivative ${{{\partial}\over{\partial E}}}(\tilde{K}(E)\phi)={{\frac{\partial \tilde K}{\partial E}}}(E)\phi$ is provided by Lemma \[le:K\_C1\]. By assumptions , , we thus see that $h_\phi$ is in $C^1(I,L^2(G\times S))$. By Theorem \[evoth\] there exists a unique solution $\phi\in C(I,W^2_{-,0}(G\times S))\cap C^1(I,L^2(G\times S))$ of (\[ecsd7\]). Then $\psi(x,\omega,E):= e^{C(E_m-E)}\phi(x,\omega,E_m-E)$ is the required solution of the problem , , for $g=0$. B. Suppose that more generally $g\in C^2(I,T^2(\Gamma_-))$ and that the compatibility condition (\[cc\]) holds. By [@tervo14 Lemma 5.10] there exists a lift $Lg\in C^2(I,\tilde W^2(G\times S))$ for which $\gamma_-(Lg)=g$, and $(Lg)(\cdot,\cdot,E_m)=0$ (follows from (\[cc\])), and furthermore $\omega\cdot\nabla_x(Lg)=0$. Substituting into the problem (\[se1\]), (\[se2\]), (\[se3\]) the function $u:=\psi-Lg$ for $\psi$ we obtain the following problem for $u$, $$\begin{gathered} -{{\frac{\partial (S_0u)}{\partial E}}}+\omega\cdot\nabla_x u+\Sigma u - Ku = \tilde{f},\nonumber\\ u_{\|\Gamma_-}=0,\nonumber\\ u(\cdot,\cdot,E_m)=0,\label{ecsd8}\end{gathered}$$ where $$\tilde{f}:=f-\Big(-{{\frac{\partial (S_0(Lg))}{\partial E}}}+\Sigma (Lg)-K(Lg)\Big),$$ and we have $\tilde{F}\in C^1(I,L^2(G\times S))$ under the assumption , , . By Part A of the proof, the problem , , (for $g=0$) has a unique solution $u\in C(I,\tilde{W}^2_{-,0}(G\times S))\cap C^1(I,L^2(G\times S))$. As argued above, $\psi:=u+Lg$ is then then desired unique solution for the problem , , , for the given, arbitrary $g\in C^2(I,T^2(\Gamma_-))$. We claim that the solution $\psi$ belongs to $W^2_1(G\times S\times I)$. Indeed, since $\psi\in C^1(I,L^2(G\times S))$, we have $\psi\in L^2(G\times S\times I)$ and ${{\frac{\partial \psi}{\partial E}}}\in L^2(G\times S\times I)$. On the other hand, since $\psi$ solves , these imply, together with the assumptions made, that $\omega\cdot\nabla_x\psi\in L^2(G\times S\times I)$, which confirms the claim. Hence, under the additional assumptions (\[ass2b\]), the estimate (\[evoest\]) follows from Corollary \[csdaco1\] (part (iii)). This completes the proof. The term compatibility condition used above for the assumption for $g$ comes from the observation that since the solution $\psi$ is to satisfy and we have $\psi\in C(I,\tilde{W}^2(G\times S))$, it follows that $\psi(E_m)=0$, and therefore $0=\psi(E_m)\|_{\Gamma_-}=g(E_m)$. Strictly speaking, in the proof of Theorem \[evoth1\] above we did not fully address the claim that $u\in C(I,\tilde{W}^2_{-,0}(G\times S))$, since, as pointed out in Remark \[re:evoth\_norm\], the common domain $D=D(A_C(E))=\tilde{W}^2_{-,0}(G\times S)$ of the operators $A_C(E)$, $E\in I$, is to be equipped with the graph norm of $A_C(0)$ when applying Theorem \[evoth\]. However, the norm ${\left\Vert \cdot\right\Vert}_{\tilde{W}^2_{-,0}(G\times S)}$ of $\tilde{W}^2_{-,0}(G\times S)$ is equivalent to the graph norm ${\left\Vert \cdot\right\Vert}_{A_C(E)}$ in $D=\tilde{W}^2_{-,0}(G\times S)$, for every $E\in I$, and in particular for $E=0$. This can be readily seen from the estimates, for $\phi\in D$, $${\left\Vert A_C(E) \phi\right\Vert}_{L^2(G\times S)}\leq{} & \frac{1}{\kappa} \big({\left\Vert \omega\cdot\nabla_x\phi\right\Vert}_{L^2(G\times S)} + C_2{\left\Vert \phi\right\Vert}_{L^2(G\times S)}\big), \\ {\left\Vert \omega\cdot\nabla_x\phi\right\Vert}_{L^2(G\times S)} \leq{} & C_1{\left\Vert A_C(E)\phi\right\Vert}_{L^2(G\times S)} + C_2{\left\Vert \phi\right\Vert}_{L^2(G\times S)},$$ where $$C_1:={\left\Vert \tilde{S}(E)\right\Vert}_{L^\infty},\quad C_2:=C C_1 + {\left\Vert \tilde{\Sigma}(E)\right\Vert}_{L^\infty} + {\left\Vert {\frac{\partial \tilde{S}}{\partial E}}(E)\right\Vert}_{L^\infty} + {\left\Vert \tilde{K}(E)\right\Vert},$$ and we have used a short hand notation $L^\infty$ for $L^\infty(G\times S)$. Notice also that ${\left\Vert \gamma_-(\phi)\right\Vert}_{T^2(\Gamma_-)}=0$ when $\phi\in \tilde{W}^2_{-,0}(G\times S)$. Let $H^m(I,X),\ m\in{\mathbb{N}}_0$ be the Sobolev space for Hilbert space valued functions $I\to X$. We have the following corollary. \[evothco\] Suppose that the assumptions , , , and , and of Theorem \[evoth1\] are valid, and $\tilde{\sigma}\geq 0$. Let $f\in H^2(I,L^2(G\times S))$ and $g\in H^3(I,T^2(\Gamma_-))$ which satisfies the compatibility condition g(E\_m)=0. Then the problem (\[se1\]), (\[se2\]), (\[se3\]) has a unique solution $\psi\in C(I,\tilde W^2(G\times S))\cap C^1(I,L^2(G\times S))$. If in addition the assumptions (with $c>0$) are also valid, the estimate (\[evoest\]) holds. By the Sobolev Embedding Theorem $$H^m(I,X)\subset C^{j}(I,X)\ {\rm for}\ m>j+{1\over 2}$$ and then the assertion follows from Theorem \[coupthev\]. The evolution equation based approach given above can be generalized for $L^p$-theory when $1\leq p<\infty$. The approach based on the Lions-Lax-Milgram Theorem (section \[esols\]) is limited to the Hilbert space structure, and can therefore be only applied for $p=2$. However, some (recent) generalizations for reflexive Banach spaces of Lions-Lax-Milgram theory might allow methods of section \[esols\] to be generalized also for $1<p<\infty$. On the Existence of Solutions for Volterra Type Collision Operators ------------------------------------------------------------------- In the previous section, we assumed that the collision operator $K$ is of the form $(K\psi)(x,\omega,E)=\int_S \sigma(x,\omega',\omega,E)\psi(x,\omega',E) d\omega'$ in order to avoid integration over $I$ with respect to $E'$. Considerations were founded on the fact that $K\psi$ had a representation $(K\psi)(E)=K(E)\psi(E)$. However, for some collision operators of special type, also integration with respect to $E'$ is possible in evolution operator based approaches. In this section, we give a short and formal description of such a technique for Volterra type collision operators. Consider the problem (\[se1\]), (\[se2\]), (\[se3\]) with $g=0$, $$\begin{gathered} -{{\frac{\partial \psi}{\partial E}}}+{1\over{S_0(E)}}\omega\cdot\nabla_x\psi+{1\over{S_0(E)}}\Sigma(E)\psi-{1\over{S_0(E)}}{{\frac{\partial S_0}{\partial E}}}(E)\psi -{1\over{S_0(E)}} K\psi= {1\over{S_0(E)}}f,\nonumber\\ \psi_{\Gamma_-}=0,\quad \psi(\cdot,\cdot,E_{\rm m})=0.\end{gathered}$$ For simplicity we denote ${1\over{S_0(E)}}f$ and ${1\over{S_0(E)}} K$ again by $f$ and $K$. Assume that $I=[0,\infty[{}=:{\mathbb{R}}_+$ and that $K$ is of the Volterra type operator (cf. [@engelnagel pp. 447-452]) $$(K\psi)(x,\omega,E)=\int_0^E\int_S \sigma(x,\omega',\omega,E-E',E)\psi(x,\omega',E') d\omega' dE'.$$ In other words, $K\psi=\int_{S\times I} \tilde{\sigma}(\cdot,\omega',\cdot,E',\cdot)\psi(\cdot,\omega',E') d\omega' dE'$ for a differential cross section $\tilde{\sigma}$ of the form $\tilde{\sigma}(x,\omega',\omega,E',E)=\chi_I(E-E')\sigma(x,\omega',\omega,E-E',E)$, where $\chi_I$ is the characteristic function of $I$. Assume additionally that $\sigma$ has a decomposition $$\sigma(x,\omega',\omega,E-E',E)=\sigma_1(x,\omega',\omega,E)\sigma_2(E-E').$$ Let $K_1(E)$ (for a fixed $E$) be a linear operator $L^2(G\times S)\to L^2(G\times S)$ defined by $$(K_1(E)\phi)(x,\omega)=\int_S\sigma_1(x,\omega',\omega,E)\phi(x,\omega') d\omega'.$$ Then we find that $$(K\psi)(x,\omega,E)=\int_0^E \sigma_2(E-E')K_1(E)\psi(E') dE'.$$ Define an [extended space]{} by ${{{\mathcal{}}}X}:=L^2(G\times S)\times L^1({\mathbb{R}}_+,L^2(G\times S))$ and a linear operator ${{{\mathcal{}}}A}(E):{{{\mathcal{}}}X}\to {{{\mathcal{}}}X}$ for a fixed $E$ by (here the argument of ${\mathbb{R}}_+$ is denoted by $s$) & D([A]{}(E))=D:=W\_[-,0]{}\^2(GS)H\^[1,1]{}(\_+,L\^2(GS)),\ & [A]{}(E):= ( A(E)&\_0B(E)&[d]{} ) , where $$H^{1,1}({\mathbb{R}}_+,L^2(G\times S)):= \{F\in L^1({\mathbb{R}}_+,L^2(G\times S))\ \|\ F'\in L^1({\mathbb{R}}_+,L^2(G\times S))\},$$ is the domain of the derivative operator $\frac{d}{ds}:L^1({\mathbb{R}}_+,L^2(G\times S))\to L^1({\mathbb{R}}_+,L^2(G\times S))$, the linear operator $A(E):L^2(G\times S)\to L^2(G\times S)$ with domain $\tilde{W}^2_{-,0}(G\times S)$ (independent of $E$) is given by, $$A(E)\phi=-\Big({1\over { S_0(E)}}\omega\cdot\nabla_x\phi+{1\over { S_0(E)}}\Sigma(E)\phi -{1\over{ S_0(E)}}{{\frac{\partial S_0}{\partial E}}}(E)\phi\Big),$$ the linear operator $\delta_0$ is bounded and defined by $$\delta_0:H^{1,1}({\mathbb{R}}_+,L^2(G\times S))\to L^2(G\times S); \quad \delta_0 F=F(0),$$ and finally $B(E)$ is the bounded linear operator defined by $$B(E):L^2(G\times S)\to L^1({\mathbb{R}}_+,L^2(G\times S));\quad (B(E)\phi)(s)=\sigma_2(s)K_1(E)\phi.$$ Let ${ \left( \begin{matrix} \phi\cr \eta\cr \end{matrix} \right) }\in {{{\mathcal{}}}X}$ and let ${ \left( \begin{matrix} \psi\cr F\cr \end{matrix} \right) }\in H^1({\mathbb{R}}_+,{{{\mathcal{}}}X})$ (note that $\psi:{\mathbb{R}}_+\to L^2(G\times S),\ F:{\mathbb{R}}_+\to L^1({\mathbb{R}}_+,L^2(G\times S))$; here the argument of $\psi$ and $F$ is denoted by $E$). In this extended setting, for a given ${ \left( \begin{matrix} F_1\cr F_2 \end{matrix} \right) }\in C({\mathbb{R}}_+,{{{\mathcal{}}}X})$, the validity of the evolution equation \[R1\] ( F ) -[A]{}(E) ( F ) = ( F\_1F\_2 ) , ( F ) (0)= ( ) , implies (from the first row of the matrix equation) that \[R4\] [E]{}-A(E)=F\_1(E)+\_0(F(E))=F\_1(E)+F(E)(0). In some cases the evolution system of the operators ${{{\mathcal{}}}A}(E)$ has properties which enable one to deduce the existence of solutions for the transport problem \[R0\] & [E]{}-A(E)- K=f,\ & \_[\_-]{}=0,(0)=. Note that carrying out the change of variables as above, one is able to replace the initial condition $\psi(0)=\phi$ with $\psi(E_{\rm m})=\phi$. The basis of the idea is as follows. Assume, for example, that $S_0$, $\Sigma$ and $\sigma_1$ are independent of $E$. Then the operators $A(E)$, $K_1(E)$, $B(E)$ and ${{{\mathcal{}}}A}(E)$ are independent of $E$ as well. Let $T(E)$ be the $C^0$-semigroups generated by $A$. Define a linear operator $R(E):L^1({\mathbb{R}}_+,L^2(G\times S))\to L^2(G\times S)$ by $$R(E)\eta: =\int_0^ET(E-s)\eta(s)ds.$$ Denote by $S(E)$ the (left) translation semigroup on $L^1({\mathbb{R}}_+,L^2(G\times S))$, \[tr\] (S(E))(s):=(E+s). Finally, let $${{{\mathcal{}}}T}(E):={ \left( \begin{matrix} T(E)&R(E)\cr 0&S(E)\cr \end{matrix} \right) }.$$ Then ${{{\mathcal{}}}T}(E)$ is the $C^0$-semigroup generated by ${ \left( \begin{matrix} A&\delta_0\cr 0&{d\over{ds}}\cr \end{matrix} \right) }:{{{\mathcal{}}}X}\to {{{\mathcal{}}}X}$ ([@engelnagel pp. 437-438]). Furthermore, letting ${{{\mathcal{}}}S}(E):{{{\mathcal{}}}X}\to {{{\mathcal{}}}X}$ be the $C^0$-semigroup generated by the perturbed operator $${{{\mathcal{}}}A}:={ \left( \begin{matrix} A&\delta_0\cr 0&{d\over{ds}}\cr \end{matrix} \right) }+ { \left( \begin{matrix} 0&0\cr B&0 \end{matrix} \right) },$$ it follows that ${ \left( \begin{matrix} \psi\cr F \end{matrix} \right) }(E):={{{\mathcal{}}}S}(E){ \left( \begin{matrix} \phi\cr \eta \end{matrix} \right) }$ is the solution of (\[R1\]) for ${ \left( \begin{matrix} F_1\cr F_2 \end{matrix} \right) }={ \left( \begin{matrix} 0\cr 0 \end{matrix} \right) }$. The above semigroups obey the following formula, for every ${ \left( \begin{matrix} \phi\cr \eta\cr \end{matrix} \right) }\in {{{\mathcal{}}}X}$: ([@engelnagel Corollary III.1.7]) (E) ( ) =& [T]{}(E) ( ) +\_0\^E[T]{}(E-s)[B]{}[S]{}(s) ( ) ds\ =& [T]{}(E) ( ) +\_0\^E[T]{}(E-s)[B]{} ( F ) \(s) ds,. where ${{{\mathcal{}}}B}:={ \left( \begin{matrix} 0&0\cr B&0\cr \end{matrix} \right) }$. Choose $\eta:=f$ and ${ \left( \begin{matrix} F_1\cr F_2 \end{matrix} \right) }={ \left( \begin{matrix} 0\cr 0 \end{matrix} \right) }$. Applying the above facts we have \[R7\] & ( F ) (E)=[S]{}(E) ( f ) \ =& ( T(E)&R(E)0&S(E) ) ( f ) + \_0\^E ( T(E-s)&R(E-s)0&S(E-s) ) ( 0&0B&0 ) ( F ) (s)ds\ =& ( T(E)&R(E)0&S(E) ) ( f ) + \_0\^E ( T(E-s)&R(E-s)0&S(E-s) ) ( 0B(s) ) ds, from which the last row gives us F(E)=S(E)f+\_0\^ES(E-s)B(s) ds, and then (recall (\[tr\]) and that $(B\psi(s))(t)=\sigma_2(t)K_1\psi(s)$) \[R3\] F(E)(0)=f(E)+\_0\^E (S(E-s)(B(s)))(0) ds = f(E)+\_0\^E\_2(E-s)K\_1(s) ds. Combining (\[R3\]) and (\[R4\]) we finally see that $\psi$ is a solution of the transport problem (\[R0\]). Finally, we note that in the case where ${ \left( \begin{matrix} \psi\cr F \end{matrix} \right) }\in D({{{\mathcal{}}}A})$, we have $\psi(E)\in D(A)=\tilde W_{-,0}^2(G\times S)$, for any $E\geq 0$, and thus in particular, $\psi_{\|\Gamma_-}=0$. We find by the first row of matrix equation (\[R7\]) that $\psi(0)=T(0)\phi+R(0)f=\phi$. It might be worth attempting to generalize this method under less restrictive assumptions, especially for the case where $S_0$, $\Sigma$, $\sigma_1$ are allowed to be $E$-dependent. Existence of Solutions for the Coupled System {#cosyst} ============================================= In this section, we consider the coupled transport problem. For simplicity denote $\Sigma_j:=\Sigma_{j,r},\ S_j:=S_{j,r},\ \sigma_{jj,r}:=\sigma_{jj}$ for $j=2,3$. Let $f=(f_1,f_2,f_3)\in L^2(G\times S\times I)^3$ and $g=(g_1,g_2,g_3)\in T^2(\Gamma_-)^3$. We deal with the following coupled system of integro-partial differential equations for $\psi=(\psi_1,\psi_2,\psi_3)$ on $G\times S\times I$, $$\begin{gathered} \omega\cdot\nabla_x\psi_1+\Sigma_1\psi_1-K_{1}\psi=f_1, \label{csda1a}\\ -{{\frac{\partial (S_{j}\psi_j)}{\partial E}}}+\omega\cdot\nabla_x\psi_j+\Sigma_{j}\psi_j-K_{j}\psi=f_j,\quad j=2,3.\label{csda1b}\end{gathered}$$ In order to guarantee uniqueness of solutions, we moreover impose the inflow boundary condition on $\Gamma_-$, \[csda2\] [\_j]{}\_[\|\_-]{}=g\_j,j=1,2,3, and initial value (or energy boundary) condition on $G\times S$, \[csda3\] \_j(,,E\_[m]{})=0,j=2,3, where $E_{\rm m}$ is the cut-off energy. As mentioned in the introduction the problem (\[csda1a\])-(\[csda3\]) is an approximation of the problem (\[intro1\]), (\[intro2\]). We assume that the [total (restricted) cross sections]{} $\Sigma_j:G\times S\times I\to{\mathbb{R}}$, for $j=1,2,3$, are functions such that \[scateh\] \_jL\^(GSI),\_j0,j=1,2,3. Furthermore, we assume that the [differential (restricted) cross sections]{} $\sigma_{kj}:G\times S^2\times I^2\to{\mathbb{R}}$, $k,j=1,2,3$, are measurable functions such that $$\begin{aligned} {3}\label{colleh} &\sum_{k=1}^3\int_{S\times I} \sigma_{kj}(x,\omega',\omega,E',E)d\omega' dE'\leq M_1,\quad && {\rm a.e.}\ G\times S\times I,\quad j=1,2,3,\nonumber\\[2mm] &\sum_{k=1}^3\int_{S\times I} \sigma_{jk}(x,\omega,\omega',E,E')d\omega' dE'\leq M_2,\quad && {\rm a.e.}\ G\times S\times I,\quad j=1,2,3,\\[2mm] &\sigma_{kj}\geq 0,\quad && {\rm a.e.}\ G\times S^2\times I^2,\quad k,j=1,2,3.\nonumber\end{aligned}$$ Define the (restricted) scattering operator $\Sigma_j$ and the (restricted) collision operator $K_j$ corresponding to the particle type $j$, for $j=1,2,3$ and $\psi_j\in L^2(G\times S\times I)$, as follows \[scat\] (\_j\_j)(x,,E)=\_j(x,,E)\_j(x,,E), and for $\psi=(\psi_1,\psi_2,\psi_3)\in L^2(G\times S\times I)^3$, \[coll3\] (K\_j)(x,,E)=\_[k=1]{}\^3\_[SI]{}\_[kj]{}(x,’,,E’,E)\_k(x,’,E’)d’ dE’. Furthermore, we define for $\psi=(\psi_1,\psi_2,\psi_3)\in L^2(G\times S\times I)^3$, \[sda1\] =(\_1\_1,\_2\_2,\_3\_3) and \[sda2\] K=(K\_1,K\_2,K\_3). One immediately sees that $\Sigma:L^2(G\times S\times I)^3\to L^2(G\times S\times I)^3$ is a bounded linear operator. In addition, by applying Hölder’s inequality we have the following (cf. [@dautraylionsv6 pp. 227-228] and [@tervo14 Theorem 5.2] for $p=1$; see also , ). \[skb\] The linear operator $K:L^2(G\times S\times I)^3\to L^2(G\times S\times I)^3$ is bounded and \[k-norm\] [K]{}& \_[j=1,2,3]{}[\_[k=1]{}\^3\_[SI]{}\_[kj]{}(,’,,E’,)d’ dE’]{}\_[L\^]{}\^[1/2]{} [\_[k=1]{}\^3\_[SI]{} \_[jk]{}(,,’,,E’)d’ dE’]{}\_[L\^]{}\^[1/2]{}\ & [M\_1]{}\^[1/2]{}[M\_2]{}\^[1/2]{}, where $L^\infty = L^\infty(G\times S\times I)$. We assume that functions $S_j:\ol G\times I\to{\mathbb{R}},\ j=2,3$, the so-called restricted stopping powers, satisfy the following assumptions: $$\begin{aligned} & S_j\in L^\infty(G\times I), \label{sda2a} \\ & {{\frac{\partial S_j }{\partial E}}}\in L^\infty(G\times I), \label{sda2a-2} \\ & \kappa_j:=\inf_{(x,E)\in \ol{G}\times I} S_j(x,E) > 0, \label{sda2b} \\ & \nabla_x S_j\in L^\infty(G\times I), \label{sda2c}\end{aligned}$$ Note that (\[sda2b\]) implies that in $\ol{G}\times I$, \[sda2d\] [1]{}. We point out that the assumption will, in fact, be needed only in section \[evo-op\] when considering (a special case of) the problem - within the context of the theory of evolution operators (see Theorem \[coupthev\], Eq. ). In order to prove some accretivity properties for the scattering-collision operator $\Sigma-K:L^2(G\times S\times I)^3\to L^2(G\times S\times I)^3$ (or, equivalently, dissipativity properties for the operator $-\Sigma+K$) we assume that the cross-sections $\Sigma_j$, $\sigma_{jk}$, satisfy the following condition: There exists $c\geq 0$ such that for a.e. $(x,\omega,E)\in G\times S\times I$, and for every $j=1,2,3$, \[sda3\] \_j(x,,E)-\_[k=1]{}\^3\_[SI]{}\_[jk]{}(x,,’,E,E’) d’ dE’ c, and \[sda4\] \_j(x,,E)-\_[k=1]{}\^3\_[SI]{}\_[kj]{}(x,’,,E’,E) d’ dE’ c. (See [@dautraylionsv6 pp. 241] for one particle, [@tervo07] for coupled system, and [@tervo14] within $L^1$-theory.) The following [accretivity]{} result holds (see Lemma \[csdale1a\]). \[sdath1\] Let the assumptions (\[scateh\]), (\[colleh\]), (\[sda3\]) and (\[sda4\]) be valid. Then for every $\psi\in L^2(G\times S\times I)^3$, \[sda5\] (-K),\_[L\^2(GSI)\^3]{}c \^2\_[L\^2(GSI)\^3]{}. Notice that the estimate is equivalent to the property that for every $\lambda >0$, and $\psi\in L^2(G\times S\times I)^3$, \[sda5a\] [(I-(-+K+cI))]{}\_[L\^2(GSI)\^3]{}\_[L\^2(GSI)\^3]{}, which means that the operator $-\Sigma+K+cI:L^2(G\times S\times I)^3\to L^2(G\times S\times I)^3$ is [dissipative]{} ([@engelnagel Section II.3.b], or [@pazy83 Section 1.4]). \[cosdare1\] In certain situations one may assume that the restricted cross-sections are of the form \_[kj]{}(x,’,,E’,E)=\_[kj]{}(x,’,,E)(E-E’), where $\delta$ is the Dirac measure at zero. Hence $\sigma_{kj}\not\in L^\infty(G\times S^2\times I^2)$, but the collision operators are simpler, \[colla3\] (K\_j)(x,,E)=\_[k=1]{}\^3\_S\_[kj]{}(x,’,,E)\_k(x,’,E)d’, where $\psi=(\psi_1,\psi_2,\psi_3)\in L^2(G\times S\times I)^3$. In the assumptions, and , the integrals $\int_{S\times I}\sigma_{jk}(x,\omega',\omega,E',E)d\omega' dE'$ (resp. $\int_{S\times I}\sigma_{jk}(x,\omega,\omega',E,E')d\omega' dE'$) over $S\times I$ are to be replaced with $\int_S\tilde\sigma_{kj}(x,\omega',\omega,E)d\omega'$ (resp. $\int_{S}\sigma_{jk}(x,\omega,\omega',E)d\omega'$) over $S$. For the coupled BTE system (\[intro1\]), (\[intro2\]) we formulate the following result which is a slight modification of results given in [@tervo14]. Note that it is valid only for Schur collision operators and hence it does not govern completely the particle transport including charged particles, such as applications in radiation therapy. \[origbte\] Suppose that the assumptions (\[scateh\]), (\[colleh\]), (\[sda3\]), (\[sda4\]) are valid, and that $c$ is strictly positive. Then for every $f\in L^2(G\times S\times I)^3$ and $g\in T^2_{\tau_-}(\Gamma_-)^3$ the following assertions hold. 1. The boundary value problem $$\begin{gathered} \omega\cdot\nabla_x\psi_j+\Sigma_j\psi_j-K_j\psi=f_j, \nonumber\\ {\psi_j}_{\|\Gamma_-}=g_j, \label{origbte1}\end{gathered}$$ for $j=1,2,3$, has a unique solution $\psi\in W^2(G\times S\times I)^3$. 2. There exists a constant $C>0$ such that \[origbte2\] \_[W\^2(GSI)\^3]{}C([f]{}\_[L\^2(GSI)\^3]{}+[g]{}\_[T\^2\_[\_-]{}(\_-)\^3]{}). 3. If $f\geq 0$ and $g\geq 0$, then $\psi\geq 0$, i.e. the solution $\psi$ is non-negative for non-negative data $f,g$. The assertions follow from the considerations expressed in [@tervo14] noting that in Lemma 5.8 (see its proof) of [@tervo14] we actually have $${\left\Vert Lg\right\Vert}_{L^2(G\times S\times I)}\leq {\left\Vert g\right\Vert}_{T^2_{\tau_-}(\Gamma_-)},\ {\rm for}\ g\in T^2_{\tau_-}(\Gamma_-).$$ We omit details here. The corresponding result for time-dependent coupled system of BTEs has been proven in [@tervo14] as well. Existence of Solutions Based on Variational Formulation {#LLM} ------------------------------------------------------- As before, we perform a change of unknown functions, by setting \_j:=e\^[CE]{}\_j,j=1,2,3, where the constant $C$ will be fixed below. This transforms the problem (\[csda1a\])-(\[csda3\]) into an equivalent form, with transport equation on $G\times S\times I$, $$\begin{gathered} \omega\cdot\nabla_x\phi_1+\Sigma_1\phi_1- K_{1,C}\phi = {\bf f}_1,\label{cosyst1}\\ -{{\frac{\partial (S_j\phi_j)}{\partial E}}}+\omega\cdot\nabla_x\phi_j+CS_j\phi_j+\Sigma_{j}\phi_j -{K}_{j,C}\phi={\bf f}_j,\quad j=2,3,\label{cosyst2}\end{gathered}$$ boundary condition on $\Gamma_-$, \[cosyst3\] \_[\|\_-]{}=[g]{}, and initial condition on $G\times S$, \[cosyst4\] \_j(,,E\_m)=0,j=2,3, where $\phi=(\phi_1,\phi_2,\phi_3)$ and $${\bf f}_j={}e^{CE} f_j,\quad {\bf g}_j={}e^{CE} g_j,\quad j=1,2,3.$$ The operator $K_C=(K_{1,C},K_{2,C},K_{3,C})$ in (\[cosyst1\]), (\[cosyst2\]) above is given by K\_[j,C]{}:=\_[k=1]{}\^3\_[SI]{}\_[kj,C]{}(x,’,,E’,E)\_k(x,’,E’) d’ dE’, where the corresponding differential cross-sections are \_[kj,C]{}(x,’,,E’,E):=\_[kj]{}(x,’,,E’,E)e\^[C(E-E’)]{},j,k=1,2,3. We could perform a more refined change of the unknown functions by setting $\phi_1:=\psi_1$, $\phi_j:=e^{C_jE}\psi_j$. In this case, the modified differential cross-sections $\sigma_{kj,C}$ would be $$\begin{gathered} \sigma_{11,C}=\sigma_{11},\quad \sigma_{1k,C}=\sigma_{1k}e^{C_k E},\quad k=2,3, \nonumber\\ \sigma_{k1,C}=\sigma_{k1}e^{-C_k E'},\quad \sigma_{kj,C}=\sigma_{kj}e^{C_2 E-C_k E'},\quad j,k=2,3.\end{gathered}$$ We omit further considerations of such refinement in this work. In the rest of the section, work under the assumptions (\[scateh\]), (\[colleh\]), (\[sda2a\]), , (\[sda2b\]) (as already mentioned, the assumption will be needed only in section \[evo-op\]) and suppose furthermore that (for $C$ given below in (\[cosyst6a\]) and) for some $c\geq 0$ the estimates $$\begin{aligned} & \Sigma_j(x,\omega,E)-\sum_{k=1}^3\int_{S\times I}\sigma_{jk,C}(x,\omega,\omega',E,E') d\omega' dE' \geq c,\label{csda3aa} \\ & \Sigma_j(x,\omega,E)-\sum_{k=1}^3\int_{S\times I}\sigma_{kj,C}(x,\omega',\omega,E',E) d\omega' dE' \geq c, \label{csda4aa}\end{aligned}$$ hold for a.e. $(x,\omega,E)\in G\times S\times I$. Note that if $\sigma_{kj}$ were (cf. Remark \[cosdare1\]) of the form $\sigma_{kj}(x,\omega',\omega,E',E)=\tilde\sigma_{kj}(x,\omega',\omega,E)\delta(E-E')$ then $$\int_{S\times I}\sigma_{kj,C}(x,\omega',\omega,E',E) d\omega' dE'= \int_S\tilde\sigma_{kj}(x,\omega',\omega,E) d\omega',$$ for any $C$, and hence the conditions (\[csda3aa\]), (\[csda4aa\]) would be independent of $C$. All the considerations below, after obvious adaptations, are valid for this simplified case. At first, we apply the variational formulations to deduce existence of solutions. Recall that the inner product in $L^2(G\times S\times I)^3$ is given by $${\left\langle}\phi,v{\right\rangle}_{L^2(G\times S\times I)^3} =\sum_{j=1}^3 {\left\langle}\phi_j,v_j{\right\rangle}_{L^2(G\times S\times I)},$$ and analogously in other products of inner product spaces. Integrating by parts and applying the Green’s formula (\[green\]) we find (similarly as in section \[esols\]) that the bilinear form ${\bf B}(\cdot,\cdot):C^1(\ol G\times S\times I)^3\times C^1(\ol G\times S\times I)^3\to{\mathbb{R}}$ and the linear form ${\bf F}: C^1(\ol G\times S\times I)^3 \to{\mathbb{R}}$ corresponding to the problem (\[cosyst1\])- (\[cosyst4\]) are \[cosyst5\] [B]{}(,v)=& \_[j=2,3]{}\_j,S\_j[E]{}\_[L\^2(GSI)]{} -,\_x v\_[L\^2(GSI)\^3]{}\ & +\_[j=2,3]{} C\_j,S\_jv\_j\_[L\^2(GSI)]{} +,(\^\-K\_C\^\) v\_[L\^2(GSI)\^3]{}\ & +\_+(), \_+(v) \_[T\^2(\_+)\^3]{} +\_[j=2,3]{} \_j(,,0),S\_j(,0) v\_j(,,0)\_[L\^2(GS)]{} , and \[cosyst6\] [F]{}(v) = [f]{},v\_[L\^2(GSI)\^3]{} +[g]{},\_-(v)\_[T\^2(\_-)\^3]{}. For $j=2,3$, let $$q_j:={1\over 2}{\mathop{\mathrm{ess\hspace{1mm}sup}}}_{(x,E)\in G\times I}{{\frac{\partial S_j}{\partial E}}}(x,E).$$ Moreover, define $$\begin{aligned} \label{cosyst6a} &C_j:={{\max\{q_j,0\}}\over\kappa_j},\quad j=2,3, \nonumber \\ &C:=\max\{C_1,C_2\}. \end{aligned}$$ The appropriate Hilbert spaces are defined as &[H]{}:=HH\_1H\_1,\ &:=\^2(GSI)H\_2H\_2, where the space $\tilde {\bf W}^2(G\times S\times I)$ was defined just before Corollary \[convg\]. Recall from section \[fs\] that the elements of ${{{\mathcal{}}}H}$ are of the form $$\tilde\phi=(\tilde\phi_1,\tilde\phi_2,\tilde\phi_3) =\big((\phi_1,q_1), (\phi_2,q_2,p_{02},p_{{\rm m}2}), (\phi_3,q_3,p_{03},p_{{\rm m}3})\big),$$ with $\phi_i\in L^2(G\times S\times I)$, $q_i\in T^2(\Gamma)$ for $i=1,2,3$, and $p_{0j}, p_{{\rm m}j}\in L^2(G\times S)$ for $j=2,3$. Moreover, $\tilde {{{\mathcal{}}}H}\subset {{{\mathcal{}}}H}$ through the continuous embedding $$\begin{aligned} \label{eq:embed_sH} & v=(v_1,v_2,v_2) \mapsto \big( (v_1,\gamma(v_1)),\varrho(v_2), \varrho(v_3) \big), \nonumber \\ & \varrho(v_j):=\big(v_j,\gamma(v_j),v_j(\cdot,\cdot,0),v_j(\cdot,\cdot,E_{\rm m})\big),\hspace{1cm} j=2,3.\end{aligned}$$ In spaces ${{{\mathcal{}}}H}$ and $\tilde {{{\mathcal{}}}H}$ we use respectively the inner products $${\left\langle}\phi,v{\right\rangle}_{{{{\mathcal{}}}H}}={\left\langle}\phi_1,v_1{\right\rangle}_{H}+\sum_{j=2,3}{\left\langle}\phi_j,v_j{\right\rangle}_{H_1}$$ and $${\left\langle}\phi,v{\right\rangle}_{\tilde {{{\mathcal{}}}H}} ={\left\langle}\phi_1,v_1{\right\rangle}_{{\tilde {\bf W}^2(G\times S\times I)}}+\sum_{j=2,3}{\left\langle}\phi_j,v_j{\right\rangle}_{H_2}.$$ The bilinear form ${\bf B}:C^1(\ol G\times S\times I)^3\times C^1(\ol G\times S\times I)^3\to{\mathbb{R}}$ has the following boundedness and coercivity properties. \[cosystth1\] Suppose that the assumptions (\[scateh\]), (\[colleh\]), (\[sda2a\]), , (\[sda2b\]) are valid and that (\[csda3aa\]), (\[csda4aa\]) hold for $c>0$ and for $C$ given in (\[cosyst6a\]). Then there exists $M>0$ such that \[cosyst12\] \|[B]{}(,v)\|M\_[[H]{}]{}[v]{}\_, vC\^1(GSI)\^3, and \[cosyst13\] [B]{}(,)c’\_[[H]{}]{}\^2C\^1(GSI)\^3, where $$\begin{aligned} \label{cocprime} c':=\min\{{1\over 2},{{\kappa_2}\over 2},{{\kappa_3}\over 2},c\}.\end{aligned}$$ In addition, for all $v\in C^1(\ol G\times S\times I)^3$, \[cosyst14\] \|[F]{}(v)\| ([f]{}\_[L\^2(GSI)\^3]{}+[g]{}\_[T\^2(\_-)\^3]{})[v]{}\_[[H]{}]{}. The boundedness (\[cosyst12\]) can be seen as in the proof of Theorem \[csdath1\]. The assumptions (\[csda3aa\]), (\[csda4aa\]) imply by Theorem \[sdath1\] that (-K\_C),\_[L\^2(GSI)\^3]{}c\^2\_[L\^2(GSI)\^3]{}. Hence we see as in the proof of Theorem \[csdath1\] that for $C=\max\{C_1,C_2\}$ the coercitivity (\[cosyst13\]) holds with the stated $c'>0$. The estimate (\[cosyst14\]) is immediate (see ) and so the proof is finished. Due to the above theorem, the bilinear form ${\bf B}(\cdot,\cdot)$ has a unique extension $\tilde {\bf B}(\cdot,\cdot):{{{\mathcal{}}}H}\times \tilde {{{\mathcal{}}}H}\to {\mathbb{R}}$ which satisfies \[cosyst15\] \|(,v)\|M\_[[H]{}]{}[v]{}\_, v, and \[cosyst16\] (v,v)c’[v]{}\_[[H]{}]{}\^2v. Likewise, implies that the linear form ${\bf F}$ has a unique bounded extension ${{{\mathcal{}}}H}\to{\mathbb{R}}$, which we still denote by ${\bf F}$. The variational equation corresponding to the problem (\[cosyst1\])-(\[cosyst4\]) is \[cosyst7\] (,v)=[F]{}(v)v. For the coupled BTE we have the following variational existence theorem, to be compared with Theorem \[csdath3\] for single particle transport. \[cosystth2\] Suppose that the assumptions (\[scateh\]), (\[colleh\]), (\[sda2a\]), , (\[sda2b\]) are valid, and that (\[csda3aa\]), (\[csda4aa\]) hold for $c>0$ and for $C$ given in (\[cosyst6a\]). Let ${\bf f}\in L^2(G\times S\times I)^3$ and ${\bf g}\in T^2(\Gamma_-)^3$. Then the following assertions hold. \(i) The variational equation \[cosyst17\] (,v)=[F]{}(v)v, has a solution $\tilde{\phi}=(\tilde{\phi}_1, \tilde{\phi}_2, \tilde{\phi}_3)\in {{{\mathcal{}}}H}$. Writing $\tilde{\phi}_1=(\phi_1,q_1)$, $\tilde \phi_j=(\phi_j,q_j,p_{0j},p_{mj})$, $j=2,3$, and $\phi=(\phi_1,\phi_2,\phi_3)\in L^2(G\times S\times I)^3$, then $\phi\in{\mathcal{H}}_{{\bf P}}(G\times S\times I^\circ)$ (see ) and it is a weak (distributional) solution of the system of equations (\[cosyst1\]), (\[cosyst2\]), and $\phi_1\in W^2(G\times S\times I)$. \(ii) Suppose that additionally the assumption ${\bf TC}$ holds (p. ). Then a solution $\phi$ of the equations (\[cosyst1\]), (\[cosyst2\]) obtained in part (i) is a solution of the problem -. \(iii) Under the assumptions imposed in part (ii), any solution $\phi$ of the problem - that further satisfies \[asscl-aa-co-a\] \_[\|\_+]{}T\^2(\_+)\^3 (,,0)L\^2(GS)\^3, is unique and obeys the estimate \[cosyst17a\] \_[[H]{}]{} (\_[L\^2(GSI)\^3]{}+\_[T\^2(\_-)\^3]{}), where $c'$ is given in . The proofs of items (i)-(iii) are analogous to the proofs of the corresponding items in Theorem \[csdath3\]. Note that the claim $\phi_1\in W^2(G\times S\times I)$ in part (i) follows from noticing that by we have $\omega\cdot\nabla_x\phi_1=-\Sigma_1\phi_1+K_{1,C}\phi+{\bf f}_1$, and the right hand side belongs to $L^2(G\times S\times I)$. The problem (\[csda1a\])-(\[csda3\]) also admits a variational formulation. Define bilinear $\tilde{\bf B}_0(\cdot,\cdot):{{{\mathcal{}}}H}\times \tilde {{{\mathcal{}}}H}\to{\mathbb{R}}$ by \[cosyst5a\] \_0(,v) =& \_[j=2,3]{} \_j,S\_j[E]{}\_[L\^2(GSI)]{} -, \_x v\_[L\^2(GSI)\^3]{} +,(\^\-K\^\) v\_[L\^2(GSI)\^3]{}\ & +q, \_+(v)\_[T\^2(\_+)\^3]{} +\_[j=2,3]{}p\_[0j]{},S\_j(,0) v\_j(,,0)\_[L\^2(GS)]{}, where $\tilde{\psi}=(\tilde{\psi}_1,\tilde{\psi}_2,\tilde{\psi}_3)$, $\tilde{\psi}_1=(\psi_1,q_1)$, $\tilde{\psi}_j=(\psi_j,q_j,p_{0j},p_{{\rm m}j})$, $j=2,3$, and $\psi=(\psi_1,\psi_2,\psi_3)$, $q=(q_1,q_2,q_3)$. In other words, $\tilde{\bf B}_0$ is the unique extension of (\[cosyst5\]) with $C=0$. Then by part (ii) of Theorem \[cosystth2\], a given $\psi\in W^2(G\times S\times I)\times W_1^2(G\times S\times I)^2 $ is a solution of the problem (\[csda1a\])-(\[csda3\]) if (and only if) the variational equation $$\tilde{\bf B}_0(\tilde{\psi},v)={\bf F}_0(v),\quad \forall v\in \tilde {{{\mathcal{}}}H},$$ holds. Here $\psi$ is identified with $\tilde{\psi}\in {{{\mathcal{}}}H}$ through the mapping given in , and \[linf\] [F]{}\_0(v) =[f]{},v\_[L\^2(GSI)\^3]{}+ [g]{},\_-(v)\_[T\^2(\_-)\^3]{}. We have the following corollary for the original problem. \[cosystco1\] Suppose that the assumptions of Theorem \[cosystth2\] are valid. Let ${ f}\in L^2(G\times S\times I)^3$ and ${ g}\in T^2(\Gamma_-)^3$. Then the following assertions hold. \(i) The variational equation \[vareqco\] \_0(,v)=[F]{}\_0(v)v has a solution $\tilde{\psi}=(\tilde{\psi}_1, \tilde{\psi}_2, \tilde{\psi}_3)\in {{{\mathcal{}}}H}$. Writing $\tilde{\psi}_1=(\psi_1,q_1)$, $\tilde{\psi}_j=(\psi_j,q_j,p_{0j},p_{mj}),\ j=2,3$, and $\psi=(\psi_1,\psi_2,\psi_3)\in L^2(G\times S\times I)^3$, then $\psi\in{\mathcal{H}}_{{\bf P}}(G\times S\times I^\circ)$ (see ) and it is a weak (distributional) solution of the system of equations (\[csda1a\]), (\[csda1b\]), and $\psi_1\in W^2(G\times S\times I)$. \(ii) Suppose that additionally the assumption ${\bf TC}$ holds (p. ). Then a solution $\psi$ of the equations (\[csda1a\]), (\[csda1b\]) obtained in part (i) is a solution of the problem (\[csda1a\])-(\[csda3\]). \(iii) Under the assumptions imposed in part (ii), any solution $\psi$ of the problem (\[csda1a\])-(\[csda3\]) that further satisfies \[asscl-aa-co\] \_[\|\_+]{}T\^2(\_+)\^3 (,,0)L\^2(GS)\^3, is unique and obeys the estimate \[csda40aaa-co\] \_[[H]{}]{} (\_[L\^2(GSI)\^3]{}+\_[T\^2(\_-)\^3]{}). (Recall that $C$ is defined in , $c'$ in and that $E_m$ is the cutoff energy.) A solution $\psi$ of the problem (\[csda1a\])-(\[csda3\]) is obtained from a solution $\phi$ of the problem - by taking $\psi=e^{-CE}\phi$. Note that if $\phi_1\in W^2(G\times S\times I)$, then $\psi_1\in W^2(G\times S\times I)$ as well. The rest of the proof proceeds in exactly the same way as that for Corollary \[csdaco1\] (of course, one uses Theorem \[cosystth2\] instead of Theorem \[csdath3\]). Existence of Solutions Based on $m$-dissipativity {#mdiss-op} ------------------------------------------------- The method of section \[m-d\] can extended to the case if coupled system in a straightforward manner. Let us state what the problem to be solved for $\phi$ is in this context, in accordance with section \[m-d\]. Given ${\bf f}=({\bf f}_1, {\bf f}_2, {\bf f}_3)\in L^2(G\times S\times I)^3$, and ${\bf g}\in T^2(\Gamma_-)\times H^1(I,T^2(\Gamma_-'))^2$, find $\phi=(\phi_1,\phi_2,\phi_3)\in L^2(G\times S\times I)^3$ which satisfies the system of equations of $G\times S\times I$, $$\begin{aligned} \omega\cdot\nabla_x\phi_1+\Sigma_1\phi_1- K_{1,C}\phi = {}& {\bf f}_1,\label{codiss1}\\ -{{\frac{\partial (S_j\phi_j)}{\partial E}}}+\omega\cdot\nabla_x\phi_j+CS_j\phi_j+\Sigma_{j}\phi_j -{K}_{j,C}\phi = {}& {\bf f}_j,\quad j=2,3,\label{codiss2}\end{aligned}$$ the boundary condition on $\Gamma_-$, \[codiss3\] \_[\|\_-]{}=[g]{}, and the initial condition on $G\times S$, \[codiss4\] \_j(,,E\_m)=0,j=2,3. Let $$P_{1}(x,\omega,E,D)\phi_1:={}&\omega\cdot\nabla_x\phi_1, \\[2mm] P_{C,j}(x,\omega,E,D)\phi_j:={}&-{{\frac{\partial (S_j\phi_j)}{\partial E}}}+\omega\cdot\nabla_x\phi_j +CS_j\phi_j,\quad j=2,3, \\[2mm] {\bf P}_{C}(x,\omega,E,D)\phi:={}&\big(P_{1}(x,\omega,E,D)\phi_1, P_{C,2}(x,\omega,E,D)\phi_2,P_{C,2}(x,\omega,E,D)\phi_3\big).$$ When $C=0$ we write ${\bf P}(x,\omega,E,D):={\bf P}_0(x,\omega,E,D)$, and define $$\begin{gathered} {{{\mathcal{}}}H}_{\bf P}(G\times S\times I^\circ) :=\{\psi\in L^2(G\times S\times I)^3\ \| \\ {\bf P}(x,\omega,E,D)\psi\in L^2(G\times S\times I)^3\ \textrm{in the weak sense}\}. \label{eq:H_bfP}\end{gathered}$$ The operator $\tilde {\bf P}_{C,0}$ is defined in the same way as $\tilde {P}_{C,0}$ in section \[m-d\], namely it is the smallest closed extension (closure) of ${\bf P}_{C,0}$, where $$D({\bf P}_{C,0}):={}&\big\{ \phi\in \tilde{W}^2(G\times S\times I)\times \big(\tilde{W}^2(G\times S\times I)\cap H^1(I,L^2(G\times S)\big)^2 \ \big\| \\[2mm] &\hspace{2.5mm} \phi_{\|\Gamma_-}=0,\ \phi_j(\cdot,\cdot,E_m)=0,\ j=2,3\big\} \\[2mm] {\bf P}_{C,0}\phi:={}&{\bf P}_{C}(x,\omega,E,D)\phi.$$ Using these notations, the problem - with ${\bf g}=0$, in the strong sense, is equivalent to $$(\tilde {\bf P}_{C,0}+\Sigma-K_C)\phi={\bf f},$$ where $\phi\in D(\tilde {\bf P}_{C,0})$. We assume that the stopping powers for $j=2,3$ satisfy $$\begin{aligned} {}& S_j\in C^2(I, L^\infty(G)), \label{Sj-ass:1} \\[2mm] {}& \kappa_j:=\inf_{(x,E)\in \ol{G}\times I}S_j(x,E)>0, \label{Sj-ass:2} \\[2mm] {}& \nabla_x S_j\in L^\infty(G\times I). \label{Sj-ass:3}\end{aligned}$$ We give here only the following concluding result, which can be proven using the methods of section \[m-d\] and those of [@tervo14 Section 5.3]. Recall that by the conventions adopted above, ${\bf f}_j=e^{CE}f_j$ and ${\bf g}_j=e^{CE}g_j$, where $C$ is as defined in . \[m-d-j-co1\] Suppose that the assumptions (\[scateh\]), (\[colleh\]), (\[csda3aa\]), (\[csda4aa\]) (with $c>0$) and , , are valid. Furthermore, suppose that ${f}\in L^2(G\times S\times I)^3$, and $g\in T^2(\Gamma_-)\times H^1(I,T^2(\Gamma_-'))^2$ is such that the compatibility condition \[comp-d-j\] g\_j(,,E\_[m]{})=0,j=2,3, holds. Then the problem - has a unique solution $\psi\in {{{\mathcal{}}}H}_{\bf P}(G\times S\times I^\circ)$. In addition, there exists a constant $C_1>0$ such that a priori estimate \[diss-co-es\] \_[L\^2(GSI)\^3]{} C\_1(\_[L\^2(GSI)\^3]{}+ \_[T\^2(\_-)H\^1(I,T\^2(\_-’))\^2]{}), holds. Existence of Solutions Based on the Theory of Evolution Equations {#evo-op} ----------------------------------------------------------------- For collision operator of special type, the existence result based on the theory of evolution operators is valid also for the coupled system. One of the important features of this approach is that it yields more regularity for the solution. We assume that the collision operator $K=(K_1,K_2,K_3)$ is of the form (see Remark \[cosdare1\]) \[ec1moda\] (K\_j)(x,,E)=\_[k=1]{}\^3\_S\_[kj]{}(x,’,,E)\_k(x,’,E)d’,j=1,2,3. For a fixed $E\in I$ we define bounded linear operators $L^2(G\times S)\to L^2(G\times S)$ by $$\begin{gathered} (\Sigma_1(E)v)(x,\omega):=\Sigma_1(x,\omega,E)v(x,\omega), \\ (\ol K_1(E)v)(x,\omega):=\int_S\tilde\sigma_{11}(x,\omega',\omega,E)v(x,\omega') d\omega'.\end{gathered}$$ In order to avoid ambiguity, below we often denote by $S'$ the sphere $S$ for the variable $\omega'$, while $S$ is reserved for $\omega$. For example, saying that $h$ is a function $S\to (S'\to{\mathbb{R}})$ (resp. $S'\to (S\to {\mathbb{R}})$) means that we would write $h(\omega)(\omega')$ (resp. $h(\omega')(\omega)$). \[coupevle2\] Suppose that $$\begin{gathered} \Sigma_1\in C^1(I,L^\infty(G\times S)), \label{ass1-11} \\[2mm] \tilde{\sigma}_{11}\in C^1(I,L^\infty(G\times S,L^1(S')))\cap C^1(I,L^\infty(G\times S',L^1(S))), \label{ass3-11} \\[2mm] \Sigma_1\geq 0,\quad \tilde\sigma_{11}\geq 0, \label{ass4-a}\end{gathered}$$ and that for some $c>0$ the following hold a.e. $(x,\omega,E)\in G\times S\times I$, &\_1(x,,E)-\_S\_[11]{}(x,,’,E) d’ c, \[ec8-a\]\ &\_1(x,,E)-\_S\_[11]{}(x,’,,E) d’ c. \[ec9-a\] Then for every $q\in C^1(I,L^2(G\times S))$ and every $E\in I$ the problem $$\begin{gathered} \omega\cdot\nabla_x v+\Sigma_1(E)v-\ol K_1(E)v=q(E), \nonumber\\ v_{\|\Gamma_-'}=0, \label{cle1}\end{gathered}$$ has a unique solution $v=v(E)\in \tilde W^2_{-,0}(G\times S)$, and \[cle2\] [v(E)]{}\_[L\^2(GS)]{} [q(E)]{}\_[L\^2(GS)]{}, EI. In addition, $v(\cdot):I\to L^2(G\times S)$ belongs to $C^1(I,L^2(G\times S))$ and for all $E\in I$, \[cle3\] [(E)]{}\_[L\^2(GS)]{} & [1c]{}( [(E)]{}\_[L\^2(GS)]{}+[1c]{} [(E)]{}\_[L\^(GS)]{}[q(E)]{}\_[L\^2(GS)]{}\ & +[1c]{} [(E)]{}[q(E)]{}\_[L\^2(GS)]{} ), where $$\Big({{\frac{\partial \ol K_1}{\partial E}}}(E)v\Big)(x,\omega) :=\int_S{{\frac{\partial \tilde\sigma_{11}}{\partial E}}}(x,\omega',\omega,E)v(x,\omega') d\omega' .$$ That the problem has, for every fixed $E\in I$, a unique solution $v=v(E)$, and that the estimate holds can be proven similarly to [@tervo14 Corollary 5.15] (for the existence of solutions, see also [@dautraylionsv6 Lemma 4, p. 241]). A. At first we show that $v\in C(I,L^2(G\times S))$. Let $E_1,\ E_2\in I$. Since $v(E)\in \tilde W^2_{-,0}(G\times S)$ for all $E\in I$, it follows that $(v(E_1)-v(E_2))_{\|\Gamma_-'}=0$. Moreover, $v(E_1)-v(E_2)$ satisfies the equation \[cle4\] & \_x(v(E\_1)-v(E\_2))+\_1(E\_1)(v(E\_1)-v(E\_2)) -K\_1(E\_1)(v(E\_1)-v(E\_2))\ =&q(E\_1)-q(E\_2)-(\_1(E\_1)-\_1(E\_2))v(E\_2) +(K\_1(E\_1)-K\_1(E\_2))v(E\_2). Therefore, one can apply the estimate , obtaining \[cle5\] &[v(E\_1)-v(E\_2)]{}\_[L\^2(GS)]{}\ & [1c]{} ([q(E\_1)-q(E\_2)]{}\_[L\^2(GS)]{} +[(\_1(E\_1)-\_1(E\_2))v(E\_2)]{}\_[L\^2(GS)]{}\ & +[(K\_1(E\_1)-K\_1(E\_2))v(E\_2)]{}\_[L\^2(GS)]{})\ & [1c]{} ([q(E\_1)-q(E\_2)]{}\_[L\^2(GS)]{} +[\_1(E\_1)-\_1(E\_2)]{}\_[L\^(GS)]{}[v(E\_2)]{}\_[L\^2(GS)]{}\ & +[K\_1(E\_1)-K\_1(E\_2)]{}[v(E\_2)]{}\_[L\^2(GS)]{}), where (see or ) \[cle5a\] [K\_1(E\_1)-K\_1(E\_2)]{}[\_[11]{}(E\_1)-\_[11]{}(E\_2)]{}\_[L\^(GS,L\^1(S’))]{}\^[1/2]{} [\_[11]{}(E\_1)-\_[11]{}(E\_2)]{}\_[L\^(GS’,L\^1(S))]{}\^[1/2]{}. The continuity of $v:I\to L^2(G\times S)$ follows immediately from these estimates and assumptions , . B. Next we verify that $v\in C^1(I,L^2(G\times S))$ and that the estimate (\[cle3\]) holds. For a map $k:G\times S\times I\to{\mathbb{R}}$, which we identify as a map $I\to (G\times S\to{\mathbb{R}})$; $k(E)(x,\omega)=k(x,\omega,E)$ whenever appropriate, define for $h\not=0$ small enough $$(\delta_h k)(E):={{k(E+h)-k(E)}\over h}$$ where at $E$-boundaries $E=0$ or $E=E_{\rm m}$ we take $h>0$ and $h<0$, respectively. Since $v(E')\in \tilde W^2_{-,0}(G\times S)$ for all $E'\in I$, we have $(\delta_h v)(E)_{\|\Gamma_-'}=0$, and from the equation (\[cle1\]) we obtain (from , with $E_1=E$, $E_2=E+h$) \[cle6\] & \_x ((\_hv)(E))+\_1(E)(\_hv)(E)-K\_1(E)(\_hv)(E)\ =&(\_h q)(E)-(\_h\_1)(E)v(E+h)+(\_h \_1)(E)v(E+h), where for any $w\in L^2(G\times S)$, $$(\delta_h \ol{K}_1)(E)w :=&\int_S{{\tilde\sigma_{11}(x,\omega',\omega,E+h)- \tilde\sigma_{11}(x,\omega',\omega,E)}\over h} w(x,\omega') d\omega' \\ =&\int_S (\delta_{h}\tilde\sigma_{11})(E)(x,\omega',\omega)w(x,\omega') d\omega'.$$ To show that the limit $\lim_{h\to 0}(\delta_h v)(E)$ exists in $L^2(G\times S)$ it suffices to verify that $\lim_{n\to\infty}(\delta_{h_n} v)(E)$ exists in $L^2(G\times S)$ for any sequence $\{h_n\}$ tending to zero as $n\to\infty$, or equivalently that $\{(\delta_{h_n}v)(E)\}$ is a Cauchy sequence in $L^2(G\times S)$ for any such sequence $\{h_n\}$. We see by (\[cle6\]) that for $n,m\in{\mathbb{N}}$, \[cle7\] & \_x ((\_[h\_n]{}v)(E)-(\_[h\_m]{}v)(E)) +\_1(E)((\_[h\_n]{}v)(E)-(\_[h\_m]{}v)(E))\ & -\_1(E)((\_[h\_n]{}v)(E)-(\_[h\_m]{}v)(E))\ =& (\_[h\_n]{}q)(E)-(\_[h\_m]{}q)(E) -((\_[h\_n]{}\_1)(E)v(E+h\_n)-(\_[h\_m]{}\_1)(E)v(E+h\_m))\ & +((\_[h\_n]{}\_1)(E)v(E+h\_n)-(\_[h\_m]{}\_1)(E)v(E+h\_m))\ =:&f\_[h\_n,h\_m]{}(E). The following estimates hold (see , ) $$&{\left\Vert (\delta_{h_n}\Sigma_1)(E)v(E+h_n)-(\delta_{h_m}\Sigma_1)(E)v(E+h_m)\right\Vert}_{L^2(G\times S)} \\ \leq &{\left\Vert (\delta_{h_n}\Sigma_1)(E)-(\delta_{h_m}\Sigma_1)(E)\right\Vert}_{L^\infty(G\times S)}{\left\Vert v(E+h_n)\right\Vert}_{L^2(G\times S)} \\ &+{\left\Vert (\delta_{h_m}\Sigma_1)(E)\right\Vert}_{L^\infty(G\times S)}{\left\Vert v(E+h_n)-v(E+h_m)\right\Vert}_{L^2(G\times S)},$$ and $$& {\left\Vert (\delta_{h_n}\ol{K}_1)(E)v(E+h_n)-(\delta_{h_m}\ol{K}_1)(E)v(E+h_m)\right\Vert}_{L^2(G\times S)} \\ \leq & {\left\Vert (\delta_{h_n}\ol{K}_1)(E)-(\delta_{h_m}\ol{K}_1)(E)\right\Vert}{\left\Vert v(E+h_n)\right\Vert}_{L^2(G\times S)} \\ &+{\left\Vert (\delta_{h_m}\ol{K}_1)(E)\right\Vert}{\left\Vert v(E+h_n)-v(E+h_m)\right\Vert}_{L^2(G\times S)},$$ where $${\left\Vert (\delta_{h_m}\ol{K}_1)(E)\right\Vert} \leq {\left\Vert (\delta_{h_m}\tilde\sigma_{11})(E)\right\Vert}_{L^\infty(G\times S,L^1(S'))}^{1/2} {\left\Vert (\delta_{h_m}\tilde\sigma_{11})(E)\right\Vert}_{L^\infty(G\times S',L^1(S))}^{1/2},$$ and $$& {\left\Vert (\delta_{h_n}\ol{K}_1)(E)-(\delta_{h_m}\ol{K}_1)(E)\right\Vert} \\ \leq &{\left\Vert (\delta_{h_n}\tilde\sigma_{11})(E_1)-(\delta_{h_m}\tilde\sigma_{11})(E_2)\right\Vert}_{L^\infty(G\times S,L^1(S'))}^{1/2} {\left\Vert (\delta_{h_n}\tilde\sigma_{11})(E_1)-(\delta_{h_m}\tilde\sigma_{11})(E_2)\right\Vert}_{L^\infty(G\times S',L^1(S))}^{1/2}.$$ Therefore, since we assume , and $q\in C^1(I,L^2(G\times S))$, and since $v\in C(I,L^2(G\times S))$ by part A of the proof, we find that $f_{h_n,h_m}(E)$ in (\[cle7\]) convergences to zero in $L^2(G\times S)$ when $n,m\to\infty$. This fact combined with the estimate obtained by applying , [(\_[h\_n]{}v)(E)-(\_[h\_m]{}v)(E)]{}\_[L\^2(GS)]{} [f\_[h\_n,h\_m]{}(E)]{}\_[L\^2(GS)]{}, shows that $\{(\delta_{h_n} v)(E)\}$ is a Cauchy sequence, and so ${{\frac{\partial v}{\partial E}}}(E)$ exists (in $L^2(G\times S)$) for every $E\in I$. Applying the estimate (\[cle2\]) to $(\delta_{h} v)(E)$ which satisfies , we get \[cle6a\] [(\_[h]{}v)(E)]{}\_[L\^2(GS)]{} & [1c]{} ([(\_h q)(E)]{}\_[L\^2(GS)]{}+[(\_h\_1)(E)]{}\_[L\^(GS)]{}[v(E+h)]{}\_[L\^2(GS)]{}\ &+[(\_h \_1)(E)]{}[v(E+h)]{}\_[L\^2(GS)]{}). Under the standing assumptions (and the fact that $v$ is continuous), the inequality gives in the limit $h\to 0$, \[cle8\] [(E)]{}\_[L\^2(GS)]{} & [1c]{}([(E)]{}\_[L\^2(GS)]{}+[(E)]{}\_[L\^(GS)]{}[v(E)]{}\_[L\^2(GS)]{}\ & +[(E)]{}[v(E)]{}\_[L\^2(GS)]{}), and thus, using (\[cle2\]) once more, we obtain (\[cle3\]) as claimed. It remains to be shown that ${\frac{\partial v}{\partial E}}\in C(I,L^2(G\times S))$. By letting $h\to 0$ in equation we get that $\omega\cdot\nabla_x \Big({{\frac{\partial v}{\partial E}}}(E)\Big)\in L^2(G\times S)$ for all $E\in I$, and $$\begin{gathered} \label{cle9} \omega\cdot\nabla_x \Big({{\frac{\partial v}{\partial E}}}(E)\Big)+\Sigma_1(E){{\frac{\partial v}{\partial E}}}(E)-\ol{K}_1(E)\big({{\frac{\partial v}{\partial E}}}(E)\big) \\ ={{\frac{\partial q}{\partial E}}}(E)-{{\frac{\partial \Sigma_1}{\partial E}}}(E)v(E)+{{\frac{\partial \ol{K}_1}{\partial E}}}(E)v(E).\end{gathered}$$ In addition, $(\delta_h v)(E)\to {{\frac{\partial v}{\partial E}}}(E)$ in $W^2(G\times S\times I)$ when $h\to 0$, and therefore, since $((\delta_h v)(E))_{\|\Gamma_-'}=0$, we conclude that $({{\frac{\partial v}{\partial E}}})(E)_{\|\Gamma_-'}=0$ (by applying inflow trace results; see e.g. Remark \[re:W2\_0\_trace\]). Finally, an application of yields, as in , (we write $L^2=L^2(G\times S)$, $L^\infty=L^\infty(G\times S)$ in order to slightly compress the formulas), \[cle10\] &[(E\_1)-(E\_2)]{}\_[L\^2]{}\ & [1c]{} ( [(E\_1)-(E\_2)]{}\_[L\^2]{} +[(E\_1)-(E\_2)]{}\_[L\^]{}[v(E\_2)]{}\_[L\^2]{}\ &+[(E\_2)]{}\_[L\^]{}[v(E\_2)-v(E\_1)]{}\_[L\^2]{} +[\_1(E\_1) - \_1(E\_2)]{}\_[L\^]{}[(E\_2)]{}\_[L\^2]{}\ & +[(E\_1)-(E\_2)]{}[v(E\_2)]{}\_[L\^2]{} +[(E\_2)]{}[v(E\_2)-v(E\_1)]{}\_[L\^2]{}\ & +[\_1(E\_1)-\_1(E\_2)]{}[(E\_2)]{}\_[L\^2]{} ), and one can further estimate (see ), $${\left\Vert {{\frac{\partial \ol{K}_1}{\partial E}}}(E_1)-{{\frac{\partial \ol{K}_1}{\partial E}}}(E_2)\right\Vert} \leq {\left\Vert {{\frac{\partial \tilde\sigma_{11}}{\partial E}}}(E_1)-{{\frac{\partial \tilde\sigma_{11}}{\partial E}}}(E_2)\right\Vert}_{L(S,S')}^{1/2} {\left\Vert {{\frac{\partial \tilde\sigma_{11}}{\partial E}}}(E_1)-{{\frac{\partial \tilde\sigma_{11}}{\partial E}}}(E_2)\right\Vert}_{L(S',S)}^{1/2},$$ where we have written $L(S,S'):=L^\infty(G\times S,L^1(S'))$ and $L(S',S):=L^\infty(G\times S',L^1(S))$. This implies the continuity of ${\frac{\partial v}{\partial E}}$ that we were to demonstrate, and completes the proof. We are ready to formulate, in the context of the theory of evolution operators, the key existence result for the coupled transport problem -, assuming the collision operator $K=(K_1,K_2,K_3)$ is of the (special) form . \[coupthev\] Assume that the cross-sections $\Sigma_j$, $\tilde\sigma_{jk}$, where $j,k=1,2,3$, satisfy $$\begin{aligned} &\Sigma_j\in C^1(I,L^\infty(G\times S)), \label{ass1-aa} \\[2mm] &\tilde\sigma_{jk}\in C^1(I,L^\infty(G\times S,L^1(S')))\cap C^1(I,L^\infty(G\times S',L^1(S))), \label{ass3-aa} \\[2mm] &\Sigma_j\geq 0,\quad \tilde\sigma_{jk}\geq 0, \label{ass4}\end{aligned}$$ and that for some $c>0$, $$\begin{aligned} &\Sigma_1(x,\omega,E)-\int_S\tilde\sigma_{11}(x,\omega,\omega',E) d\omega' \geq c, \label{ec8-aa} \\[2mm] &\Sigma_1(x,\omega,E)-\int_S\tilde\sigma_{11}(x,\omega',\omega,E) d\omega' \geq c, \label{ec9-aa}\end{aligned}$$ for a.e. $(x,\omega,E)\in G\times S\times I$. Furthermore, assume that the stopping powers $S_j$, where $j=2,3$, satisfy $$\begin{aligned} & S_j\in C^2(I,L^\infty(G)), \label{ass5} \\[2mm] & \nabla_x S_j\in L^\infty(G\times I), \label{ass5n} \\[2mm] & \kappa_j:=\inf_{(x,E)\in \ol{G}\times I} S_j(x,E) > 0. \label{ec4}\end{aligned}$$ Let $f\in C^1(I,L^2(G\times S)^3)$ and let $g\in C^2(I,T^2(\Gamma_-')^3)$ which satisfies the compatibility condition $$g_j(E_m)=0,\quad j=2,3.$$ Then the problem (\[csda1a\])-(\[csda3\]) has a unique solution $\psi\in \tilde W^2(G\times S\times I)\times \big(C(I,\tilde{W}^2(G\times S)^2)\cap C^1(I,L^2(G\times S)^2)\big)$. In particular, $\psi\in \tilde{W}^2(G\times S\times I)\times (\tilde{W}^2(G\times S\times I)\cap W_1^2(G\times S\times I))^2$. If in addition, for some $c>0$, the inequalities $$\begin{aligned} \Sigma_j(x,\omega,E)-\sum_{k=1}^3\int_S\tilde\sigma_{jk}(x,\omega,\omega',E) d\omega' \geq c, \label{ec8} \\ \Sigma_j(x,\omega,E)-\sum_{k=1}^3\int_S\tilde\sigma_{kj}(x,\omega',\omega,E) d\omega' \geq c, \label{ec9}\end{aligned}$$ hold for $j=1,2,3$ and for a.e. $(x,\omega,E)\in G\times S\times I$, then the solution $\psi$ satisfies the estimate . At first we notice that by the assumption (\[ass3-aa\]), for a.e. $(x,\omega,E)\in G\times S\times I$, $$\begin{aligned} &\sum_{k=1}^3\int_S\tilde\sigma_{kj}(x,\omega',\omega,E)d\omega' \leq \sum_{k=1}^3\sup_{E\in I}{\left\Vert \tilde\sigma_{kj}(E)\right\Vert}_{L^\infty(G\times S,L^1(S'))}=:M_1<\infty, \label{m'} \\ &\sum_{k=1}^3\int_S\tilde\sigma_{jk}(x,\omega',\omega,E)d\omega \leq \sum_{k=1}^3 \sup_{E\in I}{\left\Vert \tilde\sigma_{jk}(E)\right\Vert}_{L^\infty(G\times S',L^1(S))}=:M_1'<\infty. \label{m''}\end{aligned}$$ We begin by treating the special case where $g=0$. Recall that the system of equations of interest on $G\times S\times I$ for $\psi=(\psi_1,\psi_2,\psi_3)$ is &\_x\_1+\_1\_1-K\_[1]{}=f\_1 \[proof1\]\ &-[E]{}+\_x\_j+\_[j]{}\_j-K\_[j]{}=f\_j,j=2,3.\[proof2\] Equation can be written as \[pr5.5.1\] \_x\_1+\_1\_1-K\_1\_1-K=f\_1, where $$\hat\psi:=(\psi_2,\psi_3),$$ and $$& \ol{K}_1\psi_1 :=\int_S\tilde\sigma_{11}(x,\omega',\omega,E)\psi_1(x,\omega',E) d\omega' =\ol{K}_1(E)\psi_1(E) \\ & \ol{K}\hat\psi :=\sum_{k=2}^3\int_S\tilde\sigma_{k1}(x,\omega',\omega,E)\psi_k(x,\omega',E) d\omega' =\ol K(E)\hat\psi(E),$$ when we define $$\begin{aligned} {3} &\ol{K}_1(E)v :=\int_S\tilde\sigma_{11}(x,\omega',\omega,E)v(x,\omega') d\omega',\quad && v\in L^2(G\times S), \\ &\ol{K}(E)u :=\sum_{k=2}^3\int_S\tilde\sigma_{k1}(x,\omega',\omega,E)u_k(x,\omega') d\omega',\quad && u=(u_2,u_3)\in L^2(G\times S)^2.\end{aligned}$$ For any $q\in L^2(G\times S\times I)$ the problem $$\begin{gathered} \omega\cdot\nabla_x\psi_1+\Sigma_1\psi_1-\ol K_1\psi_1=q\nonumber \\ {\psi_1}_{\Gamma_-}=0 \label{pr5.5.3}\end{gathered}$$ has a unique solution and \[pr5.5.4\] [\_1]{}\_[L\^2(GSI)]{}[q]{}\_[L\^2(GSI)]{}. This can also be proven similarly to [@tervo14 Corollary 5.15] (for the existence of solutions cf. also [@dautraylionsv6 Lemma 4, p. 241]). Define a linear operator $T_{1,0}:L^2(G\times S\times I)\to L^2(G\times S\times I)$ with domain $D(T_{1,0})$ by \[pr5.5.5\] & D(T\_[1,0]{}):=\^2\_[-,0]{}(GSI)\ & T\_[1,0]{}\_1:=\_x\_1+\_1\_1-K\_1\_1. Then $\psi_1=T_{1,0}^{-1}q$ (exists and) is the solution of , and it follows from that \[pr5.5.6\] [T\_[1,0]{}\^[-1]{}q]{}\_[L\^2(GSI)]{}[q]{}\_[L\^2(GSI)]{},qL\^2(GSI). Equations (\[proof2\]) can be written as \[proof4\] -[E]{}+\_x\_j+\_[j]{}\_j- K\_[j]{}-K\_[1,j]{}\_1=f\_j,j=2,3 where $$&\hat{K}_j\hat\psi :=\sum_{k=2}^3 \int_S\tilde\sigma_{kj}(x,\omega',\omega,E)\psi_k(x,\omega',E) d\omega'=\hat K_j(E)\hat\psi(E) \\ &\hat{K}_{1,j}\psi_1 :=\int_S\tilde\sigma_{1j}(x,\omega',\omega,E)\psi_1(x,\omega',E) d\omega'=\hat K_{1,j}(E)\psi_1(E)$$ when one defines $$\begin{aligned} {3} &\hat{K}_j(E)u :=\sum_{k=2}^3\int_S\tilde\sigma_{kj}(x,\omega',\omega,E)u_k(x,\omega') d\omega',\quad && u=(u_2,u_3)\in L^2(G\times S)^2, \\ &\hat{K}_{1,j}(E)v :=\int_S\tilde\sigma_{1j}(x,\omega',\omega,E)v(x,\omega') d\omega',\quad && v\in L^2(G\times S).\end{aligned}$$ Therefore, if $\psi=(\psi_1,\psi_2,\psi_3)=(\psi_1,\hat{\psi})$ is a solution of , , we have by equation , \[psi1\] \_1=T\_[1,0]{}\^[-1]{}(f\_1+K) and hence by the function $\hat\psi$ satisfies the equation, for $j=2,3$, \[pr5.5.7\] -[E]{}+\_x\_j+\_j\_j-K\_j-K\_[1,j]{}(T\_[1,0]{}\^[-1]{}(K)) =f\_j+f\_j, where we wrote $$\hat{f}_j:=\hat K_{1,j}(T_{1,0}^{-1}f_1),\quad j=2,3.$$ Consider the term $\hat K_{1,j}(T^{-1}_{1,0}(\ol K\hat\psi))$. We find that for a fixed $E\in I$ $$\hat K_{1,j}(T^{-1}_{1,0}(\ol K\hat\psi))=\hat K_{1,j}(E)T_{1,0}(E)^{-1}(\ol K(E)\hat\psi(E))$$ where for every $E\in I$ the linear operator $T_{1,0}(E):L^2(G\times S)\to L^2(G\times S)$ with domain $D(T_{1,0}(E))$ is defined to be $$& D(T_{1,0}(E)):=\tilde{W}^2_{-,0}(G\times S)\nonumber\\ & T_{1,0}(E)v:=\omega\cdot\nabla_x v+\Sigma_1(E)v-\ol K_1(E)v.$$ By (the proof of) Lemma , the operator $T_{1,0}(E)$ is invertible and $$\begin{aligned} \label{eq:T10Einv_bound} {\left\Vert T_{1,0}(E)^{-1}\tilde{q}\right\Vert}_{L^2(G\times S)}\leq {1\over c}{\left\Vert \tilde{q}\right\Vert}_{L^2(G\times S)},\quad \forall\tilde{q}\in L^2(G\times S).\end{aligned}$$ Define linear operators $Q_j(E):L^2(G\times S)^2\to L^2(G\times S)$, $j=2,3$, $E\in I$, by setting $$Q_j(E)u:=\hat{K}_{1,j}(E)T_{1,0}(E)^{-1}(\ol{K}(E)u),$$ and let $Q(E):=(Q_1(E),Q_2(E))$. Then we have by and , (see e.g. ) that for all $E\in I$, [Q\_j(E)]{}[\_[1,j]{}(E)]{}[T\_[1,0]{}(E)\^[-1]{}]{}[K(E)]{} , where the operator norms used are taken, in an obvious way, with respect to the space $L^2(G\times S)$ and its product $L^2(G\times S)^2$. Using assumption and Lemma \[coupevle2\], we find that for any fixed $u\in L^2(G\times S)^2$ the mapping $h_u:I\to L^2(G\times S)^2$ given by $h_u(E):=Q(E)u$ belongs to $C^1(I,L^2(G\times S)^2)$. Similarly, we find that $\hat{f}:=(\hat{f}_1,\hat{f}_2)\in C^1(I,L^2(G\times S)^2)$, since we have $$\hat{f}_j(E)=\hat{K}_{1,j}(E)T_{1,0}(E)^{-1}(f_1(E)),\quad E\in I,\ j=2,3,$$ where $\hat{f}_j(E)(x,\omega)=\hat{f}_j(x,\omega,E)$. Let $C:=\max\{C_2,C_3\}$ where for $j=2,3$, $$\begin{gathered} C_j:={1\over 2}\kappa_j^{-2}{\left\Vert \nabla_x S_j\right\Vert}_{L^\infty(G\times I)} \\ +\kappa_j^{-1}\Big({\left\Vert \Sigma_j\right\Vert}_{L^\infty(G\times S\times I)} +{\left\Vert {{\frac{\partial S_j}{\partial E}}}\right\Vert}_{L^\infty(G\times I)}+\sqrt{M_1M_1'} +{{M_1M_1'}\over c}\Big).\end{gathered}$$ Replacing $\hat\psi$ with $\hat\phi(x,\omega,E):=e^{-CE}\hat\psi(x,\omega,E_{\rm m}-E)$ (as in section \[evcsd\]) we find that the system (\[pr5.5.7\]) is equivalent to \[proof7\] [E]{}-[A]{}\_C(E)=F(E),(0)=0, where $$\hat{\phi}&=(\phi_2,\phi_3), \\[2mm] {\bf A}_C(E)\hat\phi&=({A}_{C,2}(E)\hat\phi,{A}_{C,3}(E)\hat\phi), \\[2mm] F&=(F_2,F_3),$$ and for $j=2,3$, $$F_j(E):={1\over{\tilde S_j(E)}}e^{-CE}\big(\tilde f_j(x,\omega,E)+\tilde{\hat f}_j(x,\omega,E)\big),$$ and $$\begin{gathered} \label{proof9} A_{C,j}(E)\hat\phi :=-\Big({1\over {\tilde S_j(E)}}\omega\cdot\nabla_x\phi_j+C \phi_j+{1\over {\tilde S_j(E)}}\tilde\Sigma_j(E)\phi_j +{1\over{\tilde S_j(E)}}{{\frac{\partial \tilde S_j}{\partial E}}}(E)\phi_j \\ -{1\over{\tilde S_j(E)}} \tilde{\hat{K}}_j(E)\hat\phi -{1\over{\tilde S_j(E)}} \tilde Q_{j}(E)\hat\phi\Big).\end{gathered}$$ Here $\tilde S_j(x,E):=S_j(x,E_m-E)$ and similarly for other expressions equipped with “tilde”. Considering ${\bf A}_C(E)$ as an (unbounded) operator $L^2(G\times S)^2\to L^2(G\times S)^2$ with domain $$D({\bf A}_C(E))=\tilde W^2_{-,0}(G\times S)\times \tilde W^2_{-,0}(G\times S),$$ we get by applying Theorem \[evoth\] along with computations analogous to the ones done in section \[evcsd\], that the system (\[proof7\]) has a unique solution $\hat{\phi}\in C(I,\tilde W^2_{-,0}(G\times S)^2)\cap C^1(I,L^2(G\times S)^2)$, which satisfies the homogeneous boundary and initial conditions, $\hat{\phi}_{\|\Gamma_-}=0$, $\hat{\phi}(\cdot,\cdot,0)=0$. Then $\hat{\psi}(x,\omega,E)=e^{C(E_{\rm m}-E)}\hat\phi(x,\omega,E_{\rm m}-E)$, and $\psi_1$ is obtained from (\[psi1\]) that is, $\psi_1=T_{1,0}^{-1}(f_1+\ol K\hat\psi)$, giving us the solution $\psi=(\psi_1,\psi_2,\psi_3)=(\psi_1,\hat{\psi})$ for homogeneous (inflow) boundary, and initial condition data, that is $\psi\|_{\Gamma_-}=0$, $\psi_j(\cdot,\cdot,E_m)=0$ for $j=2,3$, which is what we were looking for. By applying the lifts, the existence of a unique solution of the problem (\[csda1a\])-(\[csda3\]) satisfying the inhomogeneous boundary condition \[proof11\] [\_j]{}\_[\|\_-]{}=g\_j,j=1,2,3, is obtained as in section \[evcsd\] as well (see the proof of Theorem \[evoth1\]). The estimate under the assumptions , follows from Corollary \[cosystco1\]. This completes the proof. \[coupcoev\] Suppose that the assumptions (\[ass1-aa\])-(\[ec9-aa\]) of Theorem \[coupthev\] are valid. Furthermore, suppose that $f\in H^2(I,L^2(G\times S)^3)$ and $g\in H^3(I,T^2(\Gamma_-')^3)$ which satisfies the compatibility condition $$g_j(E_m)=0,\quad j=2,3.$$ Then the problem (\[csda1a\])-(\[csda3\]) has a unique solution $\psi\in \tilde{W}^2(G\times S\times I)\times \big(C^1(I,L^2(G\times S)^2)\cap C(I,\tilde W(G\times S)^2)\big)$. If, in addition, (\[ec8\]) and (\[ec9\]) are valid, the estimate holds. Follows from the Sobolev Embedding Theorem as in Corollary \[evothco\]. The evolution operator based approach given above can be generalized for $L^p$-theory when $1\leq p<\infty$. Note on Reflective Boundary Conditions -------------------------------------- Let $R:=(R_1,R_2,R_3):T^2(\Gamma_+)^3\to T^2(\Gamma_-)^3$ be a linear, possibly unbounded operator with domain $D(R)$. In some problems the inflow boundary condition \[inflowbc\] \_[\|\_-]{}=g, of the problem (\[csda1a\])-(\[csda3\]) is substituted with a more general, so-called reflective boundary condition (cf. [@dautraylionsv6 Chapter XXI, Appendix of §2, p. 249-262]) \[reflexbc\] \_[\|\_-]{}=R(\_[\|\_+]{})+g, where $g\in T^2(\Gamma_-)^3$. We call $R$ a reflection operator. Note that when $R=0$, we obtain (\[inflowbc\]). We consider here one specific type of reflection operator $R=R_{\rm b}$ which is of interest in applications where backscattering from outside the region $G$ is of importance. Let $G'\subset{\mathbb{R}}^3$ be an open bounded set such that $\ol{G'}$ is a $C^1$-manifold with boundary (i.e. $G'$ has the same regularity properties as $G$) and such that $\ol{G}\subset G'$. Let $G_{\rm e}:=G'\setminus \ol G$. Then $$\partial G_{\rm e}=\partial^1 G_{\rm e}\cup \partial^2G_{\rm e},$$ where $\partial^1 G_{\rm e}:=\partial G$ and $\partial^2 G_{\rm e}:=\partial G_{\rm e}\backslash \partial^1 G_{\rm e}$. Let $$\Gamma_{\rm e}:= (\partial G_{\rm e})\times S\times I = \Gamma_{\rm e,+}\cup \Gamma_{\rm e,-}\cup \Gamma_{\rm e,0},$$ where on the right hand side, the decomposition of $\Gamma_{\rm e}$ into the three disjoint subsets corresponds to $\Gamma=\Gamma_+\cup \Gamma_-\cup\Gamma_0$ when considering $G_{\rm e}\times S\times I$ instead of $G\times S\times I$. Finally, we can decompose $\Gamma_{\rm e,+}$ and $\Gamma_{\rm e,-}$ respectively as $$\Gamma_{\rm e,\pm} = \Gamma^1_{\rm e,\pm}\cup \Gamma^2_{\rm e,\pm},$$ where for $j=1,2$, $$\Gamma^j_{\rm e,\pm}:=\big((\partial^j G_{\rm e})\times S\times I\big)\cap \Gamma_{\rm e,\pm}.$$ Notice that $\Gamma^1_{\rm e,-}=\Gamma_+$, while $\Gamma^1_{\rm e,+}=\Gamma_-$. We assume (for simplicity) that $f=0$. Consider the problem (\[csda1a\])-(\[csda3\]) that is, find $\psi=(\psi_1,\psi_2,\psi_3)$ that satisfies on $G\times S\times I$ the system of transport equations, $$\begin{gathered} \omega\cdot\nabla_x\psi_1+\Sigma_1\psi_1-K_{1}\psi=0,\label{ref2}\\ -{{\frac{\partial (S_{j}\psi_j)}{\partial E}}}+\omega\cdot\nabla_x\psi_j+\Sigma_{j}\psi_j-K_{j}\psi=0,\quad j=2,3\label{ref3}\end{gathered}$$ under the (inflow) boundary condition on $\Gamma_-$, \[ref4\] [\_j]{}\_[\|\_-]{}=g\_j,j=1,2,3, and the initial condition on $G\times S$, \[ref5\] \_j(,,E\_[m]{})=0,j=2,3. Since $\Gamma^1_{\rm e,-}=\Gamma_+$, we find that the flux $\psi_{\|\Gamma_+}$ is an inflow boundary source for the domain $G_{\rm e}$, on the part $\partial^1 G_{\rm e}=\partial G$ of its boundary. Suppose that $G_{\rm e}$ does not contain any extra internal or boundary sources. Then the transport of particles in $G_{\rm e}$ is governed by the system of equations on $G_{\rm e}\times S\times I$, $$\begin{gathered} \omega\cdot\nabla_x\Psi_1+\Sigma_{{\rm e},1}\Psi_1-K_{{\rm e},1}\Psi=0,\label{ref2a}\\ -{{\frac{\partial (S_{{\rm e},j}\Psi_j)}{\partial E}}}+\omega\cdot\nabla_x\Psi_j+\Sigma_{{\rm e},j}\Psi_j-K_{{\rm e},j}\Psi=0,\quad j=2,3,\label{ref3a}\end{gathered}$$ for $\Psi=(\Psi_1,\Psi_2,\Psi_3)$, along with the boundary conditions $$\begin{aligned} {5}\label{ref4a} & {\Psi_j}_{\|\Gamma^1_{\rm e,-}}={\psi_j}_{\|\Gamma_+}\quad && {\rm on}\ \Gamma^1_{\rm e,-}=\Gamma_+,\quad && \\ & {\Psi_j}_{\|\Gamma^2_{\rm e,-}}=0\quad && {\rm on}\ \Gamma^2_{\rm e,-},\quad && j=1,2,3, \end{aligned}$$ and the initial condition \[ref5a\] \_j(x,,E\_[m]{})=0 G\_[e]{}SI,j=2,3. Above $\Sigma_{{\rm e},j}$, $\sigma_{{\rm e},kj}$ and $S_{{\rm e},j}$ are the (restricted) cross-sections and the (restricted) stopping powers for the medium inside $G_{\rm e}$, and $$(K_{{\rm e},j}\Psi)(x,\omega,E) := \sum_{k=1}^3\int_{S\times I}\sigma_{{\rm e},kj}(x,\omega',\omega,E',E)\Psi_k(x,\omega',E')d\omega' dE',$$ for $\Psi\in L^2(G_{\rm e}\times S\times I)^3$. In this setup, we define a reflection operator $R=R_{\rm b}$ by setting (recall that $\Gamma^1_{{\rm e},+}=\Gamma_-$) $$\begin{aligned} & R_{\rm b}(\psi_{\|\Gamma_+}):=\Psi_{\|\Gamma^1_{{\rm e},+}}=\Psi_{\|\Gamma_-}, \nonumber\\ & D(R_{\rm b}):=\{\psi_{\|\Gamma_+}\ \|\ \psi\ {\rm is\ a\ solution\ of\ (\ref{ref2})-(\ref{ref5})\ for \ some}\ g\in T^2(\Gamma_-)^3\}. \label{ref6}\end{aligned}$$ If the assumptions of Theorem \[cosystth2\] hold for $G'$ in place of $G$ (and for the respective cross-sections, stopping powers, $\Sigma'_j,\sigma'_{kj},S'_j$), the operator $R_{\rm b}$ is a linear operator $T^2(\Gamma_+)^3\to T^2(\Gamma_-)^3$ with domain of definition $D(R_{\rm b})\subset T^2(\Gamma_+)^3$. In general, $R_{\rm b}$ is not bounded. The meaning of this definition is that $R_{\rm b}(\psi_{\|\Gamma_+})$ models an extra source on $\Gamma_-$ (i.e. an inflow boundary source for $G$) due to backscattering of particles from the given external region $G_{\rm e}$. The flux $u=(u_1,u_2,u_3)$ contributed by this inflow source is governed by the system of equations $$\begin{gathered} \omega\cdot\nabla_x u_1+\Sigma_1 u_1-K_{1}u=0,\label{ref9}\\ -{{\frac{\partial (S_{j}u_j)}{\partial E}}}+\omega\cdot\nabla_x u_j+\Sigma_{j} u_j-K_{j}u=0,\quad j=2,3,\label{ref10}\end{gathered}$$ on $G\times S\times I$, such that on $\Gamma_-$, \[ref11\] u\_[\|\_-]{}=R\_[b]{}(\_[\|\_+]{}), and for almost every $(x,\omega)\in G\times S$, \[ref12\] u\_j(x,,E\_[m]{})=0,j=2,3. We point out that if $\psi\in D(R_{\rm b})$ and if $u$ solves the problem -, and is such that \[nrep\] R\_[b]{}(u\_[\|\_+]{})=0, holds, i.e. no particles are backscattered repeatedly from $G_{\rm e}$ into $G$, then for $\varphi:=\psi+u$ we have $$\varphi_{\|\Gamma_-} =\psi_{\|\Gamma_-}+u_{\|\Gamma_-} =g+R_{\rm b}(\psi_{\|\Gamma_+}) =g+R_{\rm b}((\psi+u)_{\|\Gamma_+}) =g+R_{\rm b}(\varphi_{\|\Gamma_+}),$$ and hence $\varphi=(\varphi_1,\varphi_2,\varphi_3)=\psi+u$ is a solution of the following problem on $G\times S\times I$, with boundary and initial conditions holding on $\Gamma_-$ and $G\times S$, respectively, $$\begin{gathered} \omega\cdot\nabla_x \varphi_1+\Sigma_1 \varphi_1-K_{1}\varphi=0, \label{ref13}\\[2mm] -{{\frac{\partial (S_{j}\varphi_j)}{\partial E}}}+\omega\cdot\nabla_x \varphi_j+\Sigma_{j}\varphi_j-K_{j}\varphi=0,\label{ref14} \\[2mm] \varphi_{\|\Gamma_-}=R_{\rm b}(\varphi_{\|\Gamma_+})+g, \label{ref15} \\[2mm] \varphi_j(\cdot,\cdot,E_{\rm m})=0, \label{ref16}\end{gathered}$$ where $j=2,3$. This shows that solving for $\varphi$ (in place of $\psi$) the problem , , under boundary condition , is equivalent to solving first for $\psi$ the problem -, and then for $u$ the problem - under the additional condition . We will not explore the question of the existence of solutions for the problem - (or for -) in this paper. Adjoint Transport Problem {#adjoint} ========================= We will discuss briefly the adjoint version of the transport problem (\[csda1a\])-(\[csda3\]), and related operators. Write &T\_1:=\_x\_1+\_1\_1-K\_1,\ &T\_j:=-[E]{}+\_x\_j+ \_j\_j - K\_j,j=2,3,and define a (densely defined) linear operator $T:L^2(G\times S\times I)^3\to L^2(G\times S\times I)^3$ by D(T):=&{L\^2(GSI)\^3 \| T\_jL\^2(GSI), j=1,2,3},\ T:=&(T\_1,T\_2,T\_3). Let $f\in L^2(G\times S\times I)^3$ and $g\in T^2(\Gamma_-)^3$. The problem (\[csda1a\])-(\[csda3\]) can be expressed equivalently as the problem \[adjoint1\] T=f,\_[\|\_-]{}=g,\_j(,,E\_m)=0,j=2,3. As in section \[esols\], an application of integration by parts and the Green’s formula (\[green\]) implies \[adjoint2\] T,v\_[L\^2(GSI)\^3]{}=,T\^\v\_[L\^2(GSI)\^3]{}vC\_0\^1(GSI\^), where $T^v=(T_1^v,T_2^v,T_3^v)$, and \[adjoint3a\] T\_1\^\v:=&-\_x v\_1+\_1\^\v\_1-K\_1\^\v,\ T\_j\^\v:=&S\_j[E]{}-\_x v\_j +\_j\^\v\_j-K\_j\^\v,j=2,3. Moreover, we have $\Sigma_j^=\Sigma_j$ and for $v\in L^2(G\times S\times I)^3$, $j=1,2,3$, $$(K_j^v)(x,\omega,E) =\sum_{k=1}^3\int_{S\times I} \sigma_{jk}(x,\omega,\omega',E,E')\psi_k(x,\omega',E')d\omega' dE'.$$ Let $f^\in L^2(G\times S\times I)^3$ and $g^\in T^2(\Gamma_+)^3$. The adjoint problem* of (\[adjoint1\]) (or equivalently (\[desol10\])-(\[desol12\])) is defined by (cf. [@agoshkov pp. 24-28]) \[adjoint4\] T\^\\^\=f\^\,\_[\|\_+]{}=g\^\,\_j\^\(,,0)=0, j=2,3, or more explicitly -\_x\_1\^\+\_1\^\\_1\^\-K\_1\^\\^\=& f\_1\^\, \[adesol10:1\]\ S\_j[E]{}-\_x\_j\^\ +\_j\^\\_j\^\ - K\_j\^\\^\=& f\_j\^\,j=2,3,\[adesol10:2\] holding a.e. on $G\times S\times I$, together with the outflow* boundary and initial values $$\begin{aligned} {3} \psi^_{\|\Gamma_+}={}&g^ && \quad {\rm a.e.\ on}\ \Gamma_+, \label{adesol11} \\[2mm] \psi_j^(\cdot,\cdot,0)={}&0\ && \quad {\rm a.e.\ on}\ G\times S,\quad j=2,3. \label{adesol12}\end{aligned}$$ The adjoint problem has various kind of applications both in the existence theory of solutions and in computations. At the end of this section we give an example which is related to the dose calculation in radiation therapy. We refer also to computations concerning a related optimal control problem considered in [@tervo14 Section 7], where the adjoint problem plays a significant role. In addition, recall that the concept of Green distribution (or Green function) is usually founded on the theory of adjoint problem. We also point out that the adjoint field $\psi^$ obeying Eqs. , is sometimes called an importance function (cf. [@duderstadt76 Ch. 5, Sec. V]). Consider the variational formulation of the adjoint problem. As in sections \[single-eq\] and \[cosyst\] (for $C=0$) we find that the bilinear form ${\bf B}_0^(\cdot,\cdot):C^1(\ol G\times S\times I)^3\times C^1(\ol G\times S\times I)^3$ and the linear form ${\bf F}^$ corresponding to the adjoint problem are \[adjoint5\] [B]{}\_0\^\(\^\,v) =& -\_[j=2,3]{}\_j\^\,[E]{}\_[L\^2(GSI)]{} +\^\, \_x v\_[L\^2(GSI)\^3]{}\ & +\^\,(-K) v\_[L\^2(GSI)\^3]{} +\_-(\^\), \_-(v)\_[T\^2(\_-)\^3]{}\ & +\_[j=2,3]{} \_j\^\(,,E\_m),S\_j(,E\_m) v\_j(,,E\_m)\_[L\^2(GS)]{} , where $\psi^=(\psi_1^,\psi_2^,\psi_3^)$ and $v=(v_1,v_2,v_2)$, and \[adjoint6\] [F]{}\_0\^\(v)=f\^\,v\_[L\^2(GSI)\^3]{}+g\^\,\_+(v)\_[T\^2(\_+)\^3]{}. Similarly as in Theorem \[cosystth1\] we find that \[ajoint6\] \|[B]{}\_0\^\(\^\,v)\|M[\^\]{}\_[[H]{}]{}[v]{}\_\^\, vC\^1(GSI)\^3 and \|[F]{}\_0\^\(v)\|([f\^\]{}\_[L\^2(GSI)\^3]{}+[g\^\]{}\_[T\^2(\_+)\^3]{})[v]{}\_[[H]{}]{} vC\^1(GSI)\^3. The bilinear form ${\bf B}_0^(\cdot,\cdot)$ has a unique extension $\tilde {\bf B}_0^(\cdot,\cdot):{{{\mathcal{}}}H}\times \tilde {{{\mathcal{}}}H}\to {\mathbb{R}}$ (and $\tilde{\bf B}_0^(\cdot,\cdot)$ has an explicit expression in the same way as in (\[cosyst5a\])) which satisfies \[adjoint8\] \|\_0\^\(\^\,v)\|M[\^\]{}\_[[H]{}]{}[v]{}\_\^\=(\^\,q\^\,p\_0\^\,p\_[m]{}\^\), v. Moreover, the linear form ${\bf F}_0^$ has a unique extension to a bounded linear form ${{{\mathcal{}}}H}\to{\mathbb{R}}$ which we still denote by ${\bf F}_0^$. The variational equation corresponding to the adjoint problem , , , is then \[adjoint10\] \^\\_0(\^\,v)=[F]{}\_0\^\(v),v. Suppose that the assumptions (\[scateh\]), (\[colleh\]), (\[sda2a\]), and (\[sda2b\]) are valid. Using integration by parts and Green’s formula we have the following. For all $\psi,\psi^\in \tilde {{{\mathcal{}}}H}$ one has \[adjoint7\] \_0\^\(\^\,)=\_0(,\^\), where $\tilde{\bf B}_0(\cdot,\cdot)$ is the bilinear form (\[cosyst5a\]). The relation is the justification for the term adjoint problem. For the existence of solutions of the adjoint problem, we formulate the following result. \[adjointco1\] Suppose that the assumptions (\[scateh\]), (\[colleh\]), (\[sda2a\]), , (\[sda2b\]) are valid, and that (\[csda3aa\]), (\[csda4aa\]) hold for $c>0$ and for $C$ given in (\[cosyst6a\]). Let $f^\in L^2(G\times S\times I)^3$ and that $g^\in T^2(\Gamma_+)^3$. Then the following assertions hold. \(i) The variational equation \[vareqcoad\] \_0\^\(\^\,v)=[F]{}\_0\^\(v)v, has a solution $\tilde{\psi}^=(\tilde{\psi}^_1, \tilde{\psi}^_2, \tilde{\psi}^_3)\in {{{\mathcal{}}}H}$. Writing $\tilde{\psi}_1^:=(\psi_1^,q_1^)$, $\tilde{\psi}_j^:=(\psi_j^,q_j^,p_{0j}^,p_{{\rm m}j}^)$, $j=2,3$, and $\psi^=(\psi_1^, \psi_2^, \psi_3^)\in L^2(G\times S\times I)^3$, then $\psi^\in{{{\mathcal{}}}H}_{{\bf P}^}(G\times S\times I^\circ)$ (see ) is a weak (distributional) solution of the system of equations , , and $\psi^_1\in W^2(G\times S\times I)$. \(ii) Suppose that additionally the assumption ${\bf TC}$ holds (p. ). Then a solution $\psi^$ of the equations , obtained in part (i) is a solution of the problem -. \(iii) Under the assumptions imposed in part (ii), any solution $\psi^$ of the problem - that further satisfies \[asscl-aa-co-ad\] \^\\_[\|\_-]{}T\^2(\_-)\^3 (,,E\_[m]{})L\^2(GS)\^3. is unique and obeys the estimate \[adjoint11b\] [\^\]{}\_[[H]{}]{}( \_[L\^2(GSI)\^3]{}+\_[T\^2(\_+)\^3]{}). (Recall that $C$ is defined in , $c'$ in and that $E_m$ is the cutoff energy.) Note that if $\psi^$ is a solution of the problem -, then it is a solution of the variational problem (\[vareqcoad\]) and vice versa. The method founded on the $m$-dissipativity (see sections \[m-d\] and \[mdiss-op\]) can be applied also to the adjoint problem. Define $$P_{1}^(x,\omega,E,D)\psi_1^:={}&-\omega\cdot\nabla_x\psi_1^, \\[2mm] P_{j}^(x,\omega,E,D)\psi_j^:={}&S_j{{\frac{\partial \psi_j^}{\partial E}}}-\omega\cdot\nabla_x\psi_j^,\quad j=2,3, \\[2mm] {\bf P}^(x,\omega,E,D)\psi^:={}&\big(P_{1}^(x,\omega,E,D)\psi_1^, P_{2}^(x,\omega,E,D)\psi_2^,P_{3}^(x,\omega,E,D)\psi_3^\big),$$ and the space $$\begin{gathered} {{{\mathcal{}}}H}_{{\bf P}^}(G\times S\times I^\circ) :=\{\psi^\in L^2(G\times S\times I)^3\ \| \\ {\bf P}^(x,\omega,E,D)\psi^\in L^2(G\times S\times I)^3\ \textrm{in the weak sense}\}. \label{eq:H_bfP_star}\end{gathered}$$ The relevant operator here is the smallest closed extension (closure) $\tilde{\bf P}^_0$ of the operator ${\bf P}^_0$ defined by $$D({\bf P}_0^):={}&\big\{ \psi^\in \tilde{W}^2(G\times S\times I)\times \big(\tilde{W}^2(G\times S\times I)\cap H^1(I,L^2(G\times S)\big)^2 \ \big\| \\[2mm] &\hspace{2.5mm} \psi^_{\|\Gamma_+}=0,\ \psi^_j(\cdot,\cdot,0)=0,\ j=2,3\big\} \\[2mm] {\bf P}_0^\phi:={}&{\bf P}^(x,\omega,E,D)\phi.$$ When $g^=0$, the problem - is equivalent, in the strong sense, to $$(\tilde{\bf P}_0^+\Sigma^-K^)\psi^=f^,$$ where $\psi^\in D(\tilde {\bf P}_0^)$, and $\Sigma^\psi^=(\Sigma_1^\psi_1^, \Sigma_2^\psi_2^, \Sigma_3^\psi_3^)$, $K^\psi^=(K_1^\psi^, K_2^\psi^, K_3^\psi^)$. The result analogous to Theorem \[m-d-j-co1\] is the following. \[m-d-ad\] Suppose that the assumptions (\[scateh\]), (\[colleh\]), (\[csda3aa\]), (\[csda4aa\]) (with $c>0$) and , , are valid. Furthermore, suppose that ${f^}\in L^2(G\times S\times I)^3$ and $g^\in T^2(\Gamma_+)\times H^1(I,T^2(\Gamma_+'))^2$ is such that the compatibility condition \[comp-d-j-ad\] g\_j\^\(,,E\_0)=0, j=2,3, holds. Then the problem - has a unique solution $\psi^\in {{{\mathcal{}}}H}_{\bf P^}(G\times S\times I^\circ)$. In addition, there exists a constant $C_1>0$ such that a priori estimate \[diss-co-es-ad\] [\^\]{}\_[L\^2(GSI)\^3]{} C\_1(\_[L\^2(GSI)\^3]{}+ \_[T\^2(\_+)H\^1(I,T\^2(\_+’))\^2]{}), holds. The existence result analogous to Theorem \[coupthev\] (based on the theory of evolution operators) holds also for the adjoint problem, and it guarantees that $\psi^ \in\tilde W^2(G\times S\times I)\times (\tilde W^2(G\times S\times I)\cap W_1^2(G\times S\times I))^2$. In this case, one assumes that $K$ takes the form , and that consequently its adjoint version $K^=(K_1^, K_2^, K_3^)$ is, $$\begin{aligned} \label{ec1moda:ad} (K_j^\psi)(x,\omega,E)=\sum_{k=1}^3\int_S\tilde\sigma_{kj}(x,\omega,\omega',E)\psi_k(x,\omega',E)d\omega',\quad j=1,2,3.\end{aligned}$$ The adjoint version of Theorem \[coupthev\] can be formulated as follows. \[coupthevad\] Suppose that the adjoint collision operator is of the form , and that the assumptions (\[ass1-aa\])-(\[ec9-aa\]) of Theorem \[coupthev\] are valid for $\Sigma_j,\ \sigma_{jk}$ and $S_j$. Furthermore, suppose that $f^\in C^1(I,L^2(G\times S)^3)$ and $g^\in C^2(I,T^2(\Gamma_+')^3)$ which satisfies the compatibility condition $$g_j^(0)=0,\quad j=2,3.$$ Then the problem - has a unique solution $\psi^\in \tilde W^2(G\times S\times I)\times \big( C(I,\tilde W^2(G\times S)^2)\cap C^1(I,L^2(G\times S)^2)\big)$. In particular, $\psi^\in\tilde{W}^2(G\times S\times I)\times \big(\tilde{W}^2(G\times S\times I)\cap W_1^2(G\times S\times I)\big)^2$. If, in addition, the conditions , are valid, then the solution $\psi^$ satisfies the estimate (\[adjoint11b\]). It is clear that a Sobolev space version of the above theorem analogous to Corollary \[coupcoev\], holds for the adjoint problem as well. \[exgreenf1\] In radiation therapy the absorbed dose* from the particle field $\psi=(\psi_1,\psi_2,\psi_3)$ is defined by the functional D(x)=(D)(x):=\_[j=1]{}\^3\_[SI]{}\_j(x,E)\_j(x,,E) ddE, where $\psi$ is the solution of (\[desol10\])-(\[desol12\]). We see that (D)(x)=\_[j=1]{}\^3\_j(x,),\_j(x,,)\_[L\^2(SI)]{} =(x,),(x,,)\_[L\^2(SI)\^3]{}. Define for any fixed $x\in G$ distribution $T_{\varsigma_{j,x}}$ on $S\times I^\circ$ by T\_[\_[j,x]{}]{}:= \_j(x,),\_[L\^2(SI)]{},C\_0\^(SI\^),j=1,2,3. Then for all $\psi_j=\phi_j\otimes\varphi_j$, where $\phi_j\in C_0^\infty(G)$, $\varphi_j\in C_0^\infty(S\times I^\circ)$, and where $(\phi_j\otimes\varphi_j)(x,\omega,E):=\phi_j(x)\varphi_j(\omega,E)$, we find that \[gf1\] \_[j=1]{}\^3(\_xT\_[\_[j,x]{}]{})(\_j) =& \_[j=1]{}\^3\_x(\_j)T\_[\_[j,x]{}]{}(\_j) =\_[j=1]{}\^3 \_j(x)\_[SI]{}\_j(x,E)\_j(,E) ddE\ =&\_[j=1]{}\^3 \_[SI]{}\_j(x,E)(\_j\_j)(x,,E) ddE =(D)(x). Hence also for a general element $\psi=(\psi_1,\psi_2,\psi_3)\in C_0^\infty(G\times S\times I^\circ)$ one has \[gf2\] (D)(x)=\_[j=1]{}\^3(\_xT\_[\_[j,x]{}]{})(\_j)=:T\_x(), where $T_x$ is a distribution on $G\times S\times I^\circ$ with values in ${\mathbb{R}}^3$. The following discussion will be formal. Let $x\in G$ be fixed. Assume that there exists a sufficient regular solution $\Psi^_x$ to the variational problem \[gf3\] \_0\^\(\^\\_x,v)=T\_x(v),v. Furthermore, assume that $\psi$ (which depends on $f,g$) is a sufficient regular solution of (\[csda1a\])-(\[csda3\]) and that $\Psi^_x=\Psi_x^(x',\omega,E)$ is a solution of (\[gf3\]) such that (\[adjoint7\]) holds (for $\psi$ and $\Psi_x^$). Then we have \[gf4\] (D)(x)=&T\_x()=\_0\^\(\^\\_x,) =\_0(,\^\\_x) =[F]{}\_0(\^\\_x)\ =& f,\^\\_x\_[L\^2(GSI)\^3]{} + g, \_-(\_x\^\) \_[T\^2(\_-)\^3]{}, implying that the dose can be obtained with the help of $\Psi^_x$. That is why, when the equation (\[gf3\]) is solved once, one can obtain the dose $(D\psi)(x)$ at $x\in G$ for any $f$ and $g$ from (\[gf4\]). Some Notes on Computational Methods {#comp} =================================== A Decomposition of Solutions Corresponding to Primary and Secondary Particles {#desol} ----------------------------------------------------------------------------- Consider the system of transport equations as above, $$\begin{aligned} \omega\cdot\nabla_x\psi_1+\Sigma_1\psi_1-K_1\psi={}&f_1,\nonumber\\ -{{\frac{\partial (S_j\psi_j)}{\partial E}}}+\omega\cdot\nabla_x\psi_j+ \Sigma_j\psi_j - K_j\psi={}&f_j, \quad j=2,3, \label{desol10}\end{aligned}$$ holding a.e. on $G\times S\times I$, together with the inflow boundary and initial values $$\begin{aligned} {3} \psi_{\|\Gamma_-}&=g && \quad {\rm a.e.\ on}\ \Gamma_-, \label{desol11} \\[2mm] \psi_j(\cdot,\cdot,E_m)&=0\quad && \quad {\rm a.e.\ on}\ G\times S,\ j=2,3. \label{desol12}\end{aligned}$$ The solution $\psi=(\psi_1,\psi_2,\psi_3)$ for this problem can be decomposed as follows. Let $u=(u_1,u_2,u_3)$ be the solution of the problem without collisions $$\begin{aligned} \omega\cdot\nabla_x u_1+\Sigma_1 u_1={}&f_1,\nonumber\\ -{{\frac{\partial (S_ju_j)}{\partial E}}}+\omega\cdot\nabla_x u_j+\Sigma_j u_j ={}& f_j,\quad j=2,3, \label{desol13}\end{aligned}$$ together with the inflow boundary and initial values u\_[\|\_-]{}&=g, \[desol14\]\ u\_j(,,E\_m)&=0,j=2,3.\[desol15\] Furthermore, let $w=(w_1,w_2,w_3)$ be the solution of the problem $$\begin{aligned} \omega\cdot\nabla_x w_1+\Sigma_1 w_1-K_1w={}& K_1u\nonumber\\ -{{\frac{\partial (S_jw_j)}{\partial E}}}+\omega\cdot\nabla_x w_j +\Sigma_j w_j-K_jw ={}& K_ju,\quad j=2,3,\label{desol16}\end{aligned}$$ together with homogeneous* inflow boundary and initial values w\_[\|\_-]{}&=0,\[desol17\]\ w\_j(,,E\_m)&=0,j=2,3. \[desol18\] Then we find that $\psi=u+w$ is the solution of -. This corresponds to decomposing the evolution of the particle field $\psi$ obeying the full CSDA Boltzmann transport problem - in terms of the evolution of the primary (uncollided) particles, represented by $u$, and of secondary (collided) particles, represented by $w$. The method of decomposing $\psi=u+w$ in this way is useful e.g. in constructing numerical solutions, and is known, for example in neutron transport theory, under the name collided-uncollided split (cf. the recent work [@hauck2013coll] and references therein). To explain a bit this terminology, notice that the primary, uncollided field $u$ obeys which does not involve the collision operator $K=(K_1,K_2,K_3)$, and $u$ contains a direct contribution from the external (boundary) sources $g$ (through ). On the other hand, the field $u$ right after collision as modelled by the term $Ku$, acts as an internal source in the equation for the secondary field $w$, while external sources do not contribute to $w$ directly (a fact captured by ). Note especially that the system (\[desol13\])-(\[desol15\]) is uncoupled, in that the different particle species (photon, electron, positron that we consider here) evolve independently of each other. In some cases the primary component $u$ can be calculated exactly such as the following example shows. \[desolex1\] Suppose that $\Sigma_1\in L^2(G\times S\times I)$, $\Sigma_1\geq c>0$ for some constant $c$, and that, $\Sigma_j(x,\omega,E)=\Sigma_j(x,\omega)$ (i.e. $\Sigma_j$ does not depend on $E$) and $\Sigma_j\in L^2(G\times S),\ \Sigma_j\geq c>0$, for $j=2,3$. Furthermore, suppose that $S_j(x,E)=S_j(E)$, $j=2,3$ (i.e. $S_j$ is independent of $x$) and that $S_j:I\to {\mathbb{R}}_+$ are continuous, strictly positive functions. Finally, let $f_1\in L^2(G\times S\times I)$, $g_1\in T^2(\Gamma_-)$, and for $j=2,3$ let $f_j\in H^1(I,L^2(G\times S))$, $g_j\in H^1(I,T^2(\Gamma'_-))$, such that $g_j(E_{\rm m})=0$ (compatibility condition). Define $R_j:I\to{\mathbb{R}}$ by R\_j(E):=\_0\^E[1]{}d,j=2,3. Let $r_{m,j}:=R_j(E_m)$. Then $R_j:I\to [0,r_{m,j}]$ are continuously differentiable and strictly increasing bijections. Let $R_j^{-1}: [0,r_{m,j}]\to I$ be their inverses. We denote the argument of $R_j^{-1}$ on $[0,r_{m,j}]$ by $\eta$, i.e. $E=R_j^{-1}(\eta)$ (or equivalently $\eta=R_j(E)$). Consider first the (primary) uncoupled problem, $$\begin{gathered} -{{\frac{\partial (S_ju_j)}{\partial E}}}+\omega\cdot\nabla_x u_j+ \Sigma_j u_j = f_j, \nonumber\\ {u_j}_{\|\Gamma_-}=g_j,\quad u_j(\cdot,\cdot,E_m)=0, \label{desol19}\end{gathered}$$ where $j=2,3$. We perform a well-known change of variables (see e.g. [@frank10], [@rockell]) in the problem by defining a new unknowns $v_j$, for $j=2,3$, by setting v\_j(x,,):=S\_j(R\_j\^[-1]{}())u\_j(x,,R\_j\^[-1]{}()), i.e. v\_j(x,,R\_j(E))=S\_j(E)u\_j(x,,E). Then we find that =R\_j’(E)=[1]{} =[1]{}, and so, after writing $$\begin{aligned} {2} \tilde{f}_j(x,\omega,\eta):=S_j(R_j^{-1}(\eta))f_j(x,\omega,R_j^{-1}(\eta)), \quad & (x,\omega,\eta)\in G\times S\times [0,r_{m,j}], \\ \tilde{g}_j(y,\omega,\eta):=S_j(R_j^{-1}(\eta))g_j(y,\omega,R_j^{-1}(\eta)), \quad & (y,\omega,\eta)\in \tilde\Gamma_{-,j},\end{aligned}$$ where $$\tilde\Gamma_{-,j}:=\{(y,\omega,\eta)\in \partial G\times S\times [0,r_{m,j}]\ \|\ \omega\cdot\nu(y)<0\},$$ we see that the problem is equivalent to \[desol20\] &-+\_x v\_j+ \_j v\_j =\_j GS, subject to inflow boundary and initial value conditions, $$\begin{aligned} {3} {v_j}_{\|\tilde\Gamma_{-,j}}&=\tilde{g}_j\quad && {\rm a.e.\ on}\ \tilde\Gamma_{-,j},\label{desol19a} \\ v_j(\cdot,\cdot,r_{m,j})&=0\quad && {\rm a.e.\ on}\ G\times S,\ j=2,3. \label{desol19aa}\end{aligned}$$ Notice that $\tilde{g}_j(r_{m,j})=0$ since $g_j(E_{\rm m})=0$. The original unknowns $u_j$, $j=2,3$, are given in terms of $v_j$ by \[desol21\] u\_j(x,,E)=[1]{}v\_j(x,, R\_j(E)). The problem - can be solved explicitly, at least formally. The solution $v_j$ of is the sum $v_{1,j}+v_{2,j}$ of solutions $v_{1,j}$ and $v_{2,j}$ of the following problems $$\begin{gathered} -{{\frac{\partial v_{1,j}}{\partial \eta}}}+\omega\cdot\nabla_x {v_{1,j}}+ \Sigma_j v_{1,j} =\tilde f_j,\nonumber\\ {v_{1,j}}_{\|\tilde\Gamma_{-,j}}=0, \quad v_{1,j}(\cdot,\cdot,r_{m,j})=0,\label{desol20a}\end{gathered}$$ and $$\begin{gathered} -{{\frac{\partial v_{2,j}}{\partial \eta}}}+\omega\cdot\nabla_x {v_{2,j}}+ \Sigma_j v_{2,j} =0,\nonumber\\ {v_{2,j}}_{\|\tilde\Gamma_{-,j}}=\tilde g_j, \quad v_{2,j}(\cdot,\cdot,r_{m,j})=0,\label{desol20b}\end{gathered}$$ where the (partial differential) equations are to be satisfies on $G\times S\times [0,r_{m,j}]$, the (inflow) boundary conditions on $\tilde\Gamma_{-,j}$ and the initial (energy) conditions on $G\times S$. The solution of (\[desol20b\]) is (cf. [@dautraylionsv6 Ch. XXI, Sec. 3.2, pp. 233-235], or [@tervo14 proof of Theorem 6.3]; replace first $\eta$ by $r_{m,j}-\eta$) $$\begin{gathered} \label{desol20ab} v_{2,j}(x,\omega,\eta) \\ =H(r_{m,j}-\eta-t(x,\omega))e^{\int_0^{t(x,\omega)}-\Sigma_j(x-s\omega,\omega)ds} \tilde{g}_j(x-t(x,\omega)\omega,\omega,\eta+t(x,\omega)),\end{gathered}$$ where $H$ is the Heaviside function. By performing similar computations as in the proof of Lemma \[trathle1\], one sees that $v_{2,j}$ defined by is in fact a weak (distributional) solution of . Moreover, $v_{2,j}$ satisfies (in generalized sense) the inflow boundary condition, since $t(y,\omega)=0$ on $\tilde{\Gamma}_{-,j}$ (see Lemma \[le:esccont:1\]) and $r_{m,j}-\eta>0$ (therefore $H(r_{m,j}-\eta-t(y,\omega))=1$ on $\tilde{\Gamma}_{-,j}$), as well as the initial (energy) condition, since $t(x,\omega)>0$ on $G\times S$ (hence $H(r_{m,j}-\eta-t(x,\omega))=0$ for all $\eta$ close to $r_{m,j}$). The solution of , on the other hand, is obtained as follows. Let $V_{1,j}(x,\omega,\eta):=v_{1,j}(x,\omega,r_{m,j}-\eta)$. Then the problem (\[desol20a\]) is equivalent to $$\begin{gathered} {{\frac{\partial V_{1,j}}{\partial \eta}}}+\omega\cdot\nabla_x {V_{1,j}}+ \Sigma_j V_{1,j} =F_j,\nonumber\\ {V_{1,j}}_{\|\tilde\Gamma_{-,j}}=0, \quad V_{1,j}(\cdot,\cdot,0)=0, \label{desol20aa}\end{gathered}$$ where $F_j(x,\omega,\eta):=\tilde f(x,\omega,r_{m,j}-\eta)$. Let $B_0:L^2(G\times S)\to L^2(G\times S)$ be a densily defined operator (as in section \[evcsd\]) such that $$&D(B_0)=\tilde{W}_{-,0}^2(G\times S), \quad B_0\psi=-\omega\cdot\nabla_x\psi.$$ Then $B_0$ generates a contraction $C^0$-semigroup $T(\eta)$, and in fact for $h\in L^2(G\times S)$ we have (cf. [@dautraylionsv6 Ch. XXI, Sec. 2.2, p. 222], or [@tervo14 proof of Theorem 5.15]) $$(T(\eta)h)(x,\omega)=H(t(x,\omega)-\eta)h(x-\eta\omega,\omega),$$ where $H$ is the Heaviside function. The problem (\[desol20aa\]) can be put into the abstract form \[desol20aaa\] -(B\_0-\_j) [V\_[1,j]{}]{} =F\_j, V\_[1,j]{}(0)=0, where $(V_{1,j}(\eta))(x,\omega)=V_{1,j}(x,\omega,\eta)$ and $(F_j(\eta))(x,\omega)=F_j(x,\omega,\eta)$. The $C^0$-semigroup $G(\eta)$ generated by $B_0-\Sigma_j$ is (by the Trotter’s formula) given by \[infgen\] (G()h)(x,) = e\^[-\_0\^\_j(x-,) d]{}H(t(x,)-)h(x-,). Hence the solution $V_{1,j}$ is (cf. [@engelnagel p. 439], [@pazy83 pp. 105-108]) \[solV\] V\_[1,j]{}()=\_0\^G(-s)F\_j(s) ds, and thus \[solv1\] &v\_[1,j]{}(x,,)=V\_[1,j]{}(x,,r\_[m,j]{}-) =\_0\^[r\_[m,j]{}-]{} (G(r\_[m,j]{}--s)F\_j(s))(x,) ds\ =& \_0\^[r\_[m,j]{}-]{} e\^[-\_0\^[r\_[m,j]{}--s]{} \_j(x-,) d]{}\ & H(t(x,)-(r\_[m,j]{}--s))f\_j(x-(r\_[m,j]{}--s),,r\_[m,j]{}-s) ds. This can be shown to be a weak (distributional) solution of by a similar argument as in the proof of Lemma \[trathle2\]. Moreover, the weak solution of the (primary) problem $$\omega\cdot\nabla_x u_1+\Sigma_1 u_1={}&f_1, \\[2mm] {u_1}_{\|\Gamma_-}={}&g_1,$$ is given by (see Lemmas \[trathle1\] and \[trathle2\]) \[desol23\] u\_1(x,,E) =& \_0\^[t(x,)]{}e\^[-\_0\^t\_1(x-s,,E) ds]{}f\_1(x-t,,E)dt\ &+e\^[-\_0\^[t(x,)]{}\_1(x-s,,E) ds]{}g\_1(x-t(x,),,E). Hence the explicit solution of the total primary problem $$\begin{aligned} \omega\cdot\nabla_x u_1+\Sigma_1(x,\omega,E) u_1={}&f_1,\nonumber\\ -{{\frac{\partial (S_ju_j)}{\partial E}}}+\omega\cdot\nabla_x u_j +\Sigma_j u_j ={}& f_j,\quad j=2,3 \nonumber\\ u_{\|\Gamma_-}={}&g \nonumber\\[2mm] u_j(\cdot,\cdot,E_m)={}&0,\quad j=2,3, \label{desol24}\end{aligned}$$ is given by $u=(u_1,u_2,u_3)$, where $u_1$ is obtained from and $u_j$, $j=2,3$ are obtained from formulas , and , recalling that $v_j=v_{1,j}+v_{2,j}$, $j=2,3$. A Solution Based on the Neumann Series {#meta} -------------------------------------- Consider the transport problem (\[desol10\]), (\[desol11\]), (\[desol12\]). Setting $\phi=e^{CE}\psi$ as in Section \[cosyst\], we recall that the problem then takes the equivalent form (\[cosyst1\])-(\[cosyst4\]). Denote \[cmp4\] T\_[1,C]{}:=&\_x\_1+\_1\_1-K\_[1,C]{}\ T\_[j,C]{}:=&-[E]{}+\_x\_j+ CS\_j\_j+\_j\_j - K\_[j,C]{},j=2,3, and define a (densily defined) closed linear operator $T_C:L^2(G\times S\times I)^3\to L^2(G\times S\times I)^3$ by setting D(T\_C):=&{L\^2(GSI)\^3 \| T\_[j,C]{}L\^2(GSI), j=1,2,3}\ T\_C:=&(T\_[1,C]{},T\_[2,C]{},T\_[3,C]{}). Let $f\in L^2(G\times S\times I)^3$ and $g\in T^2(\Gamma_-)^3$. In the case where $\phi$ is regular enough, say $\phi\in \tilde W^2(G\times S\times I)\times (\tilde W^2(G\times S\times I)\cap W_1^2(G\times S\times I))^2$ , the problem (\[desol10\]), (\[desol11\]), (\[desol12\]) can be expressed equivalently as \[comp5\] T\_C=[f]{},\_[\|\_-]{}=[g]{},\_j(,,E\_m)=0,j=2,3, where ${\bf f}=e^{CE}f$, ${\bf g}=e^{CE}g$ as in Section \[cosyst\]. Assume that $g\in H^1(I,T^2(\Gamma'_-)^3)$ such that $g_j(E_{\rm m})=0$, $j=2,3$. Then ${\bf g}\in H^1(I,T^2(\Gamma'_-)^3)$ and ${\bf g}_j(E_{\rm m})=0,\ j=2,3$. Applying on ${\bf g}$ the lift operator $L$ given by $$\big((L{\bf g})(E)\big)(x,\omega):={\bf g}(E)(x-t(x,\omega)\omega,\omega)={\bf g}(x-t(x,\omega)\omega,\omega,E),$$ we have $L{\bf g}\in H^1(I,\tilde W^2(G\times S)^3)$, and it satisfies (cf. [@tervo14 Lemma 5.11]) $$\omega\cdot\nabla_x (L{\bf g})=0,\quad (L{\bf g})_{\|\Gamma_-}={\bf g}.$$ Furthermore, the condition $g_j(E_{\rm m})=0$ implies that \[comp7\] (L[g]{}\_j)(,,E\_[m]{})=0,j=2,3. Denoting \[comp8\] P\_1(x,,E,D)\_1:=&\_x\_1+\_1\_1\ P\_[j,C]{}(x,,E,D)\_j:=&-[E]{}+\_x\_j+ CS\_j\_j+\_j\_j,j=2,3, and $${\bf P}_C(x,\omega,E,D)\phi:={}& \big(P_1(x,\omega,E,D)\phi_1,P_{2,C}(x,\omega,E,D)\phi_2,P_{3,C}(x,\omega,E,D)\phi_3\big), \\[2mm] K_C\phi:={}&(K_{1,C}\phi,K_{2,C}\phi,K_{3,C}\phi),$$ we find that $T_C={\bf P}_C-K_C$. To simplify the notation, we shall write below $T=T_C$, $K=K_C$, and ${\bf P}={\bf P}_C$. Let ${\bf P}_0$ be the densely defined linear operator acting in $L^2(G\times S\times I)^3$ such that D([P]{}\_0):=&{W\^2(GSI)(W\^2(GSI)W\_1\^2(GSI))\^2 \|\ & \_[\|\_-]{}=0, (,,E\_[m]{})=0},\ [P]{}\_0:=&[P]{}. Furthermore, let $\widetilde{\bf P}_0:L^2(G\times S\times I)^3\to L^2(G\times S\times I)^3$ be the smallest closed extension of ${\bf P}_0$. Writing $u:=\phi-L{\bf g}$, we see that $\phi=u+L{\bf g}$ is a solution of (\[comp5\]) if and only if \[comp10\] ([P]{}-K)(u+L[g]{})=[f]{},u\_[\|\_-]{}=0,u\_j(,,E\_m)=0,j=2,3, or equivalently, \[comp11\] \_0u=Ku+[f]{}-(P-K)(L[g]{}). Now suppose that for some $k=0,1,2,\dots$ the following assumptions hold: - $\ol{\bf f}:={\bf f}-({\bf P}-K)(L{\bf g})\in H^k(I,L^2(G\times S)^3)$, - $Ku\in H^k(I,L^2(G\times S)^3)$ for all $u\in L^2(G\times S\times I)^3$, and - $\widetilde{\bf P}_0^{-1}$ exists as an operator $H^k(I,L^2(G\times S)^3)\to L^2(G\times S\times I)^3$. These assumptions can be met if the data (and the geometry) are regular enough; see the example below. Then Eq. gives \[comp12\] u=\_0\^[-1]{}Ku+\_0\^[-1]{}, which is equivalent to \[comp13\] (I-Q)u=\_0\^[-1]{}, where $$Q:=\widetilde{\bf P}_0^{-1}K.$$ If $1$ belongs to the resolvent set $\rho(Q)$ of $Q$ we thus have u=(I-Q)\^[-1]{}\_0\^[-1]{}, and therefore =(I-Q)\^[-1]{}\_0\^[-1]{}+L[g]{}, where we recall that $\ol{\bf f}={\bf f}-({\bf P}-K)(L{\bf g})$. Assuming that $Q:L^2(G\times S\times I)^3\to L^2(G\times S\times I)^3$ is bounded and that \[comp14\] [Q]{}<1, which implies in particular that $1\in \rho(Q)$, the solution $u$ of can be computed through Neumann series \[15\] u= \_[k=0]{}\^Q\^k(\_0\^[-1]{}) = \_[k=0]{}\^(\_0\^[-1]{}K)\^k(\_0\^[-1]{}). Finally, the solution $\phi$ of is then \[comp16\] =\_[k=0]{}\^(\_0\^[-1]{}K)\^k(\_0\^[-1]{})+L[g]{}, from which the solution $\psi$ of the original problem - is obtained by $$\psi=e^{-CE}\phi.$$ This method is known in the neutron transport community under the name source iteration [@Lewis-Miller]. It can also be used to prove existence, uniqueness and positivity of solutions for the standard transport equation (see e.g. [@frank-goudon]). In the case treated in Example \[desolex1\], choose $C=0$ and $k=0$. Then a bounded inverse $\widehat{\bf P}_0^{-1}:L^2(G\times S\times I)^3\to L^2(G\times S\times I)^3$ exists and, in fact, can be explicitly computed using formulas given in Example \[desolex1\] (choose $\tilde{g}=0$ in Eqs. , , ), and it is \[comp17\] & \_0\^[-1]{}h=((\_0\^[-1]{}h)\_1,(\_0\^[-1]{}h)\_2,(\_0\^[-1]{}h)\_3),\ & (\_0\^[-1]{}h)\_1(x,,E) = \_0\^[t(x,)]{}e\^[-\_0\^t\_1(x-s,,E) ds]{}h\_1(x-t,,E)dt,\ &(\_0\^[-1]{}h)\_j(x,,E) = [1]{}( \_0\^[r\_[m,j]{}-R\_j(E)]{} e\^[-\_0\^[r\_[m,j]{}-R\_j(E)-s]{}\_2(x-,) d]{}\ & H(t(x,)-(r\_[m,j]{}-R\_j(E)-s))\_j(x-(r\_[m,j]{}-R\_j(E)-s),,r\_[m,j]{}-s) ds), for $j=2,3$, and where $H$ is the Heaviside function, and $$\tilde{h}_i(x,\omega,\eta)=S_i(R_i^{-1}(\eta))h_i(x,\omega,R_i^{-1}(\eta)),\quad i=1,2,3.$$ Hence under the stated assumptions the solution $\psi$ $(=\phi)$ is obtained from (\[comp16\]) (note that ${\bf f}=f$, ${\bf g}=g$ for $C=0$) and (\[comp17\]). This shows that the assumption (A1)-(A3) can indeed be met (here for $k=0$). It is also worth noticing that the above expression for $(\widetilde{\bf P}_0^{-1}h)_j$, $j=2,3$, can be simplified into $$\begin{gathered} \label{comp17:1} (\widetilde{\bf P}_0^{-1}h)_j(x,\omega,E) \\ = {1\over{S_2(E)}}\Big( \int_0^{\min\{r_{m,j}-R_j(E),t(x,\omega)\}} e^{-\int_0^{s}\Sigma_2(x-\tau\omega,\omega) d\tau}\tilde{h}_j(x-s\omega,\omega,R_j(E)+s) ds\Big).\end{gathered}$$ The Neumann series based method enables one to compute the solution, or an approximation of it by considering in the series only finitely many terms, without any explicit inversion of matrices (coming from a chosen discretization of the problem), which tend to be large due to the dimensionality of the problem, which is 6: there are 3 spatial ($x$), 2 angular ($\omega$) and one energy ($E$) dimensions. Sufficient criteria for the condition ${{\left\Vert Q\right\Vert}}<1$ must be retrieved. The applicability of the above method for general cases remains open. One possibility to apply formulae like (\[comp17\]) for spatially inhomogeneous substance (that is, $S_j$ and $\Sigma_j$ are dependent on $x\in G$) is to apply [domain decomposition method]{} in such a way that $S_j$ are assumed to be constant in subdomains. In this example, we write out Eq. in special case of constant $S_0\geq 0$ and $\Sigma\geq 0$. Moreover, we consider a single particle CSDA transport equation only. Let $${}&P(x,\omega,E,D)u:=-{\frac{\partial (S_0u)}{\partial E}}+\omega\cdot\nabla_x u+\Sigma u,$$ In this case, $R(E)=\int_0^E \frac{1}{S_0}d\tau=\frac{1}{S_0}E$, and Eq. gives, when $S_0>0$, using the notation $\eta(E):=(E_m-E)/S_0$ and noticing that $r_m:=R(E_m)=\frac{E_m}{S_0}$, $$(\widetilde {P}_0^{-1}h)(x,\omega,E)=\int_0^{\min\{\eta(E),t(x,\omega)\}} e^{-\Sigma s} h\big(x-s\omega,\omega,E+S_0s\big)ds,$$ for $h\in L^2(G\times S\times I)$. It is clear that this last formula gives the correct (explicit) expression for $\widetilde{P}_0^{-1}$ also in the case where $S_0=0$, if we make the convention that $\eta(E)=+\infty$ for all $E\in I$ when $S_0=0$. An Approximative Solution Based on the Theory of Evolution Equations {#appevo-o} -------------------------------------------------------------------- In this section we for simplicity restrict ourselves to a single particle CSDA-equation $$\begin{gathered} -{{\frac{\partial (S_0\psi)}{\partial E}}}+\omega\cdot\nabla_x\psi+ \Sigma\psi - K\psi=f, \label{mb1} \\ \psi_{\|\Gamma_-}=g,\quad \psi(\cdot,\cdot,E_m)=0. \label{mb2}\end{gathered}$$ Suppose that the assumptions of Theorem \[coth3-dd\] are valid. Another method to compute approximately the solution of the problem (\[mb1\]), (\[mb2\]) which avoids the explicit inversions of matrices, can be (formally) described as follows. Note that we will be using throughout this section the notations of Section \[possol\]. After the change of unknown $\phi=e^{CE}\psi$ the problem is \[ps1\] T\_[C]{}=[f]{},\_[\_-]{}=[g]{},(,,E\_[m]{})=0, where $$T_C\phi:=-{{\frac{\partial (S_0\phi)}{\partial E}}}+\omega\cdot\nabla_x\phi+CS_0\phi+\Sigma\phi-K_C\phi.$$ Assume that $g\in H^1(I,T^2(\Gamma'_-))$ and that $g(\cdot,\cdot,E_{\rm m})=0$ on $G\times S$. Let $u:=\phi-L({\bf g})$. Then $u$ satisfies \[ps0\] T\_[C,0]{}u=,u\_[\_-]{}=0,u(,,E\_[m]{})=0. where $$\tilde{\bf f} := {\bf f}-T_C(L({\bf g})).$$ By the Trotter’s formula, the semigroup $G(t)$ generated by $T_{C,0}$ is given by \[calceq2\] G(t) = \_[n]{}(T\_[B\_0]{}(t/n)T\_[A\_0]{}(t/n)T\_[-(+C I)]{}(t/n)T\_[K\_C]{}(t/n))\^n, where the convergence is uniform on compact $t$-intervals $[0,T]$. Note that the individual semi-groups $T_{B_0}(t)$, $T_{A_0}(t)$, $T_{-(\Sigma+C I)}(t)$ and $T_{K_C}(t)$ contributing to this expression can be computed explicitly (see section \[possol\]). Hence we get \[calceq4\] =&\_0\^G(t) dt\_0\^T G(t) dt\ =&\_0\^T\_[n]{} (T\_[B\_0]{}(t/n)T\_[A\_0]{}(t/n)T\_[-(+C I)]{}(t/n)T\_[K\_C]{}(t/n))\^ndt\ & \_0\^T(T\_[B\_0]{}(t/n\_0)T\_[A\_0]{}(t/n\_0)T\_[-(+C I)]{}(t/n\_0)T\_[K\_C]{}(t/n\_0))\^[n\_0]{}dt, for large enought $T$ and $n_0$. On the other hand, the semi-group $T_{K_C}$ generated by the bounded operator $K_C$ can be approximately computed from $$T_{K_C}(t) \approx \sum_{k=0}^{N_0} \frac{1}{k!}(tK_C)^k,$$ for large enough $N_0$. We point out that this semi-group theory -based approach, unlike the one given in the previous section, does not require extra assumptions on cross-sections, like the ones imposed by the condition . \[evocomp\] Assume that $K$ is of the form $$(K\psi)(x,\omega,E)=\int_S\sigma(x,\omega',\omega,E)\psi(x,\omega',E)d\omega'.$$ Furthermore, as in Example \[desolex1\] we assume that $S_0=S_0(E)$ (independent of $x$) and we define $R(E):=\int_0^E{1\over{S_0(\tau)}}d \tau$, $\eta:=R(E)$, $r_m:=R(E_m)$ and $\tilde{I}:=R(I)=[0,r_{\rm m}]$. Let $$\begin{gathered} v(x,\omega,\eta):=S_0(R^{-1}(\eta))\psi(x,\omega,R^{-1}(\eta)), \\[2mm] \tilde\Sigma(x,\omega,\eta):=\Sigma(x,\omega,R^{-1}(\eta)),\\[2mm] \tilde\sigma(x,\omega',\omega,\eta):=\sigma(x,\omega',\omega,R^{-1}(\eta)), \\ (\tilde K v)(x,\omega,\eta):=\int_S\tilde\sigma(x,\omega',\omega,\eta) v(x,\omega',\eta)d\omega', \\ \tilde f(x,\omega,\eta):=S_0(R^{-1}(\eta))f(x,\omega,R^{-1}(\eta)), \\[2mm] \tilde g(x,\omega,\eta):=S_0(R^{-1}(\eta))g(y,\omega,R^{-1}(\eta)).\end{gathered}$$ As a further simplification, suppose that $g=0$. After some technical considerations, the problem (\[mb1\]), (\[mb2\]) can be cast into the abstract form \[mb4aa\] -A() V= F(),V(0)=0, where $V(\eta):=v(r_m-\eta)$, $F(\eta):=\tilde f(\cdot,\cdot,r_m-\eta)$. In the case where appropriate assumptions are valid (cf. Theorem \[evoth\]), the solution of (\[mb4aa\]) is given by \[mb7\] V()=\_0\^U(,s) F(s) ds, where $U(\eta,s):L^2(G\times S)\to L^2(G\times S)$, $0\leq \eta\leq s\leq r_m$, is the evolution family of operators $A(\eta)$, $\eta\in\tilde I$. By making use of the explicit expressions for the semi-groups $T_{B_0}(s)$ (here $B_0$ is as in the proof of Lemma \[csdale1\]), $T_{\tilde \Sigma(\eta)}(s)$, and $T_{\tilde K(\eta)}(s)$, there exist some approximative methods (e.g. [Cauchy-Peano approximations]{}) for computing the family of evolution operators $U(\eta,s)$ (cf. [@goldstein]). This gives an approach for calculating $V$ approximately by $$V(\eta)\approx \tilde V(\eta):=\int_0^\eta \tilde U(\eta,s)F(s) ds,$$ where $\tilde U(\eta,s)$ is an approximation of $U(\eta,s)$. The solution $\psi$ would then be approximated as \[mb11\] (x,,E)V(r\_m-R(E)). For general $g$ the idea remains the same. It is worth studying this approach under more general assumptions as well. An Outlook on the Inverse Radiation Treatment Problem {#irtpre} ===================================================== As mentioned above, in radiation therapy the dose absorbed from particle field $\psi=(\psi_1,\psi_2,\psi_3)$ is defined by \[irtp8\] D(x)=(D)(x):=\_[j=1]{}\^3\_[SI]{}\_j (x,E)\_j(x,,E) ddE, where $\varsigma_j\in L^\infty(G\times I)$, $\varsigma_j\geq 0$ are the so-called (total) stopping powers. Also, it is worth recalling that the component fields of $\psi$, relevant to photon and electron radiation therapy, are $\psi_1={}$photons, $\psi_2={}$electrons and $\psi_3={}$positrons. Clearly, nothing we have said above depends on the number, or designation (to certain particle species) of fields treated (or even the dimensionality of the spaces $G$, $S$ or $I$ in fact), with the exception that typically only charged particle fields (are assumed to) obey CSDA version of the transport equation (cf. ), while non-charged particles obey the standard linear BTE (cf. ). Thus with very minor modifications, and in particular if one is interested in radiation therapy, what will and has been said works, in principle, equally well in proton (and ion) therapy framework as well. We find that $D:L^2(G\times S\times I)^3\to L^2(G)$ is a bounded linear operator and its adjoint operator $D^:L^2(G)\to L^2(G\times S\times I)^3$ is simply a multiplication type operator, \[irtp14\] D\^\d=(\_1,\_2,\_3)d, dL\^2(G). We describe shortly an optimization problem related to inverse radiation treatment planning. We restrict ourselves to [external radiation therapy]{} in which the particles are inflowing through the patch(es) of patient surface. This means that in the transport problem $f=0$ (i.e. the internal particle source vanishes) and $g$ (the inflow particle flux) is the variable to be controlled. Conversely, for the internal radiation therapy problems one sets $g=0$, and $f$ would be the variable to be controlled. Anyhow, the results presented below would be analogous in this situation. We refer to [@tervo14 Section 7] and to the references therein for a more detail exposition of inverse problem (optimization) in this setting. Let $g\in T^2(\Gamma_-)^3$ and let $\psi=\psi(g)\in \tilde W^2(G\times S\times I)\times (\tilde W^2(G\times S\times I)\cap W_1^2(G\times S\times I))^2$ be the solution of the variational equation (see , and ) \[irtp10\] \_0((g),v)=[F]{}\_0(v)v, where (since $f=0$) \_0(v)=([F]{}\_0g)(v) := \_[j=1]{}\^3\_[GSI]{}()\_-g\_j v\_j dddE=g,\_-(v)\_[T\^2(\_-)\^3]{}. The deposited dose is then \[irtp12\] D(x)=(D((g)))(x),xG. We shall also denote ${{{\mathcal{}}}D}(g):=D(\psi(g))$. Denote the target region by ${\bf T}\subset G$, the critical organ region by ${\bf C}\subset G$ and the normal tissue region by ${\bf N}\subset G$. Then $G={\bf T}\cup {\bf C}\cup {\bf N}$ where the union is mutually disjoint. Suppose that $ d_{\bf T}\in L^2({\bf T})$, $d_{\bf C}\in L^2({\bf C})$, $d_{\bf N}\in L^2({\bf N})$ are given dose distributions in the respective regions (for example, they may be constants). We define a strictly convex object (cost) function $J:X\to{\mathbb{R}}$ by (see [@tervo14]) \[irtp15\] J(g)=&c\_[T]{}[d\_[T]{}-[D]{}(g)]{}\_[L\^2([T]{})]{}\^2+c\_[C]{}[d\_[C]{}-[D]{}(g)]{}\_[L\^2([C]{})]{}\^2\ & +c\_[N]{}[d\_[N]{}-[D]{}(g)]{}\_[L\^2([N]{})]{}\^2+c[g]{}\_X\^2, where $X:=T^2(\Gamma_-)\times H^1(I,T^2(\Gamma_-'))^2$ equipped with the inner product $${\left\langle}g,h{\right\rangle}_{X} := {\left\langle}g_1,h_1{\right\rangle}_{T^2(\Gamma_-)}+\sum_{j=2}^3 \int_I{\left\langle}{{\partial g_j}\over{\partial E}}, {{\partial h_j}\over{\partial E}}{\right\rangle}_{T^2(\Gamma_-')} dE.$$ Let $Y$ be a closed subspace of $X$ defined by (here we denote $g(y,\omega,E):=(g(E))(y,\omega)$) \[adhil\] Y:={gX \| g\_j(,,E\_[m]{})=0, j=2,3}. A relevant admissible set (of controls) is U\_[ad]{}={gY \| g0 [a.e. on]{} GSI}, which is a closed convex subset of $X$. Suppose that the assumptions of Theorem \[m-d-j-co1\] are valid. Furthermore, suppose that $\varsigma_j\in H^2(I,L^\infty(G))$, $j=1,2,3$. Then the minimum $\min_{g\in U_{\rm ad}}{J}(g)$ exists. In addition, a necessary and sufficient condition that $\ol{g}\in U_{\rm ad}$ is the minimum point, is that the following relations hold (cf. [@tervo14 proof of Theorem 7.7]) & -\_-(\^\),w\_[T\^2(\_-)\^3]{}+cg,w\_[X]{}0 wU\_[ad]{}, \[irtp16\]\ & -\_-(\^\),g\_[T\^2(\_-)\^3]{} +c[g]{}\_[X]{}\^2=0,\ & \_0(,v)=g,v\_[T\^2(\_-)\^3]{} v,\[irtp18\]\ & \_0\^\(\^\,v)+ 2 c\_[T]{}D,Dv\_[ L\^2([T]{})]{}+2c\_[C]{}D,Dv\_[ L\^2([C]{})]{}+2c\_[N]{}D,Dv\_[ L\^2([N]{})]{}\ & =2c\_[T]{}d\_[T]{},Dv\_[ L\^2([T]{})]{}+2c\_[C]{}d\_[C]{},Dv\_[ L\^2([C]{})]{} +2c\_[N]{}d\_[N]{},Dv\_[ L\^2([N]{})]{} v,\[irtp19\] where $\tilde{\bf B}_0(\cdot,\cdot)$ is given in , and $\tilde{\bf B}_0^(\cdot,\cdot)$ is the extension onto ${\mathcal{H}}\times\tilde{{\mathcal{H}}}$ of ${\bf B}_0^$ given in . We emphasize that here the described solution $\ol{g}$ of the optimal control problem can be used only as an initial point for the actual treatment planning where global optimization (see e.g. [@pinter]) is required. The determination of a carefully chosen initial point for a large dimensional global optimization scheme is very essential for achieving (time savings and) satisfactory results ([@pinter14]). We also notice that if we contented ourselves with so-called mild solutions ([@pazy83 p. 146]), then the existence of an optimal control $\ol{g}$ (the admissible set being a subset of $T^2(\Gamma_-)^3$), together with the explicit formula $\ol{g}={1\over c}(\gamma_-(\psi^))_+$ for it, could be proven under quite weak assumptions. Here $(h)_+$ denotes the positive part of a function $h$. In any case, the validity of estimates such as (\[diss-co-es\]) is essential for guaranteeing that ${{{\mathcal{}}}D}$ be a bounded linear operator in appropriate spaces. The use of mild solutions, however, has the drawback that the inflow boundary conditions are not necessarily satisfied by the solutions (and thus the solutions might be non-physical). We remark that, for example with respect to $x$-variable the solution is generally at most in $H^{2,(s,0,0)}(G\times S\times I^\circ)$ with $s<\frac{3}{2}$ where $H^{2,(s,0,0)}(G\times S\times I^\circ)$ is the mixed-norm Sobolev-Slobodeckij space with fractional index $s$ and Lebesgue index $2$ (with respect to $x$-variable). Finally, we notice that the system of the above variational equations can be implemented within e.g. a FEM-type numerical scheme to get numerical solutions for the optimal control $\ol{g}$. In virtue of Cea’s estimate, the boundedness and coercivity (after replacing the unknown $\psi$ by $\phi=e^{CE}\psi$) of the bilinear forms ${\bf B}_0(\cdot,\cdot)$, ${\bf B}_0^(\cdot,\cdot)$ guarantee the convergence of the scheme in principle. 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--- abstract: 'We present a complete atlas of the Cygnus Loop supernova remnant in the light of ø3 ($\lambda 5007$), , and 2 ($\lambda\lambda 6717, 6731$). We include low-resolution ($25\arcsec$) global maps and smaller fields at $6\arcsec$ resolution from observations using the Prime Focus Corrector on the 0.8-m telescope at McDonald Observatory. Despite its shell-like appearance, the Cygnus Loop is not a current example of a Sedov-Taylor blast wave. Rather, the optical emission traces interactions of the supernova blast wave with clumps of gas. The surrounding interstellar medium forms the walls of a cavity through which the blast wave now propagates, including a nearly complete shell in which non-radiative filaments are detected. We identify non-radiative shocks around half the perimeter of the Cygnus Loop, and they trace a circle of radius $R = 1\fdg 4$ (19 pc) in the spherical cavity walls. The Cygnus Loop blast wave is not breaking out of a dense cloud, but is instead running into confining walls. Modification of the shock velocity and gas temperature due to interaction of the blast wave with the surrounding medium introduces errors in estimates of the age of this supernova remnant. The optical emission of radiative shocks arises only where the blast wave encounters inhomogeneities in the ambient medium; it is not a consequence of gradual evolution to a global radiative phase. Distance measurements that rely on this uniform blast wave evolution are uncertain, but the radiative shocks can be used as distance indicators because of the spherical symmetry of the surrounding medium. The interstellar medium dominates not only the appearance of the Cygnus Loop but also the continued evolution of the blast wave. If this is a typical example of a supernova remnant, then global models of the interstellar medium must account for such significant blast wave deceleration.' author: - 'N. A. Levenson and James R. Graham' - 'Luke D. Keller and Matthew J. Richter' nocite: - '[@Oort46]' - '[@Fes82]' - '[@Min58]' - '[@Kir76]' - '[@Gre91]' - '[@Shu91]' - '[@Fes82]' - '[@McC79]' - '[@Cha85]' - '[@McK75]' - '[@Lev97]' - '[@Hes83]' - '[@Hes94]' - '[@Fes92]' - '[@Fes92]' - '[@Hes94]' - '[@Hes94]' - '[@Bla91]' - '[@Mor96e]' - '[@Cha85]' - '[@Lev97]' - '[@McK84]' - '[@Fal82]' - '[@Shu91]' - '[@Lev97]' - '[@PShu85]' - '[@Van92]' - '[@Vin97]' - '[@Lev97]' - '[@Min58]' - '[@Hub37]' - '[@Hes86]' - '[@Che80]' - '[@McK77]' - '[@Shu87]' title: Panoramic Views of the Cygnus Loop --- ø3[\[\]]{} 2[\[\]]{} \#1\#2 \#1\#2\#3[ ]{} Introduction\[secintro\] ======================== Supernova remnants greatly determine the large-scale structure of the interstellar medium. The energy of supernova remnants heats and ionizes the interstellar medium (ISM), and their blast waves govern mass exchange between various phases of the ISM. In doing so, supernova remnants (SNRs) influence subsequent star formation and the recycling of heavy elements in galaxies. Global models of the interstellar medium that include a hot ionized component ([@Cox74]; [@McK77]) are sensitive to the supernova rate, the persistence of their remnants, and the sizes they attain. A simple calculation of the last of these assumes that the blast wave expands adiabatically in a uniform medium once the blast wave has swept up mass comparable to the mass of the ejecta. During this Sedov-Taylor phase, the radius of the SNR as a function of $E_{51}$, the initial energy in units of $10^{51}$ erg, $n_o$, the ambient number density in units of ${\rm cm^{-3}}$, and $t_4$, time in units of $10^4$ yr, is $R=13 (E_{51}/n_o)^{1/5} t_4^{2/5} {\rm \,pc}$ in a medium where the mean mass per particle is $2.0 \times 10^{-24} {\rm \, g}$. This phase will last until radiative losses become important. The beginning of this subsequent phase, marked by the initial loss of pressure behind the blast wave, occurs at $t=1.9\times 10^4 E_{51}^{3/14} n_o^{-4/7} {\rm \,yr}$, when the radius is $R=16.2 E_{51}^{2/7} n_o^{-3/7} {\rm \,pc}$ ([@Shu87]), although the radiating shell is not fully formed yet. We approach these large-scale questions with analysis of complete images of a particular supernova remnant, the Cygnus Loop, in three optical emission lines. This supernova remnant appears to be a limb-brightened shell at radio ([@Keen73]), infrared ([@Bra86]), optical ([@Fes82]), and X-ray ([@Ku84]; [@Lev97]) energies, which at first glance suggests that it is presently in the transition to the radiative stage. The Cygnus Loop has the advantages of being nearby, bright, and relatively unobscured by dust. This allows us to examine in detail the evolution of various portions of the shock front and to determine physical parameters, such as shock velocity and local ambient density, as they vary throughout the remnant. Despite its appearance, the Cygnus Loop is not a current example of blast wave propagation in a uniform medium at any stage. Instead, its evolution is governed by the inhomogeneous interstellar medium, which we map using the shock as a probe. Many of the features we discuss have been noted by others. Oort (1946) first suggested that the Cygnus Loop is an expanding supernova shell. Spectroscopy of radiative shocks in selected locations (e.g., [@Mil74], [@Ray80a], and Fesen et al. 1982) combined with theoretical models of these shocks (e.g., [@Cox72], [@Dop77], [@Ray79], and [@Shu79]) has been used to derive the physical conditions of the observed shocks. We utilize the radial velocity measures of Minkowski (1958), Kirshner & Taylor (1976), Greidanus & Strom (1991), and Shull & Hippelein (1991) to discern some of the three-dimensional structure that is ambiguous from the data we present. Many non-radiative or Balmer-dominated shocks in the Cygnus Loop have been identified (e.g., [@Kir76], [@Ray80b], [@Tref81], Fesen et al. 1982, [@Fes92], and [@Han92]). Our observations qualitatively match these, and we rely on these works and others ([@Ray83]; [@Long92]; [@Hes94]) for quantitative measures of parameters such as shock velocity and preshock density. McCray & Snow (1979) and Charles, Kahn, & McKee (1985) have suggested that the Cygnus Loop is the result of a cavity explosion, and we adapt this global model to interpret the surrounding interstellar medium, as well. This paper is a companion to the soft X-ray survey presently in progress with the [ROSAT]{} High Resolution Imager ([@Gra96]; [@Lev97]). With these two surveys, we examine the Cygnus Loop as a whole, not restricting our investigation only to those regions that are exceptionally bright or that appear to be particularly interesting. We hope to understand both the global processes and the specific variations that are responsible for the emission we detect. The X-rays probe hot (temperature $T\sim 10^6$ K) gas that shocks with velocities $v_s \sim 400 \kms$ heat. The optical emission is expected from slower shocks ($v_s \lesssim 200 \kms$) in which the post-shock region cools to temperatures $T\sim 10^4$ K, yet the most prominent regions at optical wavelengths are also bright in X-rays. McKee & Cowie (1975) suggested that the broad correlation of X-ray and optical emission is the result of a blast wave propagating in an inhomogeneous medium. In this scenario, the shock is significantly decelerated in dense clumps of gas, while portions of it proceed unimpeded through the lower-density intercloud medium. We apply the principles of this basic cloud–blast-wave interaction to a range of locations in the Cygnus Loop. In particular, we refine the cavity model introduced in Levenson et al. (1997), using these optical data to constrain the current ISM in the vicinity of the Cygnus Loop and to determine how the stellar progenitor modified it in the past. We present the observations in §2. We describe them in detail, noting individual regions of interest, and we use these data to measure the physical conditions of the blast wave and the ambient medium in particular locations in §3. The purpose of the detailed examination is to combine the results in a complete map of the surrounding ISM. We present this three-dimensional model while providing a coherent explanation of the history that accounts for it in §4. We predict the future of this SNR and relate its fate to more general theories of supernova modification of the interstellar medium in §5 and summarize our conclusions in §6. Observations ============ We obtained narrow-band images of the Cygnus Loop in 1995 August and September and 1995 September on the 0.8-m telescope at McDonald Observatory with the Prime Focus Corrector (PFC; [@Cla92]). The PFC system is a 5-element catadioptric corrector located at the F/3 prime focus of the telescope and projects a flat focal plane onto a Loral-Fairchild $2048 \times 2048$, $15\micron$-pixel charge-coupled device. The camera and corrector system has a $0\farcs3$ spot size in the V-band and projects each pixel to $1\farcs35$ on the sky, producing a $46\arcmin \times 46\arcmin$ field of view. The interference filters each have a rectangular transmission profile and full width at half maximum $FWHM = 40$Å in the converging beam of the PFC. The filter centers in the F/3 beam are 5007Å, 6723Å, and 6563Å to detect line emission from ø3 ($\lambda5007$), 2 ($\lambda\lambda6717, 6731$), and +\[\], which we refer to throughout as . Each individual integration of approximately $600$ s duration was bias-subtracted, flat-fielded, and scaled by its median sky value before being combined with other images. Every region of the Cygnus Loop was observed at least twice through each filter in order to remove cosmic rays and exclude bad regions of the detector, and most regions were observed three times in . We obtained a total of 62 observations in ø3, 89 in , and 54 and 2. Over the large spatial scale of the Cygnus Loop ($3^\circ \times 3\fdg 5$), each observation is a distinct projection of the sky onto the plane of the detector, so images must be remapped to a common projection before being co-added. An astrometric solution for each integration was calculated based on the [Hubble Space Telescope]{} Guide Star Catalog. Alignment does not rely on overlapping observations of a particular region but depends instead on the astrometry of individual integrations. The final images are the result of pixel-by-pixel averaging in the combined image plane using nearest-neighbor matching and rejecting bad regions of the detector. The global maps of the Cygnus Loop with $25\arcsec$ pixels are shown with linear scaling in , ø3, and 2, and in a false-color combination with logarithmic scaling (Figure \[figwhole\]). While the observations at different wavelengths resemble each other broadly, their variation is apparent in the combined image. In this display, blends of and 2 alone are yellow, ø3 and 2 together appear as magenta, and and ø3 combine to make cyan. Only the regions that are displayed as white exhibit strong emission at all three wavelengths. In addition to the global maps, we present 5 smaller fields, each $95\arcmin$ across, at $6\arcsec$ resolution (Figs. \[figsubv\]–\[figsubz\]). Although we first describe the current physical conditions in specific regions of interest, our ultimate goal is to combine the understanding of the particular sites into a complete account of the surrounding ISM, its modification by the progenitor star, and the passage of the blast wave through it. Emission Morphology and Structure of the ISM ============================================ Observed Morphologies --------------------- The radiative shocks that are responsible for most of the optical emission from the Cygnus Loop result in two characteristic morphologies: filaments and diffuse emission. Figure \[figann\] identifies examples of these features and other structures to which we will refer by name. The morphological distinction is the result of different viewing geometry. As the “wavy sheet” model (Hester et al. 1983; [@Hes87]) describes, bright, sharp filaments are the result of long lines of sight through tangencies of the shock front. When the shock front is viewed face-on, the emission appears more diffuse. Combined with the X-ray data, these observations provide more specific information on the evolutionary state of the Cygnus Loop and the distribution of inhomogeneities of the ISM around it. The widespread X-ray emission indicates that the global shock velocity is several hundred , so the optical emission comes from regions of denser gas that have decelerated the blast wave. The correlation of the X-rays with optically-bright regions make clear that this SNR is not globally in the gradual transition to the radiative phase that is the result of blast wave propagation in a uniform medium. The blast wave has recently and suddenly decelerated where radiative shocks are detected. This rapid, environmentally-imposed evolution is distinct from the simple model prediction for incremental change in a homogeneous medium. These complete data sets demonstrate that many characteristics that have been investigated in small regions of the Cygnus Loop are widespread, and we adapt the physical explanations of these limited regions in previous work to understand the Cygnus Loop globally. For example, the bright emission from radiative shocks is found to be stratified at select observed locations ([@Fes82]), and we show this stratification to be typical of all prominent radiative shocks. The ø3 extends toward the projected exterior, and the and 2 are located toward the projected interior. The 2 is farthest to the interior and is particularly diffuse. The stratification is a function of the column density that the shock has swept up. The incomplete ø3 emission is concentrated behind shocks that have encountered relatively little gas, while where the shock lags because it has swept up a greater column of gas, and then 2 are prominent. The strong ø3 emission requires that the shock has progressed through a column density $N \approx 3\times 10^{17} {\rm \ cm^{-2}}$, while the complete recombination zone develops after the swept-up column $N \gtrsim 10^{18} {\rm \ cm^{-2}}$ for $v_s < 150 \kms$ ([@Ray88]). The apparent thickness of these zones and the extensive regions of bright emission in a combination of all the observed lines is a consequence of projection effects. The curved surface of the blast wave encounters the cloud at different projected radii, so some lines of sight intersect multiple stages of shock evolution. Around most of the perimeter of the Cygnus Loop, filaments are detected in alone. These Balmer-dominated filaments are the result of non-radiative shocks ([@Che78]; [@Che80]). Unlike the more common radiative shocks, the emission here does not come from the extended cooling zone behind the shock. Instead, it is the result of collisional ionization before excitation in the immediate post-shock region, or the downward cascade toward the ground state following charge exchange. In either case, neutral atoms are required in the post-shock region. Because these do not survive long, the emission originates at the shock front. Thus, we use the Balmer filaments to trace the shock front directly. This emission is intrinsically weak, so it is most obvious when the line of sight through the shock front is long. This is the case at the projected edge of the SNR, where the line of sight is tangent to the spherical blast wave. Projected Edges\[subsecedge\] ----------------------------- The projected edges of the Cygnus Loop are the simplest regions to understand because the geometric complications are minimized there. The observed structures may be slightly—but not significantly—in the foreground or background and still appear near the projected edge. The prominent regions, NGC 6992 at the northeast ($\alpha=\hms{20}{56}{24},\delta=31^\circ 43\arcmin$; Fig. \[figsubv\]) and NGC 6960 at the west ($\alpha=\hms{20}{45}{42},\delta=30^\circ 43\arcmin$; Fig. \[figsubw\]), are examples of interactions of the blast wave with large clouds ([@HesCox]; Hester et al. 1994; [@Gra95]; [@Lev96]), and the corresponding X-ray enhancement arises from reflected shocks that propagate through the hot, shocked SNR interior. The observed length scales of optical and X-ray emission from NGC 6992 and NGC 6960 reveal the sizes of these large clouds. At the 770 pc distance of the Cygnus Loop, the coherent networks of filaments over $20\arcmin$ imply that the clouds have lengths of about $10^{19}$ cm. Balmer filaments, high-resolution X-ray observations or both constrain the location of clouds along the line of sight. X-ray emission extends beyond both of these bright optical edges, where some portion of the three-dimensional blast wave is not yet impeded by the clouds. Thus, the clouds do not extend over the complete edge of the blast wave along the line of sight. In contrast, the southeast knot at $\alpha=\hms{20}{56}{20},\delta=30^\circ 25\arcmin$ (Fesen et al. 1992; [@Gra95]) is an example of a cloud that is clearly extended along the line of sight but is not necessarily large across the plane of the sky. The nearly-circular edge of the X-ray SNR and the Balmer filaments are distinctly concave at the southeast knot (Fig. \[figsubx\]a), and there is no X-ray emission to the exterior. Thus, this cloud must be at least 11 pc long along the line of sight in order to impede the entire projected edge of the blast wave at this location. Again, the presence of the complete optical cooling structure indicates that the shock has progressed through a column $N \gtrsim 10^{18} {\rm cm^{-2}}$ here. The breakout to the south, centered on $\alpha=\hm{20}{50},\delta=29^\circ 10\arcmin$ and extending approximately $1^\circ$ across (Fig. \[figsubz\]), is the most significant departure from circularity in the remnant. The greater advance of the blast wave in the breakout compared with the circular loop north of $\delta=29^\circ40\arcmin$ indicates that the ambient density is lower in this southern region. Balmer-dominated filaments surround much of the breakout, where the blast wave has now encountered atomic gas. There are sections of incomplete shocks to the east and west (e.g., $\alpha=\hm{20}{53},\delta=29^\circ 20\arcmin$ and $\alpha=\hm{20}{48},\delta=29^\circ 15\arcmin$) and fully radiative shocks at the east ($\alpha=\hm{20}{52},\delta=29^\circ 15\arcmin$). Emission is absent across the face of this area. In the plane of the sky, the breakout is not very extended. Either the blast wave has only recently entered this low-density region, so it has not had enough time to advance significantly, or the extent of the low-density material is limited, so the blast wave no longer continues to progress rapidly. The stratification of emission around the edge of the breakout is apparent on smaller scales than those observed elsewhere in the Cygnus Loop, and the breakout exhibits smaller regions of common overlap. In the northeast, for comparison, the distinct regions of and ø3 are typically $3\arcmin$ wide, and they are observed together over $10\arcmin$ scales, whereas in the breakout, the exclusive and common zones are both typically $1\arcmin$ wide. These characteristics are consequences of the more limited scale of interactions with clouds in the breakout. There is no evidence for indentation or blast wave progress beyond the optical-emitting regions, which could be due to either intrinsically small clouds or early stages of encounter. In either case, at least some dense material surrounds the SNR at its southern extremity. The exterior of the southwest quadrant exhibits many filaments that are nearly aligned with one another (Fig. \[figsuby\]). Some of these are part of the radiative shock structure related to the southern end of NGC 6960, while farther away from the center, these are non-radiative and incomplete shocks (Fesen et al. 1992). Along some of the filaments, the transition from non-radiative to incomplete shock structure is clear where ø3 emission arises. The multiple filaments are easily understood as tangencies of a wavy sheet. These filaments are shorter than those at the north and east of the Cygnus Loop. This indicates that the ISM is clumpier here, which would distort the blast wave in many places and account for the number of filaments that we observe. Shock diffraction through low-density ($\delta \rho/ \rho \gtrsim 1$) regions may be responsible for the large curvature of some of these filaments, as Fesen et al. (1992) suggest, although multiple interfering shock fronts are not necessary to produce the different filaments. They arise from multiple projections of the same shock front. ### Balmer Filaments and X-ray Emission The Balmer-dominated filaments around the periphery of the Cygnus Loop indicate that the fraction of atomic gas surrounding the SNR is high. The extreme circularity of these filaments requires that the blast wave remain nearly spherical with little deformation at the projected edge. With the exclusion of the breakout region, we fit them with a circle of radius $R=1\fdg 4$, which is equivalent to 19 pc at a distance of 770 pc. In Figure \[figcirc\], the model circle is drawn on the soft X-ray image obtained with the [ROSAT]{} High Resolution Imager ([@Lev97]). We calculate the correlation of radial variation as a function of angular separation and the corresponding Fourier coefficients for the Balmer filaments. Excluding the breakout, the amplitude of the $m = 2$ mode is less than 0.02, which implies that the surroundings of the Cygnus Loop are homogeneous on large scales. The individual filaments tend to be very long, typically extending over more than $40\arcmin$. The continuity and smoothness of the Balmer filaments indicate the uniformity of the medium in which these shocks propagate. For constant ram pressure $\rho v_s^2$ behind the shock, variations in density and velocity are related $\delta \rho / \rho = -2 \delta v_s / v_s$. In terms of the radius, $R$, and time, $t$, $v=\eta R/t$, where $\eta$ is of order unity; for free expansion $\eta =1$, and for an adiabatic supernova remnant in a uniform medium, $\eta = 2/5$. Thus, $\delta \rho / \rho = (-2/\eta) (\delta R/ R)(t/\delta t)$. The observed radial variations of a particular filament are $\delta R/ R \sim 0.01$. An approximate timescale of these shocks is 1000 yr, and the age of the Cygnus Loop is about 14,000 yr (cf. §\[secage\]), so the overall density variations in the present medium of the Balmer-dominated filaments are of order $\delta \rho / \rho \sim 0.3 \eta$. We expect $ 2/5 < \eta < 1 $, so $0.1 \lesssim \delta \rho / \rho \lesssim 0.3$. The Balmer-dominated filaments define the current location of the blast wave and mark the presence of neutral material. Detailed studies of particular locations at the northeast have used these non-radiative shocks as density probes ([@Ray83]; [@Long92]; Hester et al. 1994) and derive densities $n \sim 1 {\rm \ cm^{-3}}$. The total observed Balmer line intensity is sensitive to density, shock velocity, and geometric perspective. All of the filaments around the periphery share the same edge-on geometry. Because of the remnant’s circularity, it is reasonable to assume that the shock velocity is nearly constant around the extreme projected edge except at the breakout. The filaments that have been studied in detail to determine shock velocity and ambient density are the brightest ones, however, so these regions are most likely in the transition to incomplete radiative emission, and these densities are expected to be higher than elsewhere. Thus, we conclude that where typical non-radiative filaments are observed around the periphery, the density is constrained $n \lesssim 1 {\rm \, cm^{-3}}$, and most likely $n \sim 0.1 {\rm \ cm^{-3}}$. We roughly calculate the surface brightness of a non-radiative shock in the Cygnus Loop. We examine a portion of the blast wave in the southeast, near $\alpha=\hm{20}{56}, \delta=30^\circ 00\arcmin$. This is not an exceptionally bright filament, and there is no obvious interaction with a large cloud here. The whole filament extends over more than $20\arcmin$, so the blast wave edge is smooth here. We estimate a surface brightness of $4.2\times 10^{-6} {\rm \,erg\,s^{-1}\,cm^{-2}\,sr^{-1}}$ from this filament. Assuming that this line of sight passes through two folds of the spherical blast wave and the ISM does not distort it, the intensity of a single shock surface is $2.1\times 10^{-6} {\rm \,erg\,s^{-1}\,cm^{-2}\,sr^{-1}}$. The line intensity, $I$, depends on the energy of the transition, $h\nu$, and is given by $$I = {{h \nu}\over{4\pi}}n_H v_s{q_{ex}\over{q_i}} {\rm \,erg\,s^{-1}\,cm^{-2}\,sr^{-1}},$$ where $n_H$ is the preshock neutral density and the ratio of excitation and ionization rates $q_{ex}/q_i \approx 0.2$ for ([@Ray91]). If we reasonably constrain the shock velocity $200 < v_s < 400 \kms$, these observations limit the density $0.3 > n_H > 0.15 {\rm \,cm^{-3}}$ at the location of the non-radiative shock. The lower velocity limit is based on observations of Balmer filaments ([@Ray83]; [@Long92]; Hester et al. 1994), which are likely to represent the slowest non-radiative shocks, as noted above, and the upper bound assumes that the forward shock does not heat the X-ray-emitting medium to temperatures greater than $2 \times 10^6 {\rm \, K}$. The Balmer filaments are related to adjacent X-ray emission: the optical filaments form the exterior boundary of low surface brightness, limb-brightened X-rays ([@Lev97]). This characteristic morphology reveals where the blast wave is decelerated in the cavity walls. We combine the density determined from the Balmer observations with X-ray temperature measurements to calculate the unperturbed blast wave velocity and the original density of the cavity, and we compare this with the cavity density based on the observed X-ray surface brightness of the interior of the SNR. A bimodal temperature distribution has been detected in the Cygnus Loop, with peaks at $1.5 \times 10^6$ and $5 \times 10^6$ K in EXOSAT observations with the channel multiplier array (sensitive over 0.05–2 keV) and the medium energy experiment (sensitive over 1–50 keV; [@Bal89]). We suggest that the higher temperature component is due to the reflected shock and that previous low single-temperature measures are contaminated by the slow, forward shock in the cavity walls. We model these simply as a reflected shock and ignore non-equilibrium effects, which are probably important. The physical parameters temperature, pressure, and density, are related by the equations of conservation of mass, momentum, and energy flux across shock fronts. Assuming the temperature of $1.5 \times 10^6 {\rm \,K}$ corresponds to the decelerated shock in the cavity wall and $T=5 \times 10^6 {\rm \,K}$ corresponds to the reflected shock, the current blast wave velocity is around $300 \kms$, and the original, unperturbed blast wave velocity was around $500 \kms$. The reflected shock model also predicts that the original cavity density, $n_c$, is related to the shell density $n_s/n_c = 5$. Combined with the surface brightness measurement of the Balmer filaments, this yields an atomic shell density $n_s = 0.2 {\rm \,cm^{-3}}$ and a cavity density $n_c = 0.04{\rm \,cm^{-3}}$ if the shell is entirely neutral. As described below (§\[subsecmodel\]), the shell is partially ionized, so the original cavity density was $n_c = 0.08 {\rm \,cm^{-3}}$. In addition to neglect of non-equilibrium effects, other sources of error include confusion of the reflected shock and the unperturbed shock in the cavity, uncertainty of the derived X-ray temperature, and uncertainty of the Balmer intensity of a single shock surface derived from observation through the spherical edge of the blast wave. The predicted density is consistent with that derived from X-ray observations of the projected interior. In a featureless region of the southeast near $\alpha=\hm{20}{55},\delta=30^\circ 12\arcmin$, the median HRI surface brightness is $0.01 {\rm \, counts \, s^{-1} \, arcmin^{-2}}$. We compare this with a model of a Raymond-Smith plasma at $T_e = 3.5 \times 10^6$, correcting for absorbing column of $N_H = 6\times 10^{20} {\rm \, cm^{-2}}$. Assuming the emitting region is 22 pc deep (the predicted line of sight through the SNR at this radius), the observed count rate corresponds to a density $n_e = 0.2 {\rm \, cm^{-3}}$. This is the density of the shocked material; the original cavity density would have been $n_c = 0.05 {\rm \, cm^{-3}}$. ### Photoionization One unusual feature at the east is a region of very smooth emission adjacent to the bright structure of NGC 6995 at $\alpha=\hms{20}{57}{30},\delta=31^\circ 00\arcmin$. This area is obvious in and detectable in 2 but absent in ø3. There are no X-rays here ([@Lev97]), so a fast shock wave has not passed through. The X-ray map shows a clear indentation in this region, where dense gas has significantly impeded the blast wave over large scales, similar to the southeast cloud. There is no other direct evidence, such as a Balmer filament, that the blast wave is presently located toward the exterior of this region. We propose that this gas is photoionized by a shock precursor, and it has not been shocked. Radiation from the shock-heated gas of the interior of the Cygnus Loop serves as the ionization source. Typically, when the shock velocity $v_s \gtrsim 100 {\rm \ km \ s^{-1}}$, much of the subsequent radiation is at UV wavelengths ([@Ray79]; [@Shu79]; [@Dop84]), which can ionize the surrounding gas, both ahead of and behind the shock front. The hardness of the ionizing spectrum increases with shock velocity, and the resulting emission from photoionized gas is a function of shock velocity and preshock density ([@Sut93]; [@Dop96]; [@Mor96]). The absence of ø3 in the ionized precursor indicates that the shock has intermediate velocity ($v_s \lesssim 200 {\rm \ km \ s^{-1}}$), which is consistent with UV spectroscopy of Blair et al. (1991) who find $v_s \sim 170 \kms$ is typical in this region. The bright emission ahead of the shock front is probably a consequence of relatively high density; the Balmer luminosity of the precursor scales linearly with density in the steady case ([@Ray79]; [@Shu79]; [@Dop96]). A second important factor is that the shock is fully radiative, having been decelerated in the dense cloud, so the shock front does not move through the ionized region before the ions have time to radiate (cf. [@Ray88]). Finally, the column density ahead of the shock must be great for the precursor emission to be observable. The images do allow us to estimate crudely the density of the photoionized material. We assume that the length of the photoionized region along the line of sight is equal to its extent north-south in the plane of the sky and use the observed surface brightness to predict a density $n\sim 60 {\rm \, cm^{-3}}$. Spectral data would allow a more exact determination of the characteristics of the emitting region and quantitative measurements of the photoionizing source. The blast wave history causes the ambient medium to be photoionized here and not elsewhere. The penetrating X-rays of very fast shocks ($v_s \gtrsim 400 \kms$) do not efficiently ionize the surrounding medium; the optical depth to 200 eV photons $\tau_{\rm 200 eV} = N/(1.1 \times 10^{20}{\rm \, cm^{-2}})$. Only the softer UV-dominated spectra of slower shocks are effective ionizing sources. In other regions of the Cygnus Loop, the recent shock velocity has been too great to produce much photoionization, and there has not been enough time for the ionizing field of the post-shock cooling and recombination zone to develop. Thus, most of the surrounding medium has remained sufficiently neutral to exhibit Balmer-dominated filaments. The exceptional photoionization at the eastern edge is consistent with the cavity model of this SNR. The blast wave has not slowed sufficiently elsewhere for most of the shocked gas to be a useful source of ionizing photons. Over nearly all of the Cygnus Loop, the deceleration has been recent, as the blast wave has run into the walls of the cavity. Only in this region has the softer spectrum of a slower shock existed long enough to ionize the preshock medium so that its radiative transitions can be detected. Photoionized regions have been observed around other SNRs, including N132D ([@Mor95]) and N49 ([@Van92]) in the Large Magellanic Cloud, for example. One difference between the Cygnus Loop and the former SNR, however, is that while in the Cygnus Loop the photoionized precursor is the exception, detected only in a limited region, in N132D it is the rule, occurring over the majority of the younger remnant’s perimeter. Another difference is that the exceptional photoionization ahead of the Cygnus Loop blast wave is adjacent to the region that contains its brightest soft X-ray knots ([@Lev97]), while the photoionization and X-ray surface brightness variations are not correlated in N132D ([@Mor96e]). These differences are likely to be a consequence of the different sources of ionizing photons and the density of the surrounding medium. First, in N132D, the X-ray–emitting shock can account for only half of the ionization ([@Mor96e]), so another significant source must also contribute to the ionization in this case. The ubiquitous radiative shocks in clouds and O-rich filaments that Morse et al. (1996) propose as this additional source are not preferentially associated with the X-ray emission. Second, a large molecular cloud extends over the entire southern hemisphere of this SNR ([@Ban97]). While the present blast wave is not in the dense core that is observed in CO, the outer regions of the cloud are expected to be over-dense with respect to the typical ambient ISM. The higher densities that make the ionizing precursor visible are therefore characteristic of the majority of the perimeter of N132D. In contrast, we suggest that it is the unusual conditions of extreme blast wave deceleration and related high density that result in both the bright X-ray emission and the photoionized precursor in this small area at the eastern edge of the Cygnus Loop. These conditions make the Cygnus Loop precursor similar to that of N49. This middle-aged SNR is associated with a small molecular cloud on its eastern limb ([@Ban97]). The blast wave interaction with the cloud results in local optical emission and X-ray enhancement, blast wave deceleration observed in a range of shock velocities ([@Van92]), and a prominent photoionized precursor on the eastern limb of N49. Projected Interior ------------------ The interior of the Cygnus Loop contains diffuse emission and filaments. Both morphologies share some characteristics, such as exclusive ø3 emission (e.g., $\alpha =\hm{20}{48}, \delta=30^\circ 30\arcmin$ in diffuse regions and $\alpha =\hm{20}{52}, \delta=30^\circ 30\arcmin$ in filaments), and blends of and 2 (e.g., $\alpha =\hm{20}{50}, \delta=29^\circ 45\arcmin$ and $\alpha =\hm{20}{50}, \delta=29^\circ 45\arcmin$). A combination of emission in all three lines is observed only in diffuse emission (e.g., $\alpha =\hm{20}{52}, \delta=31^\circ 10\arcmin$), and alone is observed only in filaments (e.g., $\alpha =\hm{20}{50}, \delta=30^\circ 30\arcmin$). Projection effects are more severe in the interior, but these optical images and high-resolution spectroscopy ([@Kir76]; [@Gre91]; [@Shu91]) together help unravel the tangled Cygnus Loop. In all cases, the optical emission is the result of the interaction of the three-dimensional blast wave surface with dense structures in the ISM. The emitting regions are truly at the outside of the SNR, but some appear projected to the interior. The brighter filaments tend to have lower absolute radial velocities and narrower velocity dispersions than the fainter diffuse emission ([@Kir76]; [@Shu91]). This is consistent with the model in which filaments are long lines of sight through the edge of a wavy shock front, and diffuse emission results when the blast wave is observed nearly face-on. The shock velocity is primarily perpendicular to the line of sight in the former case and along the line of sight in the latter case. In some instances, adjacent filamentary and diffuse emission are related, being tangential and normal views of the same portion of the blast wave. Elsewhere, spatially separated shock regions are projected onto the same location in the observer’s view, causing significant confusion. The extensive diffuse emission at the western interior near $\alpha=\hm{20}{47}$—$49^{\rm m}$ and $\delta=30^\circ 40\arcmin$—$31^\circ 20\arcmin$ suggests that a large wall of dense gas is present across the entire face of the SNR (Figs. \[figsubw\] and \[figsuby\]). The irregularity of the diffuse emission is due to the clumpiness of the cloud. This emission is blueshifted with respect to nearby gas ([@Gre91]; [@Shu91]) so it is on the front face. The bright filaments of the “carrot” (at $\alpha =\hm{20}{49}, \delta=31^\circ 40\arcmin$), having positive radial velocities, are on the rear face. The filaments at $\alpha =\hm{20}{49}, \delta=31^\circ 00\arcmin$ (Fig. \[figsubw\]) are distinct on small scales, similar to the edges of the breakout, although here the region of common overlap is absent. The origin of the stratification is the same: the limited physical scale of the interaction with the intervening cloud, either because the cloud is intrinsically small or because the encounter is recent and the blast wave has not progressed far into the cloud. Along a particular line of sight, only a single coherent section of the blast wave is detected. Unlike the areas where strong emission is observed at all wavelengths, here there are not many projected overlapping regions of various stages of shock evolution. The filaments that are bright in and 2 illustrate an earlier section of the cloud shock than do the ø3 filaments. In this area on the western half of the Cygnus Loop, there is no planar blast wave that actually moves to the east, however. Rather, the [interaction]{} with a cloud is progressing from west to east as the blast wave surface expands radially. The intervening cloud must be oriented along the line of sight to some degree, as shown in Figure \[figcartc\]. Another region of diffuse emission and fainter filaments runs north-south across the center of the remnant, around $\alpha=\hm{20}{50}$—$52^{{\rm m}}, \delta=30^\circ 20\arcmin$—$50\arcmin$ (Figs. \[figsubx\] and \[figsuby\]). The high-resolution spectra in the vicinity are of the diffuse component alone, which includes very bright central knots. These clumps are blueshifted, so these clouds are on the near face. The westernmost of these filaments, at $\alpha=20^{{\rm h}} 50^{{\rm m}} 20^{{\rm s}}$, is detected only in . The blast wave has run into an extended wall of gas, enabling us to look through a long edge of the shock front. These filaments are broader than most of the other pure Balmer filaments, which is expected in this scenario. The broad filaments appear where the blast wave is viewed through a point of inflection, whereas the sharper, narrow ones are viewed at a tangency. For these non-radiative filaments to be observable, the wall of gas must be extremely smooth at the present projected location of the shock front. The wall of gas is likely to be related to the source of the rest of the optical emission at the center of the SNR, not the clouds that are illuminated as the “carrot,” although this is uncertain. History of the Cygnus Loop ========================== Modelling the ISM\[subsecmodel\] -------------------------------- Except in the single photoionized section, the optical emission traces dense shocked regions. Using some knowledge of shock evolution and geometry, we map the surrounding ISM. The basic principles applicable to these observations are: (1) optical emission due to radiative shocks requires that the blast wave has swept up a significant column density ($N \gtrsim 10^{18} {\rm \, cm^{-2}}$), which is likely in regions of increased density ($n\gtrsim 1 {\rm \, cm^{-3}}$); (2) sharp filaments are edge-on views of the blast wave surface, and diffuse emission is the result of face-on views of this surface; (3) the optical line emission from a radiative shock is progressively dominated by ø3, , then 2; and (4) filaments of alone are the result of non-radiative shocks in material that has a significant neutral fraction. With these data, we construct a three-dimensional model of the interstellar medium around the Cygnus Loop . Over much of its surface, the blast wave is propagating through large clouds. Long filaments, detected in around the perimeter and in other lines across the interior, indicate that a spherical shell of atomic gas surrounds the SNR. The dense clumps of gas and the atomic shell together form the walls of a cavity in which the supernova occurred. Slices through the Cygnus Loop parallel to the plane of the sky from the near side to the far side (Fig. \[figcartmap\]) show some of the three-dimensional structure of the ISM before the supernova event. The center of the Cygnus Loop is at latitude $b = -8^\circ$. At the rear of the cavity, toward the Galactic plane, is a large wall of gas that is detected in neutral hydrogen ([@DeN75]) and in the IRAS sky survey ([@Bra86]). The cloud that causes the bright northeastern emission is also on the far side. Clumps at the edge of the breakout and at the northwest are on the near side. Some clouds, including the southeast knot and the molecular gas at the west, extend along the line of sight and appear in all slices. The Cygnus Loop is not the result of a supernova explosion in an arbitrary region of the interstellar medium. Rather, the blast wave propagates through a medium that the progenitor star has processed. Several characteristics imply a massive progenitor, the sort most capable of modifying the surrounding ISM, which Charles et al. (1985) and Levenson et al. (1997) discuss. One strong piece of evidence is the paradoxical morphology of the Cygnus Loop. At optical and X-ray wavelengths, it appears to be nearly circular, yet it exhibits tremendous structure on smaller scales. The optical data alone could be interpreted as a SNR in transition from the adiabatic to the radiative phase, but the presence and correlation of X-rays require that overall the blast wave still has (or until very recently had) a high velocity, $v_s \gtrsim 400 {\rm \ km \ s^{-1}}$. A model of blast wave interaction with large clouds accounts for this correlation ([@HesCox]; [@Gra95]; [@Lev96]), but such large clouds cannot be typical of the pre-supernova interstellar medium. If they were, the SNR would not appear so circular, and X-ray emission due to clouds evaporating in the interior would be observed. Thus, although large clouds are found around the periphery of the Cygnus Loop, they were not located within the current blast wave prior to the supernova event. This structure is expected in the vicinity of an O or B star. During its main-sequence lifetime, such a massive star homogenizes the surrounding ISM, creating a uniform region at constant density, temperature, and pressure. The star’s UV emission removes clumps of denser gas within the H II region, either destroying the clouds through photoevaporation or relocating them to the exterior by the rocket effect ([@McK84]). The result is a homogeneous spherical region surrounded by dense clouds. A star of type earlier than B0 will clear a cavity with a radius of about 50 pc (McKee et al. 1984 ). The Cygnus Loop cavity is smaller, so its progenitor must be a later-type star ([@Cha85]), yet one still able to create the photoionized cavity. Thus, we suggest that a B0, $M\approx 15M_{\sun}$ star was the progenitor of the Cygnus Loop. The observer’s line of sight through shocked clouds at the projected edge is preferentially longer than through those that appear at the projected center. Photoevaporation of clouds progresses radially outward from the central star, so lines of sight tangent to the cavity are biased to be through the longer dimensions of surviving clumps. Thus, optical emission on the projected interior is not as bright as on the projected edge of the SNR, despite their common cause. The same effect is even more pronounced in the X-ray data. The atomic shell in which the non-radiative filaments appear is also a consequence of the progenitor’s evolution. While material is eroded from clouds during the star’s main sequence lifetime, increasing the density of the H II region, the radius of the ionized region decreases as the outer portions recombine, creating the shell ([@Elm76]). The atomic gas of this recombined region together with the cloud remnants on the periphery form the walls of the cavity (Fig. \[figcartmap\]). The Cygnus Loop blast wave has expanded through the uniform cavity and is now hitting the surrounding walls ([@Lev97]). The atomic shell is not actually a sharp density discontinuity. It is more likely a gradual transition from the cavity interior to the maximum shell density. At the end of the progenitor’s main sequence lifetime, the pressure in the former H II region declines as the gas cools and recombines. (See [@PShu85] for a discussion of this phenomenon applied to N49.) In the extreme case of a sudden loss of pressure at the boundary, a rarefaction wave would move back through the dense shell resulting in a density profile of the form $n \propto (1 + {3r \over {at_{RG}}})^3$ ([@Zel66]), where the distance, $r$, from the original shell edge is constrained $-3at_{RG} \le r \le 0$ in terms of sound speed, $a$, and red giant lifetime, $t_{RG}$. The drop in pressure is not so sudden, however. Over $t_{RG} \sim 10^6$ for $M\approx 15M_{\sun}$ ([@Mae89]), about half the atoms in the former H II region have recombined. Thus, the density profile of the shell edge is likely to be intermediate between a sharp discontinuity and the offset power law above. The Cygnus Loop has been suggested previously to be the result of an explosion in an incomplete cavity. Both Falle & Garlick (1982) and Shull & Hippelein (1991), for example, argue for a partial cavity at a density discontinuity, such as the edge of a molecular cloud. In these models, only half of the blast wave runs into the cavity boundary. Consequently, the bright radiative emission of NGC 6992 and NGC 6960 and the corresponding X-ray emission are physically separated by great distances along the line of sight: the former are the result of radiative shocks in the cavity wall, and the opposite hemisphere of the blast wave propagates through the low-density medium outside the cloud to cause the latter. In contrast, the model presented here and in Levenson et al. (1997) has a completely defined cavity, the atomic shell and dense clouds together forming its spherical boundary. The blast wave does not break [out]{} of the edge of the molecular cloud into lower-density material; instead, it slows down when it hits a wall of [denser]{} gas. Although NGC 6992 and NGC 6960 and the X-rays that appear to the exterior of these radiative regions are two slightly different portions of the blast wave, they are not significantly separated. Furthermore, some of the large clouds are obviously extended along the line of sight, not restricted to the rear boundary (Fig. \[figcartmap\]). These optical and previous X-ray data ([@Lev97]) demonstrate that the cavity wall is complete. All around the projected edges of the Cygnus Loop, the Balmer filaments and X-ray enhancements mark the smooth component of the cavity wall. [Even the region called the breakout illustrates where the blast wave runs into a boundary of dense gas, not continued expansion in a low-density medium.]{} Only the eastern third of the projected interior of the Cygnus Loop lacks direct evidence for a cavity wall, although the smooth atomic shell that is apparent in Balmer-dominated filaments elsewhere would not be detected across the face of the SNR. Only clumps of gas could be observed, and their distribution depends on the arrangement that existed before the ISM was processed by the progenitor. The southeast of the Cygnus Loop is farthest from the galactic plane, so we expect the original filling factor of clouds to have been lowest there. Other SNRs, including RCW 86 ([@Vin97]) and N132D ([@Mor96e]), have been identified as cavity explosions. \[Shull (1985) identifies N49 as a cavity supernova, but Vancura et al. (1992) argue that the dense ambient material associated with the single observed CO cloud accounts for the data better.\] In these cases, interaction of the blast wave with the cavity wall enhances the observable properties of the SNRs. RCW 86 shares many features of the Cygnus Loop, including similar size, a limb-brightened (though very incomplete) X-ray shell (Vink et al. 1997), and Balmer-dominated filaments to the exterior of the X-ray emission ([@Smi97]). N132D is a younger remnant, and the apparent cavity is smaller. The progenitor’s wind during its main sequence lifetime may form the cavity boundary, or the scale of photoevaporation may have been limited in this dense environment adjacent to a molecular cloud ([@Ban97]). The bright interior X-ray emission ([@Mor96e]) does imply that the cavity interior was not uniform. Age and Distance\[secage\] -------------------------- Although the shell-like visual morphology of the Cygnus Loop and its X-ray limb-brightening broadly suggest that this is an example of a supernova remnant in the Sedov-Taylor or adiabatic expansion phase, in detail this model is inconsistent with the data presented here and in Levenson et al. (1997). The observed X-ray and optical limb-brightening are a consequence of interaction with the inhomogeneous interstellar medium, not the result of evolution of the blast wave in a uniform medium. Assuming that the blast wave has slowed sufficiently during its progress through a uniform medium to produce optical line emission (i.e. to $v_s \lesssim 200 \kms$) yields an age estimate of about 50,000 yr ([@Min58]; [@Fal82]). This is much too great because the age is inversely proportional to the blast wave velocity, and only in propagating through dense clouds has the shock been decelerated to $v_s \lesssim 200 \kms$. The Balmer filaments mark the location of the unimpeded, spherical blast wave, and the X-ray emission immediately behind these filaments demonstrates that the typical shock velocity is $v_s \gtrsim 400 \kms$, which is still to great for radiative cooling of the Sedov-Taylor shell to be significant. This SNR has not made the global transition to the radiative evolutionary phase. The X-ray emission alone has been used to determine an age of around 18,000 yr ([@Rap74]; [@Ku84]), assuming that this is presently an adiabatic blast wave expanding in a uniform medium; $t = 2R/5v_s$, where $R=19{\rm \ pc}$ and $v_s = 400 \kms$. The X-ray data provide a more accurate measurement of the age, but it is still likely to be an upper limit because these calculations contain uncertainties in the shock temperature and the velocity that is derived from it. Only the current unperturbed shock velocity is relevant to the age in this model, but any observation of the SNR includes contributions from a variety of temperatures, heated by different stages of the blast wave’s history. Observing the projected edge of the SNR minimizes confusion due to overlapping projections of different temperature components, but the interaction of the blast wave with the inhomogeneous ISM greatly alters the shock velocity and the corresponding postshock gas temperature. When the blast wave hits a density discontinuity, as it does over most of the surface of the Cygnus Loop, the continuing shock is suddenly decelerated and a reflected shock propagates back toward the interior, further heating the dense, hot, shocked material. At a density contrast of 10, for example, the twice-shocked matter will have a temperature approximately 1.5 times the singly-shocked material at the same radius, which leads to an overestimate of the unperturbed shock velocity by about $20\%$. Both the reflected shock and the forward shock through the dense cloud enhance the X-ray surface brightness and thus bias the mean measured temperature to favor blast-wave–cloud interactions rather than the undisturbed blast wave. If the X-ray measurement of two temperature components ([@Bal89]) due to the decelerated cloud shock and the reflected shock is correct and the original velocity of the unperturbed blast wave was closer to $500 \kms$ (§\[subsecedge\]), the corresponding age $t = 14,000$ yr. A reliable determination of the age of the Cygnus Loop is required to determine the initial energy, $E_o$, accurately (§\[secintro\]). Adopting the parameters $n_o = 0.08 {\rm \, cm^{-3}}$, $R = 19$ pc, and $t = 14,000$ yr, and a partially-ionized cavity interior, we find $E_o=2\times10^{50} {\rm\, erg}$. Because the radiative emission shows where the blast wave has run into clumpy material and decelerated, care is required to use it to measure the distance to the Cygnus Loop. Minkowski (1958) determined the expansion velocity to be $116 \kms$, fitting an ellipse to radial velocity measurements as a function of radius. This, combined with Hubble’s (1937) observed proper motion of filaments of $0.03\arcsec {\rm \ year^{-1}}$ yields the frequently-cited distance of 770 pc. This calculation assumes that all the optical filaments are part of a common expanding surface. In fact, the optical emission reveals different regions of inhomogeneity in the ISM. A second complication is the distinct geometry of the diffuse and filamentary emission on the face of the SNR. One is the result of a face-on shock, and the other is an example of a shock viewed edge-on. The latter does not have the radial velocity structure as a function of projected radius that is assumed in the model calculation. A third source of confusion arises because the observed filaments are the cooling zone behind the shock front, so their apparent motion may be the result of further development of the postshock region, not motion of the same clump of gas. The cooling zone may be propagating, without any physical motion of a parcel of gas in the ISM. Despite these difficulties, however, the radiative shocked gas can be used to measure distance to the Cygnus Loop or other supernova remnants when the distribution of clumps in the surrounding medium is spherically symmetric. In this case, although the emission comes from distinct regions of the ISM, not a single expanding blast wave surface, the different clouds acquire the same velocity when the blast wave passes through them. This approach is not appropriate where the surrounding medium is asymmetric, however. IC 443, for example, is asymmetrical, and its corresponding distance determinations are highly uncertain ([@Gre84]). The non-radiative filaments can also be used to determine distance, as Hester, Danielson, & Raymond (1986) have done, finding a distance of about 700 pc. These structures are advantageous because proper motion and spatial velocity can be calculated from the same filament. There is some uncertainty in the shock velocity because the derived value depends on the amount of ion-electron temperature equilibration (Chevalier et al. 1980). Measuring proper motion of filaments also introduces some uncertainty. All filaments are detected where the observer’s line of sight through the shock front is long, which may be the tangent surface of a spherical bubble. If it is instead the deformed surface of the blast wave that appears as a filament, however, the apparent filament will not necessarily persist or be detectable over the timescales required to obtain proper motion measurements. Long baselines over which to measure proper motion are desirable, but the oldest observations are sensitive to the brightest filaments. These bright filaments are exactly the locations where the shock is likely slowed in denser gas, so the present velocity differs from the spatial velocity that corresponds to the proper motion. With these caveats in mind, we adopt a distance of 770 pc to the Cygnus Loop. Consequences ============ The Cygnus Loop blast wave is now in the walls of the cavity, the boundary between the low-density former H II region and the ambient ISM. Blast wave propagation in the atomic shell will be a long-lived phenomenon. The smooth material of the shell recombined as clumps were photoevaporated in the interior. We crudely calculate the thickness of the shell assuming a constant number of ions in the H II region in both the initial Stromgren sphere, in which only the less-dense intercloud material was ionized, and in the final Stromgren sphere of radius $R_f$, in which all material, having mean density $n_m$, is ionized. The typical, unprocessed ISM determines the initial conditions; we assume the intercloud density $n_{ic}=0.1 {\rm \, cm^{-3}}$, and a volume filling factor of 0.03 of clouds of density $n_{cl}=1 {\rm \, cm^{-3}}$, so $n_{m}=0.13 {\rm \, cm^{-3}}$. Conserving ion number and neglecting increased recombination as clouds are evaporated, we obtain for the shell thickness $$R_{sh}=R_f ((n_m/n_{ic})^{1/3} - 1),$$ so using the observed radius $R_f = 19{\rm \,pc}$ yields $R_{sh}=1.5 {\rm \, pc}$. A blast wave of velocity $v_s= 300\kms$ will propagate through the shell for 5000 years. Eventually the blast wave will propagate through the denser medium of the cavity walls and clouds everywhere. It will rapidly decelerate, not only in the select locations where it has already encountered dense clumps that protrude into the cavity. Some of the surrounding material consists of large dense clumps like the ones observed presently in the very extended regions of radiative shocks. These, too, will light up and will be detectable at optical wavelengths. The southeast cloud may be an example of the earliest stages of interaction with a large cloud, and it may later resemble the northeast region ([@Gra95]). We also expect the entire northern limb, which in the optical regime is now detected primarily in bright Balmer filaments, to turn into an extended radiative region. The incomplete shocks along the northern limb are the early stages of interaction with a larger structure that is perhaps directly associated with the bright northeastern region. The entire western face of the Cygnus Loop will soon be bright when the blast wave has fully encountered the wall of gas that we have noted there. If the long filaments across the center of the SNR are the result of another wall of gas, the middle will also light up with optical emission in several hundred years. Large clouds of gas will significantly affect blast wave evolution. The shock will no longer be adiabatic when the cooling timescale becomes comparable with the dynamical timescale. In large clouds, where density $n_{cl}=1 {\rm \, cm^{-3}}$, radius $R_{cl}= 5 {\rm \, pc}$, and in which the shock velocity $v_{cl}=200\kms$, the dynamical time $R_{cl}/v_{cl} \approx 2\times 10^4 {\rm \, yr}$. In these clouds, the cooling time $3kT/n_{cl}\Lambda(T)\approx 4 \times 10^4 {\rm \, yr}$, in terms of the cooling function, $\Lambda$, having units ${\rm erg\, cm^3\, s^{-1}}$. If the covering fraction of clumps in the surrounding medium is sufficiently great, they will force the Cygnus Loop into the radiative phase, and energy losses will further decelerate the blast wave. The time and radius at which this occurs are distinct from self-similar calculations, such as those of McKee & Ostriker (1977) and Shull & Draine (1987) in which the gradual transition to the radiative phase happens as the blast wave propagates through a uniform medium. We use the pressure-driven snowplow model ([@McK77]) to set an upper limit on the final size of the SNR. In this model, work of the blast wave pushing into the ambient material and radiative losses of the hot interior are taken into account. In the Cygnus Loop, the radiative losses of the outer shell also will be significant, slowing the blast wave and making the final radius smaller. During this phase, $PV^\gamma$ is constant, where $P$ is the pressure inside the SNR, $V$ is its volume, and $\gamma$ is the ratio of specific heats. For $\gamma = 5/3$, this phase ends when $R = R_{PDS}(P_{PDS}/P_{ISM})^{1/5}$, in terms of the external pressure of the interstellar medium, $P_{ISM}$, and the radius and pressure inside the supernova remnant at the onset of the pressure-driven snowplow stage, $R_{PDS}$ and $P_{PDS}$, respectively. At this final time, the pressure inside the supernova remnant is equal to the ambient pressure. For the Cygnus Loop, we take $P_{ISM} = 10^{-12} {\rm \, dyne\, cm^{-2}}$, $R_{PDS}=20 {\rm \, pc}$, and $P_{PDS} = 4\times 10^{-9} {\rm \, dyne\, cm^{-2}}$, which places an upper limit on the final radius $R < 105 {\rm \, pc}$. Assuming that blast waves propagate through uniform media over great distances yields misleading conclusions about the size of SNRs and how they distribute energy through the ISM. Global models of the interstellar medium that include the hot contribution of SNRs ([@Cox74]; [@McK77]) are sensitive to the supernova rate, the final sizes of supernova remnants, and their persistence as hot bubbles. These factors determine whether the dominant component of the ISM is hot and in which smaller regions of warm and cold gas are embedded, or whether the warm and cold components form the substrate in which the distinct hot bubbles of SNRs are located. For the former to occur, the onset of the radiative phase of the SNR must be postponed until $R > 100 {\rm \, pc}$, independent of its prior evolution ([@Cox86]). (This calculation assumes that the supernova rate is 1 per 30 yr in a disk of radius 15 kpc, yet the conclusion holds for a range of reasonable supernova rates.) Although the cavity of the Cygnus Loop is small because its progenitor is one of the least massive stars to undergo core collapse, the initial mass function favors these lower mass (and smaller cavity) progenitors. Furthermore, even more massive stars, where cavities of 50 pc radius are expected, will be forced into the radiative phase and not expand to overlap if the density of the cavity walls is sufficient. The cavity walls are indeed likely to be denser on average than the typical ISM because the walls contain much of the material that was cleared out of the cavity interior. The conditions of the ISM in general, however, constrain the mass that will be commonly available to form clouds and cavity walls around pre-supernova stars. If the hot, low-density medium is pervasive, cavity walls typically would not be sufficiently massive to induce the radiative phase early, and SNRs would then normally attain larger radii before their blast waves decelerate. In any case, the typical environments of supernova progenitors must be considered. If dense, molecular material normally surrounds these stars, the radiative phase will be induced early, even if globally the ISM has low density. The cavity structure is significant also in determination of the mass of gas that a supernova remnant heats. The onset of shell formation for a supernova blast wave expanding in a uniform medium occurs at $R \propto n^{-3/7}$, so the shocked mass is then $M \propto n^{-2/7}$. If the ambient density varies by three orders of magnitude, the heated mass changes by only a factor of 7, for example. In contrast, when the supernova environment is a cavity with dense walls, the initial size of the cavity constrains the mass of the hot interior. The cavity interior is rarefied, so the heated cavity mass is significantly lower. In the Cygnus Loop, assuming a cavity density $n_c = 0.08 {\rm \, cm^{-3}}$, the heated mass is $20 M_{\sun}$, whereas in a uniform medium of this density, the heated mass would be greater than $400 M_{\sun}$. Conclusions =========== This optical emission line atlas of the Cygnus Loop supernova remnant provides the information to render a portrait of the surrounding ISM. The ambient medium, which the massive stellar progenitor shaped as it evolved, consists of many large, dense regions. These will significantly affect the subsequent development of the blast wave. The blast wave will decelerate, and it will no longer have sufficient velocity to excite the material through which it propagates to X-ray-emitting temperatures. Although this is a study of a particular object, it has far-reaching consequences for the ISM as a whole. Supernova remnants provide the energy to heat the ISM, affect the velocity dispersion of interstellar clouds, and set the stage for future generations of star formation. Thus, any global model of the gas in the Galaxy critically depends on the evolution of SNR blast waves in their inhomogeneous environments, of which this example is typical. While optical emission-line characteristics identify supernova remnants, they may preferentially select those whose evolution the extant ISM determines. Other SNRs need to be examined over a broad range of energies in the same careful way to draw more certain conclusions about their net effect on the interstellar medium. This research was supported in part by the Packard Foundation. 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--- abstract: 'Given number fields $L \supset K$, smooth projective curves $C$ defined over $L$ and $B$ defined over $K$, and a non-constant $L$-morphism $h \colon C \to B_L$, we consider the curve $C_h$ defined over $K$ whose $K$-rational points parametrize the $L$-rational points on $C$ whose images under $h$ are defined over $K$. Our construction gives precise criteria for deciding the applicability of Faltings’ Theorem and the Chabauty method to find the points of the curve $C_h$. We provide a framework which includes as a special case that used in Elliptic Curve Chabauty techniques and their higher genus versions. The set $C_h(K)$ can be infinite only when $C$ has genus at most $1$; we analyze completely the case when $C$ has genus 1.' address: - 'Mathematical Institute, University of Oxford, 24–29 St. Giles, Oxford OX1 3LB, United Kingdom' - 'Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, United Kingdom' author: - 'E.V. Flynn' - 'D. Testa' date: '30 September, 2012' title: Finite Weil restriction of curves --- Introduction {#introduction .unnumbered} ============ Let $L$ be a number field of degree $d$ over $\mathbb{Q}$, let $f \in L[x]$ be a polynomial with coefficients in $L$, and define $$A_f := \bigl\{ x \in L ~ \mid ~ f(x) \in \mathbb{Q} \bigr\} .$$ Choosing a basis of $L$ over $\mathbb{Q}$ and writing explicitly the conditions for an element of $L$ to lie in $A_f$, it is easy to see that the set $A_f$ is the set of rational solutions of $d-1$ polynomials in $d$ variables. Thus we expect the set $A_f$ to be the set of rational points of a (possibly reducible) curve $C_f$; indeed, this is always true when $f$ is non-constant. A basic question that we would like to answer is to find conditions on $L$ and $f$ that guarantee that the set $A_f$ is finite, and ideally to decide when standard techniques can be applied to explicitly determine this set. We formalize and generalize the previous problem as follows. Let $L \supset K$ be a finite separable field extension, let $B \to {{\rm Spec}}(K)$ be a smooth projective curve defined over $K$ and let $C$ be a smooth projective curve defined over $L$. Denote by $B_L$ the base-change to $L$ of the curve $B$, and suppose that $h \colon C \to B_L$ is a non-constant morphism defined over $L$. Then there is a (possibly singular and reducible) curve $C_h$ defined over $K$ whose $K$-rational points parametrize the $L$-rational points $p \in C(L)$ such that $h(p) \in B(K) \subset B_L(L)$. In this context, Theorem \[cocha\] identifies an abelian subvariety $F$ of the Jacobian of the curve $C_h$ and provides a formula to compute the rank of the Mordell-Weil group of $F$: these are the ingredients needed to apply the Chabauty method to determine the rational points of the curve $C_h$. To see the relationship of this general problem with the initial motivating question, we let $C:=\mathbb{P}^1_L$ and $B:=\mathbb{P}^1_\mathbb{Q}$. The polynomial $f$ determines a morphism $h \colon \mathbb{P}^1_L \to (\mathbb{P}^1_{\mathbb{Q}})_L$ and the $L$-points of $\mathbb{P}^1_L$ (different from $\infty$) with image in $\mathbb{P}^1(\mathbb{Q})$ correspond to the set $A_f$. Over an algebraic closure of the field $L$, the curve $C_h$ is isomorphic to the fibered product of morphisms $h_i \colon C_i \to B_L$ obtained from the initial morphism $h$ by taking Galois conjugates (Lemma \[profi\]). Thus we generalize further our setup: we concentrate our attention on the fibered product of finitely many morphisms $h_1 \colon C_1 \to B$, …, $h_n \colon C_n \to B$, where $C_1 , \ldots , C_n$ and $B$ are smooth curves and the morphisms $h_1,\ldots,h_n$ are finite and separable. We determine the geometric genus of the normalization of $C_h$, as well as a natural abelian subvariety $J_h$ of the Jacobian of $C_h$. Due to the nature of the problem and of the arguments, it is immediate to convert results over the algebraic closure to statements over the initial field of definition. Suppose now that $K$ is a number field. We may use the Chabauty method to find the rational points on $C_h$, provided the abelian variety $J_h$ satisfies the condition that the rank of $J_h(K)$ is less than the genus of $C_h$; by Chabauty’s Theorem (see [@prolegom; @chab1; @chab2]) this guarantees that $C_h(K)$ is finite. Chabauty’s Theorem has been developed into a practical technique, which has been applied to a range of Diophantine problems, for example in [@colemanchab; @colemanpadic; @flynnchab; @lortuck]. Further, if the set $C_h(K)$ is infinite, then, by Faltings’ Theorem [@faltings], the curve $C_h$ contains a component of geometric genus at most one; thus the computation of the geometric genus of the normalization of $C_h$ is a first step towards answering the question of whether $C_h(K)$ is finite or not. Moreover, since all the irreducible components of $C_h$ dominate the curve $C$, it follows that the set $C_h(K)$ can be infinite only in the case in which the curve $C$ has geometric genus at most one. In the case in which $C$ has genus zero, results equivalent to special cases of this question have already been studied ([@az; @bt; @pa; @sch; @za]). We shall analyze completely the case in which the genus of $C$ is one and the curve $C_h$ has infinitely many rational points. This covers as a special case the method called Elliptic Curve Chabauty (which is commonly applied to an elliptic curve $E$ defined over a number field $L \supset K$, satisfying that the rank of $E(L)$ is less than $[L : K]$ and we wish to find all $(x,y) \in E(L)$ subject to an arithmetic condition such as $x \in K$); see, for example, [@bruinth; @bruinchab; @flywet1; @flywet2; @wethth] and a hyperelliptic version in [@siksekchab]). Let $K$ be a number field and let $E : y^2 = (a_2 x^2 + a_1 x + a_0)(x + b_1 + b_2 \sqrt{d})$, with $a_0,a_1,a_2,b_1,b_2,d \in K$, be an elliptic curve defined over $L = K(\sqrt{d})$; suppose also that $b_2 \not= 0$, $d \not= 0$, $d \not\in (K^)^2$, so that $E$ is not defined over $K$. We are interested in $(x,y) \in E(L)$ with $x \in K$. Let $y = r + s\sqrt{d}$, with $r,s \in K$. Equating coefficients of $1,\sqrt{d}$ gives $$r^2 + d s^2 = (a_2 x^2 + a_1 x + a_0) (x + b_1),\ \ 2 r s = (a_2 x^2 + a_1 x + a_0) b_2.$$ Let $t = s^2/(a_2 x^2 + a_1 x + a_0)$. Eliminate $r$ to obtain $b_2^2/(4 t) + d t = x + b_1$, so that $x = x(t) := b_2^2/(4 t) + d t - b_1$. Hence $s,t$ satisfy the curve $C : (st)^2 = t^3(a_2 x(t)^2 + a_1 x(t) + a_0)$, for which the right hand side is a quintic in $t$. One can check directly that this quintic has discriminant whose factors are powers of: $a_0, b_2, d$, the discriminant of $E$ and its conjugate; all of these are guaranteed to be nonzero, from our assumptions, so that $C$ has genus $2$. By Faltings’ Theorem, $C(K)$ is finite, so there are finitely many such $s,t\in K$, and hence finitely many $x = x(t)$, and so finitely many $(x,y) \in E(L)$ with $x \in K$. Furthermore, one can check directly from the induced map from $C$ to $E$ that, if $J$ is the Jacobian of $C$ then $\hbox{rank}\bigl(J(K)\bigr) = \hbox{rank}\bigl(E(L)\bigr)$; hence if $\hbox{rank}\bigl(E(L)\bigr) < [L:K] = 2$ (the condition for Elliptic Curve Chabauty) then $C$ satisfies the conditions for Chabauty’s Theorem over $K$. So, for this special case, there is a perfect match between both conditions. Let $f \in \mathbb{Q}(\sqrt{2})[x]$ be the polynomial $f(x) = x^2 (x - \sqrt{2})$ and suppose that we are interested in finding the set $A_f$ of values of $x \in \mathbb{Q}(\sqrt{2})$ such that $f(x) \in \mathbb{Q}$. Writing $x=a+b \sqrt{2}$ with $a,b \in \mathbb{Q}$ and substituting in $f$ we find that $f(a + b \sqrt{2})$ is a rational number if and only if the equality $a^2+2b^2=3a^2b+2b^3$ holds. Apart from the solution $(a,b)=(0,0)$, all the remaining solutions of the resulting equation can be determined by the substitution $a=tb$ and the set $A_f$ is the set $$\left\{ \frac{t^2+2}{(3t^2+2)} (t+\sqrt{2})~ \Bigl\| ~ t \in \mathbb{Q} \right\} \cup \bigl\{ 0 \bigr\} .$$ Let $f \in \mathbb{Q}(\sqrt{2})[x]$ be the polynomial $f(x) = \frac{x (x - \sqrt{2})}{x-1}$. Arguing similarly to the previous example, we find that the set values of $x \in \mathbb{Q}(\sqrt{2})$ such that $f(x) \in \mathbb{Q}$ consists of the solutions to the equation $a^2 b - a^2 - 2 a b + a - 2 b^3 + 2 b^2 = 0$. The projective closure of the previous equation defines an elliptic curve $E$ with Weierstrass form $y^2=x^3-x$; by a 2-descent it is easy to show that $E(\mathbb{Q}) = E(\mathbb{Q})[2]$ and we conclude that the set we are seeking is the set $\{ 0 , \sqrt{2} \}$. The last two of these examples exhibit the two qualitative behaviours that we analyze in what follows. Relative Weil Restriction {#rwr} ========================= We begin this section by recalling the definition of the Weil restriction functor. The setup is quite general, though we will only use it in a very specialized context. We prefer to adopt this formal point of view at the beginning, since it simplifies the arguments; for the cases mentioned in the introduction, it is straightforward to translate all our arguments into explicit computations that are also easy to verify. Let $s \colon S' \to S$ be a morphism of schemes and let $X'$ be an $S'$-scheme; the contravariant functor $$\begin{array}{rcl} {{\mathfrak{R}}}_{S'/S}(X') \colon ({\rm Sch}/S) ^o & \longrightarrow & {\rm Sets} \\[5pt] T & \longmapsto & {\rm Hom}_{S'} (T \times _S S' , X') \end{array}$$ is the [Weil restriction functor]{}. If the functor ${{\mathfrak{R}}}_{S'/S}(X')$ is representable, then we denote by ${{\mathfrak{R}}}_{S'/S}(X')$ also the scheme representing ${{\mathfrak{R}}}_{S'/S}(X')$; sometimes, to simplify the notation, we omit the reference to $S$ and $S'$ and write that ${{\mathfrak{R}}}(X')$ is the Weil restriction of $X'$. The scheme ${{\mathfrak{R}}}(X')$ is determined by the isomorphism $${\rm Hom}_S ( - , {{\mathfrak{R}}}(X')) \stackrel{\sim}{\longrightarrow} {\rm Hom}_{S'} (- \times_S S', X')$$ of functors $({\rm Sch}/S)^o \to {\rm Sets}$. Informally this means that the $S$-valued points of ${{\mathfrak{R}}}(X')$ are the same as the $S'$-valued points of $X'$. Suppose now that $Y \to S$ is another morphism of schemes, denote by $Y'$ the fibered product $Y \times _S S'$ with natural morphism $b \colon Y' \to Y$, and let $h \colon X \to Y'$ be any scheme. We have the diagram $$\label{diato} \begin{minipage}{150pt} \xymatrix{ X \ar[d] _h \\ Y' \ar[d] \ar[r]^b & Y \ar[d] \\ S' \ar[r] ^{s} & S } \end{minipage}$$ and $X$ is therefore both a $Y'$-scheme and an $S'$ scheme. Thus there are two possible Weil restrictions we can construct: - the Weil restriction ${{\mathfrak{R}}}_{S'/S} (X)$, using the $S'$-scheme structure of $X$, - the Weil restriction ${{\mathfrak{R}}}_{Y'/Y} (X)$, using the $Y'$-scheme structure of $X$. Below we shall use the notation ${{\rm Res}}_h(X)$ for the Weil restriction ${{\mathfrak{R}}}_{Y'/Y}(X)$ and call it the [[relative Weil restriction]{}]{}. In order to relate the Weil restriction ${{\mathfrak{R}}}_{Y'/Y}(X)$ to the discussion in the introduction, we give an alternative definition of this functor and then proceed to prove the equivalence of the two. For concreteness, suppose that in diagram the morphism $S' \to S$ is induced by a (finite, separable) field extension $L \supset K$; then there is a subset of the $L$-points of $X$ whose image under $h$ is not simply an $L$-point of $Y'$, but it is actually a $K$-point of $Y$. We would like to say that this set of $L$-points of $X$ with $K$-rational image under $h$ are the $K$-rational points of a scheme defined over $K$, and that this scheme [[is]{}]{} the relative Weil restriction of $X$. This is what we discussed in the introduction and we now formalize it. Hence, let $T$ be any $S$-scheme and denote by $T'$ the $S'$-scheme $T \times _S S'$. Pull-back by the morphism $s$ defines a function ${\rm Hom}_S (T,Y) \to {\rm Hom}_{S'} (T',Y')$; there is also a function ${\rm Hom}_{S'} (T',X) \to {\rm Hom}_{S'} (T',Y')$ determined by composition with $h$. Summing up, for any $S$-scheme $T$, we obtain a diagram $$\label{sta} \begin{minipage}{150pt} \xymatrix{{\rm Hom}(T,X/Y) \ar@{-->}[r] \ar@{-->}[d] & {\rm Hom}_{S'} (T',X) \ar[d] ^{h \circ -}\\ {\rm Hom}_S (T,Y) \ar[r] ^{s^} & {\rm Hom}_{S'} (T',Y') . } \end{minipage}$$ We denote by ${\rm Hom}(T,X/Y)$ the pull-back of diagram , and we define the [relative Weil restriction functor]{} to be the functor $$\begin{array}{rcl} {{\rm Res}}_h \colon ({\rm Sch}/S)^o & \longrightarrow & {\rm Sets} \\[5pt] T & \longmapsto & {\rm Hom}(T,X/Y). \end{array}$$ If the functor ${{\rm Res}}_h$ is representable, we denote a scheme representing it by ${{\rm Res}}_h(X)$. \[lem:rwr\] Let $s \colon S' \to S$ be a morphism of schemes. Let $X$ be an $S'$-scheme and let $Y$ be an $S$-scheme; denote by $Y_{S'}$ the $S'$-scheme $Y \times _S S'$. Let $h \colon X \to Y_{S'}$ be an $S'$-morphism and assume that both Weil restriction functors ${{\mathfrak{R}}}(X)$ and ${{\mathfrak{R}}}(Y_{S'})$ are representable (notation as in ). Then the relative Weil restriction functor ${{\rm Res}}_h \colon ({\rm Sch}/S)^o \to {\rm Sets}$ is representable and the schemes ${{\rm Res}}_h(X)$ and ${{\mathfrak{R}}}_{Y'/Y}(X)$ coincide. Moreover, there is a commutative diagram of $S$-schemes $$\label{dere} \begin{minipage}{150pt} \xymatrix{ {{\rm Res}}_h(X) \ar[r] \ar[d] & {{\mathfrak{R}}}(X) \ar[d]^{h'} \\ Y \ar[r]^{\iota~~~} & {{\mathfrak{R}}}(Y_{S'}) } \end{minipage}$$ exhibiting ${{\rm Res}}_h(X)$ as a fibered product of ${{\mathfrak{R}}}(X)$ and $Y$ over ${{\mathfrak{R}}}(Y_{S'})$. We begin by constructing an $S$-morphism $h' \colon {{\mathfrak{R}}}(X) \to {{\mathfrak{R}}}(Y_{S'})$. By representability of ${{\mathfrak{R}}}(Y_{S'})$ and of ${{\mathfrak{R}}}(X)$ there are natural bijections $$\begin{array}{rcll} {\rm Hom}_{S} ({{\mathfrak{R}}}(X) , {{\mathfrak{R}}}(Y_{S'})) & \stackrel{\sim}{\longrightarrow} & {\rm Hom}_{S'} ({{\mathfrak{R}}}(X)_{S'} , Y_{S'}) & \quad {\textrm{and}} \\[7pt] {\rm Hom}_{S} ({{\mathfrak{R}}}(X) , {{\mathfrak{R}}}(X)) & \stackrel{\sim}{\longrightarrow} & {\rm Hom}_{S'} ({{\mathfrak{R}}}(X)_{S'} , X) . \end{array}$$ Let $\gamma \in {\rm Hom}_{S'} ({{\mathfrak{R}}}(X)_{S'} , X)$ be the $S'$-morphism corresponding to the identity in ${\rm Hom}_{S} ({{\mathfrak{R}}}(X) , {{\mathfrak{R}}}(X))$ and let $h' \colon {{\mathfrak{R}}}(X) \to {{\mathfrak{R}}}(Y_{S'})$ be the $S$-morphism corresponding to $h \circ \gamma \in {\rm Hom}_{S'} ({{\mathfrak{R}}}(X)_{S'} , Y_{S'})$. Note also that there is an $S$-morphism $\iota \colon Y \to {{\mathfrak{R}}}(Y_{S'})$ corresponding to the identity in ${\rm Hom}_{S'} (Y_{S'}, Y_{S'})$. Define ${{\rm Res}}_h(X) := Y \times _{{{\mathfrak{R}}}(Y_{S'})} {{\mathfrak{R}}}(X)$ so that there is a commutative diagram as in . To check that ${{\rm Res}}_h(X)$ represents ${{\rm Res}}_h$, let $T$ be any $S$-scheme; we have $$\begin{aligned} {\rm Hom}_S (T,{{\rm Res}}_h(X)) & = & {\rm Hom}_S (T,Y) \times_{{\rm Hom}_S (T , {{\mathfrak{R}}}(Y_{S'}))} {\rm Hom}_S (T,{{\mathfrak{R}}}(X)) = \\[5pt] & = & {\rm Hom}_S (T,Y) \times_{{\rm Hom}_S (T,{{\mathfrak{R}}}(Y_{S'}))} {\rm Hom}_{S'} (T_{S'},X) = {\rm Hom} (T,X/Y)\end{aligned}$$ as required. Thus, the scheme ${{\rm Res}}_h(X)$ defined above represents the functor ${{\rm Res}}_h$ and is obtained by the fibered product . Finally, we check that the schemes ${{\rm Res}}_h(X)$ and ${{\mathfrak{R}}}_{Y'/Y}(X)$ coincide. Let $T$ be any $Y$-scheme; we have $T \times_Y Y' = T \times _Y (Y \times_S S') = (T \times _Y Y) \times_S S' = T \times_S S'$ and $$\begin{aligned} {\rm Hom}_Y (T,{{\mathfrak{R}}}(X)) & = & {\rm Hom}_{Y'} (T \times_Y Y',X) = {\rm Hom}_{Y'} (T \times _S S',X) = \\[5pt] & = & \Bigl\{ f \in {\rm Hom}_{S'} (T \times_S S',X) ~\Bigl\|~ h \circ f = s^(T_Y \to Y) \Bigr\} = \\[5pt] & = & {\rm Hom} (T,X/Y) = {\rm Hom}_Y (T, {{\rm Res}}_h(X)) .\end{aligned}$$ We conclude using Yoneda’s Lemma that ${{\rm Res}}_h(X)$ and ${{\mathfrak{R}}}_{Y'/Y}(X)$ coincide, and hence the lemma follows. Informally, we may describe the scheme ${{\rm Res}}_h(X)$ as the scheme whose $S$-valued points correspond to the points in $X(S')$ lying above the points in $Y(S)$. Observe that, by the construction of ${{\rm Res}}_h(X)$ and diagram , there is an $S'$-morphism ${{\rm Res}}_h(X)_{S'} \to X$. The morphism ${{\rm Res}}_h(X)_{S'} \to X$ will prove useful later. We now introduce some notation that is used in the following lemma. Let $L \supset K$ be a finite, separable field extension of degree $n$, and let ${K^{\rm s}}$ be a separable closure of $K$. Set $S' := {\rm Spec}(L)$, $S := {\rm Spec}(K)$, and ${S^{\rm s}}:= {\rm Spec}({K^{\rm s}})$, denote by $s \colon S' \to S$ the morphism of schemes corresponding to the extension $L \supset K$ and suppose that $\varphi_1 , \ldots , \varphi_n \colon {S^{\rm s}}\to S'$ are the morphisms corresponding to all the distinct embeddings of $L$ into ${K^{\rm s}}$ fixing $K$. Given an $S'$-scheme $X$, we denote by $X_1 , \ldots , X_n$ the ${S^{\rm s}}$-schemes obtained by pulling back $X \to S'$ under the morphisms $\varphi_1 , \ldots , \varphi_n$. \[profi\] Let $X$ be an $S'$-scheme and let $Y$ be an $S$-scheme. Let $h \colon X \to Y_{S'}$ be an $S'$-morphism and assume that both Weil restriction functors ${{\mathfrak{R}}}(X)$ and ${{\mathfrak{R}}}(Y_{S'})$ are representable. Then the isomorphism $$\label{diago} {{\rm Res}}_h (X)_{{{S^{\rm s}}}} \simeq X_1 \times_{Y_{{{S^{\rm s}}}}} X_2 \times_{Y_{{{S^{\rm s}}}}} \cdots \times_{Y_{{{S^{\rm s}}}}} X_n$$ holds. More precisely, the isomorphism holds replacing ${K^{\rm s}}$ by any normal field extension of $K$ containing $L$. To prove the result, it suffices to consider the case in which both $X$ and $Y$ are affine; thus suppose that $Y = {\rm Spec}(A)$, $X = {\rm Spec}(B)$, and that $\underline{x}$ denotes a set of generators of $B$ as an $A$-algebra. Let $\alpha_1 , \ldots , \alpha_n$ denote a basis of $K$ over $L$ and let $\underline{x}_1 , \ldots , \underline{x}_n$ denote disjoint sets of variables each in bijection with $\underline{x}$; whenever we denote a variable in $\underline{x}$ by a symbol such as $x$, we denote by the symbols $x_1 , \ldots x_n$ the variables corresponding to $x$ in $\underline{x}_1 , \ldots , \underline{x}_n$. For every $x \in \underline{x}$ define linear forms $\tilde{x}_1 := \sum _i x_i \varphi_1(\alpha_i) , \ldots , \tilde{x}_n := \sum _i x_i \varphi_n(\alpha_i)$ in $A[\underline{x}_1 , \ldots , \underline{x}_n]$. First we show that the forms $\tilde{x}_1 , \ldots , \tilde{x}_n$ are linearly independent. Indeed, let $\Delta$ be the matrix whose $(i,j)$ entry is $\varphi_j(\alpha_i)$; the $(i,j)$ entry of $\Delta \Delta^t$ is $\sum _k \varphi_k(\alpha_i \alpha_j) = {\rm Tr}_{L/K}(\alpha_i \alpha_j)$. Thus the determinant of $\Delta \Delta ^t$ is equal to the discriminant of $L$ over $K$, and it is therefore non-zero. We deduce that the matrix $\Delta$ is invertible and that the forms $\tilde{x}_1 , \ldots , \tilde{x}_n$ are independent. Thus, defining $\tilde{\underline{x}}_j := \{ \tilde{x}_j \mid x \in \underline{x} \}$ for $j \in \{1,\ldots,n\}$, we proved that there is an isomorphism ${K^{\rm s}}\otimes_K A[\underline{x}_1 , \ldots , \underline{x}_n] \simeq {K^{\rm s}}\otimes_K A[\tilde{\underline{x}}_1 , \ldots , \tilde{\underline{x}}_n]$. The relative Weil restriction of $X$ may be defined as follows. Let $g(\underline{x})$ be an element of the polynomial ring $A[\underline{x}]$ contained in the ideal defining $X$; evaluate $g(\underline{x})$ substituting for each variable $x \in \underline{x}$ the sum $\sum x_i \alpha_i$ and write the resulting polynomial in $L \otimes_K A[\underline{x}_1 , \ldots , \underline{x}_n]$ as $\sum g_i \alpha_i$, where $g_1 , \ldots , g_n$ are elements of $A[\underline{x}_1 , \ldots , \underline{x}_n]$; denote the sequence $(g_1 , \ldots , g_n)$ by $\tilde{g}$. Then the scheme ${{\rm Res}}(X)$ is the scheme in ${\rm Spec}(A[\underline{x}_1 , \ldots , \underline{x}_n])$ whose ideal is the ideal generated by the elements of $\tilde{g}$, as $g$ varies among all the elements of the ideal defining $X$. It is therefore clear that the ideal $I$ defining ${{\rm Res}}(X)$ in ${K^{\rm s}}\otimes_K A[\underline{x}_1 , \ldots , \underline{x}_n]$ contains, for every embedding $\varphi \colon L \to {K^{\rm s}}$ fixing $K$, the elements $\sum_i g_i \varphi(\alpha_i) = \varphi \bigl( \sum g_i \alpha_i) = \varphi(g)$, and conversely that the ideal containing all such elements contains $g_1 , \ldots , g_n$ and hence it contains $I$. Let $\mathscr{F} \in A[\underline{x}]$ be a set of generators of the ideal of $X$; since, for $i \in \{1,\ldots , n\}$, the scheme $X_i$ is defined by $\{\varphi_i(f) \mid f \in \mathscr{F} \}$ in $A[\tilde{\underline{x}}_i]$ the result follows. The case of curves over an algebraically closed field ===================================================== In this section we compute the geometric genus of the relative Weil restriction $\mathscr{C}$ of a curve and we also determine a natural abelian variety isogenous to a subvariety of the Jacobian of $\mathscr{C}$. To calculate the geometric genus in Theorem \[thm:fibre\] we may clearly assume that the ground field is algebraically closed; the proof reduces the computation to the étale local case, settled in Lemma \[lem:fibre\]. \[lem:fibre\] Let $n$ be a positive integer. Suppose that $k$ is an algebraically closed field and that $r_1 , \ldots , r_n$ are positive integers relatively prime with the characteristic of $k$. Let $R$ be the least common multiple of $r_1 , \ldots , r_n$ and let $C \subset \mathbb{A}_{x,y_1,\ldots,y_n}^{n+1}$ be the affine scheme defined by $$C \colon \left\{ \begin{array}{rcl} y_1^{r_1} & = & x , \\ & \vdots \\ y_n^{r_n} & = & x . \end{array} \right.$$ The scheme $C$ has $\frac{r_1 \cdots r_n}{R}$ irreducible components and the morphism induced by the projection onto the $x$-axis from the normalization of each component ramifies at the origin to order $R$. Observe that the curve $C$ carries the action $\rho$ of $\mathbb{G}_m$ defined by $$t \cdot (x,y_1,\ldots,y_n) = (t^R x , t^{R/r_1} y_1 , \ldots , t^{R/r_n} y_n).$$ First we show that the action $\rho$ has trivial stabilizer at all points of $C$ different from the origin. Indeed, let $q=(x,y_1,\ldots,y_n)$ be a point of $C$ different from the origin; it follows that all the coordinates of $q$ are non-zero and the equality $(x,y_1,\ldots,y_n) = (t^R x , t^{R/r_1} y_1 , \ldots , t^{R/r_n} y_n)$ implies that $t$ is a root of unity of order dividing $\gcd (R/r_1 , \ldots , R/r_n) = 1$. Thus the complement of the origin in $C$ is a principal homogeneous space for $\mathbb{G}_m$. There are $r_1 \cdots r_n$ points on $C$ such that $x=1$ and these are stabilized precisely by the action of the $R$-th roots of unity; we deduce that $C$ consists of $\frac{r_1 \cdots r_n}{R}$ irreducible components, each isomorphic to closure of the orbit of a point $(1,\eta_1 , \ldots , \eta_n)$, where $\eta_i$ is an $r_i$-th root of unity, for $i \in \{1,\ldots,n\}$. Therefore the normalization of $C$ consists of $\frac{r_1 \cdots r_n}{R}$ components each mapping to the $x$-axis by $(t^R,\eta_1 t^{R/r_1}, \ldots , \eta_n t^{R/r_n}) \mapsto t^R$, as required. \[thm:fibre\] Let $k$ be an algebraically closed field and let $n$ be a positive integer. Suppose that $S,C_1,\ldots,C_n$ are smooth curves; for $i \in \{1,\ldots,n\}$ let $f_i \colon C_i \to S$ be a finite separable morphism whose ramification indices are coprime to the characteristic of $k$. Let $f$ denote the morphism of the curve $C := C_1 \times _S \cdots \times _S C_n$ to $S$, and let $C'$ be the normalization of $C$; denote by $g_S$ and $g_{C'}$ the arithmetic genera of $S$ and $C'$ respectively. For any point $p \in C$ and any $i \in \{1,\ldots,n\}$ denote by $r_i=r_i(p)$ the ramification index of $f_i$ at the point corresponding to $p$ and let $R=R(p) := {\rm lcm} \{r_1(p) , \ldots , r_n(p) \}$. Then the curve $C$ is a local complete intersection and the curve $C'$ has arithmetic genus $$\begin{aligned} g_{C'} & = & 1 + (g_S-1) \prod_{i=1}^n \deg(f_i) + \frac{1}{2} \sum _{p \in C} r_1(p) \cdots r_n(p) \left( 1 - \frac{1}{R(p)} \right) = \\[5pt] & = & 1 + \frac{1}{2} (r - 2g_S - 2) \prod_{i=1}^n \deg(f_i) - \frac{1}{2} \sum _{p \in f^{-1} (R_{f})} \frac{r_1(p) \cdots r_n(p)}{R(p)} \end{aligned}$$ where $R_f \subset S$ is the union of the sets of branch points of the morphisms $f_1 , \ldots , f_n$ and $r$ is the cardinality of $R_f$. Let $\pi \colon C' \to S$ be the morphism induced by the structure morphism $f \colon C \to S$. Let $p' \in C'$ be a closed point, let $p$ be the corresponding point of $C$ and let $p_1 , \ldots , p_n$ be the points of $C_1 , \ldots , C_n$ respectively corresponding to $p$. We prove the result by finding a local model of $C$ near $p$ which is a local complete intersection, and then applying the Hurwitz formula to the morphism induced by $f$ on the normalization of such a model. Choosing a local coordinate $x$ on $S$ near $\pi(p')$ we reduce to the case in which $S$ is $\mathbb{A}^1$ and $\pi(p') = 0$. Similarly choose local coordinates $z_1 , \ldots , z_n$ on $C_1 , \ldots , C_n$ near $p_1 , \ldots , p_n$ respectively. Thus, near $p$, the curve $C$ is defined by $$C \colon \left\{ \begin{array}{rcl} z_1^{r_1} \varphi_1(z_1) & = & x , \\ & \vdots \\ z_n^{r_n} \varphi_n(z_n) & = & x , \end{array} \right.$$ where $(\varphi_1 , \ldots , \varphi_n)$ is a rational function on $C$ defined and non-zero at $p$, and $r_1 , \ldots , r_n$ are the local ramification indices. In particular, the curve $C$ is a local complete intersection near $p$. Denote by $\mathcal{O}_{p}$ the local ring of $C$ near $p$; the base-change defined by the inclusion $$\mathcal{O}_{p} \longrightarrow \mathcal{O}_{p} [t_1 , \ldots , t_n] / \bigl( t_1^{r_1} - \varphi_1(z_1) , \ldots , t_n^{r_n} - \varphi_n(z_n) \bigr)$$ is finite étale (of degree $r_1 \cdots r_n$) by the assumption that the ramification indices are relatively prime to the characteristic of the field $k$. Hence, each component of the resulting curve is locally isomorphic to the curve $C_{p} \subset \mathbb{A}^{n+1}_{x,y_1,\ldots,y_n}$ defined by $$C_{p} \colon \left\{ \begin{array}{rcl} y_1^{r_1} & = & x , \\ & \vdots \\ y_n^{r_n} & = & x , \end{array} \right.$$ where $y_1 = z_1 t_1 , \ldots , y_n = z_n t_n$. The morphism induced by $\pi$ on $C_{p}$ is the morphism induced by the coordinate $x$. Using Lemma \[lem:fibre\], we conclude that the contribution of the point $p$ to the Hurwitz formula is $\frac{r_1 \cdots r_n}{R} (R - 1) = r_1 \cdots r_n (1 - \frac{1}{R})$, and hence we obtain $$2g_C-2 = (2g_S-2) \prod \deg(f_i) + \sum _{p \in C} r_1(p) \cdots r_n(p) \left( 1 - \frac{1}{R(p)} \right)$$ and the first formula follows. To prove the second one note that the quantity $r_1(p) \cdots r_n(p) ({1 - \frac{1}{R(p)}})$ vanishes for $p \notin R_f$, since in this case all the local ramification indices equal $1$, and that for all points $s$ of $S$ we have $$\sum _{p \in f^{-1}(s)} r_1(p) \cdots r_n(p) = \prod _{i=1}^n \sum _{p \in f_i^{-1}(s)} r_i(p) = \prod \deg(f_i)$$ and we conclude. In the next results, for a projective scheme $Y$, we denote by ${{\rm Jac}}(Y)$ the [[Jacobian variety of $Y$]{}]{}, that is the connected component of the identity of the group ${{\rm Pic}}(Y)$. \[prodo\] Suppose that $X_1 , \ldots , X_n$ are smooth projective varieties defined over an algebraically closed field $k$ and let $X=X_1 \times \cdots \times X_n$. For $i \in \{1,\ldots,n\}$ let $\rho_i \colon X \to X_i$ denote the canonical projection. The morphism $$\rho_\bullet^ = (\rho_1^* , \ldots , \rho_n^) \colon {{\rm Jac}}(X_1) \times \cdots \times {{\rm Jac}}(X_n) \to {{\rm Jac}}(X)$$ is an isomorphism. Choosing a point in each variety $X_1 , \ldots , X_n$ allows us to define, for $i \in \{1,\ldots,n\}$, an inclusion $X_i \hookrightarrow X$. These inclusions in turn determine a section ${{\rm Jac}}(X) \to {{\rm Jac}}(X_1) \times \cdots \times {{\rm Jac}}(X_n)$ of the morphism $\rho_\bullet^$. We deduce that $\rho_\bullet^$ is indeed an isomorphism of Jacobian varieties ${{\rm Jac}}(X_1) \times \cdots \times {{\rm Jac}}(X_n) \simeq {{\rm Jac}}(X)$. Maintaining the notation of the previous lemma, the morphisms $\rho_1,\ldots,\rho_n$ also induce a homomorphism $\psi^ \colon {{\rm Pic}}(X_1) \times \cdots \times {{\rm Pic}}(X_n) \to {{\rm Pic}}(X)$. The morphism $\psi$ though need not be an isomorphism. Since $\psi$ induces an isomorphism on the connected component of the identity, it factors through the Néron-Severi group and in particular its cokernel is a finitely generated abelian group. For instance, if $E$ is an elliptic curve and $X_1=X_2=E$, then the three classes $\{0\} \times E$, $E \times \{0\}$ and the diagonal are independent in ${\rm NS}(E \times E)$ so that ${\rm NS}(E \times E) \supset \mathbb{Z}^3$. On the other hand, the group ${\rm NS}(E) \times {\rm NS}(E)$ is isomorphic to $\mathbb{Z}^2$. For the next theorem we need to introduce some notation. Suppose that $B,X_1 , \ldots , X_n$ are smooth projective varieties defined over an algebraically closed field $k$. For each $i \in \{1,\ldots,n\}$ let $f_i \colon X_i \to B$ be a finite morphism. Denote by $X$ the product $X_1 \times \cdots \times X_n$, by $X_B$ the fibered product $X_1 \times _B \cdots \times _B X_n$ and by $\iota \colon X_B \to X$ the natural inclusion. For each $i \in \{1,\ldots,n\}$ - let $d_i$ denote the degree of $f_i$, - let $\rho_i \colon X \to X_i$ and $\pi_i=\rho_i \circ \iota \colon X_B \to X_i$ be the canonical projections, - let $\pi \colon X \to B$ be the composition $f_1 \circ \pi_1 = \cdots = f_n \circ \pi_n$, - let $d = d_1 \cdots d_n$ denote the degree of $\pi$. We summarize the notation in the case $n=2$ in . $$\label{diauno} \begin{array}{c} X := X_1 \times X_2 \\[25pt] X_B := X_1 \times _B X_2 \\ \vphantom{\begin{array}{c}W\\W\\W\\W\\W\end{array}} \end{array} \hspace{20pt} \begin{minipage}{150pt} \xymatrix{ & X_1 \times X_2 \ar[dddl] _{\rho_1} \ar[dddr] ^{\rho_2} \\ \\ & {\hphantom{{}_B} X_1} \times_B X_2 \ar[uu] _\iota \ar[dl] ^{\pi_1} \ar[dd] ^{\pi} \ar[dr] _{\pi_2} \\ X_1 \ar[dr] ^{f_1} & & X_2 \ar[dl] _{f_2} \\ & B } \end{minipage} \hphantom{\hspace{20pt}C}$$ Even though the diagram may suggest it, the identities $f_1 \circ \rho_1 = \cdots = f_n \circ \rho_n$ [[do not]{}]{} hold necessarily. On the other hand, the identities $f_1 \circ \pi_1 = \cdots = f_n \circ \pi_n = \pi$ hold. Let $\phi \colon {{\rm Jac}}(X) \to {{\rm Jac}}(X)$ be the isogeny defined by $$\begin{array}{ccl} \phi \colon \quad {{\rm Jac}}(X) & \longrightarrow & {{\rm Jac}}(X) \\[5pt] \rho_1^D_1 + \cdots + \rho_n^D_n & \longmapsto & \frac{d}{d_1}D_1 + \cdots + \frac{d}{d_n}D_n , \end{array}$$ where we used the identification of ${{\rm Jac}}(X)$ with ${{\rm Jac}}(X_1) \times \cdots \times {{\rm Jac}}(X_n)$ of Lemma \[prodo\]. Let $M'$ denote the kernel of the multiplication map ${{\rm Jac}}(B)^n \to {{\rm Jac}}(B)$, so that $M' \simeq {{\rm Jac}}(B)^{n-1}$; define the group $M$ as the image of $M'$ under the morphism $$\begin{array}{rcl} {{\rm Jac}}(B)^n & \longrightarrow & {{\rm Jac}}(X) \\[7pt] (D_1,\ldots,D_{n-1}) & \longmapsto & \rho_1^f_1^D_1 + \cdots + \rho_n^f_n^D_n. \end{array}$$ By construction, the group $M$ is connected and contained in the kernel of $\iota^$. Moreover, it follows from Lemma \[prodo\] and the fact that the morphisms $f_1,\ldots,f_n$ are finite that the morphism $M' \to M$ is finite and hence that the dimension of the group $M$ is $(n-1) \dim ({{\rm Jac}}(B))$. \[thm:jaco\] Maintaining the notation introduced above, the group $M$ has finite index in $\ker(\iota^)$. More precisely, for each element $D$ of $\ker(\iota^)$ there are elements $D_1,\ldots,D_n$ of ${{\rm Jac}}(B)$ such that the identities $$\begin{array}{rcll} \phi(D) & = & \sum_i \rho_i^f_i^D_i & \quad {\textrm{and}} \\[7pt] d \sum_i D_i & = & 0 \end{array}$$ hold. In particular, the dimension of the kernel of $\iota^$ is $(n-1) \dim ({{\rm Jac}}(B))$. By Lemma \[prodo\], the morphism $$\begin{array}{ccl} {{\rm Jac}}(X_1) \times \cdots \times {{\rm Jac}}(X_n) & \longrightarrow & {{\rm Jac}}(X) \\[5pt] (D_1 , \ldots , D_n) & \longmapsto & \rho_1^* D_1 + \cdots + \rho_n^* D_n \end{array}$$ is an isomorphism. Thus we identify the divisor classes in ${{\rm Jac}}(X)$ with $n$-tuples of divisor classes, one in each Jacobian variety ${{\rm Jac}}(X_1), \ldots , {{\rm Jac}}(X_n)$. Let $i,j$ be distinct indices in $\{1,\ldots,n\}$ and let $P$ be a divisor on $X_i$; it is easy to check that the identities $$\begin{aligned} \pi_i{}_\pi_i^(P) & = & \frac{d}{d_i} P \, , \\[5pt] \pi_i{}_\pi_j^(P) & = & \frac{d}{d_i d_j} f_i^f_j{}_(P) \end{aligned}$$ hold, and hence the class of the divisor $\pi_i{}_\pi_j^(P)$ is contained in $f_i^* {{\rm Pic}}(B)$. Suppose that $D=\rho_1^D_1+\cdots+\rho_n^D_n$ is a divisor on $X$ representing an element of ${{\rm Jac}}(X)$. Let $i$ be an index in $\{1,\ldots,n\}$; we have $$\pi_i{}_* \iota^* (D) = \pi_i{}_* (\pi_1^D_1+\cdots+\pi_n^D_n) \in \frac{d}{d_i}D_i + f_i^* {{\rm Jac}}(B)$$ and summing over all indices $i$, we find the equivalence $$\begin{array}{rcl} \displaystyle \sum _i \pi_i{}_* \iota^* (D) & \equiv & \displaystyle \sum_i \frac{d}{d_i}D_i \\[15pt] & \equiv & \displaystyle \phi(D) \pmod{f_1^{{\rm Jac}}(B) \times \cdots \times f_n^ {{\rm Jac}}(B)} . \end{array}$$ In particular, if $D$ is contained in the kernel of $\iota^$, then $\phi(D)$ is contained in $f_1^{{\rm Jac}}(B) \times \cdots \times f_n^* {{\rm Jac}}(B)$, establishing the first of the two identities. Finally, let $D_1 , \ldots , D_n \in {{\rm Jac}}(B)$ be divisor classes such that the element $\rho_1^f_1^D_1 + \cdots + \rho_n^f_n^D_n$ of ${{\rm Jac}}(X)$ lies in $\ker (\iota^)$. Then, the element $D_1 + \cdots + D_n$ of ${{\rm Jac}}(B)$ lies in the kernel of $\pi^$, so that $d(D_1 + \cdots + D_n) = \pi_* \pi^(D_1 + \cdots + D_n) = 0$, proving the second identity. It follows from what we just proved that the equalities $$\dim (\ker(\iota^)) = \dim M = ({n-1}) \dim ({{\rm Jac}}(B))$$ hold, proving the final assertion of the theorem. Mordell-Weil groups and relative Weil restriction ================================================= From now on, we shall be in the following set up (specializing the assumptions of Lemma \[lem:rwr\]): - $L$ is a number field and $S':={{\rm Spec}}(L)$, - $K \subset L$ is a subfield and $S:={{\rm Spec}}(K)$, - $C$ is a smooth projective curve over $L$, - $B$ is a smooth projective curve over $K$, and - $h \colon C \to B_{S'}=B_L$ is a finite morphism. To simplify the notation, for any variety $Z$ defined over a number field $k$ denote by $mw_k(Z)$ the rank of the Mordell-Weil group of the Jacobian of $Z$; we are only going to apply this notation with $k \in \{K,L\}$ to varieties $Z$ that are either reduced curves or products of smooth integral curves. \[cocha\] Suppose that $C$ is a smooth projective curve defined over a number field $L$. Suppose that $B$ is a smooth projective curve defined over a subfield $K$ of $L$, and let $h \colon C \to B_L := B \times_{{{\rm Spec}}(K)} {{\rm Spec}}(L)$ be a finite morphism. Denote by $n$ the dimension of $L$ as a vector space over $K$ and by $g(C)$ and $g(B)$ the genera of $C$ and $B$ respectively. The Jacobian of ${{\rm Res}}_h(C)$ contains an abelian subvariety $F$ of dimension $n g(C) - (n - 1) g(B)$ defined over $K$ and with Mordell-Weil group over $K$ of rank $mw_L(C) - \bigl( mw_L (B_L) - mw_K(B) \bigr)$. The $L$-morphism $h \colon C \to B_L$ induces a $K$-morphism ${{\mathfrak{R}}}(C) \to {{\mathfrak{R}}}(B_L)$ which in turn induces a pull-back $K$-morphism ${{\rm Jac}}({{\mathfrak{R}}}(B_L)) \to {{\rm Jac}}({{\mathfrak{R}}}(C))$. Furthermore, from the inclusion $\iota \colon {{\rm Res}}_h(C) \subset {{\mathfrak{R}}}(C)$ we obtain a sequence of $K$-morphisms $${{\rm Jac}}({{\mathfrak{R}}}(B_L)) \longrightarrow {{\rm Jac}}({{\mathfrak{R}}}(C)) \stackrel{\iota^}{\longrightarrow} {{\rm Jac}}({{\rm Res}}_h(C)) .$$ From the representability of ${{\mathfrak{R}}}(B_L)$, there is a $K$-morphism $\kappa \colon B \to {{\mathfrak{R}}}(B_L)$ associated to the identity $B_L \to B_L$ using the bijection ${{\rm Hom}}_K(B,{{\mathfrak{R}}}(B_L)) = {{\rm Hom}}_L(B_L,B_L)$. The morphism $\kappa$ induces a pull-back $K$-morphism $\kappa^ \colon {{\rm Jac}}({{\mathfrak{R}}}(B_L)) \to {{\rm Jac}}(B)$; we denote by $M$ the kernel of the morphism of $\kappa^$. Geometrically, the Jacobian of ${{\mathfrak{R}}}(B_L)$ is isomorphic to the product of $n$ copies of the Jacobian of $B$ and the morphism $\kappa^$ corresponds to the addition of the line bundles in the various components using the isomorphisms between them (defined over an algebraic closure of $K$). We obtain that the group $M$ is the specialization to our setting of the group denoted also by $M$ in Theorem \[thm:jaco\]. Therefore $M$ is geometrically isomorphic to ${{\rm Jac}}(B)^{n-1}$, and it is connected of dimension $(n-1) g(B)$. Thus we obtain the diagram $$\xymatrix{ M \ar@{^(->}[d] \ar[r] & {{\rm Jac}}({{\mathfrak{R}}}(C)) \ar[r]^{\iota^} & {{\rm Jac}}({{\rm Res}}_h(C)) \\ {{\rm Jac}}({{\mathfrak{R}}}(B_L)) \ar[d]^{\kappa^} \ar[ur] \\ {{\rm Jac}}(B) }$$ of $K$-morphisms and it follows from Theorem \[thm:jaco\] that the group $M$ has finite index in $\ker(\iota^)$. We let $F$ be the connected component of the identity of the image of $\iota^$ and we show that it has the required properties. First of all, $F$ is an abelian variety over $K$, isogenous over $K$ to ${{\rm Jac}}({{\mathfrak{R}}}(C))/M$, and hence the dimension of $F$ is $$\dim (F) = \dim \left( \frac{{{\rm Jac}}({{\mathfrak{R}}}(C))}{M} \right) = n g(C) - (n-1) g(B)$$ as needed. Next, we prove the statement about the Mordell-Weil rank of $F$. For a curve $D$ defined over $L$ we have $$\begin{aligned} mw_K \bigl( {{\mathfrak{R}}}(D) \bigr) & = & {{\rm rk}}\Bigl( {{\rm Jac}}\bigl( {{\mathfrak{R}}}(D) \bigr) (K) \Bigr) \\ & = & {{\rm rk}}\biggl( {{\rm Hom}}_K \Bigl( {{\rm Spec}}(K) , {{\rm Jac}}\bigl( {{\mathfrak{R}}}(D) \bigr) \Bigr) \biggr) \\ & = & {{\rm rk}}\biggl( {{\rm Hom}}_K \Bigl( {{\rm Spec}}(K) , {{\mathfrak{R}}}\bigl( {{\rm Jac}}(D) \bigr) \Bigr) \biggr) \\ & = & {{\rm rk}}\Bigl( {{\rm Hom}}_L \bigl( {{\rm Spec}}(L) , {{\rm Jac}}(D) \bigr) \Bigr) \\ & = & mw_L(D) .\end{aligned}$$ Since the abelian varieties $F$ and ${{\rm Jac}}({{\mathfrak{R}}}(C))/M$ are $K$-isogenous, the ranks of their Mordell-Weil groups are the same. By the previous computation we conclude that $$\begin{aligned} {{\rm rk}}\bigl( F(K) \bigr) & = & {{\rm rk}}\left( \frac{{{\rm Jac}}\bigl( {{\mathfrak{R}}}(C) \bigr)}{M} (K) \right) \\ & = & mw_K \bigl( {{\mathfrak{R}}}(C) \bigr) - \Bigl( mw_K \bigl( {{\mathfrak{R}}}(B_L) \bigr) - mw_K(B) \Bigr) \\ & = & mw_L(C) - \bigl( mw_L (B_L) - mw_K(B) \bigr) ,\end{aligned}$$ as required, and the result follows. The theorem just proved opens the way to applications of the Chabauty method to find the $L$-rational points of the curve $C$ with $K$-rational image in $B$. We show how this method works on an example. Let $d$ be a squarefree integer; we let $K = \mathbb{Q}$ and $L = \mathbb{Q}(\sqrt{d})$. Denote by $g(x) \in \mathbb{Q}[x]$ the polynomial $$g(x) = x^3 + a x + b$$ and by $f(x) \in \mathbb{Q}(\sqrt{d})[x]$ the polynomial $$f(x) = g(x^2+\sqrt{d}) .$$ Let $E$ be the elliptic curve over $\mathbb{Q}$ with Weierstrass equation $y^2=g(x)$ and let $C$ be the smooth projective model over $\mathbb{Q}(\sqrt{d})$ of the genus two hyperelliptic curve with affine equation $y^2=f(x)$. By construction, there is a morphism $\phi \colon C \to E$ given by $$\begin{array}{rcl} \phi \colon C & \longrightarrow & E \\[5pt] (x,y) & \longmapsto & ( x^2 + \sqrt{d} , y ). \end{array}$$ Suppose we wish to find all points $P$ in $C\bigl( \mathbb{Q}(\sqrt{d}) \bigr) $ such that $\phi(P)$ is in $E(\mathbb{Q})$. Such points are the rational points of the curve $D = {{\rm Res}}_\phi(C)$ over $\mathbb{Q}$, for which we now determine an explicit model. Let $x = x_1 + x_2 \sqrt{d}$ and $y = y_1 + y_2 \sqrt{d}$, where $ x_1, x_2,y_1,y_2$ are $\mathbb{Q}$-rational variables. Substituting $x = x_1+x_2\sqrt{d}$ and $y = y_1+y_2\sqrt{d}$ in the polynomial defining $C$ we find the polynomial $$r(x_1,x_2,y_1,y_2) = (y_1 + y_2 \sqrt{d})^2 - f(x_1 + x_2 \sqrt{d})$$ in $\mathbb{Q}(\sqrt{d})$, whose vanishing represents the condition that the point $P = ( x_1 + x_2 \sqrt{d}, y_1 + y_2 \sqrt{d})$ lies on $C$. Define polynomials with rational coefficients $$r_1 := \frac{r + \overline{r}}{2} \quad {\textrm{and}} \quad r_2 := \frac{r - \overline{r}}{2 \sqrt{d}}$$ where $\overline{r}$ denotes the polynomial obtained from $r$ by applying the nontrivial element of the Galois group of $\mathbb{Q}(\sqrt{d})/\mathbb{Q}$; we have the identity $r = r_1 + \sqrt{d} r_2$. Thus, the two equations $r_1=r_2=0$ in $ x_1,y_1, x_2,y_2$ correspond to $P$ lying on $C$; these are also the equations defining (an affine model of) the Weil restriction of $C$ from $\mathbb{Q}(\sqrt{d})$ to $\mathbb{Q}$. To determine $D = {{\rm Res}}_\phi(C)$, we also wish $\phi(P)$ to be in $E(\mathbb{Q})$. Note that the coordinates of $\phi(P)$ are $$\phi(P) = \bigl( x_1^2 + d x_2^2 + \sqrt{d} (2 x_1 x_2 + 1) , y_1+\sqrt{d} y_2 \bigr)$$ and the condition that the point $\phi(P)$ lies in $\mathbb{Q}$ translates to the equations $2x_1 x_2 + 1 = y_2 = 0$. We have therefore obtained the four equations $$\label{rima} \left\{ \begin{array}{rcl} y_1^2 & = & (x_1^6 + d^3 x_2^6) + 3(x_1^2 + d x_2^2) \bigl( 5 d x_1^2 x_2^2 + 4 d x_1 x_2 + d + \frac{a}{3} \bigr) + b \\[7pt] 0 & = & (2 x_1 x_2+1) (3 x_1^4+10 x_1^2 x_2^2 d+4 x_1 x_2 d +3 x_2^4 d^2+a+d) \\[7pt] 0 & = & 2 x_1 x_2+1 \\[5pt] y_2 & = & 0 \end{array} \right.$$ in the variables $ x_1,y_1, x_2,y_2$. But of course the second equation is divisible by the third equation, and we may ignore it (this is not a coincidence, but it is a consequence of the fact that the curve $E$ is defined over $\mathbb{Q}$). Multiplying the first equation in by $2^{12}x_1^6$, we can use the relation $2 x_1 x_2=-1$ to eliminate the variable $x_2$, obtaining the single equation $(2^6x_1^3 y_1)^2 = \overline{\rho} (x_1)$ in $x_1, y_1$ for $D$. After the birational substitution $x=2x_1$ and $y=2^6x_1^3 y_1$, we obtain that the curve $D$ is birational to the genus 5 curve with equation $$D \colon y^2 = x^{12} + 4(3 d + 4 a) x^8 + 64 b x^6 + 16d (3 d + 4 a) x^4 + 64 d^3 .$$ The curve $D$ admits the non-constant map $(x,y) \mapsto (x^2,y)$ to the genus 2 curve $F$ with equation $$F \colon y^2 = x^6 + 4(3 d + 4 a) x^4 + 64 b x^3 + 16d (3 d + 4 a) x^2 + 64 d^3.$$ Summarizing, the curve $D$ is a genus 5 curve defined over $\mathbb{Q}$ and it admits two morphisms defined over $\mathbb{Q}(\sqrt{d})$ to the curves $$C \colon y^2=f(x) = g(x^2+\sqrt{d}) \quad {\textrm{ and }} \quad \overline{C} \colon y^2=\overline{f}(x) = g(x^2-\sqrt{d}) .$$ Each of the two curves $C$ and $\overline{C}$ admits a $\mathbb{Q}(\sqrt{d})$-morphism to the elliptic curve $E$, and the corresponding two compositions $D \to E$ coincide, and are defined over $\mathbb{Q}$. We deduce that the Jacobians of $C$ and of $\overline{C}$ are contained, up to isogeny, in the Jacobian of $D$, and they have an isogenous copy of $E$ in common. This implies that the five-dimensional Jacobian of $D$ contains a further two-dimensional abelian variety: up to isogeny this is the Jacobian of the curve $F$. It is now easy to provide examples where the inequality of the Chabauty method is not satisfied for the curve $C$, nor for the curve $E$, but it satisfied for the curve $F$, so that we can still apply the Chabauty method to find the points on $D$. For example, setting $a = 1$, $b = 3$, and $d = 13$, we find ${{\rm rk}}\bigl( E(\mathbb{Q}) \bigr) = 1$ and ${{\rm rk}}\bigl( {{\rm Jac}}(F)(\mathbb{Q}) \bigr) = 1$, and moreover $${{\rm rk}}\left( {{\rm Jac}}(C) \bigl( \mathbb{Q}(\sqrt{d}) \bigr) \right) \geq {{\rm rk}}\left( E \bigl( \mathbb{Q}(\sqrt{d}) \bigr) \right) \geq 2 .$$ Thus, in this example the Chabauty method is not applicable to $C$ or $E$, but it is only applicable to $F(\mathbb{Q})$. Cases where Faltings’ Theorem does not apply ============================================ In this section we analyze the cases where the relative Weil restriction of a morphism of curves contains a component of geometric genus at most one. In such cases, Faltings’ Theorem cannot be applied to deduce the finiteness of rational points of the relative Weil restriction and we find explicit non-tautological examples in which these sets of rational points are infinite. The following remark is an immediate consequence of Faltings’ Theorem and guides the choice of cases we handle in this section. \[gamu\] Let $L/K$ be an extension of number fields, and suppose that $C$ is a curve defined over $L$, $B$ is a curve defined over $K$ and $h \colon C \to B$ is a non-constant morphism. If the curve ${{\rm Res}}_h(C)$ has infinitely many $K$-rational points, then the genus of $C$ is at most one. In the case of the relative Weil restriction of a morphism from a curve of genus one, we completely characterize the cases where the set of rational points is not finite. \[p:quo\] Let $C , X_1 , \ldots , X_n , B$ be smooth projective curves (not necessarily connected), and let $$\xymatrix{ & C \ar[dl] _{r_1} \ar[dr] ^{r_n} \\ X_1 \ar[dr] & \cdots & X_n \ar[dl] \\ & B }$$ be a commutative diagram, where all morphisms are finite and flat. There is a smooth projective curve $X_C$ and a commutative diagram $$\xymatrix{ & C \ar[dl] _{r_1} \ar[dr] ^{r_n} \\ X_1 \ar[dr] & \cdots & X_n \ar[dl] \\ & X_C }$$ such that for every projective curve $D$ fitting in the commutative diagram of solid arrows $$\xymatrix{ & C \ar[dl] _{r_1} \ar[dr] ^{r_n} \\ X_1 \ar[dr] \ar[ddr] & \cdots & X_n \ar[dl] \ar[ddl] \\ & X_C \ar@{-->}[d] \\ & D }$$ the dashed arrow $\xymatrix{X_C \ar@{-->}[r] & D}$ exists uniquely. Let $X$ be the disjoint union $X := X_1 \sqcup \cdots \sqcup X_n$. The morphisms $r_1,\ldots,r_n$ determine an equivalence relation $\sim_C$ on $X$, where $x \sim_C y$ if there is a sequence $(i_1 , \ldots , i_t)$ of elements of $\{1,\ldots,n\}$, and points $x_1 = x \in X_{i_1}$, $x_2 \in X_{i_2}$, …, $x_{t-1} \in X_{i_{t-1}}$, $x_t = y \in X_{i_t}$ and a sequence $c_1 , \ldots , c_{t-1}$ of points of $C$ such that for all $j \in \{1,\ldots,t-1\}$ we have $r_{i_j} (c_i) = r_{i_{j+1}} (c_i)$. Thus, a pair $(x,y) \in X \times X$ is in the relation determined by the morphisms $r_1,\ldots,r_n$ if and only if there is a sequence $I := (i_1 , \ldots , i_t)$ of elements of $\{1,\ldots,n\}$ such that the pair $(x,y)$ is in the image of the composition $$C \times _{X_{i_2}} \times \cdots \times _{X_{i_{t-1}}} C \stackrel{\pi}{\longrightarrow} C \times C \stackrel{(r_{i_1},r_{i_t})}{\longrightarrow} X_{i_1} \times X_{i_t} \subset X \times X$$ where $\pi$ is the projection to the first and last factor; denote by $C_I \subset X \times X$ the image of this morphism. Observe that, for every finite sequence $I$ of elements of $\{1 , \ldots , n\}$, the scheme $C_I$ is a closed subscheme of $X \times X$ that is finite and flat over each factor $X$ and hence also over $B$; in particular, each scheme $C_I$ has non-zero degree over $B$. Moreover, if the pair $(x,y)$ is in the relation $\sim_C$, then $x$ and $y$ have the same image in $B$. From this it follows that pairs $(x,y)$ in the relation $\sim_C$ are covered by at most $\bigl( \sum _i \deg(f_i) \bigr)^2$ of the schemes $C_I$ defined above since they are contained in $X \times _B X$. It follows that the whole graph $R_C \subset X \times X$ of the relation $\sim_C$ is a subscheme of finite type of $X \times X$ that is flat and proper over each factor. The hypotheses of [@SGA3], Théorème V.7.1, are therefore satisfied and the result follows. \[coge1\] With the notation of the previous proposition, assume further that the curves $C,X_1 , \ldots , X_n$ all have genus one. Then the curve $X_C$ is a torsor under ${{\rm Jac}}(C)/K$, where $K$ is the subgroup of ${{\rm Jac}}(C)$ generated by the kernels of the morphisms ${{\rm Jac}}(C) \to {{\rm Jac}}(X_1)$, …, ${{\rm Jac}}(C) \to {{\rm Jac}}(X_n)$. We can clearly assume that the ground field is algebraically closed, and further reduce to the case in which the morphisms $r_1, \ldots , r_n$ are all homomorphisms of elliptic curves. Thus $K \subset C$ is identified with the subgroup generated by the kernels of all the morphisms $r_1, \ldots , r_n$ and let $C' := C/K$ denote the quotient of $C$ by the subgroup $K$. Clearly, the curves $X_1 , \ldots , X_n$ all admit a morphism to $C'$ making the diagram of Proposition \[p:quo\] commute. It follows from the previous proposition that $X_C$ admits a morphism to $C'$ that is necessarily non-constant, and we conclude that $X_C$ has genus one. Moreover, it is also clear that the curve $X_C$ is isomorphic to the curve $C'$: if $D$ is any curve making the diagram of Proposition \[p:quo\] commute, then the fiber in $C$ of the morphism $C \to D$ over the image of the origin in $C$ contains all the kernels of the morphisms $r_1 , \ldots , r_n$, and hence it contains the subgroup $K$, since the morphism factors through the elliptic curve $X_C$. Thus we see that if the curves $X_1,\ldots,X_n$ have genus one and if the fibered product $X_1 \times _B \cdots \times _B X_n$ contains a geometrically integral curve of geometric genus one defined over the ground field, then the morphism $X_1 \to B$ factors through a morphism $E \to B$ defined over the ground field, where $E$ is a smooth geometrically integral curve of geometric genus one and the morphism $X_1 \to E$ is an isogeny defined over the extension field. Genus one --------- We specialize what we just proved to the case of the relative Weil restriction from an elliptic curve. Let $L/K$ be an extension of number fields; let $E$ be an elliptic curve defined $L$ and let $B$ be a smooth projective integral curve defined over $K$. Suppose that $h \colon E \to B_L$ is a non-constant morphism. The relative Weil restriction ${{\rm Res}}_h(E)$ is a curve defined over $K$. Fix an algebraic closure $\overline{K}$ of $K$, denote by $\sigma_1 , \ldots , \sigma_n$ the distinct embeddings of $L/K$ in $\overline{K}$, and let $E_1 , \ldots , E_n$ be the corresponding Galois conjugates of $E$. Over the field $\overline{K}$, the curve ${{\rm Res}}_h(E)$ is isomorphic to $E_1 \times _B \cdots \times_B E_n$ (Lemma \[profi\]). Suppose that the curve ${{\rm Res}}_h(E)$ contains infinitely many $K$-rational points. It follows from Faltings’ Theorem that there is a component $C$ of ${{\rm Res}}_h(E)$ of geometric genus at most one, defined over $K$; if $C$ is not normal, we replace it by its normalization. Since the morphism ${{\rm Res}}_h(E) \to B$ is flat, all its fibers are finite and therefore the curve $C$ is also finite over $B$. In particular, the $L$-morphisms $C \to E_1 , \ldots , C \to E_n$ are all finite, since all the curves are smooth. Let $E_C$ denote the universal curve fitting in the diagram $$\xymatrix{ & C \ar[dl] _{r_1} \ar[dr] ^{r_n} \\ E_1 \ar[dr] & \cdots & E_n \ar[dl] \\ & E_C }$$ of Proposition \[p:quo\]. Since the curves $C,E_1,\ldots,E_n$ all have genus one, we may therefore apply Corollary \[coge1\] to deduce that the curve $E_C$ is also a torsor under an elliptic curve, and therefore also $E_C$ has genus one. In the case in which $L$ is a number field, we obtain the following corollary. Let $L/K$ be an extension of number fields and suppose that $E$ is an elliptic curve over $L$, $B$ is a curve over $K$, and $h \colon E \to B_L$ is a non-constant morphism. If the set of $K$-rational points of ${{\rm Res}}_h(E)$ is infinite, then the curve $E$ is $L$-isogenous to an elliptic curve defined over $K$ having positive rank over $K$. By Faltings’ Theorem we deduce that ${{\rm Res}}_h(E)$ contains a geometrically integral component $E'$ of genus at most one defined over $K$ and having infinitely many $K$-rational points. Since $E'$ admits a non-constant $L$-morphism to $E$, it follows that $E'$ has genus one and that it is $L$-isogenous to $E$, as required. In the case in which $E$ and $B$ have genus one, then all the geometric components of ${{\rm Res}}_h(E)$ have genus one. Hence, finding the $K$-rational points of ${{\rm Res}}_h(E)$ is equivalent to finding the $K$-rational points of finitely many elliptic curves that are $K$-isogenous to $B$. This completes our analysis in the case in which the curve $C$ of Remark \[gamu\] has genus one. We next discuss the case in which $C$ has genus zero. We are not able to give a treatment of this case that is as detailed as the case of genus one. Genus zero ---------- We specialize to the case in which the curve $C$ is isomorphic to $\mathbb{P}^1_L$ and hence $B$ is isomorphic to $\mathbb{P}^1_K$. The morphism $h \colon \mathbb{P}^1_L \to \mathbb{P}^1_L = (\mathbb{P}^1_K)_L$ is therefore determined by a rational function $F \in L(x)$. The set of $K$-rational points of ${{\rm Res}}_h(\mathbb{P}^1_L)$ is essentially the set $A_F \subset L$ of values of $x \in L$ such that $F(x)$ lies in $K$, mentioned in the introduction. Let $G \colon C \to D$ be a finite morphism between smooth curves, let $p \in D$ be a geometric point. The [[type]{}]{} of $p$ is the partition $\lambda_p$ of $\deg(G)$ determined by the fiber $G^{-1}(p)$. We extend this definition to the case in which the curve $C$ is reduced, but not necessarily smooth, by replacing $G$ with the morphism $G^\nu \colon C^\nu \to D$, where $C^\nu$ is the normalization of $C$ and $G^\nu$ is the morphism induced by $G$. In this section we restrict our attention to field extensions $L \supset K$ of degree at most three. ### Degree two {#degree-two .unnumbered} Let $L \supset K$ be a field extension of degree two, let $F \colon \mathbb{P}^1_L \to \mathbb{P}^1_L$ be a morphism of degree three. We are interested in the values $\ell \in L$ such that $f(\ell) \in K$. Assume that $L = K(\sqrt{d})$ where $d \in K \setminus K^2$; write $\alpha = a + b \sqrt{d}$, for $a,b \in K$. Parameterizing the rational curve ${{\rm Res}}_F(F)$ we find that, for every element $t$ in $K$, the evaluation $$p \left( \frac{d b t^2+2 a t+b}{t (d t^2 + 3)} (\sqrt{d} t + 1) \right)$$ is also in $K$. ### Degree three {#degree-three .unnumbered} Suppose that $F \colon \mathbb{P}^1_L \to \mathbb{P}^1_L$ is a morphism of degree three, and suppose that the characteristic of $K$ is neither two nor three. As usual, we are interested in the values $\ell \in L$ such that $f(\ell) \in K$. Denote by $\overline{F}$ the morphism conjugate to $F$ under the Galois involution of $L/K$. We construct examples of morphisms $F$ such that ${{\rm Res}}_F(\mathbb{P}^1_L)$ is a geometrically integral curve of geometric genus zero. Note that the curve ${{\rm Res}}_F(\mathbb{P}^1_L)$ has a line bundle of degree nine given by pull-back of $\mathcal{O}_{\mathbb{P}^1_K}(1)$; since this curve is geometrically rational and it has a line bundle of odd degree, it follows that it is rational over $K$. Denote by ${{\rm Res}}_F(F)^\nu$ the composition of the normalization map of ${{\rm Res}}_F(\mathbb{P}^1_L)$ and ${{\rm Res}}_F(F)$. Applying the Hurwitz formula to the morphisms $F$, $\overline{F}$ and ${{\rm Res}}_F(F)^\nu$, we find that the respective total degrees of the ramification divisors are $4$, $4$ and $16$. We begin by analyzing the ramification patterns. For a geometric point $p \in \mathbb{P}^1$, Table \[fihu\] shows the possibilities of the types of the fibers of the three morphisms $F$, $\overline{F}$ and ${{\rm Res}}_F(F)^\nu$ and the contributions of each to the Hurwitz formula. In our setup, the Galois involution of $L/K$ induces a bijection between fiber types of $F$ and of $\overline{F}$: this is recorded in the last column of Table \[fihu\]. \[fihu\] ------------------------------------------- --------------------------- --------------------------------------------- ------------------ Fiber type of Fiber type of Hurwitz contribution to Symmetrized the pair $F,\overline{F}$ ${{\rm Res}}_F(F)$ $[F,\overline{F}]$ and $[{{\rm Res}}_F(F)]$ contribution \[5pt\] $\bigl( (1,1,1) , (1,1,1) \bigr)$ $(1,1,1 , 1,1,1 , 1,1,1)$ \[0,0\] , \[0\] \[0,0\] , \[0\] \[5pt\] $\bigl( (2,1) , (1,1,1) \bigr)$ $(2,2,2 , 1,1,1)$ \[1,0\] , \[3\] \[1,1\] , \[6\] \[5pt\] $\bigl( (3) , (1,1,1) \bigr)$ $(3,3,3)$ \[2,0\] , \[6\] \[2,2\] , \[12\] \[5pt\] $\bigl( (2,1) , (2,1) \bigr)$ $(2,2,2,2,1)$ \[1,1\] , \[4\] \[1,1\] , \[4\] \[5pt\] $\bigl( (2,1) , (3) \bigr)$ $(6,3)$ \[1,2\] , \[7\] \[3,3\] , \[14\] \[5pt\] $\bigl( (3) , (3) \bigr)$ $(3,3,3)$ \[2,2\] , \[6\] \[2,2\] , \[6\] \[5pt\] ------------------------------------------- --------------------------- --------------------------------------------- ------------------ : Fiber types and contributions to the Hurwitz formula for fiber products of morphisms of degree three It is now easy to check that the only possibilities for the fiber types of the morphisms $F,\overline{F}$ are $$\bigl( (3) , (3) \bigr) + \bigl( (2,1) , (1,1,1) \bigr) + \bigl( (1,1,1) , (2,1) \bigr) + \bigl( (2,1) , (2,1) \bigr) \label{pollo}$$ and $$\bigl( (2,1) , (2,1) \bigr) + \bigl( (2,1) , (2,1) \bigr) + \bigl( (2,1) , (2,1) \bigr) + \bigl( (2,1) , (2,1) \bigr) . \label{simme}$$ We only analyze the case . ### Fiber types {#fiber-types .unnumbered} The fiber type $((3),(3))$ in implies that the coordinate on $\mathbb{P}^1$ can be chosen so that the morphism $F$ is a polynomial and the fiber type $((2,1) , (2,1))$ shows that one of the ramification points in defined over $K$. This case is realized by morphisms $F \colon \mathbb{P}^1_L \to \mathbb{P}^1_L$ of the form $p(x) = x^2(x-\alpha)$, for $\alpha \in L$. Since the derivative of $p$ vanishes at $0$ and at $\frac{2\alpha}{3}$, it follows that the ramification types of $F,\overline{F}$ are of the form when $\alpha \notin K$.	ArXiv
--- abstract: 'The maximality property was introduced in [@T:WS95] in orthomodular posets as a common generalization of orthomodular lattices and orthocomplete orthomodular posets. We show that various conditions used in the theory of effect algebras are stronger than the maximality property, clear up the connections between them and show some consequences of these conditions. In particular, we prove that a Jauch–Piron effect algebra with a countable unital set of states is an orthomodular lattice and that a unital set of Jauch–Piron states on an effect algebra with the maximality property is strongly order determining.' author: - \| Josef Tkadlec\ Department of Mathematics, Faculty of Electrical Engineering,\ Czech Technical University, 16627 Praha, Czech Republic,\ [email protected] title: Effect algebras with the maximality property --- Basic notions ============= Effect algebras as generalizations of orthomodular posets (quantum logics) are studied in the axiomatics of quantum systems—see, e.g., [@DP:NewTrends; @FB:Effect]. An effect algebra is an algebraic structure $(E,\oplus,\0,\1)$ such that $E$ is a set, $\0$ and $\1$ are different elements of $E$ and $\oplus$ is a partial binary operation on $E$ such that for every $a,b,c \in E$ the following conditions hold: $a \oplus b = b \oplus a$ if $a \oplus b$ exists, $(a \oplus b) \oplus c = a \oplus (b \oplus c)$ if $(a \oplus b) \oplus c$ exists, there is a unique $a'\in E$ such that $a \oplus a' = \1$ (orthosupplement), $a=\0$ whenever $a \oplus \1$ is defined. For simplicity, we use the notation $E$ for an effect algebra. A partial ordering on an effect algebra $E$ is defined by $a \le b$ iff there is a $c \in E$ such that $b = a \oplus c$. Such an element $c$ is unique (if it exists) and is denoted by $b \ominus a$. $\0$ ($\1$, resp.) is the least (the greatest, resp.) element of $E$ with respect to this partial ordering. For every $a,b \in E$, $a''=a$ and $b' \le a'$ whenever $a \le b$. It can be shown that $a \oplus \0 = a$ for every $a \in E$ and that a cancellation law is valid: for every $a,b,c \in E$ with $a \oplus b \le a \oplus c$ we have $b \le c$. An orthogonality relation on $E$ is defined by $a \perp b$ iff $a \oplus b$ exists (iff $a \le b'$). (See, e.g., [@DP:NewTrends; @FB:Effect].) For $a \le b$ we denote $[a,b] = \{c \in E \st a \le c \le b \}$. A chain in $E$ is a nonempty linearly (totally) ordered subset of $E$. Obviously, if $a \perp b$ and $a \lor b$ exist in an effect algebra, then $a \lor b \le a \oplus b$. The reverse inequality need not be true (it holds in orthomodular posets). Let $E$ be an effect algebra. An element $a \in E$ is principal if $b \oplus c \le a$ for every $b,c \in E$ such that $b,c \le a$ and $b \perp c$. An orthoalgebra is an effect algebra $E$ in which, for every $a \in E$, $a=\0$ whenever $a \oplus a$ is defined. An orthomodular poset is an effect algebra in which every element is principal. An orthomodular lattice is an orthomodular poset that is a lattice. Every orthomodular poset is an orthoalgebra. Indeed, if $a \oplus a$ is defined then $a \oplus a \le a = a \oplus \0$ and, accordig to the cancellation law, $a \le \0$ and therefore $a=\0$. Orthoalgebras are characterized by the following conditions: the orthosupplementation is an orthocomplementation (i.e., $a \lor a' = \1$ for every $a$) or $a \oplus b$ is a minimal upper bound of $a,b$ for every $a,b$. Orthomodular posets are characterized as effect algebras such that $a \oplus b = a \lor b$ for every orthogonal pair $a,b$. (See [@FB:Effect; @FGR:Filters].) Let us remark that an orthomodular poset is usually defined as a bounded partially ordered set with an orthocomplementation in which the orthomodular law is valid. Let us present a special class of orthomodular posets that we will use in some examples. \[T:concreteOMP\] Let $X \neq \emptyset$, $E \subset \exp X$ be nonempty such that the following conditions are fulfilled: $X \setminus A \in E$ whenever $A \in E$, $A \cup B \in E$ whenever $A,B \in E$ are disjoint. Then $(E,\oplus,\emptyset,X)$ with $A \oplus B = A \cup B$ for disjoint $A,B \in E$ is an orthomodular poset such that the orthosupplement is the set-theoretic complement and the partial ordering is the inclusion. Since $E$ is nonempty, there is an element $A \in E$. According to the condition (1), $X \setminus A \in E$. According to the condition (2), $X = A \cup (X \setminus A) \in E$ and, according to the condition (1), $\emptyset = X \setminus X \in E$. It is easy to see that the axioms of an effect algebra are fulfilled, that the orthosupplement is the set-theoretic complement, that the partial ordering is the inclusion and that every element of $E$ is principal. An orthomodular poset of the form of \[T:concreteOMP\] is called concrete. Let us present two important notions we will use in the sequel. A system $(a_i)_{i \in I}$ of (not necessarilly distinct) elements of an effect algebra $E$ is orthogonal if $\bigoplus_{i \in F} a_i$ is defined for every finite set $F \subset I$. An effect algebra $E$ is orthocomplete if for every orthogonal system $(a_i)_{i \in I}$ of elements of $E$ the supremum $\bigvee \{\bigoplus_{i\in F} a_i \st F \subset I \ \text {is finite}\}$ exists. An effect algebra $E$ has the maximality property if $\ab$ has a maximal element for every $a,b \in E$. Obviously, every finite effect algebra has the maximality property and every lattice effect algebra has the maximality property—$a \land b$ is a maximal (even the greatest) element of $\ab$ for every $a,b$. States ====== Let $E$ be an effect algebra. A state $s$ on $E$ is a mapping $s \st E \to [0,1]$ such that: $s(\1)=1$, $s(a \oplus b) = s(a) + s(b)$ whenever $a \oplus b$ is defined. A set $S$ of states on $E$ is unital, if for every $a \in E \setminus \{\0\}$ there is a state $s \in S$ such that $s(a) = 1$. A set $S$ of states on $E$ is strongly order determining, if for every $a,b \in E$ with $a \not\le b$ there is a state $s \in S$ such that $s(a) = 1 > s(b)$. Obviously, for every state $s$ we have $s(\0)=0$, $s(a')=1-s(a)$ for every $a \in E$, $s(a) \le s(b)$ for every $a,b \in E$ with $a \le b$. There are special two-valued states on concrete orthomodular posets (it is easy to verify that they are indeed states): Let $E \subset \exp X$ be a a concrete orthomodular poset, $x \in X$. The state $s_x$ on $E$ defined by $$s_x(A) = \begin {cases} 0 \,,& x \notin A \,,\\ 1 \,,& x \in A \,, \end {cases} \qquad A \in E \,,$$ is called carried by the point $x$. It is easy to see that for concrete orthomodular posets the set of states carried by points is strongly order determining. \[L:StronglyFull-Unital\] Every strongly order determining set of states on an effect algebra is unital. Let $S$ be a strongly order determining set of states on an effect algebra $E$, $a$ be a nonzero element of $E$. Then $a \not\le \0$ and therefore there is a state $s \in S$ such that $s(a) = 1 > s(\0)$. Let us present two observations describing the impact of a sufficiently large state spaces to the properties of the algebraic structure. \[T:unital-OA\] Every effect algebra with a unital set of states is an orthoalgebra. Let $E$ be an effect algebra with a unital set $S$ of states. Let $a \in E$ be such that $a \oplus a$ is defined. Then $1 \ge s (a \oplus a) = 2 \, s(a)$ and therefore $s(a) < \frac 12$ for every state $s \in S$. Since $S$ is unital, we obtain that $a=\0$. \[T:SOD->OMP\] Every effect algebra with a strongly order determining set of states is an orthomodular poset. Let $E$ be an effect algebra with a strongly order determining set $S$ of states. Let us prove that every element of $E$ is principal. Let $a,b,c \in E$ such that $b,c \le a$ and $b \perp c$. Then for every state $s \in S$ with $s(a')=1$ we consecutively obtain: $0 = s(a) = s(b) = s(c) = s(b\oplus c)$, $s\bigl((b \oplus c)'\bigr) = 1$. Since the set $S$ is strongly order determining, we obtain that $a' \le (b \oplus c)'$ and therefore $b \oplus c \le a$. Jauch–Pironness =============== Let $E$ be an effect algebra. A state $s$ on $E$ is Jauch–Piron if for every $a,b \in E$ with $s(a)=s(b)=1$ there is a $c \in E$ such that $c \le a,b$ and $s(c)=1$. An effect algebra is Jauch–Piron if every state on it is Jauch–Piron. The following statement was proved in [@T:CEOEA Proposition 2.6], we will generalize it later (\[T:JPCU->OML\]). \[T:JPCU->M\] Every Jauch–Piron effect algebra with a countable unital set of states has the maximality property. Let $E$ be a Jauch–Piron effect algebra with a countable unital set $S$ of states. Let $a,b \in E$. If $\ab = \{\0\}$ then $\0$ is a maximal element of $\ab$. Let us suppose that $\ab \neq \{\0\}$. Then there is an element $c \in \ab \setminus \{\0\}$ and, since the set $S$ is unital, there is a state $s \in S$ such that $s(c)=1$. Hence $s(a)=s(b)=1$ and the set $S_{a,b} = \{ s \in S \st s(a)=s(b)=1 \}$ is nonempty and countable. Let $s_0$ be a $\sigma$-convex combination (with nonzero coefficients) of all states from $S_{a,b}$. Then $s_0(a) = s_0(b) = 1$. Since the state $s_0$ is Jauch–Piron, there is an element $c \in \ab$ such that $s_0(c)=1$. It remains to prove that $c$ is a maximal element of $\ab$. Indeed, if $d \in \ab$ with $d \ge c$ then $e = d \ominus c \in \ab$ and $e \perp c$. Hence $s_0(e) = 0$ and therefore there is no state $s\in S$ such that $s(e)=1$. Due to the unitality of $S$, $e=0$ and therefore $d=c$. \[T:UJP->OMP\] Every effect algebra with the maximality property and with a unital set of Jauch–Piron states is an orthomodular poset. Let $E$ be an effect algebra with the maximality property and with a unital set $S$ of Jauch–Piron states. Let us suppose that $E$ is not an orthomodular poset and seek a contradiction. There are elements $a,b,c \in E$ such that $b, c \le a$, $b \perp c$ and $b \oplus c \not\le a$. Let us denote $d = b \oplus c$. Since $E$ has the maximality property, there is a maximal element $e$ in $[\0,a'] \cap [\0,d']$. Since $d \not\le a$, we obtain that $a' \not\le d'$ and therefore $e < a'$ and $a' \ominus e \neq \0$. Since the set $S$ is unital, there is a state $s \in S$ such that $s(a' \ominus e) = 1$. Hence $s(a')=1$, $0 = s(e) = s(a) = s(b) = s(c) = s(d)$, $s(d') = 1$, $s(d'\ominus e) = 1$. Since the state $s$ is Jauch–Piron, there is an element $f \in E$ such that $f \le (a' \ominus e), (d' \ominus e)$ and $s(f) = 1$. Hence $f \neq \0$ and $e < e \oplus f \le a',d'$—this contradicts to the maximality of $e$. Let us remark that there are effect algebras with the maximality property that are not orthoalgebras—e.g., the 3-chain $C_3 = \{\0,a,\1\}$ with $a \oplus a = \1$ and $x \oplus \0 = x$ for every $x \in C_3$. It seems to be an open question whether the assumption of the maximality property in \[T:UJP->OMP\] might be omitted (it is not a consequence of the existence of a countable unital set of Jauch–Piron states—see \[E:OMPnotM\].) \[T:UJP->OMP\] cannot be improved to orthomodular lattices—see \[E:UnotSOD\] ($\{\frac12\,(s_x+s_y)\st x,y \in X, \ x \neq y\}$ is a unital set of Jauch–Piron states). It is well-known and easy to see that every state on a Boolean algebra is Jauch–Piron and that a unital set of states on a Boolean algebra is strongly order determining. Let us generalize the latter statement. \[T:Unital-StronglyFull\] A set of Jauch–Piron states on an effect algebra with the maximality property is unital if and only if it is strongly order determining. $\Leftarrow$: See \[L:StronglyFull-Unital\]. $\Rightarrow$: Let $E$ be an effect algebra with the maximality property and with a unital set $S$ of Jauch–Piron states. Let $a,b \in E$ such that $a \not\le b$. Let $c \in E$ be a maximal element of $\ab$. Then $c<a$ and therefore $a \ominus c \neq \0$. Since the set $S$ is unital, there is a state $s \in S$ such that $s(a \ominus c) = 1$ and therefore $s(a)=1$. Let us suppose that $s(b)=1$ and seek a contradiction. Since $s$ Jauch–Piron, there is an element $d \in E$ such that $d \le a \ominus c$, $d \le b$ and $s(d)=1$. Hence $d \neq \0$ and $c < c \oplus d \le a$. According to \[T:UJP->OMP\], $b$ is principal and therefore $c \oplus d \le b$—this contradicts to the maximality of $c$. Let us remark that \[T:UJP->OMP\] is a consequence of \[T:Unital-StronglyFull\] and \[T:SOD->OMP\]. Let us present examples that the assumptions in \[T:Unital-StronglyFull\] cannot be omitted. \[E:UnotSOD\] Let $X=\{a,b,c,d\}$, $E$ be the family of even-element subsets of $X$ with the $\oplus$ operation defined as the union of disjoint sets. Then $(E,\oplus,\emptyset,X)$ is a finite (hence with the maximality property) concrete orthomodular poset and the set $S = \{s_a, s_b, s_c\}$ of states carried by points $a,b,c$ is a unital set of (two-valued) states on $E$ that is not strongly order determining: $\{a,d\} \not\le \{a,b\}$ but there is no state $s \in S$ such that $s(\{a,d\}) = 1 > s(\{a,b\})$. (States in $S$ are not Jauch–Piron.) \[E:OMP-UnotSOD\] Let $X_1,X_2,X_3,X_4$ be nonempty mutually disjoint sets, $X_1, X_3$ be infinite, $X = \bigcup_{i=1}^4 X_i$, $$\begin{aligned} E_0 &= \{\emptyset, X_1 \cup X_2, X_2 \cup X_3, X_3 \cup X_4, X_4 \cup X_1, X\}\,,\\ E &= \{(A \setminus F) \cup (F \setminus A) \st F \subset X_1 \cup X_3 \ \text {is finite},\ A \in E_0 \}\,, \end{aligned}$$ $A \oplus B = A \cup B$ for disjoint $A,B \in E$. Then $(E,\oplus,\emptyset,X)$ is a concrete orthomodular poset and the set $S=\{s_x \st x \in X_1 \cup X_3\}$ of states carried by points from $X_1 \cup X_3$ is a unital set of (two-valued) Jauch–Piron states on $E$. The set $S$ is not strongly order determining because $X_1 \cup X_4 \not\le X_1 \cup X_2$ and for every $s \in S$ with $s(X_1 \cup X_4) = 1$ there is an $x \in X_1$ such that $s = s_x$ and therefore $s(X_1 \cup X_2) = 1$. ($E$ does not have the maximality property.) \[T:JPCU->OML\] Every Jauch–Piron effect algebra with a countable unital set of states is an orthomodular lattice. Let $E$ be a Jauch–Piron effect algebra with a countable unital set $S$ of states. According to \[T:JPCU->M\], $E$ has the maximality property. According to \[T:Unital-StronglyFull\], the set $S$ is strongly order determining. According to \[T:SOD->OMP\], $E$ is an orthomodular poset. Let us show that $a \land b$ exists for every $a,b \in E$. (Then also $a \lor b = (a' \land b')'$ exists for every $a,b \in E$.) If $\ab=\{\0\}$ then $\0 = a \land b$. Let us suppose that there is a nonzero element $c \in E$ such that $c \le a,b$. Then there is a state $s \in S$ such that $s(c)=1$. Hence $s(a)=s(b)=1$ and the set $S_{a,b} = \{ s \in S \st s(a)=s(b)=1 \}$ is nonempty and countable. Let $s_0$ be a $\sigma$-convex combination (with nonzero coefficients) of all states from $S_{a,b}$. Then $s_0(a)=s_0(b)=1$. Since the state $s_0$ is Jauch–Piron, there is an element $c_0 \in E$ such that $c_0 \le a,b$ and $s_0(c_0)=1$. Hence $s(c_0)=1$ for every $s \in S_{a,b}$. For every $c \in \ab$ and every $s \in S$ with $s(c)=1$ we have $s \in S_{a,b}$ and therefore $s(c_0)=1$. Since $S$ is strongly order determining, $c \le c_0$ for every $c \in \ab$. Hence $c_0 = a \land b$. Let us present examples that the conditions in \[T:JPCU->OML\] cannot be omitted. There is a concrete (hence with a strongly order determining set of two-valued states) Jauch–Piron orthomodular poset that is not a lattice—see [@Muller] (every unital set of states on it is uncountable). As the following example shows there is an orthomodular poset with a countable strongly order determining set of (two-valued) Jauch–Piron states that does not have the maximality property and therefore it is not a lattice (there are non-Jauch–Piron states). \[E:OMPnotM\] Let $X_1,X_2,X_3,X_4$ be mutually disjoint countable infinite sets, $X = \bigcup_{i=1}^4 X_i$, $$\begin{aligned} E_0 &= \{\emptyset, X_1 \cup X_2, X_2 \cup X_3, X_3 \cup X_4, X_4 \cup X_1, X\}\,,\\ E &= \{(A \setminus F) \cup (F \setminus A) \st F \subset X \ \text {is finite},\ A \in E_0 \}\,, \end{aligned}$$ $A \oplus B = A \cup B$ for disjoint $A,B \in E$. Then $(E, \oplus, \emptyset, X)$ is a concrete orthomodular poset and the set $S=\{s_x \st x \in X\}$ of states carried by points is a countable strongly order determining set of two-valued Jauch–Piron states on $E$. The set $[\emptyset, X_1 \cup X_2] \cap [\emptyset, X_4 \cup X_1]$ consists of finite subsets of $X_1$, hence $E$ does not have the maximality property. As an example of a non-Jauch–Piron state we can take a $\sigma$-convex combination (with nonzero coefficients) of states from $\{s_x \st x \in X_1\}$. Relationship of various conditions ================================== \[T:MaximalityProperties\] Let $E$ be an effect algebra. Consider the following poperties: $E$ is finite. $E$ is chain finite. $E$ is orthocomplete. $E$ is Jauch–Piron with a countable unital set of states. $E$ is a lattice. For every $a,b \in E$, every chain in $\ab$ has an upper bound in $\ab$. $E$ has the maximality property. Then the following implications hold: (F)(CF)(OC)(CU)(M), (JPCU)(L)(CU). (F)(CF): Obvious. (CF)(OC): Every orthogonal system in a chain finite effect algebra is finite. Hence $E$ is orthocomplete. (OC)(CU): Let $C$ be a chain in $\ab$. According to [@JP:Orthocomplete Theorem 3.2], every chain in an orthocomplete effect algebra has a supremum. This supremum obviously belongs to $\ab$. (CU)(M): Let $a,b \in E$. Since $\ab \supset \{\0\}$, the family of chains in $\ab$ is nonempty. According to Zorn’s lemma, there is a maximal chain $C$ in $\ab$. According to the assumption, there is an upper bound $c \in \ab$ of $C$. Since the chain $C$ is maximal, $c \in C$ is a maximal element of $\ab$. (JPCU)(L): See \[T:JPCU->OML\]. (L)(CU): Let $a,b \in E$. The element $a \land b$ is an upper bound for every chain in $\ab$. Let us present examples that the scheme of implications in the previous theorem cannot be improved. Let $X$ be an infinite set, $y \notin X$, $E = \{\emptyset\} \cup \bigl\{ \{x,y\} \st x \in X \bigr\} \cup \bigl\{ X \setminus \{x\} \st x \in X\} \cup \bigl\{ X \cup \{y\} \bigr\}$, $A \oplus B = A \cup B$ for disjoint $A,B \in E$. Then $(E,\oplus,\emptyset,X\cup\{y\})$ is an infinite chain finite concrete orthomodular lattice. Let $X$ be an uncountable set, $E=\exp X$ with $A \oplus B = A \cup B$ for disjoint $A,B \in E$. Then $(E,\oplus,\emptyset,X)$ is an orthocomplete concrete orthomodular lattice (it forms a Boolean algebra) such that there is an uncountable set of mutually orthogonal elements. Hence it is not chain finite and every unital set of states on $E$ is uncountable. Let $X$ be a countable infinite set. Let $E$ be a family of finite and cofinite subsets of $X$ with the $\oplus$ operation defined as the union of disjoint sets. Then $(E,\oplus,\emptyset,X)$ is a concrete orthomodular lattice (it forms a Boolean algebra) fulfilling the condition (JPCU) (every state on a Boolean algebra is Jauch–Piron, there is a countable unital set of states carried by points) that is not orthocomplete. Let $X$ be a 6-element set. Let $E$ be the family of even-element subsets of $X$ with the $\oplus$ operation defined as the union of disjoint sets from $E$. Then $(E,\oplus,\emptyset,X)$ is a finite concrete orthomodular poset that is not a latice. Let $X, Y$ be disjoint infinite countable sets, $$\begin{aligned} E_0 &= \{A \subset (X \cup Y) \st \card (A \cap X) = \card (A \cap Y) \ \text {is finite}\}\,,\\ E &= E_0 \cup \{(X \cup Y) \setminus A \st A \in E_0 \}\,, \end{aligned}$$ $A \oplus B = A \cup B$ for disjoint $A,B \in E$. Then $(E, \oplus, \emptyset, X \cup Y)$ is a concrete orthomodular poset with the maximality property. Let $X = \{x_n \st n \in \N\}$, $y_0 \in Y$, $f \st X \to Y \setminus \{y_0\}$ be a bijection, $A = (X \cup Y) \setminus \{x_1, f(x_1)\}$, $B = (X \cup Y) \setminus \{x_1, y_0\}$. Then the chain $\bigl\{ \{ x_2, \dots, x_n, f(x_2), \dots, f(x_n) \} \st n \in \N \setminus \{1\} \bigr\}$ in $[\emptyset, A] \cap [\emptyset, B]$ does not have an upper bound in $[\emptyset, A] \cap [\emptyset, B]$, hence the condition (CU) from \[T:MaximalityProperties\] is not fulfilled. Let us remark that not all effect algebras have the maximality property (see \[E:OMPnotM\]). Acknowledgements {#acknowledgements .unnumbered} ================ The work was supported by the grant of the Grant Agency of the Czech Republic no. 201/07/1051 and by the research plan of the Ministry of Education of the Czech Republic no. 6840770010. [9]{} Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. Kluwer Academic Publishers, Bratislava, 2000. Foulis, D. J., Bennett, M. K.: Effect algebras and unsharp quantum logics, Found. Phys. (1994) [24]{}, 1331–1352. Foulis, D., Greechie, R., Rüttimann, G.: Filters and supports in orthoalgebras. Internat. J. Theoret. Phys. [31]{} (1992), 789–807. Jenča, G., Pulmannová, S.: Orthocomplete effect algebras. Proc. Amer. Math. Soc. [131]{} (2003), 2663–2671. Müller, V.: Jauch–Piron states on concrete quantum logics. Internat. J. Theoret. Phys. [32]{} (1993), 433–442. Tkadlec, J.: Central elements of effect algebras. Internat. J. Theoret. Phys. [43]{} (2004), 1363–1369. Tkadlec, J.: Conditions that force an orthomodular poset to be a Boolean algebra. Tatra Mt. Math. Publ. [10]{} (1997), 55–62.	ArXiv
--- abstract: 'This paper presents the natural extension of Buckley-Feuring method proposed in [@BuckleyFeuring99] for solving fuzzy partial differential equations (FPDE) in a non-polynomial relation, such as the operator $\varphi(D_{x_1}, D_{x_2})$, which maps to the quotient between both partials. The new assumptions and conditions proceedings from this consideration are given in this document.' address: - 'Facultad de Matemáticas. Universidad de Sevilla, 41012 Sevilla, Spain' - 'Keywords: Fuzzy differential equations, Buckley-Feuring solution, non-polynomial' - '2000 Mathematics Subject Classification: 03E72, 46S40' author: - \| D. Gálvez and J. L. Pino\ [Departamento de Estadística e Investigación Operativa]{}\ [Universidad de Sevilla]{} title: 'The extension of Buckley-Feuring solutions for non-polynomial fuzzy partial differential equations' --- Introduction {#introduction .unnumbered} ============ Many approaches for obtaining non-numerical solutions of fuzzy differential equations (FDE) have been developed from the introduction of fuzzy set concept by Zadeh [@Zadeh65] . These ones give a diversity of definitions for FDE solution based on different notions of fuzzy derivative, such as Seikkala derivative, Buckley-Feuring derivative, Puri-Ralescu derivative, Kandel-Friedman-Ming derivative, Goetschel-Voxman derivative, or Dubois-Prade derivative . Some relations between these derivatives are presented by Buckley and Feuring in [@BuckleyFeuring00] . However, only a few of these fuzzy derivatives are valid in some contexts as FDE solution. For example, the Goetschel-Voxman derivative, or the Dubois-Prade derivative provide solutions that cannot be a fuzzy number, whereas Puri-Ralescu derivative, and Kandel-Friedman-Ming derivative, always exist and provide a fuzzy number as solution of the FDE, but making use of abstract subtractions of fuzzy concepts in their definitions, making difficult the interpretation of this solutions in some real applications. This paper uses the Buckley-Feuring derivative for solving FPDE. This derivative does not always exist, but if it does, provides a fuzzy number solution easily understandable in the context in which a specific FPDE has been developed. The authors which proposed this concept of derivative, developed a methodology for solving constant coefficients polynomial FPDE in [@BuckleyFeuring99] . This paper present the extension of this methodology to a non-polynomial expression in partial fuzzy derivatives.\ \ In the following lines, the components of a FPDE are enumerated: - $x_i,\quad i=1, 2,\quad x_1\in S_1\subset I_1=(0, M_1],\quad x_2\in S_2\subset I_2=(0, M_2]$. Other domain limits can be established in this subsets, such as $ x_1>x_2$. - $\tilde{\boldsymbol{\beta}}= (\tilde{\beta}_1, \tilde{\beta}_2,...,\tilde{\beta}_k)$, a triangular fuzzy number vector. - $\mu(\beta_j)$ is the membership function of $\beta_j\in \tilde{\beta_j}.$ - $\mu_{\beta_j}(\alpha)=\{\beta_j\mid \mu(\beta_j)\geq\alpha,\quad\alpha\in(0,1)\}$ set called $\alpha$-cut.\ \ These sets are closed and bounded, so that is possible to define, for a fuzzy number $\tilde{\beta_j}: \tilde{\beta_j}[\alpha]= [b_1(\alpha), b_2(\alpha)]$, where: - $b_1(\alpha)$ is the lower value $\beta_j$ in which $\mu(\beta_j)\geq \alpha,\quad \beta_j\in \tilde{\beta}_j$. - $b_2(\alpha)$ is the higher value $\beta_j$ in which $\mu(\beta_j)\geq \alpha,\quad \beta_j\in \tilde{\beta}_j$. and $\tilde{\boldsymbol{\beta}}[\alpha]= \prod_j\tilde{\beta_j}[\alpha]$ - $\tilde{V}(x_1,x_2,\tilde{\boldsymbol{\beta}})$ is a positive and continuous function in $(x_1, x_2)\in S_1 \times S_2$ with partials $D_{x_1}, D_{x_2}$. This function must be also strictly increasing or strictly decreasing in $x_2\in S_2$, that is $\tilde{V}(k,x_2,\tilde{\boldsymbol{\beta}})$ is strictly increasing or strictly decreasing for all constant $k\in\mathbb{R}$.\ The fuzzy character of $\tilde{V}(x_1,x_2,\tilde{\boldsymbol{\beta}})$ shown by the tilde placed over $V$ is fixed by $\tilde{\boldsymbol{\beta}}$, and support the use of Buckley-Feuring derivative for solving FPDE. - $\varphi(D_{x_1}, D_{x_2})$ is an expression with constant coefficients in $(D_{x_1}, D_{x_2})$ applied to $\tilde{V}(x_1,x_2,\tilde{\boldsymbol{\beta}})$. - $F(x_1, x_2,\tilde{\boldsymbol{\beta}})$ continuous function in $(x_1, x_2)\in S_1 \times S_2.$ The specific FPDE treated in this paper has the following form according with this notation: $$\varphi(D_{x_1}, D_{x_2})\tilde{V}(x_1,x_2,\tilde{\boldsymbol{\beta}})= \frac{\partial \tilde{V}/\partial x_1}{\partial \tilde{V}/\partial x_2}=F(x_1, x_2,\tilde{\boldsymbol{\beta}})$$ The Buckley-Feuring (B-F) solution {#sec:B-F} ================================== The Buckley-Feuring (B-F) solution uses a solution of the crisp partial differential equation: $$V(x_1,x_2)= G(x_1,x_2,\boldsymbol{\beta}),$$ with $G$ continuous $\forall (x_1,x_2)\in S_1\times S_2$.\ The next step is the fuzzification of $G$: $$\tilde{Y}(x_1,x_2)= \tilde{G}(x_1,x_2,\tilde{\boldsymbol{\beta}}),$$ with $\tilde{G}$ continuous $\forall (x_1,x_2)\in S_1 \times S_2$ and strictly monotone for $x_2\in S_2$. Note that $\tilde{Y}_i$ is only the fuzzy representation of $G$, but not necessary the solution of the fuzzy partial differential equation. If it finally happens and $\tilde{Y}(x_1,x_2)$ is a B-F solution, then $\tilde{V}(x_1,x_2,\tilde{\boldsymbol{\beta}})=\tilde{Y}(x_1,x_2)$. With this notation, it is possible to see that:\ \ $\tilde{Y}(x_1,x_2)[\alpha]= [y_1(x_1,x_2,\alpha), y_2(x_1,x_2,\alpha)]$, and\ \ $\tilde{F}(x_1,x_2,\tilde{\boldsymbol{\beta}})[\alpha]= [f_1(x_1,x_2,\alpha), f_2(x_1,x_2,\alpha)], \forall\alpha$.\ and, by definition: $y_1(x_1,x_2,\alpha)= \min\{G(x_1,x_2,\boldsymbol{\beta}),\quad \boldsymbol{\beta}\in\tilde{\boldsymbol{\beta}}[\alpha]\}$,\ $y_2(x_1,x_2,\alpha)= \max\{G(x_1,x_2,\boldsymbol{\beta}),\quad \boldsymbol{\beta}\in\tilde{\boldsymbol{\beta}}[\alpha]\}$ ,and\ \ $f_1(x_1,x_2,\alpha)= \min\{F(x_1,x_2,\boldsymbol{\beta}),\quad \boldsymbol{\beta}\in\tilde{\boldsymbol{\beta}}[\alpha]\}$,\ $f_2(x_1,x_2,\alpha)= \max\{F(x_1,x_2,\boldsymbol{\beta}),\quad \boldsymbol{\beta}\in\tilde{\boldsymbol{\beta}}[\alpha]\}$,\ $\forall x_1, x_2, \alpha$.\ If it is possible to apply the $\varphi(D_{x_1}, D_{x_2})$ operator to $y_i, i=1, 2 $, getting continuous expressions $\forall(x_1, x_2)\in S_1\times S_2,\forall\alpha$, then it will be feasible to define the following expression in this domain $\Gamma(x_1,x_2,\alpha)$: $$\Gamma(x_1,x_2,\alpha)=[\Gamma_1(x_1,x_2,\alpha),\Gamma_2(x_1,x_2,\alpha)]$$ with: $\Gamma_1(x_1,x_2,\alpha)= \varphi(D_{x_1}, D_{x_2})y_1(x_1,x_2,\alpha)$ $\Gamma_2(x_1,x_2,\alpha)=\varphi(D_{x_1}, D_{x_2})y_1(x_1,x_2,\alpha)$\ \ For a B-F solution $Y$, it is necessary to be a fuzzy number for this one. If, for each pair $(x_1, x_2)\in S_1\times S_2$, $\Gamma(x_1,x_2,\alpha)$ defines an $\alpha$-cut of a fuzzy number, then, Buckley and Feuring [@BuckleyFeuring99] call to $\tilde{Y_i}(x_1,x_2)$ differentiable, and we can write: $$\varphi(D_{x_1}, D_{x_2})\tilde{Y}(x_1,x_2)[\alpha]=\Gamma(x_1,x_2,\alpha),$$ $\forall(x_1, x_2)\in S_1\times S_2,\quad\forall\alpha$.\ \ So that, it is necessary to test if $\Gamma(x_1,x_2,\alpha)$ really define an $\alpha$-cut of a fuzzy number and verify the differentiability of $\tilde{Y}(x_1,x_2)$. For a triangular fuzzy number, the conditions are [@GoetschelVoxman86] : 1. $\varphi(D_{x_1}, D_{x_2})y_1(x_1,x_2,\alpha)$ is an increasing function of $\alpha$, for each $(x_1, x_2)\in S_1\times S_2$. 2. $\varphi(D_{x_1}, D_{x_2})y_2(x_1,x_2,\alpha)$ is a decreasing function of $\alpha$, for each $(x_1, x_2)\in S_1\times S_2$. 3. $\varphi(D_{x_1}, D_{x_2})y_1(x_1,x_2,1)\leq\varphi(D_{x_1}, D_{x_2})y_2(x_1,x_2,1)$ for each $(x_1, x_2)\in S_1\times S_2$. Once delimited the differentiability concept of $\tilde{Y}(x_1,x_2)$, it is possible to define the Buckley-Feuring solution. $\tilde{Y_i}(x_1,x_2)$ is a Buckley-Feuring solution if the following conditions hold:\ 1. $\tilde{Y_i}(x_1,x_2)$ is differentiable. 2. $\varphi(D_{x_1}, D_{x_2})\tilde{Y}(x_1,x_2)= \tilde{F}(x_1, x_2,\boldsymbol{\tilde{\beta}})$.\ Obviously, if differentiability conditions hold by the candidate to B-F solution $\tilde{Y}(x_1,x_2)$, this one will be a fuzzy number. To complete the conditions for B-F solutions only is necesary to test that: $$\varphi(D_{x_1}, D_{x_2})\tilde{Y}(x_1,x_2)= \tilde{F}(x_1, x_2,\tilde{\boldsymbol{\beta}}),$$ or the equivalent condition: 1. $\varphi(D_{x_1}, D_{x_2})y_1(x_1,x_2,\alpha)= F_1(x_1, x_2,\alpha).$ 2. $\varphi(D_{x_1}, D_{x_2})y_2(x_1,x_2,\alpha)= F_2(x_1, x_2,\alpha).$ $\forall(x_1, x_2)\in S_1\times S_2,\quad\forall\alpha$.\ \ In this case we can identify $\tilde{Y}(x_1,x_2)$ with $\tilde{V}(x_1,x_2,\tilde{\boldsymbol{\beta}})$. Boundary conditions {#sec:Bound. cond.} =================== The FPDE can be subject to certain boundary conditions in a big variety of forms depending on a constant vector $\boldsymbol{c}=(c_1,...c_n)\in C_1\times ...\times C_n$. The inclusion of this ones has not great consequences in the methodology exposed. The crisp solution $G$ acquires under boundary conditions the form $G(x_1,x_2,\boldsymbol{\beta},\boldsymbol{c})$. The fuzzification of $G$ can take $\boldsymbol{c}$ in a triangular fuzzy vector $\tilde{\boldsymbol{c}}=(\tilde{c_1},...\tilde{c_n})\in C_1\times ...\times C_n$ with $\tilde{\boldsymbol{c}}[\alpha]= \prod_i\tilde{C_i}[\alpha],\quad i=1,...n$ and: $$\tilde{Y}(x_1,x_2)= \tilde{G}(x_1,x_2,\tilde{\boldsymbol{\beta}},\tilde{\boldsymbol{c}})$$ In this environment with boundary conditions, it is necessary to add a new condition for a B-F solution: $\tilde{Y}(x_1,x_2)$ must satisfier these conditions. In this form $\tilde{Y_i}(x_1,x_2)$ is a Buckley-Feuring solution if the following conditionshold: 1. $\tilde{Y_i}(x_1,x_2)$ is differentiable. 2. $\varphi(D_{x_1}, D_{x_2})\tilde{Y}(x_1,x_2)= \tilde{F}(x_1, x_2,\tilde{\boldsymbol{\beta}})$. 3. $\tilde{Y}(x_1,x_2)$ satisfies the boundary conditions. Example {#sec:example} ======= The following FPDE is proposed: $$\dfrac{\partial {\tilde{V}}/\partial{x_1}}{\partial \tilde{V}/\partial x_2}= \tilde{\beta} x_1^{ -1}x_2,\quad \tilde{\beta}\in (0, 1),x_1\geq 1, x_2>0$$ In this example: $\tilde{F}(x_1, x_2,\boldsymbol{\tilde{\beta}})=\tilde{\beta} x_1^{ -1}x_2,\quad \tilde{\beta}\in (0, 1),x_1\geq 1, x_2>0$\ and, by definition, the operator\ $\varphi(D_{x_1}, D_{x_2})V(x_1,x_2) \longrightarrow\dfrac{\partial \tilde{V}/\partial x_1}{\partial \tilde{V}/\partial x_2},$\ \ A possible solution to this FPDE in a crisp environment is, without special boundary conditions: $$G(x_1,x_2,\boldsymbol{\beta})= x_1^{\beta}x_2+\gamma,\beta\in (0, 1),x_1\geq 1, x_2>0$$ with $\boldsymbol{\beta}=(\beta,\gamma)$ Applying the fuzzification in $\beta$ and $\gamma$, these ones acquire a triangular fuzzy number form and $\boldsymbol{\tilde{\beta}} =(\tilde{\beta},\tilde{\gamma}),\quad\tilde{\beta}\in (0, 1)$. While $G$ holds all the conditions required, the Buckley-Feuring solution candidate is: $$\tilde{Y}(x_1,x_2)=G(x_1,x_2,\tilde{\beta},\tilde{\gamma})= x_1^{\tilde{\beta}}x_2+\tilde{\gamma},\tilde{\beta}\in (0, 1),x_1\geq 1, x_2>0$$ The fuzzy parameters have a membership function associated $\mu(\beta)$ and $\mu(\gamma)$ respectiveness. From the $\alpha$-cuts, it is possible to define: $\tilde{\beta}[\alpha]= [b_1(\alpha), b_2(\alpha)]$,\ $\tilde{\gamma}[\alpha]= [g_1(\alpha), g_2(\alpha)]$, and\ $\tilde{\boldsymbol{\beta}}[\alpha]= \tilde{\beta}[\alpha] \times \tilde{\gamma}[\alpha]$\ And from this ones: $\tilde{Y}(x_1,x_2)[\alpha]= [y_1(x_1,x_2,\alpha), y_2(x_1,x_2,\alpha)]$, and\ $\tilde{F}(x_1,x_2,\tilde{\boldsymbol{\beta}})[\alpha]= [f_1(x_1,x_2,\alpha), f_2(x_1,x_2,\alpha)], \forall\alpha $.\ Where: $y_1(x_1,x_2,\alpha)= \min\{G(x_1,x_2,\boldsymbol{\beta}),\quad \boldsymbol{\beta}\in\boldsymbol{\tilde{\beta}}[\alpha]\}= G(x_1,x_2,g_1(\alpha), b_1(\alpha)) $,\ $y_2(x_1,x_2,\alpha)= \max\{G(x_1,x_2,\boldsymbol{\beta}),\quad \boldsymbol{\beta}\in\boldsymbol{\tilde{\beta}}[\alpha]\}=G(x_1,x_2, g_2(\alpha), b_2(\alpha))$\ \ and,\ \ $f_1(x_1,x_2,\alpha)= \min\{F(x_1,x_2,\boldsymbol{\beta}),\quad \boldsymbol{\beta}\in\boldsymbol{\tilde{\beta}}[\alpha]\}=F(x_1,x_2, g_1(\alpha), b_1(\alpha))$,\ $f_2(x_1,x_2,\alpha)= \max\{F(x_1,x_2,\boldsymbol{\beta}),\quad \boldsymbol{\beta}\in\boldsymbol{\tilde{\beta}}[\alpha]\}= F(x_1,x_2, g_2(\alpha), b_2(\alpha))$,\ \ $\forall\alpha\quad x_1\geq 1, x_2>0 $\ \ In the $\tilde{G}$ function proposed, $\forall(x_1, x_2)x_1\geq 1, x_2>0$:\ $y_1(x_1,x_2,\alpha)=x_1^{b_1(\alpha)}x_2+{g_1(\alpha)},\quad b_1(\alpha)\in (0, 1)$\ $y_2(x_1,x_2,\alpha)=x_1^{b_2(\alpha)}x_2+{g_2(\alpha)},\quad b_2(\alpha)\in (0, 1)$\ \ $f_1(x_1,x_2,\alpha)=b_1(\alpha)x_1^{-1}x_2,\quad b_1(\alpha)\in (0, 1)$\ $f_2(x_1,x_2,\alpha)=b_2(\alpha)x_1^{-1}x_2,\quad b_2(\alpha)\in (0, 1)$\ \ Testing the differentiability of $\tilde{G}(x_1,x_2,\tilde{\boldsymbol{\beta}})$, from $\Gamma(x_1,x_2,\alpha)=[\varphi(D_{x_1}, D_{x_2})y_1(x_1,x_2,\alpha), \varphi(D_{x_1}, D_{x_2})y_2(x_1,x_2,\alpha)]$ verifying if $\Gamma(x_1,x_2,\alpha)$ defines an $\alpha$-cut of a triangular fuzzy number for each pair $(x_1, x_2), x_1\geq 1, x_2>0$, the following conditions must be hold: 1. $\varphi(D_{x_1}, D_{x_2})y_1(x_1,x_2,\alpha)= b_1(\alpha)x_1^{-1}x_2,\quad b_1(\alpha)\in (0, 1)$ is an increasing function of $\alpha$, for each pair $(x_1, x_2), x_1\geq 1, x_2>0$. It will happen if $\varphi(D_{x_1}, D_{x_2})y_1(x_1,x_2,\alpha)$ has positive derivative on $\alpha$: $$d_{\alpha}\left(b_1(\alpha)x_1^{-1}x_2\right)= b_1(\alpha)'x_1^{-1}x_2$$ While $\beta$ is a triangular fuzzy number and $b_1(\alpha)$is defined from its $\alpha-cuts$, $b_1(\alpha)$ satisfies the condition and is an increasing function with $b_1(\alpha)'>0$, and $x_1^{-1}x_2>0$ for $ x_1\geq 1, x_2>0$, $d_{\alpha}\left(b_1(\alpha)x_1^{-1}x_2\right)>0$ and the condition holds.\ \ 2. $\varphi(D_{x_1}, D_{x_2})y_2(x_1,x_2,\alpha)= b_2(\alpha)x_1^{-1}x_2,\quad b_2(\alpha)\in (0, 1)$ is a decreasing function of $\alpha$, for each pair $(x_1, x_2), x_1\geq 1, x_2>0$. Again, it will happen if $\varphi(D_{x_1}, D_{x_2})y_2(x_1,x_2,\alpha)$ has negative derivative on $\alpha$: $$d_{\alpha}\left(b_2(\alpha)x_1^{-1}x_2\right)= b_2(\alpha)'x_1^{-1}x_2$$ We can now use that $\beta$ is a triangular fuzzy number and $b_2(\alpha)$ is defined from its $\alpha-cuts$, so that, in analogy, $b_2(\alpha)$ satisfies the condition and is a decreasing function with $b_2(\alpha)'<0$, and $x_1^{-1}x_2>0$ for $ x_1\geq 1, x_2>0$, $d_{\alpha}\left(b_1(\alpha)x_1^{-1}x_2\right)<0$ and the condition holds.\ \ 3. $\varphi(D_{x_1}, D_{x_2})y_1(x_1,x_2,1)\leq \varphi(D_{x_1}, D_{x_2})y_2(x_1,x_2,1) \forall x_1, x_2)\in I_1\times I_2$.\ In the example: $$b_1(1)x_1^{-1}x_2 \leq b_2(2)x_1^{-1}x_2$$\ This condition is holds automatically if we realize that $b_1(\alpha)$ and $b_2(\alpha)$ are defined from a triangular fuzzy number and $b_2(1)>b_1(1)$ At this point, it is possible to say that $\tilde{Y}(x_1,x_2)=G(x_1,x_2,\tilde{\beta},\tilde{\gamma})= x_1^{\tilde{\beta}}x_2+\tilde{\gamma},\tilde{\beta}\in (0, 1),x\geq 1$ is differentiable and good candidate for B-F solution. But it is necessary the last step, and it must satisfier that:\ \ $\varphi(D_{x_1}, D_{x_2})\tilde{Y}(x_1,x_2)= \tilde{F}(x_1, x_2,\tilde{\boldsymbol{\beta}}) \quad \beta, \gamma\in (0, 1), \forall(x_1, x_2), x_1\geq 1, x_2>0$.\ For $G(x_1,x_2,\tilde{\beta},\tilde{\gamma})= x_1^{\tilde{\beta}}x_2+\tilde{\gamma},\tilde{\beta}\in (0, 1),x\geq 1, x_2\geq 0$:\ \ $\varphi(D_{x_1}, D_{x_2})\tilde{Y}(x_1,x_2)= \tilde{\beta} x_1^{ -1}x_2,\quad \tilde{\beta}\in (0, 1),x_1\geq 1, x_2>0 $\ and,\ $\tilde{F}(x_1, x_2,\tilde{\boldsymbol{\beta}})=\tilde{\beta} x_1^{ -1}x_2,\quad \tilde{\beta}\in (0, 1),x_1\geq 1, x_2>0. $\ Thus $G(x_1,x_2,\tilde{\beta},\tilde{\gamma})= x_1^{\tilde{\beta}}x_2+\tilde{\gamma},\tilde{\beta}\in (0, 1),x\geq 1, x_2\geq 0$ is a Buckley-Feuring solution for this non-polynomial fuzzy partial differential equation. [99]{} Allahviranloo, T. Diference methods for fuzzy partial differential equations, Computational Methods in Applied Mathematics, Vol.2 (nº3) (2002), pp. 233-242. Buckley, J.J. and Feuring,T. Introduction to fuzzy partial differential equations, Fuzzy Sets and Systems, 105 (1999), pp.241-248. Buckley, J.J. and Feuring,T. Fuzzy differential equations, Fuzzy Sets and Systems, 110 (2000), pp.43-54. Goetschel, R. and Voxman, W. Elementary fuzzy calculus, Fuzzy Sets and Systems, 18 (1986), pp.319-330. Quan, H., Hua, Y. and Jones, J.D. A general method for calculating functions of fuzzy numbers, Applied Mathematics Letters, Vol.5 (nº6) (1992), pp.51-55. Seikkala, s. On the fuzzy initial value problem, Fuzzy Sets and Systems, 24 (1987), pp.31-43. Zadeh, L.A. Fuzzy Sets. Inf. Control, 8 (1965), pp.338-353.	ArXiv
--- abstract: 'We first prove a one-to-one correspondence between finding Hamiltonian cycles in a cubic planar graphs and finding trees with specific properties in dual graphs. Using this information, we construct an exact algorithm for finding Hamiltonian cycles in cubic planar graphs. The worst case time complexity of our algorithm is O$(2^n)$.' author: - \| Bohao Yao\ Charl Ras, Hamid Mokhtar\ The University of Melbourne** title: 'An algorithm for finding Hamiltonian Cycles in Cubic Planar Graphs*' --- Introduction ============ A [Hamiltonian cycle]{} is a cycle which passes through every vertex in a graph exactly once. A [planar graph]{} is a graph which can be drawn in the plane such that no edges intersect one another. A [cubic graph]{} is a graph in which all vertices have degree 3. Finding a Hamiltonian cycle in a cubic planar graph is proven to be an $\mathcal{NP}$-Complete problem [@garey1976planar]. This implies that unless $\mathcal{P}=\mathcal{NP}$, we could not find an efficient algorithm for this problem. Most approaches to finding a Hamiltonian cycle in planar graph utilises the [divide-and-conquer]{} method, or its derivation, the [separator theorem]{} which partitions the graph in polynomial time [@lipton1979separator]. Exact algorithms using such methods were found to have the complexity of O$(c^{\sqrt{n}})$ [@klinz2006exact] [@dorn2005efficient], where [n]{} denotes the number of vertices and [c]{} is a constant. In this paper, we consider only cubic planar graphs and attempt to find a new algorithm to provide researchers with a new method to approaching this problem. The Expansion Algorithm ======================= We first start by introducing our so-called [Expansion Algorithm]{} which increases the number of vertices in a cycle at each iteration. A cycle can be first found by taking the outer facial cycle of the planar graph. We define it as the [base cycle]{}, $\sigma_0$. This base cycle is then expanded by the Expansion Algorithm, which will be described in detail later. \[def1\] Consider a planar graph $G=(V,E)$ A complementary path, $P_e^\sigma$, is a path between 2 adjacent vertices, $v_1, v_2 \in \sigma$ connected by the edge, $e$, s.t. $P_e^\sigma$ is internally disjoint from $\sigma$. Furthermore, $P_e^\sigma$ and $e$ together will form the boundary of a face in $G$. Assuming we are not dealing with multigraphs, the complementary path will always have at least one other vertex besides $v_1, v_2$. The restriction that $P_e^\sigma$ and $e$ have to form the boundary of a face will be used later to prove Corollary \[cor1\]. \[def2\] Let $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$. Then, $G_1+G_2 := (V_1 \cup V_2,E_1 \cup E_2)$ \[alg1\] [Let]{} $\sigma_0$ be the outer facial cycle.\ [Let]{} $i=0$ At each iteration, the algorithm removes an edge, $e$ and adds a path $P_e^\sigma$. Since there are no vertices on $e$ and there is at least 1 vertex on $P_e^\sigma$, the number of vertices on the cycle will always increase at each iteration. Since there is only a finite amount of vertices in the graph, the algorithm will have to terminate eventually. ![Example of utilizing the Expansion Algorithm with the base cycle in blue](Expansion.eps) The interior* of a cycle, $C$, is the connected region lying to the left of an anticlockwise orientation of $C$. \[lem2\] At each iteration of the Algorithm \[alg1\], all the vertices of G either lie in the interior of $\sigma$, or on $\sigma$. Let $P_i$ be the statement that all the vertices of G either lie in the interior of $\sigma_i$ - $P_0$ is true as $\sigma_0$ is the outer facial cycle, with all the vertices either lie in the interior or on $\sigma_0$. - Assume $P_k$ true. Then all the vertices either lie in the interior or on $\sigma_k$. Assume $\exists P_{e_k}^{\sigma_k}$, then $\sigma_{k+1}$ exists. Let $f_k$ be the face bounded by $P_{e_k}^{\sigma_k}$ and $e_k$. The interior of $f_k$ originally lies inside $\sigma_k$. But by an iteration of Algorithm \[alg1\], the interior of $f_k$ now lies outside the new base cycle, $\sigma_{k+1}$. However, since there are no vertices in the interior of a face, all the vertices in $\sigma_{k+1}$ also lies either in the interior or on $\sigma_{k+1}$. Therefore, if $\sigma_{k+1}$ exists, then $P_k \Rightarrow P_{k+1}$. Hence, by mathematical induction, $P_i$ is true $\forall i$ as long as $\sigma_i$ exists. \[cor1\] If $P_e^\sigma$ exists for an edge $e \in \sigma$, then $P_e^\sigma$ is unique. By Lemma \[lem2\], none of the vertices can lie outside $\sigma$, thus, $P_e^\sigma$ also lies in the interior of $\sigma$. Any edge $e$ lies on the boundary of 2 faces. If $e \in \sigma$, then $\exists$ only 1 possible $P_e^\sigma$ such that $e$ and $P_e$ forms the boundary of a face that lies in the interior of $\sigma$ (the other face that $e$ is a boundary of lies outside $\sigma$) \[lem1\] If $G$ have a Hamiltonian cycle, then $\exists$ a choice of complementary paths that algorithm \[alg1\] can use to find that Hamiltonian cycle. Consider a Hamiltonian cycle, $C$, in $G$. If $C = \sigma_0$ then the case is trivial. If $C \ne \sigma_0$, then by Lemma \[lem2\], all vertices in $C$ lies either on or in the interior of $\sigma_0$ since $C$ contains all the vertices of the graph. Since $C$ contains all the vertices and $\sigma_0$ is a cycle and therefore must contain at least 3 vertices, $C \cap \sigma_0 \ne \emptyset$. Suppose $C \ne \sigma_0$. Let $v_1,v_2,...,v_n$ be consecutive vertices on $\sigma_0$ about the clockwise rotation. Let $P_i$ be the path connecting $v_i,v_{i+1}$ that is the subpath of $C$. Let the edge between $v_i$ and $v_{i+1}$ be $e_i$ which also lies on $\sigma_0$. Since $C$ contains all the vertices in the graph, by iteratively finding $P_e^\sigma$, starting with $e = e_i$, more subpath of $P_i$ will lie on $\sigma$ at each iteration. The algorithm will only move on when $P_i \subset \sigma$. Repeating this process $\forall i$, we will eventually end up with $\sigma = C$, in which the algorithm will terminate. ![Illustration of the proof for Lemma \[lem1\] with $C$ in red](Graph1.eps) The Problem in the Dual Graph ============================= Given a cubic planar graph, $G$, and the corresponding dual graph, $\overline{G}=(\overline{V},\overline{E})$. A corresponding face, $f_{\overline{v}} \in G$, is the face corresponding to the vertex $\overline{v} \in \overline{G}$. The outer vertex, $\overline{v}^$, is the vertex in $\overline{G}$ that corresponds to the outer face of $G$. Let $e \in G$ be the shared boundary between $f_{\overline{v}_1},f_{\overline{v}_2}$. A dual edge, $\overline{e} \in \overline{G}$, is defined as an edge between $\overline{v}_1, \overline{v}_2 \in \overline{G}$. ![Example of $\overline{e}$ and $\overline{v}^$](v.eps) Consider Algorithm \[alg1\]. Since $P_e^\sigma$ is unique for an edge $e$ by Corollary \[cor1\], let $e_0,e_1,...,e_n$ be the edges chosen in the [Expansion Algorithm]{} in order. Let $\overline{e}_0,\overline{e}_1,...,\overline{e}_n$ be the corresponding dual edges in $\overline{G}$. ![Expansion Algorithm in the Dual Graph with $\overline{T}$ in Purple](Expansion_Dual.eps) \[lem3\] $\bigcup\limits_{i=0}^n \overline{e}_i$ is a tree ($\overline{T}$). Let $P_a$ be the statement that $\bigcup\limits_{i=0}^a \overline{e}_i$ is a tree. - $P_0$ is true as 2 vertices connected by an edge, $\overline{e}_0$ is considered a tree. - Assume $P_k$ is true. Then $\bigcup\limits_{i=0}^k \overline{e}_i$ is a tree. From proof of Corollary \[cor1\], $e_{k+1}$ is a boundary of 2 faces, one that lies inside $\sigma_{k+1}$ ($f_1$) and another that lies outside $\sigma_{k+1}$ ($f_2$). From proof of Lemma \[lem2\], by including path $P_{e_i}^{\sigma_i}$, the face bounded by $e_i$ and $P_{e_i}^{\sigma_i}$ which originally lies in the interior of $\sigma_i$ will now lie on the exterior of $\sigma_{i+1}$. Therefore, the vertex in $\overline{G}$ that corresponds to $f_1$ will lie outside $\overline{T}$ while the vertex that corresponds to $f_2$ will lie inside $\overline{T}$. Thus, $\overline{e}_{k+1}$ will connect a vertex on $\overline{T}$ to a vertex that is outside of $\overline{T}$, thus expanding the tree. Hence $\bigcup\limits_{i=0}^{k+1} \overline{e}_i$ is also a tree and therefore $P_{k+1}$ is true. Thus, $P_k \Rightarrow P_{k+1}$. By mathematical induction, $\bigcup\limits_{i=0}^n \overline{e}_i$ is a tree. From Lemma \[lem2\], we know that faces $f_k$, bounded by $e_k$ and $P_{e_k}^{\sigma_k}$, will be on the exterior of $\sigma$ if $e_k$ was used in the algorithm. Conversely, if $e_k$ was not used in the algorithm, then $f_k$ will be in the interior of the cycle $\sigma$. This is equivalent to stating that all the vertices in $\overline{T}$ lies outside $\sigma$ while all the vertices not in $\overline{T}$ lies in the interior or $\sigma$. \[lem4\] If $\overline{v} \in \overline{G}$ lies on $\overline{T}$, then all the vertices in $G$ on $f_{\overline{v}}$ lies on $\sigma_n$. If $\overline{v}$ lies on the tree, then $\exists P_e^\sigma$ s.t. all its vertices in $G$ on $f_{\overline{v}}$ lies on $P_e^\sigma$ since the face is bounded by $P_e^\sigma$ and $e$. Since each iteration of the algorithm will only remove an edge and not any vertices, all the vertices already on the cycle will remain on the cycle at each iteration. \[thm1\] $G$ has a Hamiltonian cycle if and only if $\exists \overline{T}$ found by Algorithm \[alg1\], with $\overline{v}^$ at the root, that satisfy the following properties: 1. For each vertex $v \in G$ that lies on the boundaries of $f_{\overline{v}_1},f_{\overline{v}_2},f_{\overline{v}_3}$, at least one of the vertices, $\overline{v}_1,\overline{v}_2,\overline{v}_3 \in \overline{G}$, lies on $\overline{T}$. 2. No two vertices on $\overline{T}$ can be joined by an edge $\overline{e}$ unless $\overline{e} \in \overline{T}$. By first taking the outer facial cycle, $\sigma_0$, only $\overline{v}^$ lies outside our base cycle. Hence, $\overline{T}$ starts with $\overline{v}^$ as its root. ($\Rightarrow$) We will begin by proving each of the following property: 1. If $G$ has a Hamiltonian cycle, then $\exists$ a cycle in $G$ that uses every vertex in $G$. If none of the vertices, $\overline{v}_1,\overline{v}_2,\overline{v}_3$, lies on $\overline{T}$, then none of the $P_e^\sigma$ found by Algorithm \[alg1\] could contain $v$ from negation of Lemma \[lem4\]. This contradicts the definition of a Hamiltonian cycle. Therefore, at least one of the vertices lies on $\overline{T}$. ![A vertex lies on the cycle (blue) as $\overline{v}_1 \in \overline{T}$ (purple)](Thm1_1.eps) 2. If 2 vertices, $\overline{v}'_1, \overline{v}'_2 \in \overline{T}$ are adjacent, $\exists$ 2 vertices shared by $f_{\overline{v}'_1},f_{\overline{v}'_2} \in G$. However, the edge between them is not used on $\overline{T}$. This implies that the 2 vertices will lie on the corresponding path in $G$ twice, once from each face $f_{\overline{v}'_1},f_{\overline{v}'_2} \in G$. This is a contradiction from the definition of a Hamiltonian cycle. [.5]{} ![$\overline{v}_1,\overline{v}_2 \in \overline{T}$ and $\overline{e} \in \overline{T}$](Thm1_21.eps "fig:"){width=".4\linewidth"} [.5]{} ![$\overline{v}_1,\overline{v}_2 \in \overline{T}$ and $\overline{e} \in \overline{T}$](Thm1_22.eps "fig:"){width=".4\linewidth"} ($\Leftarrow$) Suppose that $\overline{T}=\bigcup\limits_{i=0}^n \overline{e}_i$ is a tree found by Algorithm \[alg1\] that $\overline{T}$ satisfies those properties. From 2, we apply restriction on $\overline{T}$ such that the path in $G$ passes through each vertex at most once (as proven previously). From 1, we know the path passes through every vertex in $G$ at least once. Since there is a path in $G$ that passes through every vertex at most once and at least once, we know the path passes through every vertex in $G$ once. Furthermore, we know that Algorithm \[alg1\] will always output a cycle, $\sigma$. Therefore, we know that $\overline{T}$ satisfying those properties will correspond to a Hamiltonian cycle in $\overline{G}$ Note that Theorem \[thm1\] is a corollary of a theorem by Skupie[ń]{} [@skupien2002hamiltonicity]. Finding $\overline{T}$ ====================== We start solving the problem by modifying a backtracking algorithm (procedure* `solve`) with restrictions (procedure `update`) to limit the amount of guessing required. Restrictions from procedure `update` may cause the tree to form 2 disjoint graphs, in which case a new backtracking algorithm (procedure `disjoint`) is called to trace a path between the graphs. If a particular choice of paths does not work, the algorithm will negate that choice (procedure `backtrack`). The algorithm will terminate if it successfully find $\overline{T}$ (all verties assigned) or if all other choices are exhausted, concluding that the graph has no Hamiltonian cycle. Search Algorithm ---------------- \[alg2\] To find $\overline{T}$ with properties in Theorem \[thm1\], we use a modified backtracking algorithm. [procedure]{} update($\overline{G},\overline{v}$):\ [procedure]{} backtrack($\overline{G},\overline{v}$):\ $V(\overline{T}):= V(\overline{T}) - \overline{v}$ $\overline{S}:=\overline{S}+\overline{v}$ [procedure]{} solve($\overline{G}$):\ disjoint($\overline{G},\overline{v}$):\ \ $\overline{S} := \emptyset$ $\overline{T} := \left\{\overline{v}^\right\}$ call solve($\overline{G}$) Restrictions on vertices does not apply in disjoint graphs where the 2 vertices $\in \overline{T}$ belongs to different sets due to the necessity of connecting the disjoint graphs into one tree. Complexity ---------- In the worst case scenario, we expect to force at least one vertex into $\overline{S}$ at each guess. This reduces the number of faces in which we have to check to $\frac{f}{2}$ where $f$ is the number of faces in $G$. Since we are working in planar graph, $e=\frac{3}{2}n$ where $n$ is the number of vertices. Using Euler’s formula: $$\begin{aligned} n-e+f&=2 \\ n-\frac{3}{2}n+f&=2 \\ f&=2+\frac{n}{2}\end{aligned}$$ Since we have 2 choices ($\overline{T}$ or $\overline{S}$) that we have to guess, in the worst case, our algorithm runs in $2^{\frac{f}{2}}$ or $2^{1+\frac{n}{4}}$. Hence, our algorithm has a time complexity of O$(2^n)$. Future Research =============== We could modify Theorem \[thm1\] such that it encompasses any planar graph, $G_2=(V_2,E_2)$ However, caution is advised dealing with faces that share a single vertex, but not a boundary, as this is not reflected in the dual graph, $\overline{G}_2=(\overline{V}_2,\overline{E}_2)$. To overcome this, we introduce the imaginary edge. An imaginary edge, $\overline{e}'$ between vertices in $\overline{G}_2$ if the corresponding faces in $G_2$ share a single vertex but not a boundary. \[thm2\] If $G_2$ has a Hamiltonian cycle then $\exists \overline{T}$ found by Algorithm \[alg1\], with $\overline{v}^$ at the root, that at least satisfy the following properties: 1. For each n degree vertex $v \in G_2$ that lies on the boundaries of $f_{\overline{v}_1},f_{\overline{v}_2}, ..., f_{\overline{v}_n}$, at least one of the vertices, $\overline{v}_1,\overline{v}_2, ..., \overline{v}_n \in \overline{G}_2$, lies on $\overline{T}$ 2. No two vertices on $\overline{T}$ can be joined by an edge $\overline{e}$ unless $\overline{e} \in \overline{T}$. 3. No two vertices joined by an imaginary edge can lie on $\overline{T}$ Properties 1 and 2 are proven in Theorem \[thm1\]. The proof of property 3 follows from the proof of Property 2: If 2 vertices, $\overline{v}_1, \overline{v}_2 \in \overline{T}$ are joined by an imaginary edge, then the vertex in which $f_{\overline{v}_1}$ and $f_{\overline{v}_2}$ shares will be on the cycle twice. Since this contradicts the definition of a Hamiltonian cycle, property 3 holds. Acknowledgments =============== The author wishes to thank his supervisors, Dr Charl Ras and Hamid Mokhtar, for their continued guidance and providing useful insights to this problem. This paper was supported by the Vacation Research Scholarship awarded by the Australian Mathematical Sciences Institute. [1]{} F. Dorn, E. Penninkx, H. L. Bodlaender, and F. V. Foin. Efficient exact algorithms on planar graphs: Exploiting sphere cut branch decompositions. In [Algorithms-ESA 2005]{}, pages 95-106. Springer, 2005. M. R. Garey, D. S. Johnson, and R. E. Tarjan. The planar hamiltonian circuit problem is NP-complete. [SIAM Journal of Computing]{}, 5(4):704-714, 1976. B. Klinz, G. J. Woeginger, et al. Exact algorithms for the Hamiltonian cycle problem in planar graphs. [Operations Research Letters]{}, 34(3):269-274, 2006. R. J. Lipton and R. E. Tarjan. A separator theorem for planar graphs. [SIAM Journal on Applied Mathematics]{}, 36(2):177-189, 1979. Z. Skupie[ń]{}. Hamiltonicity of planar cubic multigraphs. [Discrete mathematics]{}, 251(1):163-168, 2002.	ArXiv
--- abstract: 'Motivated by the recent experiments on periodically modulated, two dimensional electron systems placed in large transversal magnetic fields, we investigate the interplay between the effects of disorder and periodic potentials in the integer quantum Hall regime. In particular, we study the case where disorder is larger than the periodic modulation, but both are small enough that Landau level mixing is negligible. In this limit, the self-consistent Born approximation is inadequate. We carry extensive numerical calculations to understand the relevant physics in the lowest Landau level, such as the spectrum and nature (localized or extended) of the wave functions. Based on our results, we propose a qualitative explanation of the new features uncovered recently in transport measurements.' author: - Chenggang Zhou - Mona Berciu - 'R. N. Bhatt' title: Effects of large disorder on the Hofstadter butterfly --- Introduction {#sec1} ============ Two-dimensional electron systems (2DES) placed in a uniform perpendicular magnetic field exhibit a rich variety of phenomena, such as the integer[@IQHE] and fractional[@FQHE] quantum Hall effects. [@Springer] Another well-studied problem is that of a 2DES in a uniform perpendicular magnetic field subjected to a periodic potential. Even before the discovery of the quantum Hall effects, Hofstadter[@Hofst] showed that in this case, the electronic bands split into a remarkable fractal structure of subbands and gaps, the so-called Hofstadter butterfly. Two “asymptotic” regimes are usually considered: (i) if the magnitude of the periodic potential is very large compared to the cyclotron energy and the Zeeman splitting, then one can use lattice models to describe the hopping of electrons between Wannier-like states localized at the minima of the periodic potential, whereas (ii) if the magnitude of the periodic potential is small compared to the cyclotron energy, the periodic potential lifts the degeneracy of each Landau level. In both cases, the resulting butterfly structure is a function only of the ratio between the flux $\phi=B{\cal A}$ of the magnetic field through the unit cell of the periodic lattice, and the elementary magnetic flux $\phi_0=hc/e$. Remarkably, if $\phi/\phi_0$ of the first asymptotic case is equal to $\phi_0/\phi$ of the second case, their electronic structures are solutions of the same Harper’s equation.[@equiv] If the periodic potential is comparable to the cyclotron energy, Landau level mixing must be taken into account; although Landau levels still split into subbands, the structure is no longer universal, but depends also on the ratio of the periodic potential amplitude and the cyclotron energy.[@germans] Experimentally, the case with a small periodic modulation can be realized more easily. This is because the periodic potential is usually imprinted at some distance from the 2DES layer; as a result, its magnitude in the 2DES is considerably attenuated. The interesting cases to study experimentally also correspond to small values of $\phi/\phi_0$ (of order unity), where the butterfly structure shows a small number of subbands separated by large gaps, and should therefore be easier to identify. Periodic modulations have been created using lithography[@Holland; @Weimann; @Weiss] and holographic illumination. [@Wulf] The lattice constants of the resulting square lattices are of order 100 nm. As a result, the condition $\phi/\phi_0 \approx 2$ (for instance) is satisfied for $B\approx 0.8$ T. This is a very low value, in the Shubnikov-de Haas (SdH) regime, not the high-$B$ quantum regime. Significant Landau level mixing and complications from the fact that the Fermi level is inside one of the higher Landau levels for such small $B$-values make the identification of the Hofstadter structure difficult. Recently, a new method for lateral periodic modulation has been developed using a self-organized ordered phase of a diblock copolymer deposited on a GaAs/AlGaAs heterostructure.[@Sorin] The polymer spheres create a 2D triangular lattice with a lattice constant of about 39 nm. The corresponding unit cell area is almost an order of magnitude smaller than those achieved in previous experiments, implying that the condition $\phi/\phi_0 \approx 2$ is now satisfied for very strong magnetic fields, $B\approx 6$ T. At such high magnetic fields the system is in the strong quantum regime, and Landau level mixing can be safely ignored. For the experimental 2DES electron concentrations, the Fermi level is in the spin-down lowest Landau level.[@Sorin] As a result, this experimental setup appears more promising for the successful observation of the butterfly. Nevertheless, one must take into account the disorder which is present in the system (without disorder, there is no integer Quantum Hall Effect – IQHE – to begin with). If the disorder is very small compared to the periodic potential amplitude, one expects that the subbands of the Hofstadter structure are “smeared” on a scale $\hbar/\tau$, where $\tau$ is the scattering time, and $\tau \rightarrow \infty$ as disorder becomes vanishingly small. As a result, the larger gaps in the Hofstadter structure should remain open at the positions predicted in the absence of disorder, and one expects a series of minima in the longitudinal conductivity as the Fermi level traverses such gaps. The experiment indeed shows a very non-trivial modification of the longitudinal resistivity, with many peaks and valleys appearing in what is (in the absence of the periodic modulation) a smooth Lorentz-like peak.[@Sorin] However, the position of the minima in $\rho_{xx}$ do not track the positions of the main gaps in the corresponding Hofstadter butterfly structure. Instead, the data suggests that in this experimental setup, disorder is not small, but rather large compared to the estimated amplitude of the periodic potential. This is not a consequence of poor samples, since these 2DES have high mobilities. It is due to the fact that the periodic modulation is considerably attenuated in the 2DES, leading to a small energy scale for the Hofstadter butterfly spectrum as compared to $\hbar/\tau$. As a result, the Hofstadter structure predicted in the absence of disorder is of little use for interpreting the experimental data. One might expect that in this case the periodic potential should have basically no effect on the disorder-broadened Landau level. This is indeed true for the strongly localized states at the top and bottom of the Landau level. However, states in the center of the Landau level extend over many unit cells of the periodic potential, and, as we demonstrate in the following, are non-trivially modified by its presence. In this paper, we investigate numerically the behavior of a 2DES subject to a perpendicular magnetic field, a periodic potential and a disorder potential, under conditions applicable to the experimental system. The effective electron mass in GaAs is $0.067m_e$ while the magnetic fields of interest are on the order of 10 T. Under these conditions, the cyclotron energy $\hbar \omega_c$, of the order of 200 K, is the largest energy scale in the problem. The Zeeman energy $g^\mu_B B$ for these fields is roughly 3 K, but electron interaction effects lead to a considerable enhancement of the spin splitting between the (spin polarized) Landau levels, which has been measured to be 20 K.[@Sorin2] The amplitude of the periodic potential’s largest Fourier components is estimated to be of the order of 1 K, and the scattering rate from the known zero field mobility is estimated to be $\hbar/\tau \sim 8$ K. [@Chaikin] As a result of this ordering of energy scales, we neglect Landau level inter-mixing and study non-perturbatively the combined effects of a periodic and a large smooth disorder potential on the electronic structure of the lowest Landau level. Previously, the effects of small disorder on a Hofstadter butterfly have been perturbatively investigated using the self-consistent Born approximation (SCBA), [@MacDonald] and the combined effect of white-noise disorder and periodic modulation on Hall resistance was studied following the scaling theory of IQHE.[@Huckestein] Our results reveal details of the electronic structure not investigated previously. The two-lead geometry we consider is schematically shown in Fig. \[fig1\]: the finite 2DES is assumed to have periodic boundary conditions in the $y$-direction (along which the Hall currents flow), and is connected to metallic leads at the $x=-L_x/2$ and $x=+L_x/2$ edges. In particular, in this paper we study the effects of the periodic potential on the extended states carrying longitudinal currents between the two leads, and identify a number of interesting properties, in qualitative agreement with simple arguments provided by a semi-classical picture. Our main conclusion is that while the beautiful Hofstadter structure is destroyed by large disorder, the system still exhibits very interesting and non-trivial physics. ![ The two-lead geometry considered: the finite-size 2DES has periodic boundary conditions in the $y$-direction, and is attached to metallic leads at the $x=\pm L_x/2$ ends. []{data-label="fig1"}](Fig1.eps){width="0.9\FigWidth"} The paper is organized as follows: in Section \[sec2\] we briefly review the computation of the Hofstadter structure for a small-amplitude periodic potential. In Section \[sec3\] we describe the type of disorder potentials considered. Section \[sec4\] describes the numerical methods used to analyze the spectrum and the nature of the electronic states, with both semi-classical and fully quantum-mechanical formalisms. Results are presented in Section \[sec5\], while Section \[sec6\] contains discussions and a summary of our conclusions. Periodic Potential {#sec2} ================== To clarify our notation, we briefly review the problem of a free electron of charge $-e$ moving in a 2D plane (from now on, the $xy$-plane, of dimension $L_x \times L_y$) in a magnetic field ${{\bf B}}=B{{\bf e}}_z$ perpendicular to the plane, as described by $${\cal H} = { 1 \over 2m} \left({{\bf p}}+{e \over c}{{\bf A}}\right)^2 - {1 \over 2} g\mu_B\vec{\sigma}\cdot {{\bf B}}$$ In the Landau gauge ${{\bf A}} = (0, Bx,0)$, the eigenfunctions of the Schrödinger equation ${\cal H}\|n,k_y,\sigma\rangle= E_{n,\sigma}\|n,k_y,\sigma\rangle$ are: $${ \label{2.1} } \langle {{\bf r}} \| n, k_y, \sigma \rangle = { e^{-ik_yy} \over \sqrt{L_y}} e^{-{1 \over 2}\left( {x \over l} - lk_y\right)^2 } { H_n\left({x \over l} - lk_y\right) \over \sqrt{2^nn!\sqrt{\pi}l}}\chi_\sigma,$$ with eigenenergies $${ \label{2.2} } E_{n,\sigma} = \hbar\omega_c\left( n + { 1 \over2 }\right) - {1 \over2} g \mu_B B\sigma.$$ Here $l= \sqrt{ \hbar c/ eB}$ is the magnetic length, $\omega_c ={eB/ mc}$ is the cyclotron frequency, $H_n(x)$ are the Hermite polynomials and $\chi_{+1}^T =(1\; 0)$, respectively $\chi_{-1}^T =(0\; 1)$ are the eigenspinors of $\sigma_z$: $\sigma_z \chi_\sigma =\sigma \chi_\sigma$. The degeneracy of a Landau level is given by the number of distinct $k_y$ values allowed. Imposing cyclic boundary conditions in the $y$-direction, we find $${ \label{2.3} } k_y = { 2\pi \over L_y} j,$$ where $j$ is an integer. The allowed values for $j$ are found from the condition that the electron wave-functions, which are centered at positions $x_j = l^2 k_y = l^2 2\pi j/L_y $ \[see Eq. (\[2.1\])\] are within the boundary along the $x$-axis, i. e. $-L_x/2 < x_j \le L_x/2$. It follows that the degeneracy of each Landau level is $N = L_xL_y B/ \phi_0$, with $\phi_0 = hc/e$. Consider now the addition of a periodic potential, with a lattice defined by two non-collinear vectors ${{\bf a}}_1$ and $ {{\bf a}}_2$, such that $V({{\bf r}}) = V({{\bf r}} + n {{\bf a}}_1 + m{{\bf a}}_2)$ for any $n, m \in {\cal Z}$. The periodic potential has non-vanishing Fourier components only at the reciprocal lattice vectors ${{\bf g}} = h {{\bf g}}_1 + k {{\bf g}}_2$, where ${{\bf g}}_i\cdot{{\bf a}}_j = 2 \pi \delta_{ij}$ and $h,k$ are integers. Thus: $${ \label{2.4} } V({{\bf r}}) = \sum_{{{\bf g}}} V_{{{\bf g}}} e^{i{{\bf r\cdot g}}}.$$ Further, since $V({{\bf r}})$ is real, it follows that $V_{{{\bf g}}} = V^_{-{{\bf g}}}$. In the absence of Landau level mixing, the Hofstadter spectrum for both square[@Hofst] $${ \label{2.6} } V_s(x,y) = 2A\left[ \cos{2 \pi \over a}x +\cos{2 \pi \over a}y \right],$$ and triangular[@Wannier] $$V_t(x,y) = -2A\left[\cos {4 \pi \over \sqrt{3}a}x + \cos {2 \pi \over \sqrt{3}a}\left(x -y \sqrt{3} \right) \right.$$ $${ \label{2.7} } \left. + \cos {2 \pi \over \sqrt{3}a}\left(x +y \sqrt{3} \right)\right]$$ periodic potentials, with nonzero Fourier components only for [the shortest reciprocal lattice vectors]{}, have been studied extensively in the literature.[@Hofst; @Wannier; @Geisel; @Gerhardts] The parameter defining the spectrum is the ratio between the flux $\phi={{{\bf B}}}\cdot({{\bf a}}_1 \times {{\bf a}}_2)$ of the magnetic field through a unit cell and the elementary flux $\phi_0$. For $\phi/\phi_0=q/p$, where $p$ and $q$ are mutually prime integers, the original Landau level is split into $q$ sub-bands. We would like to emphasize a qualitative difference between the two types of potentials: the square potential in Eq. (\[2.6\]) is particle-hole symmetric, since $V_s(x,y) = -V_s(x+{a\over 2},y+{a \over 2})$. As a result, the sign of its amplitude is irrelevant. On the other hand, the triangular potential does not have this symmetry. With the sign chosen in Eq. (\[2.7\]) and $A>0$, $V_t$ has deep local minima at the sites of the triangular lattice, whereas the maxima are relatively flat and located on a (displaced) honeycomb lattice. As a result, the sign of $V_t$ is highly relevant. The second fact that must be mentioned is that the choice made in Eqs. (\[2.6\]) and (\[2.7\]) is rather simple, since it aligns the periodic potential with the edges of the sample in a very specific way. In general, however, one could consider the case where the periodic lattice is rotated by some finite angle with respect to the sample edges; study of such cases will be discussed in future work. Finally, it may seem that this choice of periodic potentials is very restrictive also because only the shortest lattice vectors have been kept in the Fourier expansion. In fact, the methods we employ can be directly used for potentials with more Fourier components, but their inclusion leads to no new physics. Disorder Potential {#sec3} ================== Real samples always have disorder. The current consensus is that high-quality GaAs/AlGaAs samples exhibit a slowly varying, smooth disorder potential. In a semi-classical picture, the allowed electron trajectories in the presence of such disorder follow its equipotential lines.[@Springer; @Trugman] Closed trajectories imply localized electron states, while extended trajectories connecting opposite edges of the sample are essential for current transport through the sample (for more details, see Sec. \[classical\]). In typical experimental setups,[@Sorin] dopant Si impurities with a concentration of $\sim10^{13}$ cm$^{-2}$ are introduced in a thin layer of 6 nm in thickness, located 20 nm above the GaAs/AlGaAs interface. Typically, up to 10% of the Si atoms are ionized. A small fraction of the ionized electrons migrate to the GaAs/AlGaAs interface where they form the 2D electron gas. The electrostatic potential created by the ionized impurities left behind is the major source of disorder in the 2DES layer. On the length-scale we are interested in, there are $10^4$ to $10^5$ such ionized Si impurities per $\mu m^2$. The resulting disorder potential must be viewed as a collective effect of the density fluctuation of the ionized impurities[@Nixon] rather than a simple summation of the Coulomb potential of a few impurities. The electrostatic potential from Si impurities is compensated and partially screened by other mobile negative charges in the system such as, for example, the surface screening effect by mirror charges considered by Nixon and Davies.[@Nixon] An exact treatment of this problem is difficult, since one should consider the spatial correlation of the ionized impurities. [@Stopa; @DasSarma] One model used to describe such disorder consists of randomly placed Gaussian scatterers.[@Ando] This model captures the main feature of a smooth disorder potential and supports classical trajectories on equipotential contours, but it has no natural energy/length scales associated with it. As a result, here we choose to also investigate a different model of the disorder, which incorporates the smooth character of the Coulomb potential in real space. We generate a realization of the disorder potential in the following way: positive and negative charges, corresponding to a total concentration of $10^{3}\,\mu m^{-3}$ are randomly distributed within a volume $[-L_x/2,L_x/2]\times[-L_y/2,L_y/2]\times[20\mbox{nm}+d,26\mbox{nm}+d]$ above the electron gas which is located in the $z=0$ plane. Here, we choose $d=4$ nm as an extra spacer since the electronic wave-functions are centered about 3-5 nm below the GaAs/AlGaAs interface. Since we are not simulating single impurities but density fluctuations, these charges are not required to be elementary charges. Instead, we use a uniform distribution in the range $[-e,e]$ for convenience (a Gaussian distribution would also be a valid choice), and sum up all Coulomb potentials from these charges, using the static dielectric constant in GaAs $\epsilon = 12.91$.[@Ralph] The resulting disorder potential has energy and length scales characteristic of the real samples. Typical contours for such potentials are shown in Sec. \[sec5\]. In an infinite system, in the quantum Hall regime, the existence of quantum Hall steps implies the existence of critical energies at which the localization length diverges.[@Halperin] This is the quantum analog of the two dimensional percolation problem in a smooth random landscape, for which there exists a single critical energy.[@Trugman] In the case of potentials with electron-hole symmetry $\left<V({{{\bf r}}})\right>=0$, the critical energy lies in the middle of the band ($E_c=0$), leading to percolating path at half filling. For a finite mesoscopic sample, however, not only does the percolating path (critical energy) $E_c$ deviate from this value, but in samples without a periodic boundary condition one need not have a percolating path traversing the system in the desired direction. This arises from the fluctuations near the edge of a mesoscopic system with free boundary conditions. We circumvent such a possibility by adding an extra smooth potential $V'(x,y)$ to the impurity-induced disorder potential $V_i(x,y)$, such that the total potential $V = V_i + V'$ is zero on the opposite edges $ x= \pm L_x/2$ of the sample where the metallic leads are attached. The supplementary contribution $V'(x,y)$ can be thought of as simulating the effect of the leads on the disorder potential, since the metallic leads hold the potential on each edge constant by accumulating extra charges near the interface. Therefore, physically we expect that the extra potential $V'$ decays exponentially over the screening length $\lambda$ inside the sample. This implies: $$V'(x,y) = -{V_i\left(-{L_x/2},y\right)+V_i(L_x/2, y) \over 2 } {\cosh(x/\lambda)\over \cosh(L_x /2\lambda)}$$ $$+{V_i(-L_x/2,y)-V_i(L_x/2,y) \over 2 } {\sinh(x/\lambda)\over\sinh(L_x /2\lambda)}$$ where $\lambda$ is taken to be $100$ nm in our calculation. ![Averaged Fourier amplitudes of two types of disorder potential as a function of wavevector $q = \|{{{\bf q}}}\|$. For both Coulomb and Gaussian model, $V(q)^2$ is averaged over 116 disorder realizations. The relation between $V(q)$ and $V({{\bf r}})$ and relevant parameters are discussed in the text. The standard deviation, $(L_xL_y)^{-1} \left<\int d{{\bf r}} V^2({{\bf r}})\right>$ for the Coulomb model is $3.2\times10^{-7}$eV$^2$, and $2.1\times10^{-7}$eV$^2$ for the Gaussian model.[]{data-label="fig1.5"}](Fig2.eps){width="\FigWidth"} In Fig. \[fig1.5\], we plot the average of Fourier transform of the magnitude of the random potential $\sqrt{\left< \|V(q)\|^2\right>}$ versus $q = \|{{{\bf q}}}\|$ for the Coulomb model and the Gaussian model. The Gaussian model is generated by adding 100 randomly placed Gaussian scatterers on an area of $3\mu m \times 3\mu m$, each contributing $A_d e^{-r^2/d^2}$, where $A_d$ is uniformly distributed in $[-2,2]$ meV, and $d$ is uniformly distributed in $[0,0.2]$ $\mu$m. $V({{\bf q}})$ is related to $V({{\bf r}})$ by $V({{\bf r}}) = \sum_{{{\bf q}}} V({{\bf q}}) e^{i{{{\bf q}} \cdot r}}$, where the summation is over all the wavevectors involved in the fast Fourier transformation. The Gaussian model has an arbitrary energy scale which is fixed by the maximum value of the distribution $A_m$. Here $A_m = 2$ meV. As can be seen, $V(q)$ of both models are decreasing functions of $q$. The trends of decay are exponential at large $q$. At small $q$, the two models behave differently. Despite the difference, both models lead to the same qualitative results, although, as expected, minor quantitative differences are present. This shows that the physics we uncover is independent of the particular type of slowly-varying disorder potential considered, and therefore should be relevant for the real samples. Numerical Calculations {#sec4} ====================== In this section we discuss the numerical methods we use, including derivations of some relevant formulas. As already stated, we focus on the case where the amplitudes of the periodic and disorder potentials are very small compared to the cyclotron energy and the Zeeman splitting, and therefore inter-level mixing is ignored. Semi-classical treatment {#classical} ------------------------ The semi-classical approach is valid[@Trugman] for the integer quantum Hall effect in the presence of a slowly varying, smooth disorder potential and large magnetic fields (such as we consider), so that the magnetic length $l$ which determines the spatial extent of the electron wave-functions is much smaller than the length scale of variation of the smooth disorder potential, $\|\nabla V({{\bf r}})\| \ll \hbar\omega_c/l$. Then, semi-classically the electron moves along the equipotential contours of the disorder potential $V({{\bf r}})$, in the direction parallel to $\nabla V({{\bf r}}) \times {{\bf B}} $. Since the kinetic energy is quenched in the lowest Landau level, the total energy of the electron simply equals the value of the disorder potential on the equipotential line on which its trajectory is located. As a result, the density of states in the semi-classical approach is directly given by the probability distribution for the disorder potential, which can be calculated by randomly sampling the potential energy and plotting a histogram of the obtained values.[@Trugman; @note1] In Sec. \[sec5\] we compare the results obtained within this semi-classical approach with fully quantum mechanical results. As expected, the agreement is good if only the disorder potential is present. However, if the periodic modulation is also included, the lattice constant $a$ provides a new length-scale which is comparable to the magnetic length $l$, and the semi-classical picture breaks down. Quantum mechanical calculations are absolutely necessary to quantitatively treat this case. Quantum Mechanical Treatment ---------------------------- As shown in Sec. \[sec2\], for a finite sample of size $L_x \times L_y$ at a given magnetic field $B$, the degeneracy of the unperturbed Landau level is $N = L_xL_yB/\phi_0 = L_xL_y/(2\pi l^2)$. Since the disorder varies very slowly, we need to consider systems with $L_x,L_y \gg l$ to properly account for its effects. As a result, the number of states in a Landau level can be as large as $10^4$ in our calculations. Storage of the Hamiltonian as a dense matrix requires considerable amount of computer memory and its direct diagonalization is prohibitively time-consuming. Sparse matrix diagonalization techniques could be employed, but they are less efficient when all eigenvectors are needed, and also have some stability issues. Here we introduce the numerical methods we use to compute densities of states and infer the nature (localized or extended) as well as the spatial distribution of the wave-functions, while avoiding direct diagonalization. ### Matrix elements Since inter-level mixing is ignored, the Hilbert subspaces corresponding to different spin-polarized Landau levels do not hybridize. Each Hilbert subspace $(n,\sigma)$ has a basis described by Eq. (\[2.1\]), containing $N$ orthonormal vectors indexed by different $k_y$ values. In order to compute matrix elements of the total Hamiltonian in such a basis, we use the following identity derived in Ref. (notice their different sign convention for $k_y$. If $\sigma\ne \sigma'$, the overlap is zero): $${ \label{2.5} } {\left< n', k'_y \right\|} e^{i{{\bf q\cdot r}}}{\left\|n,k_y \right>} = \delta_{k'_y,k_y-q_y} {\cal L}_{n',n}({{\bf q}}) e^{{il^2\over 2}q_x(k'_y+k_y)},$$ where $$\begin{aligned} {\cal L}_{n',n} ({{\bf q}}) = \left(m! \over M! \right)^{1 \over 2}i^{\|n'-n\|}\left[ q_x+iq_y \over \sqrt{q_x^2+q_y^2}\right]^{n-n'} && \\ \times e^{-{1 \over 2} Q}Q^{{1 \over 2}\|n'-n\|}L_m^{(\|n'-n\|)}(Q), &&\end{aligned}$$ with $Q = {1 \over 2}l^2(q_x^2+q_y^2)$, $m$ and $M$ the minimum and the maximum of $n'$ and $n$ respectively, and $L^{(\|n'-n\|)}_m(Q)$ the associated Laguerre polynomial. When band-mixing is neglected $n = n'$ and ${\cal L}_{n,n} ({{\bf q}}) = e^{-{1 \over 2}Q} L_n(Q)$. For the first Landau level, $L_0(x) =1$. Eq. (\[2.5\]) gives us the matrix elements for the square \[Eq. (\[2.6\])\] or triangular \[Eq. (\[2.7\])\] periodic potentials. In either case, there are Fourier components corresponding to $q_y=\pm 2\pi/a$ and $q_y=0$. Since only basis vectors for which the difference $k_y-k_y'=q_y$ give non-vanishing matrix elements, we must choose the length $L_y$ of the sample to be a multiple integer of $a$, the lattice constant. The matrix elements of the disorder potential are computed in a similar way. We use a grid of dimension $N_x\times N_y$ to cover the sample and generate the values of the disorder potential on this grid. Then, fast Fourier transform (FFT)[@FFT] is used to find the long wavelength components of the disorder potential corresponding to the allowed values $q_{x,y}=0,\pm{2\pi\over L_{x,y}}, ..., \pm \left[{N_{x,y}\over 2}\right]{ 2\pi\over L_{x,y}}$ (proper care is taken to define Fourier components so that $V_{{{\bf q}}} = V^_{-{{\bf q}}}$). The matrix elements of this discretized disorder potential are then computed using Eq. (\[2.5\]). In principle, finer grids (increased values for $N_x$ and $N_y$) will improve accuracy. However, they also result in longer computation times, since they add extra matrix elements in the sparse matrix, corresponding to large wave-vectors. We have verified that a grid size of dimension $N_x=N_y=72$ is already large enough to accurately capture the landscape of a $3\mu m \times 3\mu m$ sample and the computed quantities have already converged, with larger grids leading to hardly noticeable changes. This procedure is also justified on a physical basis. First, the neglected large wave-vector components describe very short-range spatial features, which are probably not very accurately captured by our disorder models to begin with, and which are certainly not believed to influence the basic physics. Secondly, this procedure insures that the actual disorder potential we use is periodic in the $y$-direction, since each Fourier component retained has this property. This is consistent with our use of a basis of wave-functions which are periodic along $y$. The matrix elements of the Hamiltonian within a given Landau level $(n,\sigma)$ are then $\langle n, k_y,\sigma \| {\cal H} \| n, k'_y,\sigma\rangle= E_{n,\sigma} + \langle n, k_y \| {\cal V} \| n, k'_y\rangle$, where $E_{n,\sigma}$ are given by Eq. (\[2.2\]) and the matrix elements of both the periodic and the disorder part of the potential ${\cal V}$ are computed as already discussed. This produces a sparse matrix, which is stored efficiently in a column compressed format. ### Densities of States and Filling Factors {#subs2} A quantity that can be computed without direct diagonalization is the filling factor $\nu_{n,\sigma}(E_F)$ as a function of Fermi energy. The filling factor is defined as: $${ \label{4.6} } \nu_{n,\sigma}(E_F) = { 1 \over N} \sum_{\alpha} \Theta(E_F- E_{n,\alpha,\sigma}),$$ where $\Theta(x)$ is the Heaviside function and $N$ is the total number of states in the $(n,\sigma)$ Landau level. (Since we neglect Landau-level mixing, we can define this quantity for individual levels.) The filling factor tells us what fraction of the states in the given Landau level are occupied at $T=0$, for a given value of the Fermi energy. This corresponds to the average filling factor measured in experiment and is also proportional to the integrated total (as opposed to local) density of states. The filling factor is straightforward to compute if the eigenenergies $E_{n,\alpha,\sigma}$ are known. However, we want to avoid the time-consuming task of numerical brute force diagonalization. The strategy we follow is a generalization to Hermitian matrices of the method used in Ref. . We restate the problem in the following way: assume we have a Hermitian matrix of size $N \times N$ (no Landau level mixing), given by the matrix elements of $M = {\cal H} - E_F {\bf 1} $ in the basis $\|n, k_y, \sigma\rangle$ (${\bf 1}$ is the unit matrix). Then, $\nu_{n,\sigma}(E_F)$ is proportional to the number of negative eigenvalues of the matrix $M$. We now generate the quadratic form ${\cal M} = \sum_{i,j = 1}^{N} \zeta_i \zeta_j^ M_{ij}$, and transform it into its standard form ${\cal M} = \sum_{i= 1}^{N} d_i\|\chi_i\|^2$ using the Jacobian method described below. Here, $d_i$’s are all real numbers, and the $\chi_i$’s are linear combinations of the $\zeta_i$’s. This is a similarity transformation which retains the signature of the matrix. As a result, even though the numbers $d_i$ are not eigenvalues of $M$, the number of negative eigenvalues equals the number of negative $d_i$ values. It follows that $\nu_{n,\sigma}(E_F)$ is obtained by simply counting the number of negative $d_i$ values for the given $E_F$. The Jacobian method is iterative in nature. First, all terms containing $\zeta_1$ and $\zeta_1^$ are collected and the needed complementary terms are added to form the first total square $d_1\|\chi_1\|^2$, so that $\zeta_1$ and $\zeta_1^$ are eliminated from the rest of the quadratic form ${\cal M}$. The procedure is then repeated for all $\zeta_2$ and $\zeta_2^$ terms (producing $d_2$) etc., until all $N$ values $d_i$ are found. Computationally, this can be done by scanning the lower or upper triangle of the Hermitian matrix $M$ only once. The total number of operations is proportional to the number of nonzero elements of the matrix, meaning that for a dense matrix it scales with $N^2$ (sparse matrices require much fewer operations). As a result, this procedure is much faster than brute force diagonalization which scales with $N^3$ (for us, $N\sim 10^4$). The filling factor $\nu_{n,\sigma}(E)$ is a sum of step-like functions, with steps located at the eigenvalues. By scanning $E$ and identifying the position of these steps we can also find the true eigenvalues $E_{n,\alpha,\sigma}$, with the desired accuracy. Finally, the total density of states is given by $ \rho_{n\sigma}(E) = d \nu_{n,\sigma}(E)/dE$. ### Green’s functions: extended vs. localized states {#subs3} The advanced/retarded Green’s functions are the solutions of the operator equation $${ \label{4.1} } \left(\hbar\omega - {\cal H} \pm i \delta\right) {\hat G}^{R,A}(\omega) = {\bf 1},$$ where $\delta \rightarrow 0^{+}$. (In practice we use a set of small positive numbers, and use the dependence on $\delta$ to obtain results.) If the exact eigenstates and eigenvalues of the total Hamiltonian ${\cal H}$ are known, ${\cal H} \| n, \alpha, \sigma\rangle = E_{n,\alpha,\sigma} \| n, \alpha, \sigma\rangle, $ (no Landau level mixing), it follows: $${ \label{4.2} } {\hat G}^{R,A}(\omega) = \sum_{n,\alpha,\sigma}^{}{ \| n,\alpha,\sigma\rangle\langle n, \alpha, \sigma\| \over \hbar\omega - E_{n,\alpha,\sigma} \pm i\delta} = \sum_{n,\sigma}^{} {\hat G}^{R,A}_{n,\sigma}(\omega).$$ The exact eigenstates can be expanded in terms of the basis states $\|n, k_y,\sigma\rangle$ as $${ \label{4.3} } \| n, \alpha, \sigma\rangle = \sum_{k_y} c_{n,\alpha}(k_y) \|n, k_y,\sigma\rangle.$$ Since the states $\|n, k_y,\sigma\rangle$ are localized near $x=k_yl^2$ \[see Eq. (\[2.1\])\], the coefficients $c_{n,\alpha}(k_y)$ describe the probability amplitude for an electron in the state $\| n, \alpha, \sigma\rangle$. Knowledge of these coefficients allows us to infer whether such states are extended or localized in the $x$-direction, i.e. whether they can carry currents between the leads. However, as already stated, we wish to avoid direct diagonalization. We can still infer whether the Hamiltonian has extended or localized wave-functions near a given energy $\hbar\omega$ in the following way. We introduce the matrix elements: $$G^{R,A}_{n,\sigma}(k_y, k'_y; \omega) = \langle n, k_y,\sigma \| {\hat G}^{R,A}(\omega) \| n, k'_y,\sigma\rangle$$ $${ \label{4.4} } = \sum_{\alpha}^{} { c_{n,\alpha}(k_y) c^_{n,\alpha}(k'_y) \over \hbar\omega - E_{n,\alpha,\sigma} \pm i\delta}.$$ If Landau level mixing is neglected, Eq. (\[4.1\]) can be rewritten in the basis $\|n, k_y,\sigma\rangle$ as: $$\sum_{k''_y}^{} \left[(\hbar\omega \pm i\delta) \delta_{k_y,k''_y} - \langle n, k_y,\sigma \| {\cal H} \| n, k''_y,\sigma\rangle \right]$$ $${ \label{4.5} } \times G^{R,A}_{n,\sigma} (k''_y,k'_y; \omega)=\delta_{k_y,k'_y}.$$ We use the popular numerical library SuperLU,[@SuperLU] based on LU decomposition and Gaussian reduction algorithm for sparse matrices, to solve these linear equations. Consider now the matrix element $ G^{R,A}_{n,\sigma}(k_{\min},k_{\max}; \omega) $ corresponding to the smallest $k_y=k_{\min}$ and the largest $k_y=k_{\max}$ values. If all wave-functions with energies close to $\hbar\omega$ are localized in the $x$-direction, it follows that $\|G^{R,A}_{n,\sigma}(k_{\min},k_{\max}; \omega)\| $ is a very small number, of the order $e^{-L_x/\xi(\omega)}$, where $\xi(\omega)$ is the localization length at the given energy. On the other hand, we expect to see a sharp peak in the value of $\|G^{R,A}_{n,\sigma}(k_{\min},k_{\max}; \omega)\| $ if $\hbar\omega$ is in the vicinity of an extended state eigenvalue, since \[see Eqs. (\[4.3\],\[4.4\])\] both $c_{n,\alpha}(k_{\min})$ and $c_{n,\alpha}(k_{\max})$ are non-vanishing for an extended wave-function with significant weight near both the $-L_x/2$ and the $L_x/2$ edges. Moreover, the height of this peak scales like $1/\delta$, so by varying $\delta$ we can easily locate the energies of the extended states. ### Green’s functions: local densities of states {#subs4} We can also use Green’s functions techniques to image the local density of states at a given energy $E$. By definition (and neglecting Landau level mixing), the local density of states in the level $(n,\sigma)$ is: $$\rho_{n,\sigma}({{\bf r}};E) = \sum_{\alpha}^{} \|\langle {{\bf r}}\| n, \alpha, \sigma\rangle \|^2 \delta\left(E-E_{n,\alpha,\sigma}\right)$$ $${ \label{5.1} } = { 1 \over \pi} \mbox{Im}\langle {{\bf r}}\| {\hat G}^{A}_{n,\sigma}(E)\|{{\bf r}}\rangle,$$ where the second equality follows from Eq. (\[4.2\]). This function traces the contours of probability $\|\phi_{n,\alpha,\sigma}({{\bf r}})\|^2$ for electrons with the given energy $E$. Its direct computation, however, is difficult and very time-consuming. For the rest of this subsection, the discussion is restricted to the Lowest Landau level $n=0$ (the value of $\sigma$ is irrelevant). We know that in the lowest Landau level, electronic wave-functions cannot be localized in any direction over a length-scale shorter that the magnetic length $l$. As a result, it suffices to compute a projected local density of states on a grid with $l\times l$ (or larger) spacings. The projection is made on maximally localized wave-function, defined as follows. Let ${{\bf r}}_0=(x_0,y_0)$ be a point on the grid. We associate it with a vector: $${ \label{5.2} } \|x_0, y_0\rangle = \sum_{k_y}^{} \|k_y\rangle \langle k_y\| x_0,y_0\rangle,$$ where we use the simplified notation $\|k_y\rangle \equiv \|n=0,k_y,\sigma\rangle$ for the basis states of the first Landau level (see Eq. (\[2.1\])) and we take $${ \label{5.3} } \langle k_y\| x_0,y_0\rangle = \sqrt{{2l\pi^{1 \over2}} \over L_y} e^{-{x_0^2 \over 2l^2} - {k_y^2l^2\over 2} + k_y(x_0+iy_0)}.$$ It is then straightforward to show that $${ \label{5.4} } \langle {{\bf r}}\| x_0, y_0\rangle = { 1\over \sqrt{2\pi}l} e^{- {(x-x_0)^2 \over 4l^2 } - {(y-y_0)^2 \over 4l^2 }}e^{-{i \over 2l^2}(x+x_0)(y-y_0)}.$$ In other words, $\|x_0,y_0\rangle$ is an eigenstate of the first Landau level strongly peaked at ${{\bf r}}= {{\bf r}}_0$. (The phase factor is due to the proper magnetic translation). We then define the projected density of states \[compare with Eq. (\[5.1\])\]: $${ \label{5.5} } \rho_P (x_0,y_0; E) = { 1 \over \pi} \mbox{Im} \langle x_0,y_0\|\hat{G}^{A}(E)\|x_0,y_0\rangle,$$ and use it to study the spatial distribution of the electron wave-functions at different energies. Strictly speaking, the local density of states defined in Eq. (\[5.1\]) cannot be projected exactly on the lowest Landau level, because the lowest Landau level does not support a $\delta$-function ($\langle {{\bf r}}\| n,k_y,\sigma\rangle \ne 0$, $\forall n$). However, the coherent states $\|x_0,y_0\rangle$ we select are the maximally spatially-localized wave functions in the lowest Landau level, and have the added advantage that they can be easily stored as sparse vectors, because of their Gaussian profiles \[see Eq. (\[5.3\])\]. Moreover, in the limit $l \rightarrow 0$ ($B \rightarrow \infty$) where $\|\langle{{\bf r}} \| x_0, y_0\rangle\|\rightarrow \delta(x-x_0)\delta(y-y_0)$, the projected density of states $\rho_P(x_0,y_0; E)\rightarrow \rho_{0,\sigma}({{\bf r}}; E)$. Therefore, for the large $B$ values that we consider here, the projected density of states $\rho_P$ should provide a faithful copy of the local density of states. We compute the projected local density of states following the method of Ref. . Let ${{\bf u}}_0$ be the vector with elements $\langle k_y\| x_0, y_0\rangle$ obtained from the representation of $\|x_0,y_0\rangle$ in the $\|k_y\rangle$ basis \[see Eq. (\[5.2\])\], and let $H$ be the matrix of the Hamiltonian ${\cal H}$ in the $\|k_y\rangle$ basis. We generate the series of orthonormal vectors ${{\bf u}}_0, {{\bf u}}_1, ...$ using: $$\begin{aligned} {{\bf v}}_1 &=& H {{\bf u}}_0 ,\\ a_0 &=& {{\bf u}}_0^\dagger {{\bf v}}_1 ,\\ {{\bf u}}_1 &=& { {{\bf v}}_1 - a_0{{\bf u}}_1 \over \sqrt{{{\bf v}}_1^\dagger {{\bf v}}_1-a_0^2}},\end{aligned}$$ and for $n \ge 2$, $$\begin{aligned} {{\bf v_n}} &=& H {{\bf u}}_{n-1} ,\\ a_{n-1} &=& {{\bf u}}_{n-1}^\dagger {{\bf v_n}} ,\\ b_{n-2} &=& {{\bf u}}_{n-2}^\dagger {{\bf v_n}} ,\\ {{\bf u}}_n &=& {{{\bf v}}_n - a_{n-1} {{\bf u}}_{n-1} - b_{n-2}{{\bf u}}_{n-2} \over \sqrt{{{\bf v}}_n^\dagger {{\bf v}}_n - a_{n-1}^2 - b_{n-2}^2 }}.\end{aligned}$$ The numbers $a_n$ and $b_n$ can be shown to be real. We do not have a “terminator”[@Haydock] to end this recursive series. Instead, our procedure ends when the orthonormal set of vectors ${{\bf u}}_0, {{\bf u}}_1, ...$ exhausts a subspace of the lowest Landau level containing all states coupled through the disorder and/or periodic potential to the state $\|x_0,y_0\rangle$ (i.e., all states that contribute to the projected DOS at this point). In the presence of disorder, this usually includes the entire lowest Landau level. Then, the projected density of states is given by Eq. (\[5.5\]), where the matrix element of the Green’s function is the continued fraction: $$\langle x_0,y_0\|G^A(E)\|x_0,y_0\rangle=$$ $${ \label{5.6} } \left[E-i\delta-a_0 -b_0^2\left[E-i\delta - a_1 -b_1^2\left[ \ldots\right]^{-1} \right]^{-1} \right]^{-1}$$ Because the Hamiltonian is a sparse matrix, the generation of these orthonormal sets and computation of $\rho_p(E)$ for all the grid points is a relatively fast procedure. Moreover, this computation is ideally suited for parallelization, with different grid points assigned to different CPUs. Numerical Results {#sec5} ================= In this section we present numerical results obtained using these methods. We have analyzed over 20 different disorder realizations for samples of different sizes, and all exhibit the same qualitative physics. Here, we show results for several typical samples. The lattice constant is always $a=39$ nm if periodic potential is present, as defined by the experimental system.[@Sorin] ![\[fig2\] Profile of the disorder potential obtained from our Coulomb model on a $3.11\mu$m$\times 2.96\mu$m sample, without (upper panel) and with (lower panel) the $V'({{\bf r}})$ correction at the $x=\pm L_x/2$ edges. The disorder potential varies between $-3$ meV and 3 meV, on a spatial length-scale much larger than $l=12.03$ nm. The critical region containing extended states is in the vicinity of $E=0.06$ meV. The contours are shown for $E=$0.0575 meV (dashed), 0.17 meV (thick solid) and 0.31 meV (thin solid). These energy values correspond to classical filling factors $\nu$=0.47, 0.58 and 0.68 in the upper panel and $\nu$=0.45, 0.56, 0.66 in the lower panel. The difference is due to the supplementary smooth potential $V'$.](Fig3a_s.eps "fig:"){width="\FigWidth"} ![\[fig2\] Profile of the disorder potential obtained from our Coulomb model on a $3.11\mu$m$\times 2.96\mu$m sample, without (upper panel) and with (lower panel) the $V'({{\bf r}})$ correction at the $x=\pm L_x/2$ edges. The disorder potential varies between $-3$ meV and 3 meV, on a spatial length-scale much larger than $l=12.03$ nm. The critical region containing extended states is in the vicinity of $E=0.06$ meV. The contours are shown for $E=$0.0575 meV (dashed), 0.17 meV (thick solid) and 0.31 meV (thin solid). These energy values correspond to classical filling factors $\nu$=0.47, 0.58 and 0.68 in the upper panel and $\nu$=0.45, 0.56, 0.66 in the lower panel. The difference is due to the supplementary smooth potential $V'$.](Fig3b_s.eps "fig:"){width="\FigWidth"} ![ Semi-classical (dashed line) and quantum (solid line) filling factors for the disorder potential shown in Fig. \[fig2\], but different amplitudes of the triangular periodic potential (a) $A$=0; (b) $A$=0.05meV; (c) $A$=0.5meV and (d) $A$=5 meV. As expected, agreement exists only in the limit $A \rightarrow 0$. []{data-label="fig3"}](Fig4.ps){width="\FigWidth"} ![ Semi-classical (dashed line) and quantum (full line) density of states calculated from corresponding filling factors in Fig. \[fig3\]. We show only the center of the disorder-broadened lowest Landau level, where the density of states is large. []{data-label="fig4"}](Fig5.ps){width="\FigWidth"} For the first sample, we consider $\phi/\phi_0=3/2$ ($B = 4.71$ T). The magnetic length is $l= 12.03$ nm, and we choose a sample size $L_x=3.11\mu$m and $L_y= 76a=2.964\mu$m. With these choices, the Landau level contains $N=10108$ states. The disorder potential obtained with our scheme described in Sec. \[sec3\] is shown in Fig. \[fig2\], both with and without the correction $V'({{\bf r}})$. An extended equipotential line appears, as expected, at $\nu \approx 0.5$. In Figs. \[fig3\] and \[fig4\] we plot the filling factor $\nu(E)$ and the corresponding total density of states $\rho(E)$ as a function of $E$ (computation details were given in Sec. \[subs2\]). These quantities are obtained in the semi-classical limit (dashed line) and with the full, quantum-mechanical treatment (solid line). Results are shown for 4 different cases: (a) only disorder potential and (b, c, d) disorder plus a triangular periodic potential with amplitudes $A=0.05$, 0.5 and 5 meV, respectively. We only plot a relatively small energy interval where the DOS is significant, and ignore the asymptotic regions with long tails of localized states. While the agreement between the semi-classical and quantum-mechanical treatment is excellent in the limit $A\rightarrow 0$, the two methods give more and more different results as the periodic potential amplitude is increased. As already explained, this is a consequence of the fact that the magnetic length $l$ is comparable to the lattice constant $a$, leading to a failure of the semi-classical treatment when this extra length-scale is introduced. In particular, in the case with the largest periodic potential \[panel (d) of Figs. \[fig3\] and \[fig4\]\] we can clearly see the appearance of the 3 subbands expected for the Hofstadter butterfly at $\phi/\phi_0=3/2$, although the disorder leads to broadened and smooth peaks, and partially fills-in the gap between the lower two subbands. This picture \[panel (d)\] is quite similar to the density of states that Ref. calculated using the self-consistent Born approximation. This is expected since the SCBA approach is valid in the limit of strong periodic potential with weak disorder. However, the SCBA approach is not appropriate in the limit of moderate or strong disorder, where the higher order terms neglected in SCBA are no longer small. For disorder varying on a much longer length-scale than the periodic potential, one still expects that [locally]{}, on relative flat regions of disorder, the system exhibits the Hofstadter-type spectrum. However, these spectra are shifted with respect to one another by the different local disorder values. If disorder variations are small, then the total spectrum shows somewhat shifted subbands with partially filled-in gaps, but overall the Hofstadter structure is still recognizable. On the other hand, for moderate and large disorder, the detailed structure of the local density of states from various flat regions are hidden in the total density of states. All one sees are some broadened, weak peaks and gaps superimposed on a broad, continuous density of states. We now analyze the nature of the electronic states for these configurations. We start with the case which has only disorder. In Fig. \[fig5\] we plot $\|G^R(k_{\min},k_{\max};E)\|^2$ as a function of the energy $E$, for different values of $\delta$ (computation details were given in Sec. \[subs3\]). As already discussed, extended states are indicated by large values of this quantity, as well as a strong (roughly $1/\delta^2$) dependence on the value of the small parameter $\delta$. Figure \[fig5\] reveals that as $\delta$ is reduced, resonant behavior appears in a narrow energy interval $E= 0.02-0.36$ meV. Panel (a) shows that results corresponding to $\delta = 10^{-7}$ eV and $\delta = 10^{-8}$ eV indeed differ by roughly 2 orders of magnitude, with $\delta = 10^{-8}$ eV showing sharper resonance peaks. The difference between results for $\delta = 10^{-8}$ eV and $\delta =10^{-9}$ eV shown in panel (b), is no longer so definite. The reason is simply that for such small $\delta$, the denominator in the Green’s function expression is usually limited by $\|E-E_{n,\alpha,\sigma}\|$ and not by $\delta$ \[see Eq. (\[4.4\])\], and the dependence on $\delta$ is minimal. Only if $E$ is such that $\|E-E_{n,\alpha,\sigma}\| <\delta$ can we expect to see a $\delta$ dependence, and indeed this is observed at some energies. Finally, in panel (c) we show the comparison with a larger energy interval. The value of the Green’s function decreases exponentially fast on both sides of the critical region, indicating strongly localized states. Here, data for $\delta = 10^{-6}$ eV is a smooth curve, whose magnitude is much less than that of the other three values even for localized states. This is due to the fact that this $\delta$ is larger than typical level spacings. As a result, several levels contribute significantly to Green’s function at each $E$ value, and the destructive interference of the random phases of different eigenfunctions lead to the supplementary $\delta$-dependence. We conclude that the disorder potential has a critical energy regime of approximately $0.3$ meV width, covering less than 5% (in energy) and 20% (in number of states) of the disorder-broadened band with total width $\sim 6$ meV. The position of the critical energy interval is in agreement with the semi-classical results which suggest an extended state in the vicinity of $E=0.06$meV. By comparison with Fig. \[fig3\], we can also see that this critical regime corresponds to a roughly half-filled band, in agreement with the experiment. The effect of an additional triangular periodic potential is shown in Fig. \[fig6\], where we plot the same quantity shown in Fig. \[fig5\] for a fixed $\delta = 10^{-7}$ eV and different amplitudes $A=0$, 0.05, 0.5 and 5 meV, respectively. These results correspond to a different Coulomb disorder potential (not shown), as can be seen from the different location of its extended states. Here we see how the narrow critical interval of extended states grows gradually as the amplitude of periodic potential is increased and finally exhibits the three well-separated extended subbands expected for $\phi/\phi_0 = 3/2$ in the limit of vanishing disorder. The three subbands can already be resolved for the moderate case $A = 0.5~$meV, although they are very wide and exhibit significant overlap. ![ Semi-log plot of the amplitude of Green’s function matrix element between the two edge states near $x=\pm L_x/2$, as a function of energy. Only the disorder potential of Fig. \[fig2\] is present. (a) comparison between $\delta=10^{-7}$ and $\delta=10^{-8}$ results; (b) comparison between $\delta=10^{-8}$ and $\delta=10^{-9}$ results; (c) comparison between results corresponding to $\delta =\delta=10^{-6},10^{-7},10^{-8}$ and $10^{-9}$. (the last three curves are indistinguishable to the eye on this scale.) All $\delta$ values are in eV units. []{data-label="fig5"}](Fig6_s.eps){width="\FigWidth"} ![ The effect of a triangular periodic potential on the critical energy regime. The disorder potential used here (not shown) supports a narrow interval of extended states centered at about $-0.6$ meV. As the amplitude $A$ of the periodic potential increases, the range of extended states increases dramatically. The left panel shows results for disorder-only and two relatively weak periodic potentials, while the right panel shows two larger periodic potentials where the three-subband structure expected for $\phi/\phi_0 = 3/2$ is clearly seen.[]{data-label="fig6"}](Fig7_s.eps){width="\FigWidth"} Qualitatively similar behavior is obtained if we use the Gaussian scatterers model for disorder. A typical realization of this disorder is shown in Fig. \[fig7bis\]. Results for the Green’s function’s values with such disorder are shown in Fig. \[fig7\], for cases with pure disorder, and also cases with either a triangular or a square periodic potential. The magnetic field has been doubled, such that $\phi/\phi_0=3$. Similar to the case shown in Fig. \[fig6\], the periodic potential leads to a widening of the critical regime. For large periodic potentials, the expected Hofstadter-like three-subband structure emerges again. We conclude that Coulomb and Gaussian disorder models show qualitatively similar behavior. ![A disorder potential of Gaussian type on a roughly $3\mu m \times 3\mu m$ square. The three lines are equipotential contours close to the critical regime, with energies of -0.1 meV (dashed), 0 meV (thick solid) and 0.1 meV (thin solid). Cyclic boundary condition are applied in the $y$ direction.[]{data-label="fig7bis"}](Fig8_s.eps){width="\FigWidth"} ![ Green’s functions for a sample with Gaussian disorder and various periodic potentials. The calculation included 20216 states with $\phi/\phi_0=3$. Similar to results shown in Fig. \[fig7\], we see that the periodic potentials widen the critical region.[]{data-label="fig7"}](Fig9.eps){width="\FigWidth"} We now analyze the projected local density of states $\rho_P(E)$ discussed in Sec. \[subs4\], in order to understand the reason for this substantial widening of the critical region by even small periodic potentials. We consider a smaller sample, of size approximately $1.6\mu$m$\times1.6\mu$m, and compute the projected density of states for 500 equally-spaced energy values, on a $60\times60$ square grid and for a value $\delta = 10^{-8}$ eV. This $\delta$ value is comparable or smaller than the level spacing, so we expect to see sharp resonances from the contribution of individual eigenfunctions as we scan the energy spectrum. Each computation generates a large amount of data (roughly 24M), corresponding to the 500 plots of the local density of states at the 500 values of $E$. Since we cannot show all this data, we select a couple of representative cases and some statistical data to interpret the overall results. Figures \[fig9\] and \[fig10\] show some of our typical results. The two figures are calculated for the same Coulomb-disorder potential, for values of $E=-0.504$ meV (at the bottom of the band) and $E=-0.124$ meV (close to, but below the band center) respectively. Each figure contains four panels, panel (a) shows the profile of the disorder potential as well as an equipotential line (solid black) corresponding to the value $E$ considered; the other three panels show the projected density of states $\rho_P(E)$ for (b) pure disorder; (c) disorder plus triangular periodic potential with $A=0.1$ meV; (d) disorder plus square periodic potential with $A=0.1$ meV. In Fig. \[fig9\], this equipotential line (which traces the semi-classical trajectory of electrons with the same energy $E$) ![ (color online) Projected local density of states $\rho_P({{\bf r}};E)$ for $E=-0.504$ meV. Panel (a) shows the profile of the disorder potential, and the equipotential contour (black line) corresponding to $E=-0.504$ eV. The other three panels show $\rho_P({{\bf r}};E)$ for (b) disorder only; (c) disorder plus triangular periodic potential with $A = 0.1$ meV; (d) disorder plus square periodic potential with $A = 0.1$ meV. The width and length of the sample are both $1.6\,\mu$m, and $\phi/\phi_0=3/2$. Increased brightness corresponds to larger values.[]{data-label="fig9"}](Fig10_s.eps){width="\FigWidth"} surrounds local minima of the disorder potential, suggesting localized electron states at such low energies. Indeed, this is what panels (b), (c) and (d) show. The projected density of states $\rho_P(E)$ is large (bright color) at the positions where electrons of energy $E$ are found with large probabilities. For pure disorder, we observe only closed trajectories (localized states), whose shape is in excellent agreement with the semi-classical trajectory, as expected. If a moderate periodic potential is added, the wave-functions spread over a larger area, and nearby contours sometimes merge together. Instead of sharp lines, as seen in panel (b), the contours now show clear evidence of interference effects of the wave-functions on the periodic potential decorating the electron reservoirs. Some periodic modulations can also be observed in the background of panels (c) and (d), especially for the square potential. These are [not ]{} the direct oscillations of the periodic potentials, since the grid we use to compute these figures has a linear size equal to $7/10$ of the period $a=39$ nm of the periodic potential. Capturing detailed behavior inside each unit cell would require a much smaller grid, which is not only time consuming, but also violates the requirement that the grid size be of order $l$ or larger. Figure \[fig10\] for an energy close to the band center, shows the same characteristics. For pure disorder, the electrons at this energy trace a sharp contour very similar ![ (color online) The same as in Fig. \[fig9\], but for an energy $E=-0.124$ meV close to the band center. []{data-label="fig10"}](Fig11_s.eps){width="\FigWidth"} to the corresponding equipotential line shown in panel (a). Electrons are still not delocalized, since this contour does not connect either pair of opposite edges. However, addition of the periodic potential now leads to extended states for both types of periodic potentials \[(c) and (d)\] at roughly $E=-0.124$ meV, demonstrating the widening of the critical region with the addition of a periodic potential. Physically, one can understand this spread of the wave function in the presence of the periodic potential using the semi-classical picture.[@Sorin] If only a smooth disorder potential is present, the equipotential at any energy $E$ must be a smooth, continuous line. However, if a periodic potential with minima $-V_m$ and maxima $V_M$ is superimposed over disorder, the new equipotential line now breaks into a series of small “bubbles” surrounding the disorder-only contour. This happens throughout the area defined by the equipotentials $E-V_M$ and $E+V_m$ of the disorder potential, since the addition of the periodic potential leads new regions in this area to have a total energy $E$. Quantum mechanically, we expect some tunneling inside this wider area and this is indeed what we observe in Figs. \[fig9\] and \[fig10\]. This mechanism suggests [enhanced delocalization ]{} on both sides of the critical region as localized wave functions spread out over larger areas, as well as a widening of the critical region itself, in agreement with our numerical results. This spreading of the wave functions in the presence of the periodic potential can also be characterized by counting, at a given energy $E$, the number of grid points ${{\bf r}}$ which have a value $\rho_P({{\bf r}}; E) > \rho_c$, where $\rho_c$ is some threshold value. For sufficiently large $\rho_c$, this procedure counts grid points where electrons with energy $E$ are found with large probabilities, thus, in effect it characterizes the “spatial extent” of the wave functions. The results of such counting are shown in Fig. \[fig11\] for 500 energy values corresponding to the disorder potential analyzed in Figs. \[fig9\] and \[fig10\]. There are a total of $60\times60=3600$ grid points on the sample. For the case of pure disorder (black line) we see that the largest values are found at energies just below 0, where the extended states (the critical region) are found for this particular realization of disorder. Because it is a smooth, sharp line, even the most extended trajectory has significant probabilities at only about 10% of the grid sites. For both higher and lower energies, this number decreases very fast, indicating wave functions localized more and more about maxima or minima of the disorder potential, as expected. Addition of a small periodic potential increases this number substantially, clearly showing the supplementary spreading of the wave functions in the presence of the periodic potential. Figure \[fig11\] shows this effect for three types of periodic potential: triangular lattices with $A > 0$ and $A<0$ (upper panel), and square lattice in the lower panel. All three cases show significant enhancement, as compared to the pure disorder case. In addition, we see that while the square potential gives a fairly symmetrical enhancement, the triangular potential does not, with curves for $\pm A$ not overlapping. This is a consequence of the asymmetric shape of the periodic potential, which has different values for its minima and maxima $\|V_m\|\ne \|V_M\|$, as well as different arrangements for the points where minima/maxima appear (triangular lattice vs. honeycomb lattice). Fig. \[fig11\] clearly shows that $A >0$ favors increased delocalization below the critical energy regime, while $A <0$ favors increased delocalization above it. The reason for this different response to the two signs of the triangular potential can be nicely explained within the semi-classical framework. In Fig. \[fig12\] we show the equipotential lines corresponding to filling factors $\nu=0.3$ (well below critical region) and $\nu=0.7$ (well above the critical region) for a realization of Coulomb disorder (not shown) plus a triangular potential with $A>0$. Areas with energy below the equipotential value are shaded. In this case we can clearly see that instead of the continuous, smooth trajectory expected for disorder-only cases, there are also extra “bubbles” regions connecting the areas between such contours. Since the choice $A>0$ leads to deep minima at $-V_m=-6A$ with triangular arrangement and relatively flat maxima at $+V_M=3A$ with honeycomb arrangement \[see Eq. (\[2.7\])\], it follows that the triangular (honeycomb) “bubbles” region appear roughly in the area bounded by the equipotentials $E$ and $E+V_m$ (respectively, $E-V_M$ and $E$) of the pure disorder potential. At low filling factors, the pure disorder $E$ equipotential is a collection of closed contours surrounding local minima \[see panel (d) of Fig. \[fig9\] for an illustration\]. It follows that for the choice $V_m > V_M$, the more extended region with triangular “bubbles” will be found outside these “islands” and will lead to a spread of the wave function over considerably larger areas, as indeed seen in the upper panel of Fig. \[fig12\]. On the other hand, at large filling factors the disorder-only contours are “islands” surrounding the maxima of the disorder potential. In this case, contours between $E$ and $E+V_m$ are inside the $E$ contour, so the triangular “bubbles” region does not help to connect various “islands” as before. The honeycomb “bubbles” regions does this, but because $V_M < V_m$ the extension of the wave function between “islands” is significantly smaller in this case. In the quantum-mechanical case one expects interference (due to tunneling) effects between the small “bubbles” regions, and therefore a wave function which is extended over their entire area, as indeed we observe to be the case in Figs. \[fig9\] and \[fig10\]. In other words, one expects that a triangular potential with $A>0$ will lead to considerable increase of the localization length, and respectively widening of the critical energy region, at filling factors below one-half, whereas $A<0$ will favor delocalization at filling factors above one-half, as seen in Fig. \[fig11\]. This asymmetry is therefore clearly a consequence of the asymmetry of the triangular potential, and is absent for a square potential with only lowest order Fourier coefficients, which possesses electron-hole symmetry. This should have clear implications for the transport properties of the system. Summary and Conclusions {#sec6} ======================= In this study we have investigated the effects of moderate-to-large smooth disorder on the Hofstadter butterfly expected for 2DES in a perpendicular magnetic field and a pure periodic modulation. The parameters of our study are chosen so as to be suitable for the interpretation of recent experiments on a two dimensional electron system in GaAs/AlGaAs heterostructure with periodic modulation provided by a diblock copolymer, [@Sorin] The experiment shows that (i) the longitudinal resistance $R_{xx}$ is still peaked approximately at half filling; (ii) there are many reproducible oscillations in $R_{xx}$, indicating non-trivial electronic structures in the patterned sample; (iii) the distribution of these oscillatory features is asymmetric, with most of them appearing on the high magnetic fields (i.e. low filling factors $\nu < 0.5$) side of the peak of $R_{xx}$; and (iv) the temperature dependence of $R_{xx}$ indicates that the asymmetric off-peak resistance is thermally excited, whereas the central $R_{xx}$ peak (close to half filling) has metallic behavior. These observations cannot be explained on the basis of the Hofstadter structure.[@Sorin] This is not surprising, since one expects that large disorder will modify this structure considerably. Effects of small disorder on the Hofstadter butterfly had been investigated previously using SCBA,[@MacDonald] but this basically perturbational approach is not appropriate for the case of moderate-to-large smooth disorder. Instead, we identify and use a number of techniques which give the exact solution (if electron-electron interactions, as well as inelastic scattering are neglected) while avoiding brute force numerical diagonalizations. Our results demonstrate that while the Hofstadter butterfly is destroyed by large disorder, the effects of the periodic potential are non-trivial for states near the critical regime. Firstly, they lead to a significant increase in localization lengths of the localized states at mesoscopic ($\mu m$) length scale and induce an effective widening of the critical regime near the critical regime. This is achieved through a spreading of the electron wave-function on the flat regions of the slowly varying disorder potential, where their behavior is dominated by the periodic modulation. This regime shows an interesting transition between the pure disorder and the pure periodic potential cases. In the case of pure disorder, the semi-classical approach tells us that at finite filling factors, some areas of the sample are fully occupied by electrons with the maximum possible density of $1/(2\pi l^2)$ (these are the areas where the disorder potential has minima) whereas other areas are fully devoid of electrons (areas where the potential has maxima) and the boundary between such regions is very sharp. On the other hand, for a pure periodic modulation all wave functions have translational invariance with the proper symmetry, and therefore electron densities are uniform over the entire sample (up to small periodic modulations inside each unit cell). When both types of potential are present, with disorder being dominant, our results show [three]{} types of areas. There are regions which are fully occupied and regions which are completely devoid of electrons, as in the case of pure disorder. However, the periodic potential leads to a widening of the boundary between the two, where the wave functions interact with several oscillations of the periodic modulation and therefore have some partial local filling. As the critical regime containing wave functions percolating throughout the sample is approached, this spreading of the wave function becomes dominant in establishing the transport properties of the system. An equivalent way to say this is that the main effect of the periodic potential is to provide bridging between the fully-occupied electron “puddles” created by the disorder potential. Since the connecting areas are relatively flat, the local wave functions respond to the local periodic potential, and therefore locally have a Hofstadter-butterfly like structure. If the partial filling factor in such a region is inside the gap of the local Hofstadter butterfly structure, one expects no transport through this local area. This should result in a dip in the longitudinal transport, since in such cases the periodic potential will not transport electrons from one “puddle” to another one. By contrast, if the local filling factor in such a region is inside a subband of a local Hofstadter structure, this area will establish a link between different “puddles” and thus help enhance the transport through the sample. Transport in this regime should show strong thermal activated behavior, in contrast to metallic transport in the critical regime where the wave functions connect opposite edges of the sample. As a result, one expects a series of local minima and maxima in the longitudinal resistivity on either side of the central peak induced by the extended states (critical regime). Furthermore, for an asymmetric triangular potential, this response should be strongly asymmetric, with the effect most visible on one side of the central peak. (One must keep in mind that since tunneling leads to exponential dependencies, even small differences in the extent of the wave functions can have rather large effects on $\rho_{xx}$). Such an asymmetry should also be present in longitudinal conductance of finite but low temperature, e. g. in the hopping regime which is sensitively dependent on the nature of the localized wavefunctions, as is indeed seen experimentally.[@Sorin] To summarize, our qualitative explanation for the various experimental features are as follows: \(i) The $R_{xx}$ peak is roughly at the center of the band because the weak periodic potential cannot establish a Hofstadter-like structure over the whole band. Instead, low and high $\nu$ states are strongly localized and do not transport longitudinal currents. \(ii) New extended states induced by the periodic potential are responsible for the reproducible peaks and valleys appearing in $R_{xx}$. \(iii) The periodic potential also leads to the expansion of localized wave functions, which contribute to the thermally activated conduction at lower filling factors. The detailed structure of the wave functions gives rise to the oscillations of the off-peak $R_{xx}$, similar to conductance fluctuations.[@Shayegan] Finally, \(iv) the asymmetry in $R_{xx}$ is a manifestation of the asymmetry of the triangular potential, which has a stronger effect at low filling factors than at high filling factors for $A>0$. We predict that this asymmetry should be absent for a symmetric square periodic potential. The weak point in our calculation is that we are unable to accurately model the potential in the real samples, because various screening effects have not been properly taken into account. Also, we have no quantitative information about the magnitude of the periodic potential in the 2DES layer, because of the additional strain[@Larkin] contribution induced by the periodic decoration. As a result, we only claim qualitative agreement with the experiment, although our investigations show the same type of behavior for various types of disorder potentials and various (small-to-moderate) strengths of the periodic potential. The most direct check of this work would be an experimental demonstration that thermally activated conduction appears symmetrically on both sides of the $R_{xx}$ peak for a periodic potential with square symmetry and primarily lowest Fourier coefficients. Limited computer resources restrict our calculations to samples no larger than $3 \mu$m$ \times 3 \mu $m, while the sample used in experiment has a size of $20 \mu$m$ \times 20\mu$m. From a theoretical point of view, it is interesting to ask what is the thermodynamic limit. For pure disorder, it is believed that in this limit the typical size of wavefunction diverges at a single critical energy. Since we cannot pursue size-dependent analysis for samples larger than $3 \mu$m$ \times 3 \mu $m, we do not know whether the small periodic potential will lead to a finite size critical regime in the thermodynamic limit, although this seems likely. From an experimental point of view, the interesting question is whether the Hofstadter structure can be observed at all. Our studies suggest that this may be possible for small mesoscopic samples, where the slowly-varying disorder has less effect. Alternatively, one must find a way to boost the strength of the periodic modulations inside the 2DES. Acknowledgements {#acknowledgements .unnumbered} ================ We thank Sorin Melinte, Mansour Shayegan, Paul M. Chaikin and Mingshaw W. Wu for valuable discussions. We also thank Prof. Li Kai’s group in Computer Science Department of Princeton University for sharing their computer cluster with us. This research was supported by NSF grant DMR-213706 (C.Z. and R.N.B.) and NSERC (M.B.). 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--- abstract: 'The relative density of visible points of the integer lattice ${\mathbb{Z}}^d$ is known to be $1/\zeta(d)$ for $d\geq 2$, where $\zeta$ is Riemann’s zeta function. In this paper we prove that the relative density of visible points in the Ammann-Beenker point set is given by $2(\sqrt{2}-1)/\zeta_K(2)$, where $\zeta_K$ is Dedekind’s zeta function over $K={\mathbb{Q}}(\sqrt{2})$.' author: - Gustav Hammarhjelm bibliography: - 'bibl.bib' title: 'The density of visible points in the Ammann-Beenker point set' --- Introduction {#secIntro} ============ A locally finite point set $\mathcal{P}\subset {\mathbb{R}}^d$ has an asymptotic density (or simply density) $\theta(\mathcal{P})$ if $$\lim_{R\to\infty}\frac{\#(\mathcal{P}\cap RD)}{{\mathrm{vol}}(RD)}=\theta(\mathcal{P})$$ holds for all Jordan measurable $D\subset {\mathbb{R}}^d$. The density of a set can be interpreted as the asymptotic number of elements per unit volume. For instance, for a lattice $\mathcal{L}\subset {\mathbb{R}}^d$ we have $\theta(\mathcal{L})=\frac{1}{{\mathrm{vol}}({\mathbb{R}}^d/\mathcal{L})}$. Let $\widehat{\mathcal{P}}=\{x\in \mathcal{P}\mid tx\notin \mathcal{P}, \forall t\in (0,1)\}$ denote the subset of the visible points of $\mathcal{P}$. If $\mathcal{P}$ is a regular cut-and-project set (see below) then it is known that $\theta(\mathcal{P})$ exists. In [@marklof2014visibility Theorem 1], J. Marklof and A. Strömbergsson proved that $\theta(\widehat{\mathcal{P}})$ also exists and that $0<\theta(\widehat{\mathcal{P}})\leq\theta(\mathcal{P})$ if $\theta(\mathcal{P})>0$. In particular, for such $\mathcal{P}$ the relative density of visible points $\kappa_\mathcal{P}:=\frac{\theta(\widehat{\mathcal{P}})}{\theta(\mathcal{P})}$ exists, but is not known explicitly in most cases. For $d\geq 2$ we have $\widehat{{\mathbb{Z}}^d}=\{(n_1,\ldots,n_d)\in {\mathbb{Z}}^d\mid \gcd(n_1,\ldots,n_d)=1\}$ and $\theta(\widehat{{\mathbb{Z}}^d})=1/\zeta(d)$ gives the probability that $d$ random integers share no common factor. This can be derived in several ways, see for instance [@nymann1972probability]; we sketch another proof in below. More generally, $\theta(\widehat{\mathcal{L}})=\frac{1}{{\mathrm{vol}}({\mathbb{R}}^d/\mathcal{L})\zeta(d)}$ for a lattice $\mathcal{L}\subset {\mathbb{R}}^d$, see e.g. [@baake2000diffraction Prop. 6]. A well-known point set, which can be realised both as the vertices of a substitution tiling and as a cut-and-project set, is the Ammann-Beenker point set. The goal of this paper is to prove that the relative density of visible points in the Ammann-Beenker point set is $2(\sqrt{2}-1)/\zeta_K(2)$. This density was computed by B. Sing in the presentation [@singppt1], but he has not published a proof of this result. The density of the visible points of ${\mathbb{Z}}^d$ {#secDensZn} ===================================================== In this section we show that $\theta(\widehat{{\mathbb{Z}}^d})=1/\zeta(d)$. We shall see that a lot of inspiration can be drawn from this example when calculating the density of the visible points in the Ammann-Beenker point set. Fix $R>0$, a Jordan measurable $D\subset {\mathbb{R}}^d$ and let ${\mathbb{P}}\subset {\mathbb{Z}}_{>0}$ denote the set of prime numbers. For each invisible point $n\in {\mathbb{Z}}^d\setminus \widehat{{\mathbb{Z}}^d}$, there is $p\in{\mathbb{P}}$ such that $\frac{n}{p}\in{\mathbb{Z}}^d$. Setting ${\mathbb{Z}}^d_={\mathbb{Z}}^d{\setminus\{(0,\ldots,0)\}}$ there are only finitely many $p_1,\ldots,p_n\in{\mathbb{P}}$ such that $p_i{\mathbb{Z}}^d_\cap RD\neq \emptyset$. By inclusion-exclusion counting we have $$\begin{aligned} \#(\widehat{{\mathbb{Z}}^d}\cap RD)&=\#\left(({\mathbb{Z}}^d_\cap RD)\setminus \bigcup_{p\in{\mathbb{P}}}(p{\mathbb{Z}}^d_\cap RD)\right)=\#\left(({\mathbb{Z}}^d_\cap RD)\setminus \bigcup_{i=1}^n(p_i{\mathbb{Z}}^d_\cap RD)\right)\\ &=\#({\mathbb{Z}}^d_\cap RD)+\sum_{k=1}^{m}(-1)^{k}\left(\sum_{1\leq i_1<\ldots <i_k\leq m}\#(p_{i_1}{\mathbb{Z}}^d_\cap\cdots\cap p_{i_k}{\mathbb{Z}}^d_\cap RD)\right). \end{aligned}$$ The last sum can be rewritten to $$\sum_{n\in{\mathbb{Z}}_{>0}}\mu(n)\cdot\#(n{\mathbb{Z}}^d_\cap RD),$$ where $\mu$ is the Möbius function. Hence $$\frac{\#(\widehat{{\mathbb{Z}}^d}\cap RD)}{{\mathrm{vol}}(RD)}=\sum_{n\in{\mathbb{Z}}_{>0}}\frac{\mu(n)\cdot\#(n{\mathbb{Z}}^d_\cap RD)}{{\mathrm{vol}}(RD)}=\sum_{n\in{\mathbb{Z}}_{>0}}\frac{\mu(n)}{n^d}\frac{\#({\mathbb{Z}}^d_\cap {n^{-1}}RD)}{{\mathrm{vol}}({n^{-1}}RD)}.$$ Letting $R\to\infty$, switching order of limit and summation (for instance justified by finding a constant $C$ depending on $D$ such that $\#({\mathbb{Z}}_^d\cap RD)\leq C{\mathrm{vol}}(RD)$ for all $R$), using $\theta({\mathbb{Z}}^d_)=1$ and $1/\zeta(s)=\sum_{n\in{\mathbb{Z}}_{>0}}\frac{\mu(n)}{n^s}$ for $s>1$, we find that $$\theta(\widehat{{\mathbb{Z}}^d})=\lim_{R\to\infty}\frac{\#(\widehat{{\mathbb{Z}}^d}\cap RD)}{{\mathrm{vol}}(RD)}=1/\zeta(d).$$ Cut-and-project sets and the Ammann-Beenker point set {#secCPS} ===================================================== The Ammann-Beenker point set can be obtained as the vertices of the Ammann-Beenker tiling, a substitution tiling of the plane using a square and a rhombus as tiles, see e.g. [@baake2013aperiodic Chapter 6.1]. In this paper however, the Ammann-Beenker set is realised as a cut-and-project set, a certain type of point set which we will now define. Cut-and-project sets are sometimes called (Euclidean) model sets. We will use the same notation and terminology for cut-and-project sets as in [@marklof2014free Sec. 1.2]. For an introduction to cut-and-project sets, see e.g. [@baake2013aperiodic Ch. 7.2]. If ${\mathbb{R}}^n={\mathbb{R}}^d\times {\mathbb{R}}^m$, let $$\begin{aligned} {2} \pi:& ~{\mathbb{R}}^n\longrightarrow {\mathbb{R}}^d & \pi_{\mathrm{int}}: & ~ {\mathbb{R}}^n\longrightarrow {\mathbb{R}}^m \\ & (x_1,\ldots,x_n)\longmapsto (x_1,\ldots,x_d)\hspace{1cm}& & (x_1,\ldots,x_n)\longmapsto (x_{d+1},\ldots, x_n) \end{aligned}$$ denote the natural projections. \[defnCPS\] Let $\mathcal{L}\subset {\mathbb{R}}^n$ be a lattice and $\mathcal{W}\subset \overline{\pi_{\mathrm{int}}(\mathcal{L})}$ be a set. Then the cut-and-project set of $\mathcal{L}$ and $\mathcal{W}$ is given by $\mathcal{P}(\mathcal{W},\mathcal{L})=\{\pi(y)\mid y\in \mathcal{L},\pi_{\mathrm{int}}(y)\in\mathcal{W}\}$. If $\partial W$ has measure zero with respect to any Haar measure on $\overline{\pi_{\mathrm{int}}(\mathcal{L})}$ we say that $\mathcal{P}(\mathcal{W},\mathcal{L})$ is regular. If the interior of $\mathcal{W}$ (the window) is non-empty, $\mathcal{P}(\mathcal{W},\mathcal{L})$ is relatively dense and if $\mathcal{W}$ is bounded, $\mathcal{P}(\mathcal{W},\mathcal{L})$ is uniformly discrete (cf. [@marklof2014free Prop. 3.1]). To realise the Ammann-Beenker point set in this way, let $K$ be the number field ${\mathbb{Q}}(\sqrt{2})$, with algebraic conjugation $x\mapsto \overline{x}$ (we will also write $\overline{x}=(\overline{x_1},\ldots,\overline{x_n})$ for $x=(x_1,\ldots,x_n)\in K^n$) and norm $N(x)=x\overline{x}$. The ring of integers $\mathcal{O}_K={\mathbb{Z}}[\sqrt{2}]$ of $K$ is a Euclidean domain with fundamental unit $\lambda:=1+\sqrt{2}$. With $\zeta:=e^{\tfrac{\pi i}{4}}$ and $\star:K\longrightarrow K$, $x\mapsto x^\star$ the automorphism generated by $\zeta \mapsto \zeta^3$, the Ammann-Beenker point set is in [@baake2013aperiodic Example 7.7] realised as $$\{x=x_1+x_2\zeta\mid x_1,x_2\in\mathcal{O}_K,x^\star\in W_8\},$$ where $W_8\subset {\mathbb{C}}$ is the regular octagon of side length $1$ centered at the origin, with sides perpendicular to the coordinate axes. Let $$\mathcal{L}=\{(x,\overline{x})\mid x=(x_1,x_2)\in \mathcal{O}_K^2\}\subset {\mathbb{R}}^4$$ be the Minkowski embedding of $\mathcal{O}_K^2$ and let $$\widetilde{\mathcal{L}}=\{(x,\overline{x})\in \mathcal{L}\mid (x_1-x_2)/\sqrt{2}\in \mathcal{O}_K\}.$$ Then, after a straight-forward translation it is seen that the Ammann-Beenker point set $\mathcal{A}$ can be realised in ${\mathbb{R}}^2$ as $\mathcal{A}=\frac{1}{\sqrt{2}}\mathcal{P}(\mathcal{W}_\mathcal{A},\widetilde{\mathcal{L}})$, where $\mathcal{W}_\mathcal{A}:=\sqrt{2}W_8$, i.e. $\mathcal{A}$ is the scaling of a cut-and-project set according to . The density of visible points of $\mathcal{A}$ {#secDensAB} ============================================== All notation used in this section is defined in and taken from . Since, for any $\mathcal{P}\subset {\mathbb{R}}^d$ whose density exists, and any $c>0$ it holds that $\theta(c\mathcal{P})=c^{-d} \theta(\mathcal{P})$ and $c\widehat{\mathcal{P}}=\widehat{c\mathcal{P}}$, finding $\theta(\widehat{\mathcal{A}'})$ with $\mathcal{A}':=\sqrt{2}\mathcal{A}=\mathcal{P}(\mathcal{W}_\mathcal{A},\widetilde{\mathcal{L}})$ will give the value of $\theta(\widehat{\mathcal{A}})$. As a first step, in , the asymptotic density of the visible points of the simpler set $\mathcal{B}=\mathcal{P}(\mathcal{W}_\mathcal{A},\mathcal{L})=\{x\in\mathcal{O}_K^2\mid \overline{x}\in\mathcal{W}_\mathcal{A}\}\subset\mathcal{O}_K^2$ will be calculated. In this result will be used to obtain $\theta(\widehat{\mathcal{A}})$. The density of visible points of $\mathcal{B}$ {#subsecDensAB'} ---------------------------------------------- The following general counting formula for bounded subsets of visible points of a point set $\mathcal{P}$ will be needed. Let $\mathcal{P}_=\mathcal{P}\setminus\{(0,\ldots,0)\}$. \[propInclExcl\] Let $\mathcal{P}\subset {\mathbb{R}}^d$ be locally finite and fix a set $C\subset{\mathbb{R}}_{>1}$ such that for each $x\in \mathcal{P}\setminus\widehat{\mathcal{P}}$ there exists $c\in C$ with $x/c\in \mathcal{P}$. Let $R>0$ and a bounded set $D\subset {\mathbb{R}}^d$ be given. Then $$\#(\widehat{\mathcal{P}}\cap RD)=\sum_{\substack{F\subset C\\\#F<\infty}}(-1)^{\#F}\#\left(\left(\mathcal{P}_\cap \bigcap_{c\in F}c\mathcal{P}_\right)\cap RD\right).$$ The set $C_R:=\{c\in C\mid \mathcal{P}_\cap c\mathcal{P}_\cap RD\neq \emptyset\}$ is finite. Indeed, suppose this is not true and pick distinct $c_1,c_2,\ldots\in C_R$ and corresponding $x_i\in \mathcal{P}_\cap c_i\mathcal{P}_\cap RD$. Since $\mathcal{P}$ is locally finite, the sequence $x_1,x_2,\ldots$ contains only finitely many distinct elements. Thus, a subsequence $x_{k_1},x_{k_2},\ldots$ which is constant can be extracted, so that $x_{k_i}/c_{k_i}\in \mathcal{P}_\cap \frac{RD}{c_{k_i}}\subset \mathcal{P}_\cap RD$ are all distinct, contradiction to $\mathcal{P}$ being locally finite. Thus, we can write $C_R=\{c_1,\ldots,c_n\}$ for some $c_1,\ldots,c_n\in C$. Then $$\begin{aligned} &\#(\widehat{\mathcal{P}}\cap RD)=\#\left((\mathcal{P}_\cap RD)\setminus\bigcup_{c\in C}(\mathcal{P}_\cap c\mathcal{P}_\cap RD)\right)\\ &=\#(\mathcal{P}_\cap RD)-\#\left(\bigcup_{i=1}^n(\mathcal{P}_\cap c_i\mathcal{P}_\cap RD)\right), \end{aligned}$$ from which the result follows from the inclusion-exclusion counting formula for finite unions of finite sets. A set $C$ as in for $\mathcal{B}$ will be needed, and to this end a visibility condition for the elements of $\mathcal{B}$ is required. Given $x_1,x_2\in \mathcal{O}_K$, let $\gcd(x_1,x_2)$ be a fixed generator of the ideal generated by $x_1,x_2$ and write $\gcd(x_1,x_2)=1$ when $x_1,x_2$ are relatively prime. In the following proposition a visibility condition of the complex realisation of the Ammann-Beenker point set given in [@baake2014radial p. 477] is adapted to our situation. \[propVisCond\] The visible points of $\mathcal{B}$ are given by $$\widehat{\mathcal{B}}=\{x=(x_1,x_2)\in \mathcal{B}\mid \gcd(x_1,x_2)=1,\lambda \overline{x}\notin \mathcal{W}_\mathcal{A}\}.$$ First the necessity of the visibility conditions is established. Take $x=(x_1,x_2)$ and suppose that $\gcd(x_1,x_2)\neq 1$ so that there exists $c\in\mathcal{O}_K$ with $\|N(c)\|>1$ and $c\mid x_1,x_2$. Scaling $c$ by units we may assume that $1<c<\lambda$. Suppose first that $\|N(c)\|=\|\overline{c}\|c\geq 3$, which implies $\|\overline{c}\|>1$. By noting that $\mathcal{W}_\mathcal{A}$ is star-shaped with respect to the origin and $\mathcal{W}_\mathcal{A}=-\mathcal{W}_\mathcal{A}$ it follows that $x/c\in\mathcal{B}$, so $x$ is invisible. If $\|N(c)\|=2$, then each prime factor of $c$ must divide $2=\sqrt{2}\cdot \sqrt{2}$, so it can be assumed that $c=\sqrt{2}$ and hence $x$ is occluded by $x/\sqrt{2}$. If $\lambda \overline{x}\in\mathcal{W}_\mathcal{A}$ it follows immediately that $x/\lambda\in\mathcal{B}$. We now turn to the sufficiency of the visibility conditions. Take $x=(x_1,x_2)\in\mathcal{B}\setminus\widehat{\mathcal{B}}$ and $c>1$ such that $x/c\in\mathcal{B}$. As $\mathcal{B}$ is uniformly discrete, we may assume that $y:=x/c\in \widehat{\mathcal{B}}$. This implies, by necessity above, that $\gcd(y_1,y_2)=1$. Now, since $x_i=cy_i$ it follows that $c\in K$. Write $c=a/b$ with $a,b\in\mathcal{O}_K$ relatively prime. If $b$ is not a unit, $\gcd(y_1,y_2)=1$ is contradicted, hence $c\in\mathcal{O}_K$. If $\|N(c)\|\neq 1$ then $\gcd(x_1,x_2)\neq1$. Otherwise, $c>1$ is a unit, i.e. $c=\lambda^k$ for some integer $k>0$. Thus $\frac{\overline{x}}{\overline{c}}=\frac{\overline{x}}{\overline{\lambda}^k}\in\mathcal{W}_\mathcal{A}$. Since $\frac{1}{\overline{\lambda}}=-\lambda$ we get $(-\lambda)^k\overline{x}\in\mathcal{W}_\mathcal{A}$ and thus also $\lambda \overline{x}\in\mathcal{W}_\mathcal{A}$ as $\mathcal{W}_\mathcal{A}$ is star-shaped with respect to the origin and $-\mathcal{W}_\mathcal{A}=\mathcal{W}_\mathcal{A}$. This establishes sufficiency of the visibility conditions. Note that the proof works just as well for more general windows, that is, $\widehat{\mathcal{P}(\mathcal{W},\mathcal{L})}=\{x\in \mathcal{P}(\mathcal{W},\mathcal{L})\mid \gcd(x_1,x_2)=1,\lambda \overline{x}\notin \mathcal{W}\}$ if $\mathcal{W}\subset{\mathbb{R}}^2$ is bounded with non-empty interior, star-shaped with respect to the origin and $-\mathcal{W}=\mathcal{W}$. Let now $${\mathbb{P}}=\{\pi\in\mathcal{O}_K\mid \pi ~\mathrm{prime}, 1<\pi<\lambda\}~\text{and}~ C={\mathbb{P}}\cup \{ \lambda\}$$ so that ${\mathbb{P}}$ is a set that contains precisely one associate of every prime of $\mathcal{O}_K$. Then we have the following proposition. \[propOcclSet\] For each $x\in\mathcal{B}\setminus\widehat{\mathcal{B}}$ there is $c\in C$ such that $x/c\in \mathcal{B}$. Fix $x\in\mathcal{B}\setminus\widehat{\mathcal{B}}$. As seen in the proof of there is $c\in \mathcal{O}_K$, $c>1$, such that $x/c\in \mathcal{B}$. If $c$ is not a unit, fix $\pi\in {\mathbb{P}}$ so that $\pi\mid c$. It can be verified by hand that $\{(x,\overline{x})\mid x\in\mathcal{O}_K\}\cap ((1,\lambda)\times (-1,1))=\emptyset$, hence $\|\pi\|>1$ and $x/\pi\in \mathcal{B}$. If $c$ is a unit, $x/\lambda\in\mathcal{B}$ is immediate. Given a finite set $F\subset \mathcal{O}_K$ let $I_F$ be the (principal) ideal generated by the elements of $F$ if $F\neq \emptyset$ and $I_F=\mathcal{O}_K$ otherwise. Let $\ell_F$ denote a fixed least common multiple* of $F$, that is, a generator of the ideal $\bigcap_{c\in F}c\mathcal{O}_K$. Let also $m_F=\min\{1,\min_{c\in F}\|\overline{c}\|\}$ and $\mathcal{L}_F=\{(\ell_Fx,\overline{\ell_Fx})\mid x\in \mathcal{O}_K^2\}$. Write $I\triangleleft \mathcal{O}_K$ when $I\subset \mathcal{O}_K$ is an ideal and define the absolute norm $N(I)$ of $I$ by $\|N(x)\|$, where $x$ is any generator of $I$. Recall Dedekind’s zeta function $\zeta_K(s)=\sum_{I\triangleleft \mathcal{O}_K}\frac{1}{N(I)^s}$ for $s\in {\mathbb{C}}$ with $\mathrm{Re}(s)>1$. Given a finite set $F\subset C$ it is verified that $\mathcal{B}_\cap \bigcap_{c\in F}c\mathcal{B}_=\mathcal{P}(m_F\mathcal{W}_\mathcal{A},\mathcal{L}_F){\setminus\{0\}}$. For any $R>0$ and bounded $D\subset {\mathbb{R}}^2$, Propositions \[propInclExcl\], \[propOcclSet\] imply that $$\label{eqA'1} \#(\widehat{\mathcal{B}}\cap RD)=\sum_{\substack{F\subset C\\\#F<\infty}}(-1)^{\#F}\#\left((\mathcal{P}(m_F\mathcal{W}_\mathcal{A},\mathcal{L}_F){\setminus\{0\}})\cap RD\right).$$ Since $\ell_F\mathcal{O}_K^2\subset\pi_{\mathrm{int}}(\mathcal{L}_F)\subset{\mathbb{R}}^2$ is dense we have $$\theta(\mathcal{P}(m_F\mathcal{W}_\mathcal{A},\mathcal{L}_F){\setminus\{0\}})=\frac{{\mathrm{vol}}(m_F\mathcal{W}_\mathcal{A})}{{\mathrm{vol}}({\mathbb{R}}^4/\mathcal{L}_F)}$$ from [@marklof2014free Prop. 3.2]. Dividing by ${\mathrm{vol}}(RD)$, letting $R\to\infty$ and switching order of limit and summation (to be justified in below) we find that $$\theta(\widehat{\mathcal{B}})=\sum_{\substack{F\subset C\\\#F<\infty}}(-1)^{\#F}\frac{{\mathrm{vol}}(m_F\mathcal{W}_\mathcal{A})}{{\mathrm{vol}}({\mathbb{R}}^4/\mathcal{L}_F)}=\sum_{\substack{F\subset C\\\#F<\infty}}(-1)^{\#F}\frac{m_F^2(1+\sqrt{2})}{2N(\ell_F)^2},$$ since ${\mathrm{vol}}(\mathcal{W}_\mathcal{A})=4(1+\sqrt{2})$ and ${\mathrm{vol}}({\mathbb{R}}^4/\mathcal{L}_F)=8N(\ell_F)^2$. The value of the right hand sum will be shown to be $1/\zeta_K(2)$ in below. The following lemma gives a bound on the number of points in the intersection of a lattice and a box in terms of the volume of the box, provided that the box is “not too thin”. \[lemLatticeBoxBound\] Let $\mathcal{L}\subset {\mathbb{R}}^d$ be a lattice and let $c>0$ be given. For any $a_i,b_i\in{\mathbb{R}}$ with $b_i-a_i>c$ set $B=\prod_{i=1}^{d}[a_i,b_i]$. Then there is a constant $L$ depending only on $\mathcal{L}$ and $c$ such that $\#(B\cap\mathcal{L})\leq L{\mathrm{vol}}(B)$. Let $n_i={\lceil\frac{b_i-a_i}{c}\rceil}\in{\mathbb{Z}}_+$. Then $\frac{b_i-a_i}{c}\leq n_i< \frac{b_i-a_i}{c}+1=\frac{b_i-a_i+c}{c}<\frac{2(b_i-a_i)}{c}.$ Hence, with $n=\prod_{i=1}^dn_i$ it follows that $n\leq \frac{2^d{\mathrm{vol}}(B)}{c^d}$. From $b_i\leq a_i+cn_i$ also $B\subset\prod_{i=1}^d[a_i,a_i+cn_i].$ Let $a=(a_1,\ldots,a_d)$ and consider $-a+\prod_{i=1}^d[a_i,a_i+cn_i]=\prod_{i=1}^d[0,cn_i]$. We have $\prod_{i=1}^d[0,cn_i]=\bigcup_{m\in{\mathbb{N}}^d,0\leq m_i<n_i}(mc+[0,c]^d).$ Hence, $B\subset \bigcup_{m\in{\mathbb{N}}^d,0\leq m_i<n_i}(a+mc+[0,c]^d)=:B'$. Find now $D>0$ depending on $\mathcal{L}$ and $c$ such that $\sup_{t\in{\mathbb{R}}^d}\#(\mathcal{L}\cap (t+[0,c]^d))=D$. Hence $\#(B\cap \mathcal{L})\leq \#(B'\cap\mathcal{L})\leq n D\leq \frac{2^dD}{c^d}{\mathrm{vol}}(B)$, so one can take $L=\frac{2^dD}{c^d}$. The following bound will be crucial in the justification of interchanging limit and summation in after division by ${\mathrm{vol}}(RD)$. \[lem-Estimate1\] Let $D\subset {\mathbb{R}}^2$ be Jordan measurable. Then there is a constant $\widetilde{L}>0$ depending only on $D$ such that for every $R>0$ and $F\subset C$ with $\#F<\infty$, $$\#((\mathcal{P}(m_F\mathcal{W}_\mathcal{A},\mathcal{L}_F)\cap RD){\setminus\{0\}})\leq \frac{\widetilde{L}R^2}{N(\ell_F)^2}.$$ By definition $$\#((\mathcal{P}(m_F\mathcal{W}_\mathcal{A},\mathcal{L}_F){\setminus\{0\}})\cap RD)=\#(\{x\in \ell_F\mathcal{O}_K^2\mid\overline{x}\in m_F\mathcal{W}_\mathcal{A}\}{\setminus\{0\}})\cap RD).$$ Note that this number is independent of the choice of $\ell_F$. There is a bijection $$(\{x\in \ell_F\mathcal{O}_K^2\mid \overline{x}\in m_F\mathcal{W}_\mathcal{A}\}{\setminus\{0\}})\cap RD\longrightarrow(\{x\in\mathcal{O}_K^2\mid \overline{x}\in \tfrac{m_F\mathcal{W}_\mathcal{A}}{\|\overline{\ell_F}\|}\}{\setminus\{0\}})\cap \tfrac{RD}{\ell_F}$$ given by $x\mapsto \tfrac{x}{\ell_F}$, so it suffices to estimate the number of elements in the latter set. Since $m_F\leq 1$ it follows that $\left(\mathcal{L}\cap \left(\tfrac{RD}{\ell_F}\times\tfrac{m_F\mathcal{W}_\mathcal{A}}{\|\overline{\ell_F}\|} \right)\right){\setminus\{0\}}\subset \left(\mathcal{L}\cap \left(\tfrac{RD}{\ell_F}\times\tfrac{\mathcal{W}_\mathcal{A}}{\|\overline{\ell_F}\|} \right)\right){\setminus\{0\}}$. Fix real numbers $m_1,m_2>1$ so that $D\subset [-m_1,m_1]^2=:B_1$ and $\mathcal{W}_\mathcal{A}\subset [-m_2,m_2]^2=:B_2$. Fix a number $c$ so that $c'<c$ implies $(\mathcal{L}\cap(\lambda D\times c'\mathcal{W}_\mathcal{A})){\setminus\{0\}}=\emptyset$. This can be done, for otherwise $(\mathcal{L}\cap(\lambda D\times c'\mathcal{W}_\mathcal{A})){\setminus\{0\}}$ would be non-empty for each $c'>0$, hence $\mathcal{L}\cap (\lambda D\times\mathcal{W}_\mathcal{A})$ would contain infinitely many points, contradiction, since $\mathcal{L}$ is a lattice and $\lambda D\times\mathcal{W}_\mathcal{A}$ is bounded. Suppose first that $\frac{R}{\|\ell_F\overline{\ell_F}\|}<c$. Scale $\ell_F$ by units so that $1\leq \tfrac{R}{\ell_F}<\lambda$ which gives $\tfrac{1}{\|\overline{\ell_F}\|}<c$. Hence $\left(\mathcal{L}\cap\left(\tfrac{RD}{\ell_F}\times \tfrac{\mathcal{W}_\mathcal{A}}{\|\overline{\ell_F}\|}\right)\right){\setminus\{0\}}\subset \left(\mathcal{L}\cap\lambda D\times\left( \tfrac{\mathcal{W}_\mathcal{A}}{\|\overline{\ell_F}\|}\right)\right){\setminus\{0\}}=\emptyset$ and therefore $\#\left(\left(\mathcal{L}\cap\left(\tfrac{RD}{\ell_F}\times \tfrac{\mathcal{W}_\mathcal{A}}{\|\overline{\ell_F}\|}\right)\right){\setminus\{0\}}\right)=0$. Suppose now that $\frac{R}{\|\ell_F\overline{\ell_F}\|}\geq c$. Scale $\ell_F$ so that $\sqrt{c}\leq \tfrac{R}{\ell_F}<\lambda \sqrt{c}$. This implies that $\tfrac{1}{\|\overline{\ell_F}\|}\geq \tfrac{1}{\lambda}\sqrt{c}>\sqrt{c}$. Thus, $[0,\sqrt{c}]^4\subset \tfrac{RB_D}{\ell_F}\times \tfrac{B_\mathcal{W}}{\|\overline{\ell_F}\|}=:B$. From we get a constant $L$ only depending on $\mathcal{L}$, $\sqrt{c}$ such that $ \#(B\cap \mathcal{L})\leq L{\mathrm{vol}}(B)=L\cdot 16m_1^2m_2^2\frac{R^2}{N(\ell_F)^2}.$ Now, since $\tfrac{RD}{\ell_F}\times \tfrac{\mathcal{W}_\mathcal{A}}{\|\overline{\ell_F}\|}\subset B$ we get that $$\#\left(\left(\mathcal{L}\cap\tfrac{RD}{\ell_F}\times \left(\tfrac{\mathcal{W}}{\|\overline{\ell_F}\|}\right)\right){\setminus\{0\}}\right)\leq\frac{\widetilde{L}R^2}{N(\ell_F)^2}$$ with $\widetilde{L}:=16m_1^2m_2^2L$. \[propInterchange\] The equality $$\lim_{R\to\infty}\sum_{\substack{F\subset C\\\#F<\infty}}\frac{(-1)^{\#F}\#\left((\mathcal{P}(m_F\mathcal{W}_\mathcal{A},\mathcal{L}_F){\setminus\{0\}})\cap RD\right)}{{\mathrm{vol}}(RD)}=\sum_{\substack{F\subset C\\\#F<\infty}}\frac{(-1)^{\#F}m_F^2(1+\sqrt{2})}{2N(\ell_F)^2}$$ holds for all Jordan measurable $D\subset {\mathbb{R}}^2$. For a finite $F\subset C$ let $N(R,F)=\#\left((\mathcal{P}(m_F\mathcal{W}_\mathcal{A},\mathcal{L}_F){\setminus\{0\}})\cap RD\right)$. We know that $\lim\limits_{R\to\infty}\frac{N(R,F)}{{\mathrm{vol}}(RD)}=\frac{m_F^2(1+\sqrt{2})}{2N(\ell_F)^2}$ so $$\label{eqnSwitchLimitsJustification} \lim\limits_{R\to\infty}\sum_{\substack{F\subset C\\\#F<\infty}}\frac{(-1)^{\#F}N(R,F)}{{\mathrm{vol}}(RD)}=\sum_{\substack{F\subset C\\\#F<\infty}}\lim\limits_{R\to\infty}\frac{(-1)^{\#F}N(R,F)}{{\mathrm{vol}}(RD)}$$ must be justified. In view of $$\sum_{\substack{F\subset C\\\#F<\infty}}\left\|\frac{(-1)^{\#F}N(R,F)}{{\mathrm{vol}}(RD)}\right\|\leq \frac{\widetilde{L}}{{\mathrm{vol}}(D)}\sum_{\substack{F\subset C\\\#F<\infty}}\frac{1}{N(\ell_F)^2}$$ and we note that $$\sum_{\substack{F\subset C\\\#F<\infty}}\frac{1}{N(\ell_F)^2}=\sum_{\substack{F\subset C\\\#F<\infty\\\lambda\in F}}\frac{1}{N(\ell_F)^2}+\sum_{\substack{F\subset C\\\#F<\infty\\\lambda\notin F}}\frac{1}{N(\ell_F)^2}\leq 2\sum_{I\triangleleft \mathcal{O}_K}\frac{1}{N(I)^2},$$ hence the sums of both sides of are absolutely convergent. Fix $\Delta>0$. We claim that there is only a finite number of non-empty $F\subset C$, $\#F<\infty$, such that $\|N(\ell_F)\|<\Delta$. Given such $F$ let $\ell_F=\prod_{c\in F}c>1$. Also, since $\|\overline{\pi}\|>1$ for all $\pi\in {\mathbb{P}}$ we have $\|\overline{\ell_F}\|\geq \|\overline{\lambda}\|$. Hence, $\|N(\ell_F)\|=\ell_F\|\overline{\ell_F}\|\leq \Delta$ implies $\ell_F\leq \frac{\Delta}{\|\overline{\ell_F}\|}\leq \lambda \Delta$ and $\|\overline{\ell_F}\|\leq \frac{\Delta}{\ell_F}\leq \Delta<\lambda\Delta$ so $(\ell_F,\overline{\ell_F})\in \{(x,\overline{x})\mid x\in\mathcal{O}_K\}\cap \lambda[-\Delta,\Delta]^2$ which is a finite set, thus elements of $F$ can only contain prime factors that occur as factors in the components of elements in this finite set, giving only finitely many possibilities for $F$. It follows that $$\begin{aligned} &\left\|\lim\limits_{R\to\infty}\sum_{\substack{F\subset C\\\#F<\infty}}\frac{(-1)^{\#F}N(R,F)}{{\mathrm{vol}}(RD)}-\sum_{\substack{F\subset C\\\#F<\infty}}\frac{m_F^2(1+\sqrt{2})}{2N(\ell_F)^2}\right\|\\ &\leq \left(\frac{\widetilde{L}}{{\mathrm{vol}}(D)}+\frac{1+\sqrt{2}}{2}\right)\sum_{\substack{F\subset C\\\#F<\infty\\\|N(\ell_F)\|\geq \Delta}}\frac{1}{N(\ell_F)^2} \end{aligned}$$ where the right hand side tends to $0$ as $\Delta\to\infty$ since $\sum\limits_{F\subset C,\#F<\infty,\|N(\ell_F)\|\geq \Delta}\frac{1}{N(\ell_F)^2}$ is the tail of an absolutely convergent sum, hence has been justified. From it follows that $\theta(\widehat{\mathcal{B}})=\sum_{\substack{F\subset C\\\#F<\infty}}\frac{(-1)^{\#F}m_F^2(1+\sqrt{2})}{2N(\ell_F)^2}$, and it will now be shown that the right hand side is equal to $1/\zeta_K(2)$. Define the function $\omega:\mathcal{O}_K\longrightarrow{\mathbb{C}}$, $\omega(x)=\#\{\pi\in{\mathbb{P}}\mid x/\pi\in\mathcal{O}_K\}$, so that $\omega(x)$ is the number of non-associated prime divisors of $x$. Given $I\triangleleft \mathcal{O}_K$, let $\omega(I)=\omega(x)$ for any generator $x$ of $I$ and define a Möbius function on the ideals of $\mathcal{O}_K$ by $$\mu(I)=\begin{cases}0 & \text{ if }\exists \pi\in{\mathbb{P}}\text{ such that } I\subset \pi^2\mathcal{O}_K,\\ (-1)^{\omega(I)} &\text{ otherwise.} \end{cases}$$ One verifies that $\mu(I_1I_2)=\mu(I_1)\mu(I_2)$ for relatively prime ideals $I_1,I_2$. The function $\zeta_K$ can be expressed as an Euler product for $s$ with $\mathrm{Re}(s)>1$ as $$\zeta_K(s)=\prod_{P\triangleleft \mathcal{O}_K,\,P\,\mathrm{prime}}\frac{1}{1-N(P)^{-s}}$$ and in analogy with the reciprocal formula for Riemann’s zeta function we have $$\label{eqnRecDedekind} \frac{1}{\zeta_K(s)}=\sum_{I\triangleleft \mathcal{O}_K}\frac{\mu(I)}{N(I)^s}.$$ \[thmAsDensA’\] The density of visible points of $\mathcal{B}$ is given by $$\theta(\mathcal{B})=\frac{1}{\zeta_K(2)}=\frac{48\sqrt{2}}{\pi^4}.$$ By we have $$\theta(\mathcal{B})=\sum_{\substack{F\subset C\\\#F<\infty}}\frac{(-1)^{\#F}m_F^2(1+\sqrt{2})}{2N(\ell_F)^2}.$$ Splitting the sum into two depending on whether $\lambda \in F$ or not, and using that $m_F$ is $1$ unless $\lambda\in F$, in which case $m_F=\|\overline{\lambda}\|=\sqrt{2}-1$, we get $$\begin{aligned} \theta(\mathcal{B})&=\sum_{\substack{F\subset C\\\#F<\infty\\\lambda\notin F}}\frac{(-1)^{\#F}(1+\sqrt{2})}{2N(\ell_F)^2}+\sum_{\substack{F\subset C\\\#F<\infty\\\lambda\in F}}\frac{(-1)^{\#F}\|\overline{\lambda}\|^2(1+\sqrt{2})}{2N(\ell_F)^2}\\ &=\frac{(1-\|\overline{\lambda}\|^2)(1+\sqrt{2})}{2}\sum_{I\triangleleft\mathcal{O}_K}\frac{\mu(I)}{N(I)^2}=\frac{1}{\zeta_K(2)}, \end{aligned}$$ last equality by . From [@washington1997introduction Theorem 4.2] one can calculate $\zeta_K(-1)=\frac{1}{12}$ and by the functional equation for Dedekind’s zeta function (cf. e.g. [@washington1997introduction p. 34]) one finds that $\zeta_K(2)=\frac{\pi^4}{48\sqrt{2}}$ which proves the claim. The density of visible points of $\mathcal{A}$ {#subsecDensAB} ---------------------------------------------- Observe that $\mathcal{A}'=\sqrt{2}\mathcal{A}\subset \mathcal{B}$. It is now shown that $C$ is also an occluding set for $\mathcal{A}'$. \[propOcclSet2\] For each $x\in\mathcal{A}'\setminus\widehat{\mathcal{A}'}$ there is $c\in C$ such that $x/c\in \mathcal{A}'$. Since $\mathcal{A}'\subset \mathcal{B}$ we have $\mathcal{A}'\setminus\widehat{\mathcal{A}'}\subset \mathcal{B}\setminus\widehat{\mathcal{B}}$ and so for each $x\in \mathcal{A}'\setminus\widehat{\mathcal{A}'}$ there exists $c\in C$ such that $x/c\in \mathcal{B}$. If $c\neq \sqrt{2}$ then $\sqrt{2}\mid \frac{x_1-x_2}{c}$ so $x/c\in\mathcal{A}'$. Take now $x\in \mathcal{A}'\setminus\widehat{\mathcal{A}'}$ such that for all $c\in C{\setminus\{\sqrt{2}\}}$ we have $x/c\notin \mathcal{B}$. Then $x/\sqrt{2}\in\mathcal{B}$, hence $\gcd(x_1,x_2)=\sqrt{2}^n$ for some $n\geq1$. Since $x\in \mathcal{A}'\setminus\widehat{\mathcal{A}'}$ there is $c\in {\mathbb{Q}}(\sqrt{2})\cap{\mathbb{R}}_{>1}$ such that $x/c\in \mathcal{A}'$. Writing $c=a/b$ with $\gcd(a,b)=1$, the only possible $\pi\in{\mathbb{P}}$ with $\pi\mid a$ is $\pi=\sqrt{2}$. If $\sqrt{2}\mid a$, then it follows that $x/\sqrt{2}\in\mathcal{A}'$. It remains to check the case where $a$ is a unit, i.e. $c=\frac{\lambda^n}{\prod_{p\in {\mathbb{P}}}\pi^{m(\pi)}}$ for some $m:{\mathbb{P}}\longrightarrow{\mathbb{Z}}_{\geq 0}$ with finite support. The facts that $c>1$ and $\pi>1$ for all $\pi\in{\mathbb{P}}$ imply $n>0$. We have $x/\lambda\notin\mathcal{B}$, hence $\overline{x}\notin \|\overline{\lambda}\| \mathcal{W}_\mathcal{A}$. Since $x/c\in\sqrt{2}\mathcal{A}$ it follows that $\overline{x}\in \|\overline{c}\|\mathcal{W}_\mathcal{A}$ and hence $\|\overline{c}\|>\|\overline{\lambda}\|$. However $$\|\overline{c}\|=\frac{\|\overline{\lambda}\|^n}{\prod_{\pi\in {\mathbb{P}}}\|\overline{\pi}\|^{k(\pi)}}\leq \|\overline{\lambda}\|^n\leq \|\overline{\lambda}\|,$$ contradiction. We have $\theta(\widehat{\mathcal{A}'})=\frac{1}{2\zeta_K(2)}$, hence $\theta(\widehat{\mathcal{A}})=\frac{1}{\zeta_K(2)}$. Propositions \[propInclExcl\], \[propOcclSet2\] imply $$\label{eqnB1} \frac{\#(\widehat{\mathcal{A}'}\cap RD)}{{\mathrm{vol}}(RD)}=\sum_{\substack{F\subset C\\ \#F<\infty}}\frac{(-1)^{\#F}\#\left(\left(\mathcal{A}_'\cap \bigcap_{c\in F}c\mathcal{A}_'\right)\cap RD\right)}{{\mathrm{vol}}(RD)}$$ and it is straight-forward to verify that $\mathcal{A}_'\cap \bigcap_{c\in F}c\mathcal{A}_'=\mathcal{P}(m_F\mathcal{W}_\mathcal{A},\widetilde{\mathcal{L}}_F){\setminus\{0\}}$ with $\widetilde{\mathcal{L}}_F=\{(\ell_Fx,\overline{\ell_Fx})\mid x\in \mathcal{O}_K^2,(x_1-x_2)/\sqrt{2}\in\mathcal{O}_K\}$ a sublattice of $\mathcal{\mathcal{L}}_F$ of index $2$. Hence, by [@marklof2014free Prop. 3.2], when letting $R\to\infty$ inside the sum one obtains $$\sum_{\substack{F\subset C\\ \#F<\infty}}\frac{(-1)^{\#F}{\mathrm{vol}}(m_F\mathcal{W}_\mathcal{A})}{16N(\ell_F)},$$ whence $\theta(\widehat{\sqrt{2}\mathcal{A}})=\frac{1}{2\zeta_K(2)}$ follows by and , and the other result is immediate as $\sqrt{2}\mathcal{A}=\mathcal{A}'$. The data of Table 2 of [@baake2014radial] shows that $\#(\widehat{\mathcal{A}}\cap RD)/\#(\mathcal{A}\cap RD)\approx 0.577$ for a particular $D$ and fairly large $R$. This agrees with our results, since $$\kappa_\mathcal{A}=\lim_{R\to\infty}\frac{\#(\widehat{\mathcal{A}}\cap RD)}{\#(\mathcal{A}\cap RD)}=\frac{\theta(\widehat{\mathcal{A}})}{\theta(\mathcal{A})}=\frac{\tfrac{1}{\zeta_K(2)}}{\tfrac{2{\mathrm{vol}}(\mathcal{W}_\mathcal{A})}{16}}=\frac{2(\sqrt{2}-1)}{\zeta_K(2)}=0.5773\ldots$$	ArXiv
--- abstract: 'We present an exact collapsing solution to 2+1 gravity with a negative cosmological constant minimally coupled to a massless scalar field, which exhibits physical properties making it a candidate critical solution. We discuss its global causal structure and its symmetries in relation with those of the corresponding continously self-similar solution derived in the $\Lambda=0$ case. Linear perturbations on this background lead to approximate black hole solutions. The critical exponent is found to be $\gamma = 2/5$.' author: - \| Gérard Clément$^{a}$ [^1] and Alessandro Fabbri$^{b}$ [^2]\ \ [$^{(a)}$Laboratoire de Physique Théorique LAPTH (CNRS),]{}\ [B.P.110, F-74941 Annecy-le-Vieux cedex, France]{}\ [$^{(b)}$Dipartimento di Fisica dell’Università di Bologna]{}\ [and INFN sezione di Bologna,]{}\ [Via Irnerio 46, 40126 Bologna, Italy]{} date: 'March 23, 2001' title: \| Analytical treatment of critical collapse in 2+1 dimensional AdS spacetime:\ a toy model --- Introduction ============ Since its discovery, the BTZ black hole solution \[1\] of 2+1 dimensional AdS gravity has attracted much interest because it represents a simplified context in which to study the classical and quantum properties of black holes. A line of approach which has been opened only recently [@CP; @HO; @gar; @burko] concerns black hole formation through collapse of matter configurations coupled to 2+1 gravity with a negative cosmological constant. As first discovered in four dimensions by Choptuik [@chop], collapsing configurations which lie at the threshold of black hole formation exhibit properties, such as universality, power-law scaling of the black hole mass, and continuous or discrete self-similarity, which are characteristic of critical phenomena [@gund]. In the case of a spherically symmetric massless, minimally coupled scalar field, a class of analytical continously self-similar (CSS) solutions was first given by Roberts [@rob; @brady; @oshiro]. These include critical solutions, lying at the threshold between black holes and naked singularities, and characterized by the presence of null central singularities. Linear perturbations of these solutions [@fro; @hay] lead to approximate black hole solutions with a spacelike central singularity. Numerical simulations of circularly symmetric scalar field collapse in 2+1 dimensional AdS spacetime were recently performed by Pretorius and Choptuik [@CP] and Husain and Olivier [@HO]. Both groups observed critical collapse, which was determined in [@CP] to be continuously self-similar near $r=0$. In [@gar], Garfinkle has found a one-parameter family of exact CSS solutions of 2+1 gravity without cosmological constant, and argued that one of these solutions should give the behaviour of the full critical solution ($\Lambda\neq 0$) near the singularity. The purpose of this paper is to present a new CSS solution to the field equations with $\L=0$ which can be extended to a threshold solution of the full $\L \neq 0$ equations. The new $\L = 0$ solution is derived in Sect. 3. It presents a null central singularity and, besides being CSS, possesses four Killing vectors. In Sect. 4 we address the extension of this CSS solution to a quasi-CSS solution of the full $\L < 0$ problem, and show that the requirement of maximal symmetry selects a unique extension. This inherits the null central singularity of the $\L = 0$ solution, and has the correct AdS boundary at spatial infinity. Finally, we perform in Sect. 5 the linear perturbation analysis in this background, find that it does lead to black hole formation, and determine the critical exponent. CSS solutions ============= The Einstein equations for cosmological gravity coupled to a massless scalar field in (2+1) dimensions are G\_ - Łg\_ = T\_, with the stress-energy tensor for the scalar field T\_ = \_\_- 12 g\_ \^\_. The signature of the metric is (+ - -), and the cosmological constant $\L$ is negative for AdS spacetime, $\L = -l^{-2}$. Static solutions of these equations include the BTZ black hole solutions [@BTZ] with a vanishing scalar field $\phi = 0$, and singular solutions when a non-trivial scalar field is coupled with the positive sign for the gravitational constant $\k$ [@sigcosm]. We shall use for radial collapse the convenient parametrisation of the rotationally symmetric line element in terms of null coordinates $(u,v)$: ds\^2 = e\^[2]{}dudv - r\^2d\^2, with metric functions $\s(u,v)$ and $r(u,v)$. The corresponding Einstein equations and scalar field equation are & r\_[,uv]{} = r e\^[2]{}, &\ & 2\_[,uv]{} = e\^[2]{} - \_[,u]{}\_[,v]{}, &\ & 2\_[,u]{} r\_[,u]{} -r\_[,uu]{} = r\_[,u]{}\^2, &\ & 2\_[,v]{} r\_[,v]{} -r\_[,vv]{} = r\_[,v]{}\^2, &\ & 2r\_[,uv]{} + r\_[,u]{}\_[,v]{} + r\_[,v]{}\_[,u]{} = 0 & . From the Einstein equations, the Ricci scalar is R = -6Ł+ 4e\^[-2]{}\_[,u]{}\_[,v]{}. It follows from (\[ric1\]) and (\[ein2\]) that the behavior of the solutions near the singularity is governed by the equations (\[ein1\])-(\[phi\]) with vanishing cosmological constant $\L = 0$ (see also [@burko]). Assuming $\L = 0$, Garfinkle has found [@gar] the following family of exact CSS solutions to these equations ds\^2 & = & -A()\^[c\^2]{}dudv - 14(v+u)\^2d\^2,\ & = & -2c( + ), depending on an arbitrary constant $c$ and a scale $A > 0$. In (\[gar1\]), $u$ is retarded time, and $-v$ is advanced time. These solutions are continuously self-similar with homothetic vector $(u\part_u + v\part_v)$. An equivalent form of these CSS solutions, obtained by making the transformation -u = (-\|[u]{})\^[2q]{}, v = (\|[v]{})\^[2q]{} (1/2q = 1 - c\^2) to the barred null coordinates $(\bar{u},\bar{v})$, is ds\^2 & = & -\|[A]{}(\|[v]{}\^q + (-\|[u]{})\^q)\^[2(2q-1)/q]{}d\|[u]{}d\|[v]{} - 14(\|[v]{}\^[2q]{} - (-\|[u]{})\^[2q]{})\^2d\^2,\ & = & -2c(\|[v]{}\^q + (-\|[u]{})\^q). The corresponding Ricci scalar is R = (\|[v]{}\^q + (-\|[u]{})\^q)\^[2(1-3q)/q]{}(-\|[u]{})\^[q-1]{} (\|[v]{})\^[q-1]{}. Garfinkle suggested that the line element (\[gar1\]) describes critical collapse in the sector $r = -(u+v)/2 \ge 0$, near the future point singularity $r = 0$ (where the Ricci scalar behaves, for $v \propto u$, as $u^{-2}$). The corresponding Penrose diagram (Fig. 1) is a triangle bounded by past null infinity $u \to -\infty$, the other null side $v = 0$, and the central regular timelike line $r = 0$. For $\k c^2 \ge 1$ ($q < 0$), the Ricci scalar R \~(\|[v]{})\^[q-1]{} \~(v)\^[(q-1)/2q]{} is regular near $v = 0$, which moreover turns out to be at infinite geodesic distance. To show this, we consider the geodesic equation (e\^[2]{}) = -2rr\_[,u]{}\^2 = -2l\^2r\^[-3]{}r\_[,u]{} ($l$ constant) near $v = 0$, $u$ constant, which gives $v \propto (ls)^{4q}$ for $l \neq 0$, or $s^{2q}$ for $l=0$, so that in all cases the affine parameter $s \to \infty$ for $v \to 0$, and the spacetime is geodesically complete. For $\k c^2 < 1$ ($q > 0$), we see from (\[ric2\]) that the null line $v = 0$ is a curvature singularity if $\k c^2 < 1/2$ ($q < 1$). If $1/2 \le \k c^2 < 1$ ($q \ge 1$), the surface $v = 0$ is regular. However, as discussed by Garfinkle, the metric (\[gar2\]) can be extended through this surface only for $q = n$, where $n$ is a positive integer. For $n$ even, the extended spacetime is made of two symmetrical triangles joined along the null side $\bar{v} = 0$, and has two coordinate singularities $r = 0$, one timelike ($\bar{u} - \bar{v} = 0$) and one spacelike ($\bar{u} + \bar{v} = 0$), but no curvature singularity. For $n$ odd, one of the $r = 0$ sides becomes a future spacelike curvature singularity ($e^{2\s} = 0$), similar to that of Brady’s supercritical solutions for scalar field collapse in (3+1) dimensions [@brady], except for the fact that in the present case the singularity is not hidden behind a spacelike apparent horizon (Fig. 2). Let us point out that, besides the solutions (\[gar1\]), the system (\[ein1\])-(\[phi\]) also admits for $\L = 0$ another family of CSS solutions ds\^2 & = & A()\^[c\^2]{}dudv - 14(v+u)\^2d\^2,\ & = & \|[A]{}(\|[v]{}\^q - (-\|[u]{})\^q)\^[2(2q-1)/q]{}d\|[u]{}d\|[v]{} - 14(\|[v]{}\^[2q]{} - (-\|[u]{})\^[2q]{})\^2d\^2, with $\phi = -2c\ln(\sqrt{v} - \sqrt{-u})$, and we choose $A > 0$ and consider the sector $0 \le v \le -u$. These solutions have a future spacelike central ($r = 0$) curvature singularity at $(-\bar{u})^q = \bar{v}^q$ (where the Ricci scalar (\[ric2\]) diverges) for all $q < 0$ or $q > 0$ (implying $q > 1/2$). For $q < 0$, the Penrose diagram is a triangle bounded by past null infinities $\bar{u} \to -\infty$ and $\bar{v} = 0$ (which is at infinite geodesic distance). For $q > 0$, geodesics terminate at $\bar{v} = 0$, unless $q = n$ integer. For $n$ even, the extended spacetime has two central curvature singularities $r = 0$, one spacelike and the other timelike. The extended spacetime for $n$ odd is more realistic. In this case the extension from $\bar{v} > 0$ to $\bar{v} < 0$ amounts to replacing (\[gar3\]) with $A > 0$ by the original Garfinkle solution (\[gar1\]) with $A > 0$, the resulting Penrose diagram being that of Fig. 2. A new CSS solution for $\L = 0$ =============================== Among the one-parameter ($c$ or $q$) family of CSS solutions (\[gar1\]), the special solution, corresponding to $\k c^2 = 1$, ds\^2 = A( + )\^4 - 14(v+u)\^2d\^2, is singled out by the fact that the transformation (\[q\]) breaks down for this value. The transformation appropriate to this case, -u = 2e\^[-U]{}, v = 2e\^V = 2e\^[U-2T]{} (with $T \ge U$ for $u+v \le 0$) transforms the solution (\[gar4\]) to ds\^2 & = & e\^[-2U]{}\[-4A(1 + e\^[U-T]{})\^4dUdV - (1 - e\^[2(U-T)]{})\^2 d\^2\],\ & = & U - 2(1 + e\^[U-T]{}) (we use from now on units such that $\k = 1$, and have dropped an irrelevant additive constant from $\phi$). Starting from this special CSS solution of the Garfinkle class, we now derive, by a limiting process, a new CSS solution which, as we shall see, exhibits a null singularity. We translate $T$ to $T-T_0$, and take the late-time limit $T_0 \to -\infty$, leading to the new CSS solution (written for $A = -1/2$) ds\^2 = e\^[-2U]{}(2dUdV - d\^2), = U, with a very simple form which is reminiscent of the Hayward critical solution for scalar field collapse in 3+1 dimensions [@hay], ds\^2 = e\^[2]{}(2d\^2 - 2d\^2 - d\^2), = . The transformation \|[u]{} = -e\^[-2U]{}, \|[v]{} = V leads from (\[crit0\]) to the even more simple form of this solution ds\^2 = d\|[u]{}d\|[v]{} + \|[u]{}d\^2, = -12(-\|[u]{}), which is reminiscent of the other form of the Hayward solution ds\^2 = 2d\|[u]{}d\|[v]{} + \|[u]{}\|[v]{}d\^2, = -12(-\|[u]{}/\|[v]{}). The solution (\[crit0\]) or (\[crit1\]) is continuously self-similar, with homothetic vector K = \_U = -2\|[u]{}\_[\|[u]{}]{}. It also has a high degree of symmetry, with 4 Killing vectors L\_1 & = & \_U + 2V\_V + \_,\ L\_2 & = & \_V + U\_,\ L\_3 & = & \_V,\ L\_4 & = & \_, generating the solvable Lie algebra & \[L\_1, L\_2\] = L\_4-L\_2, & \[L\_2,L\_3\] = 0,\ & \[L\_1, L\_3\] = -2L\_3, & \[L\_2,L\_4\] = -L\_3,\ & \[L\_1, L\_4\] = -L\_4, & \[L\_3,L\_4\] = 0. The Ricci scalar (\[ric1\]) is identically zero for the solution (\[crit0\]), for which the sole nonvanishing Ricci tensor component is $R_{UU} = 1$. It follows that this metric is devoid of curvature singularity. However there is an obvious coordinate singularity at $U \to +\infty$, or $\bar{u} = 0$ (where $r=0$). To determine the nature of this singularity, we study geodesic motion in the spacetime (\[crit1\]). The geodesic equations are integrated by = , \|[u]{} = l, + l = , where $\pi$ and $l$ are the constants of the motion associated with the Killing vectors $L_3$ and $L_4$, and the sign of $\varepsilon$ depends on that of $ds^2$ along the geodesic. The null line $\bar{u} = 0$ can be reached only by those geodesics with $\pi \neq 0$. Then, the third equation (\[geos\]) integrates to \|[v]{} = \|[u]{} - + [const.]{} = \|[u]{} - (-\|[u]{}) + [const.]{}. It follows that nonradial geodesics ($l \neq 0$) terminate at $\bar{u} = 0, \bar{v} \to +\infty$, while radial geodesics ($l = 0$), which behave as in cylindrical Minkowski space, can be continued through the null line $\bar{u} = 0$ to $\bar{u} \to +\infty$ . So in this sense only the endpoint $\bar{v} \to +\infty$ of the null line $\bar{u} = 0$ is singular. However formal analytic continuation of the metric (\[crit1\]) from $\bar{u} < 0$ to $\bar{u} > 0$ involves a change of signature from (+ - -) to (+ - +), leading to the appearance of closed timelike curves. So the null line $\bar{u} = 0$ corresponds to a singularity in the causal structure of the spacetime, analogous to the central singularity in the causal structure of the BTZ black holes [@BTZ]. The resulting Penrose diagram, reminiscent of that of the Hayward critical solution [@hay], is a diamond bound by three lines at null infinity ($\bar{v} = -\infty, \bar{u} = -\infty, \bar{v} = +\infty$) and the null singularity $\bar{u} = 0$ (Fig. 3). Extending the new solution to $\L \neq 0$ ========================================= In the preceding section we have found an exact solution for scalar field collapse with $\L = 0$, which presents a central null singularity. This property makes it a candidate threshold solution, lying at the boundary between naked singularities and black holes. However black holes exist only for $\L < 0$, so the solution (\[crit1\]) can only represent the behavior of the true threshold solution near the central singularity, where the cosmological constant can be neglected. This hypothetical $\L < 0$ solution cannot be self-similar, essentially because the scale is fixed preferentially by the cosmological constant [@CP]. So what we need is to find some other way to extend (\[crit1\]) to a solution of the full system of Einstein equations with $\L < 0$. A first possible approach is to expand this solution in powers of $\L$, with the zeroth order given by the CSS solution (\[crit1\]). In the parametrisation (\[an\]), this zeroth order is (dropping the bars in (\[crit1\]) ) r\_0 = (-u)\^[1/2]{}, \_0 = 0, \_0 = -12 \|u\|. We look for an approximate solution to first order in $\L$ of the form r = (-u)\^[1/2]{} + Łr\_1, \_ = Ł\_1, = -12 \|u\| + Ł\_1 , \[lmk\]with the boundary condition that the fonctions $r_1$, $\sigma_1$ and $\phi_1$ vanish on the central singularity $u = 0$ . Eq (\[ein1\]) gives \[cmm\] r\_1 = (-u)\^[1/2]{}(13uv + f(u)), with $f(0) = 0$. Then, the linearized Eq. (\[ein4\]) gives 2r\_0\^[1/2]{}(r\_0\^[1/2]{}[\_1]{}\_[,v]{})\_[,u]{} = -[r\_1]{}\_[,v]{}[\_0]{}\_[,u]{} = 16(-u)\^[1/2]{}, which is solved by \[zzw\] \_1 = (1[15]{}uv + g(u)). The linearized Eq. (\[ein2\]) 2[\_1]{}\_[,uv]{} = 1 - [\_0]{}\_[,u]{}[\_1]{}\_[,v]{} = then gives \[nxn\] \_1 = 4[15]{}uv + h(u). Finally Eq. (\[ein2\]) leads to the relation between the arbitrary functions $f$, $g$, $h$ uf”(u) + f’(u) = g’(u) + h’(u). Not only does this first order solution break the continuous self-similarity generated by (\[K\]), as expected, but it also breaks the isometry group generated by the Killings (\[kill\]) down to $U(1)$ (generated by $L_4 = \part_{\theta}$), except in the special case $f = g = h = 0$, where the Killing subalgebra $(L_1, L_4)$ remains. This suggests looking for an exact $\L < 0$ extension of the $\L = 0$ CSS solution of the form ds\^2 = e\^[2(x)]{}dudv + u\^2(x)d\^2, = -12\|u\| + (x), with $x = uv$. This will automatically preserve to all orders the Killing subalgebra $(L_1, L_4)$. Inserting this ansatz into the field equations (\[ein1\])-(\[phi\]) leads to the system x” + 32’ & = & e\^[2]{},\ 2(x” + ’) + ’(x’-12) & = & e\^[2]{},\ x\^2(-” + 2’’- ’\^2) + x(- ’ + (’+’)) & = & 0\ - ” + 2’’- ’\^2 & = & 0\ 2x(’)’ + 52’ & = & 12’. ($' = d/dx$). The unique, maximally symmetric extension of the CSS solution (\[crit1\]) reducing to (\[crit1\]) near $u = 0$ is the solution of the system (\[einx1\])-(\[phix\]) with the boundary conditions (0) = 1, (0) = 0, (0) = 0. The comparison of (\[einx3\]) and (\[einx4\]) yields = e\^[+]{}. The combination $(\ref{einx1}) + x(\ref{einx4})$ then gives, together with (\[rhospsi\]), x(2’\^2 + 2’’ - ’\^2) + 32(’ + ’) = e\^[2]{}. The third independent equation is for instance (\[einx2\]): 2(x” + ’) + ’(x’ - 12) = e\^[2]{}. Using these last two equations with the boundary conditions (\[boundx\]), one can in principle write down series expansions for $\s(x)$ and $\psi(x)$. Another simple relation, deriving from (\[einx4\]) and (\[rhospsi\]), is ” + ” - ’\^2 + 2’\^2 = 0. We are interested in the behavior of this extended solution in the sector $u < 0$, $v > 0$, i.e. $x < 0$. In this sector, Eqs. (\[einx1\]), (\[phix\]) and (\[einx2\]) can be integrated to (-x)\^[3/2]{}’ & = & \_x\^0 (-x)\^[1/2]{}e\^[2]{}dx,\ (-x)\^[5/4]{}’ & = & 14 \_x\^0 (-x)\^[1/4]{}’ dx,\ -x’ & = & 12 \_x\^0 ( e\^[2]{} + ’(12 - x’))dx . As long as $\rho > 0$, Eq. (\[sum1\]) (with $x < 0$, $\L < 0$) implies $\rho' < 0$, so that $\rho(x)$ decreases to 1 when $x$ increases to 0. It then follows from (\[sum2\]) that $\psi' <0$. Also, (\[sum2\]) can be integrated by parts to x’ = 14 - \_x\^0 (-x)\^[-3/4]{}dx, showing that $x\psi' < 1/4$. It then follows from (\[sum3\]) that $\s' < 0$. So, as $x$ decreases, the functions $\rho$ and $e^{2\s}$ increase and possibly go to infinity for a finite value $x = x_1$. If this is the case, the behavior of these functions near $x_1$ must be & = & \_1(1[\|[x]{}]{} + 1[4x\_1]{} - + ... )\ e\^[2]{} & = & (1 + + ...)\ & = & \_1 + - (\|[x]{}) + ... ($\bar{x} = x - x_1$). These expectations are borne out by the actual numerical solution of the system x” + ’ &=& - e\^[2]{},\ -”+4’’ &=& ’\^2 +\^2’\^2, \[sist\] (this last equation comes from (\[einx4\]) where $\psi'$ is given by derivation of (\[rhospsi\]) ) where we have set $\L=-2$, with the boundary counditions $\rho(0)=1$, $\rho'(0)=-2/3$ (see eqs. (\[cmm\]) and (\[lmk\]) ), $\sigma(0)=0$. The plots of the functions $ \rho (x)$ , $\sigma (x)$ and $\psi' (x)$ are given in Figs. (4,5,6,). The value of $x_1$ is found to be approximately $-1.94$ (i.e. $\L x_1 = +3.88$). The coordinate transformation[^3] u = Ł\^[-1]{}e\^[-\|[U]{}]{}, v = e\^[\|[V]{}]{} (\|[U]{} = \|[T]{} - \|[R]{}, \|[V]{} = \|[T]{}+\|[R]{}) leads to $x = \L^{-1} e^{2\bar{R}}$ and, on account of (\[anext\]) and (\[rhospsi\]), to the form of the metric ds\^2 = -Ł\^[-1]{}e\^[2((\|[R]{})+\|[R]{})]{}(d\|[U]{}d\|[V]{} - e\^[2(\|[R]{}) -\|[V]{}]{}d\^2). Near the spacelike boundary $\bar{R} = \bar{R}_1$ of the spacetime, the collapsing metric and scalar field behave, from (\[bound1\]), as ds\^2 -Ł\^[-1]{}(\|[R]{}\_1-\|[R]{})\^[-2]{}(d\|[T]{}\^2-d\|[R]{}\^2-e\^[\|[T]{}\_1 -\|[T]{}]{}d\^2), = \_1 + \|[T]{}/2 ($\bar{R}-\bar{R}\simeq \bar{x}/2x_1$). This metric is asymptotically AdS, as may be shown by making the further coordinate transformation, \|[R]{}-\|[R]{}\_1 = -2/XT, \|[T]{}-\|[T]{}\_1 = 2(T/2), leading to ds\^2 -Ł\^[-1]{}(X\^2 dT\^2 - - X\^2d\^2 ), = \_1 + (T/2). The next-to-leading terms in the metric containing logarithms, this asymptotic behavior differs from that of BTZ black holes. It follows from this discussion that the Penrose diagram of the $\L < 0$ threshold solution in the sector $v > 0$, $u < 0$ is a triangle bounded by the null line $v = 0$, the null causal singularity $u = 0$, and the spacelike AdS boundary $X \to \infty$. The null singularity $u = 0$ remains naked, i.e. is not hidden behind a trapping horizon, which would correspond to \_v r = -(-u)\^[3/2]{}’(x) = 0, because $\rho' <0$ (as discussed above) implies that the only solution of this equation is $u = 0$. For the sake of completeness, let us also discuss the behavior of the solution of the system (\[einx1\])-(\[phix\]) in the sector $x > 0$. In this case, one can write down integro-differential equations similar to (\[sum1\])-(\[sum3\]), from which one again derives that $\rho' < 0$, $\psi' < 0$ and $\s'< 0$. It follows that the metric function $e^{2\s}$ decreases as $x$ increases, eventually vanishing for a finite value $x = x_0$, corresponding to a spacelike curvature singularity (this has been confirmed numerically). The behavior of the solution near this singularity is found to be (x\_0-x), (x\_0-x), (x\_0-x) (= 3 - 1), and the coordinate transformation $u = e^U, v = e^V (x = e^{2T})$ leads to the form of the metric near the singularity ds\^2 (T\_0-T)\^[\^2]{}(dT\^2-dR\^2) +e\^[R\_0-R]{}(T\_0-T)\^2d\^2. Perturbations ============= To check whether the quasi-CSS solution (\[anext\]) of the full $\L \neq 0$ problem determined in the preceding section is indeed a threshold solution, we now study linear perturbations of this solution. Our treatment will follow the analysis of perturbations of critical solutions in the case of scalar field collapse in 3+1 dimensions [@fro; @hay]. The relevant time parameter in critical collapse being the retarded time $U = -(1/2)\ln(-u)$ (the “scaling variable” of [@fro]), we expand these perturbations in modes proportional to $e^{kU} = (-u)^{-k/2}$, with $k$ a complex constant. We recall that only the modes with $Re\ k>0$ grow as $U \to +\infty$ ($u \to -0$) and lead to black hole formation, whereas those with $Re\ k<0$ decay and are irrelevant. The other relevant variable is the “spatial” coordinate $x = uv$, and the perturbations are decomposed as r & = & (-u)\^[1/2]{}((x) + (-u)\^[-k/2]{}(x)),\ & = & -12\|u\| + (x) + (-u)\^[-k/2]{}(x),\ & = & (x) + (-u)\^[-k/2]{}(x). Then, the Einstein equations (\[ein1\])-(\[phi\]) are linearized in $\tilde{r}$, $\tilde{\phi}$, $\tilde{\sigma}$, using \_[,u]{} =-(-u)\^[[-k/2]{}-1]{}(x’-2), \_[,v]{} =-(-u)\^[[-k/2]{}+1]{}’ . The resulting equations are homogeneous in $u$, which drops out, and the linearized system reduces to & & x” + ([-k/2]{}+3/2)’ = e\^[2]{}( + 2),\ & & 2x” +([-k]{}+2)’ = Łe\^[2]{} - (2x’-1/2)’ + (k/2)’,\ & & -(-k+1)x’ + ((-k+1)x’-(k\^2-1)/4) + x’ - k(x’ + /2) =\ & & -(x’ - k(1/2-x’)) + (1/4-x’),\ & & 2(’’ + ’’) - ” = ’(2’ + ’),\ & & 2x” + (2x’ + (-k+5/2))’ - (k/2)’ + (2x’-1/2)’\ & & + (2x” + ([-k/2]{}+5/2)’) = 0. What is the number of the independent constants for this system? The perturbed Klein-Gordon equation (\[pphi\]) is clearly redundant, while Eqs. (\[pein3\]) and (\[pein4\]) are constraints. So, as in the (3+1)-dimensional case [@fro; @hay], the order of the system is four, and the general solution depends on four integration constants. However, one of these four independent solutions corresponds to a gauge mode and is irrelevant. The parametrisation (\[anext\]) is invariant under infinitesimal coordinate transformations $v \to v + f(v)$. For $f(v) = -\alpha v^{1+k/2}$, these lead to $x \to x - \alpha (-u)^{-k/2}(-x)^{1+k/2}$, giving rise to the gauge mode \_k(x) & = & (-x)\^[1+k/2]{}’(x),\ \_k(x) & = & (-x)\^[1+k/2]{}’(x),\ \_k(x) & = & , which solves identically the system (\[pein1\])-(\[pphi\]). So, up to gauge transformations, the general solution of this system depends only on three independent constants. These will be determined, together with the possible values of $k$ (the eigenfrequencies) by enforcing appropriate and reasonable boundary conditions. We shall use here the “weak boundary conditions” of [@hay] on the boundaries $u = 0$ and $x = x_1$ ($X \to \infty$) \_[u 0]{}r\^[-1]{} 0, \_[x x\_1]{}r 0, together with the condition (0) = 0, which guarantees that the singularity of the perturbed solution starts smoothly from that of the unperturbed one. On the third boundary $v = 0$, we shall impose a stronger condition by requiring that the perturbations are analytic in $v$, in order for the perturbed solution to be extendible beyond $v=0$ to negative values of $v$ at finite $u$. First, we consider the region $x\to 0$ where, according to Eqs. (\[zwx\]), (\[cmm\]), (\[zzw\]) and (\[nxn\]), \[wxc\] 1 + x, e\^[2]{}1+x, x. Let us assume a power-law behavior r(x) \~a (-x)\^[p]{} where $p$ is a constant to be determined. Then Eqs. (\[pein1\]), (\[pein2\]) and (\[pein4\]) can be approximated near $x = 0$ as & & x” + ([-k/2]{}+3/2)’ Ł,\ & & x” + ([-k/2]{}+1)’ 14’\ & & 2’’ - ” 2’’. Eliminating the functions $\tilde{\s}$ and $\tilde{\phi}$ between these three equations and using Eq. (\[wxc\]), we obtain the fourth-order equation 4x\^2”” + (-4k + 13)x”’ + (k/2-1)(2k - 5)” 0, which implies the power-law behavior (\[pow\]) with the exponent $p$ constrained by p(p-1)(p-k/2-3/4)(p-k/2-1)=0. Obviously the root $p = k/2 + 1$ corresponds to the gauge mode (\[gauge\]) and must be discarded as irrelevant. As a consequence the general solution near $x = 0$ can be given in terms of three independent constants as & &r(x) \~A +B(-x) +ŁC(-x)\^[3/4 +k/2]{},\ & &(x) \~- 2 + Ł\^[-1]{}2 - (k + )(-x)\^[-1/4 +k/2]{},\ & & (x) \~2 - Ł\^[-1]{}2 + (k + )(-x)\^[-1/4 +k/2]{}.\[pti\] Let us note that this solution remains valid in the limit $\L \to 0$, leading to the limiting solution $\tilde{r} \sim A + B(-x)$ (with $B = 0$ for $k \neq 3$), which could also be obtained directly by solving the equation $\tilde{r}'' = 0$ which results from (\[pein4\]) in the limit $\L \to 0$, together with the stronger condition (from Eq. (\[pein1\])) $(k-3)\tilde{r}' = 0$. Now we enforce the boundary conditions at $x = 0$. For $k > 0$, $\tilde{r}$ is dominated by its first constant term in (\[rti\]), so that the condition (\[bound0\]) can only be satisfied for $u \to 0$ if A = 0. Then, for $k > 1/2$, $\tilde{r}$ is dominated by its second term $-Bx$, leading to a perturbation $(-u)^{1/2-k/2}\tilde{r}(x)$ which blows up as $u \to 0$ and violates (\[bound\]) unless \[rang\] k 3. Then we impose the condition of analyticity in $v$ at fixed $u$. This is satisfied if \[unn\] k = 2n - 3/2, where $n$ is a positive integer. Combining eqs. (\[rang\]) and (\[unn\]) we find that $k$ has only two positive eigenvalues k = 1/2, k = 5/2. However, in the above analysis we have disregarded the fact that $k = 1/2$ is a double root of the secular equation (\[sec\]). For $k = 1/2$ the correct behavior of the general solution near $x = 0$ is & &r(x) \~A +B(-x) +ŁC(-x)\|x\|,\ & &(x) \~- 2 - Ł\^[-1]{} 4 - 4 - 4 \|x\|\ & & (x) \~4 + Ł\^[-1]{} 4 + 4 + 4 \|x\| , which satisfies the condition of analyticity only if $C = 0$. At the AdS boundary ($x\to x_1$) the leading behaviour of the background is, from Eqs. (\[bound1\]), \[cdfg\] , e\^[2]{}(), \_1. We again assume a power-law behavior \~b\|[x]{}\^q ($\bar{x} = x-x_1$). Then Eq. (\[pein2\]), where $\tilde{\phi}$ can be neglected, gives q(q-1) = 2, i.e. $q = -1$ or $q = 2$. Then, Eq. (\[pein1\]) reduces near $\bar{x} = 0$ to ” - 2\|[x]{}\^[-2]{} 4b\_1\|[x]{}\^[q-3]{}. If $q = -1$, the behavior of the solution is governed by the right-hand side, i.e. $\tilde{r} \propto \bar{x}^{-2}$, which violates the boundary condition (\[bound\]) for $x \to x_1$. So the behavior $\tilde{\s} \sim b\bar{x}^{-1}$ must be excluded, which fixes another integration constant $D=0$ (where $D$ is a linear combination of $B$ and $C$). Then, the generic behavior of the solution of Eq. (\[asr\]) with $q = 2$ is governed by that for the homogeneous equation, i.e. r \~. This is consistent with the boundary condition (\[bound\]), and is an admissible small perturbation if its amplitude is small enough, $E \ll \rho_1$. For $k = 1/2$, we have seen that two of the three integration constants in (\[rtii\])-(\[ptii\]) are fixed ($A = C = 0$) by condition (\[bound0\]) and the analyticity condition, while the weak boundary condition at the AdS boundary fixes a third constant $D=0$. However this is impossible, as the perturbation amplitude must remain as a free parameter. So the mode $k = 1/2$ cannot satisfy all our boundary conditions, and we are left with a single eigenmode, k = 5/2, completely determined up to an arbitrary amplitude by the two conditions $A=D=0$. The corresponding perturbed metric function $r$ behaves near $x = 0$ as r (-u)\^[1/2]{}\[1 + Łx - (-u)\^[-5/4]{}Bx\]. For $B < 0$, the central singularity $r = 0$ is approximately given by (-u)\^[1/4]{} -Bv. Our boundary conditions guarantee that it starts at $u=v=0$ (as for the unperturbed solution) and then becomes spacelike in the region $v>0$. This singularity is hidden behind a trapping horizon (defined by Eq. (\[trap\])) which, near $x = 0$, is null, (-u)\^[5/4]{}= (a null trapping horizon was also found in [@hay]). Let us point out the crucial role played by the cosmological constant $\Lambda$ in the formation of this trapping horizon. For $\L = 0$, $\rho(x) = 1$, while, as discussed after Eq. (\[pti\]), the perturbation $\tilde{r}$ with the boundary condition (\[bound0\]) vanishes for $\L = 0$, so that the perturbed radial function $r$ is (as in [@gar]) identical to the CSS one, and the trapping horizon does not exist. Near the AdS boundary $x \to x_1$, it follows from (\[cdfg\]) and (\[asol\]) that both the central singularity and the trapping horizon are tangent to the null line (-u)\^[5/4]{} = -E ()\^[-1/2]{}. Thus, perturbations of the quasi-CSS solution lead to black hole formation, showing that this solution is indeed a threshold solution, and is a candidate to describe critical collapse. Near-critical collapse is characterized by a critical exponent $\gamma$, defined by the scaling relation $Q \propto \|p - p^\|^{s\gamma}$, for a quantity $Q$ with dimension $s$ depending on a parameter $p$ (with $p = p^$ for the critical solution). Choosing for $Q$ the radius $r_{AH}$ of the apparent horizon, and identifying $p - p^$ with the perturbation amplitude $B$, we obtain from (\[trap1\]) r\_[AH]{} ()\^[2/5]{}, leading to the value of the critical exponent $\gamma = 2/5$, in agreement with the renormalization group argument [@kha] leading to $\gamma = 1/k$. Conclusion ========== We have discussed in detail the causal structure of the Garfinkle CSS solutions (\[gar1\]) to the $\L= 0$ Einstein-scalar field equations. From a special solution of this class, we have derived by a limiting process a new CSS solution, which we have extended to a unique solution of the full $\L < 0$ equations, describing collapse of the scalar field onto a null central singularity. This is not a curvature singularity (all the curvature invariants remain finite), but a singularity in the causal structure similar to that of the BTZ black hole. Finally, we have analyzed linear perturbations of the $\L < 0$ solution, found a single eigenmode $k = 5/2$, checked that this mode does indeed give rise to black holes, and determined the critical exponent $\gamma = 2/5$. For comparison, Choptuik and Pretorius [@CP] derived, by analysing the observed scaling behavior of the maximum scalar curvature, the value $1.15 < \gamma < 1.25$ for the critical exponent. This value is different from the value $\gamma\sim 0.81$ obtained in the numerical analysis of Husain and Olivier [@HO] from the scaling behavior of the apparent horizon radius. Our value $\gamma = 0.4$, while significantly smaller than these two conflicting estimates, is of the order of the theoretical value $\gamma = 1/2$ derived either from the analysis of dust-ring collapse [@ps], of black hole formation from point particle collisions [@bir], or of the $J = 0$ to $J \neq 0$ transition of the BTZ black hole [@kg]. It is worth mentioning here that, even though they were obtained for a vanishing cosmological constant and thus solve the $\L\neq 0$ equations only near the singularity, the Garfinkle CSS solutions are, for the particular value (chosen in order to better fit the numerical curves) $c = (7/8)^{1/2} \simeq 0.935$, in good agreement [@gar] with the numerical results of [@CP] at an intermediate time. The fact that this value is close to 1 suggests that the $c = 1$ CSS solution (\[gar5\]) approximately describes near-critical collapse at intermediate times. If this the case, then it would not be surprising if its late-time limit, our new CSS solution Eq. (\[crit0\]), gives a good description of exactly critical collapse near the singularity. A fuller understanding of the relationship between the numerically observed near-critical collapse and these various $\L = 0$ CSS solutions could be achieved by extending them to $\L < 0$, as done in the present work for the special solution (\[crit1\]). [20]{} M. Bañados, C. Teitelboim and J. Zanelli, [Phys. Rev. Lett.]{} [69]{}, 1849 (1992); M. Bañados, M. Henneaux, C. Teitelboim and J. Zanelli, [Phys. Rev.]{} D [48]{}, 1506 (1993) F. Pretorius and M.W. Choptuik, [Phys. Rev.]{} D [62]{}, 124012 (2000) V. Husain and M. Olivier, [Class. Quant. Grav.]{} [18]{}, L1 (2001) D. Garfinkle, [Pys. Rev.]{} D [63]{}, 044007 (2001) L. Burko, [Phys. Rev.]{} D [62]{}, 127503 (2000) M. Choptuik, [Phys. Rev. Lett.]{} [70]{}, 2980 (1993) C. Gundlach, [Living Rev. Rel.]{} [2]{}, 4 (1999) M.D. Roberts, [Gen. Rel. 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--- abstract: 'This paper introduces application of Reflexive Game Theory to the matter of multistage decision making processes. The idea behind is that each decision making session has certain parameters like “when the session is taking place", “who are the group members to make decision", “how group members influence on each other", etc. This study illustrates the consecutive or sequential decision making process, which consist of two stages. During the stage 1 decisions about the parameters of the ultimate decision making are made. Then stage 2 is implementation of Ultimate decision making itself. Since during stage 1 there can be multiple decision sessions. In such a case it takes more than two sessions to make ultimate (final) decision. Therefore the overall process of ultimate decision making becomes multistage decision making process consisting of consecutive decision making sessions.' author: - Sergey Tarasenko title: Modeling multistage decision processes with Reflexive Game Theory --- Introduction ============ The Reflexive Game Theory (RGT) [@lef2; @lef5] allows to predict choices of subjects in the group. To do so, the information about a group structure and mutual influences between subjects is needed. Formulation and development of RGT was possible due to fundamental psychological research in the field of reflexion, which had been conducted by Vladimir Lefebvre [@lef4]. The group structure means the set of pair-wise relationships between subjects in the group. These relationships can be either of alliance or conflict type. The mutual influences are formulated in terms of elements of Boolean algebra, which is built upon the set of universal actions. The elements of Boolean algebra represent all possible choices. The mutual influences are presented in the form of Influence matrix. In general, RGT inference can be presented as a sequence of the following steps [@lef2; @lef5]: 1\) formalize choices in terms of elements of Boolean algebra of alternatives; 2\) presentation of a group in the form of a fully connected relationship graph, where solid-line and dashed-line ribs (edges) represent alliance and conflict relationships, respectively; 3\) if relationship graph is decomposable, then it is represented in the form of polynomial: alliance and conflict are denoted by conjunction ($\cdot$) and disjunction (+) operations; 4\) diagonal form transformation (build diagonal form on the basis of the polynomial and fold this diagonal form); 5\) deduct the decision equations; 6\) input influence values into the decision equations for each subject. Let us call the process of decision making in a group to be a session. Therefore, in RGT models a single session. Model of two-stage decision making: formation of points of view =============================================================== This study is dedicated to the matter of setting mutual influences in a group by means of reflexive control [@lef1]. The influences, which subjects make on each other, could be considered as a result of a decision making session previous to ultimate decision making (final session). We will call the influences, obtained as a result of a previous session(s), a set-up influences. The set-up influences are intermediate result of the overall decision making process. The term set-up influences is related to the influences, which are used during the final session, only. Consequently, the overall decision making process could be segregated into two stages. Let the result of such discussion (decision making) be a particular decision regarding the matter under consideration. We assume the actual decision making regarding the matter (final session - Stage 2) is preceded by preliminary session (Stage 1). Stage 1 is about a decision making regarding the influences (points of view), which each subject will support during the final session. We call such overall decision making process to be a two-stage decision making process. The general schema of a two-stage decision making is presented in Fig.\[twostage\]. ![The general schema of the two-stage decision making.[]{data-label="twostage"}](twostages.png){height="2cm"} To illustrate such model we consider a simple example. Example 1. Let director of some company has a meeting with his advisors. The goal of this meeting is to make decision about marketing policy for the next half a year. The background analysis and predictions of experts suggest three distinct strategies: aggressive (action $\alpha$), moderate (action $\beta$) and soft (action $\gamma$) strategies. The points of view of director and his advisors are formulated in terms of Boolean algebra of alternatives. Term point of view implies that a subject makes the same influences on the others. Director supports moderate strategy ($\{\alpha\}$), the 1st and the 2nd advisors are supporting aggressive strategy ($\{\beta\}$), and the 3rd advisor defends the idea of soft strategy ($\{\gamma\}$). The matrix of initial influences is presented in Table \[infMtx\]. [\|c\|c\|c\|c\|c\|]{} &a&b&c&d\ ------------------------------------------------------------------------ a&a&$\{\alpha\}$&$\{\alpha\}$&$\{\alpha\}$\ ------------------------------------------------------------------------ b&$\{\alpha\}$&b&$\{\alpha\}$&$\{\alpha\}$\ ------------------------------------------------------------------------ c&$\{\beta\}$&$\{\beta\}$&c&$\{\beta\}$\ ------------------------------------------------------------------------ c&$\{\gamma\}$&$\{\gamma\}$&$\{\gamma\}$&d\ \[infMtx\] Let director is in a conflict with all his advisors, but his advisors are in alliance with each other. Variable $c$ represents the Director, variables $a$, $b$ and $d$ correspond to the 1st , the 2nd and the 3rd advisor, respectively. The relationship graph is presented in Fig.\[polyn1\]. Polynomial $abd+c$ corresponds to this graph. ![Relationship graph for a director-advisors group.[]{data-label="polyn1"}](polyn1.png){height="2cm"} After diagonal form transformation the polynomial does not change: $$\begin{array}{{20}{c}} {} & {} & {[a][b][d]} & {} & {} & {} & {} & {} \\ {} & {[abd] } & {} &{+[c]} & {} & {} & {1 + [c]} & {} \\ {[abc+d]} & {} & {} & {} & = & {[abd+c]} & {} & { = abd+c.} \\ \end{array}$$ Then we obtain four decision equations and their solutions (decision intervals) (Table \[decInt\]). [\|c\|c\|c\|]{} &[Decision Equations]{}&[Decision Intervals]{}\ ------------------------------------------------------------------------ a&$a=(bd+c)a+c\overline{a}$&$(bd+c)\supseteq a \supseteq c$\ ------------------------------------------------------------------------ b&$b=(ad+c)b+c\overline{b}$&$(ad+c)\supseteq b \supseteq c$\ ------------------------------------------------------------------------ c&$c=c+abd\overline{c}$&$1\supseteq c \supseteq abd$\ ------------------------------------------------------------------------ d&$d=(ab+c)d+c\overline{d}$&$(ab+c)\supseteq d \supseteq c$\ \[decInt\] Next we calculate the decision intervals by using information from the influence matrix: subject a: $(bd+c)\supseteq a \supseteq c$ $\Rightarrow$ $(\{\alpha\} \{\gamma\}+\{\beta\})\supseteq a \supseteq \{\beta\}$ $\Rightarrow$ $a=\{\beta\}$; subject b: $(ad+c)\supseteq b \supseteq c$ $\Rightarrow$ $(\{\alpha\} \{\gamma\}+\{\beta\})\supseteq b \supseteq \{\beta\}$ $\Rightarrow$ $b=\{\beta\}$; subject c: $1\supseteq c \supseteq abd$$\Rightarrow$ $1\supseteq c \supseteq \{\alpha\}\{\alpha\}\{\gamma\} $ $\Rightarrow$ $1\supseteq c \supseteq 0 $ $\Rightarrow$ $c = c$; subject d: $(ab+c)\supseteq d \supseteq c$ $\Rightarrow$ $(\{\alpha\} c+\{\beta\})\supseteq b \supseteq \{\beta\}$ $\Rightarrow$ $\{\alpha, \beta\} \supseteq d \supseteq \{\beta\}$. Therefore, after the preliminary sessions, the point of view of the subjects have changed. Director has obtained a freedom of choice, since he can choose any alternative: $1 \supseteq c \supseteq 0$ $\Rightarrow$ $c = c$. At the same time, the 1st and the 2nd advisors support moderate strategy ($a$ = $b$ = $\{\beta\}$). Finally, the 3rd advisor now can choose between points of view $\{\alpha,\beta\}$ (aggressive of moderate strategy) and $\{\beta\}$ (moderate strategy): $\{\alpha, \beta\} \supseteq d \supseteq \{\beta\}$. Thus, the point of view of the 1st and the 2nd advisors is strictly determined, while the point of view of 3rd advisor is probabilistic. Next we calculate choice of each subject during the final session, considering the influences resulting from the preliminary session. The matrix of set-up influences is presented in Table \[setupInf\]. The intervals in the matrix imply that a subject can choose either of alternatives from the given interval as a point of view. [\|c\|c\|c\|c\|c\|]{} &a&b&c&d\ ------------------------------------------------------------------------ a&a&$\{\beta\}$&$\{\beta\}$&$\{\beta\}$\ ------------------------------------------------------------------------ b&$\{\beta\}$&b&$\{\beta\}$&$\{\beta\}$\ ------------------------------------------------------------------------ c&$1 \supseteq c \supseteq 0$ &$1 \supseteq c \supseteq 0$ &c&$1 \supseteq c \supseteq 0$\ ------------------------------------------------------------------------ c& $\{\alpha, \beta\} \supseteq d \supseteq \{\beta\}$ & $\{\alpha, \beta\} \supseteq d \supseteq \{\beta\}$ & $\{\alpha, \beta\} \supseteq d \supseteq \{\beta\}$ &d\ \[setupInf\] Subject $a$: $d=\{\alpha,\beta\}$: $(bd+c)\supseteq a \supseteq c$ $\Rightarrow$$(\{\beta\}\{\alpha,\beta\}+c)\supseteq a \supseteq c$ $\Rightarrow$$(\{\beta\}+c)\supseteq a \supseteq c$; $d=\{\beta\}$: $(bd+c)\supseteq a \supseteq c$ $\Rightarrow$$(\{\beta\}\{\beta\}+c)\supseteq a \supseteq c$ $\Rightarrow$$(\{\beta\}+c)\supseteq a \supseteq c$; Subject $b$: $d=\{\alpha,\beta\}$: $(ad+c)\supseteq b \supseteq c$ $\Rightarrow$$(\{\beta\}\{\alpha,\beta\}+c)\supseteq b \supseteq c$ $\Rightarrow$$(\{\beta\}+c)\supseteq b \supseteq c$; $d=\{\beta\}$: $(ad+c)\supseteq b \supseteq c$ $\Rightarrow$$(\{\beta\}\{\beta\}+c)\supseteq b \supseteq c$ $\Rightarrow$$(\{\beta\}+c)\supseteq b \supseteq c$; Subject $c$: $d=\{\alpha,\beta\}$: $1 \supseteq c \supseteq abd$ $\Rightarrow$$1 \supseteq c \supseteq \{\beta\}\{\beta\}\{\alpha,\beta\}$ $\Rightarrow$$1 \supseteq c \supseteq \{\beta\}$; $d=\{\beta\}$: $1 \supseteq c \supseteq abd$ $\Rightarrow$$1 \supseteq c \supseteq \{\beta\}\{\beta\}\{\beta\}$ $\Rightarrow$$1 \supseteq c \supseteq \{\beta\}$; Subject $d$: $(ab+c)\supseteq d \supseteq c$ $\Rightarrow$$(\{\beta\}\{\beta\}+c)\supseteq d \supseteq c$ $\Rightarrow$$(\{\beta\}+c)\supseteq d \supseteq c$. Now we compare the results of a single session with the ones of the two-stage decision making. The single session case has been considered above. If the final decision has to be made after the single session, then the 3rd advisor would be able to choose alternative $\{\alpha,\beta\}$ and realize action $\alpha$. This option implies that each advisor is responsible for a particular part of the entire company and can take management decisions on his own. Next we consider the decision made after the two-stage decision making. In such a case, regardless of influence of the 3rd advisor (subject $d$), choice of advisors $a$ and $b$ is defined by the interval $\{\beta\}+c \supseteq x \supseteq c$, where $x$ is either $a$ or $b$ variable. Thus, if director is inactive ($c=0$), subjects $a$ and $b$ can choose either moderate strategy ($\{\beta\}$) or make no decision ($0=\{\}$). The same is true for subject $d$. If the director makes influence $\{\beta\}$, then all advisors will choose alternative $\{\beta\}$. The director himself can choose from the interval $1 \supseteq c \supseteq \{\beta\}$ after the final session. This means that the director can choose any alternative, containing action $\beta$. Thus, occasionally the director can realize his initial point of view as moderate strategy. This example illustrates how using the two-stage decision making it is possible to make one’s opponents choose the one’s point of view. Meanwhile a person interested in such reflexive control can still sustain the initial point of view. The obtained results are applicable in both cases when 1) only the director makes a decision; or 2) the decision are made individually by each subject. A Model of a multi-stage decision making: set-up parameters of the final session ================================================================================ Now we consider the two-stage model in more details. In the considered example, during the preliminary session only the decision regarding the influences has been under consideration. In general case, however, before the final session has begun, there can be made decisions regarding any parameters of the final session. Such parameters include but are not limit to: 1\) group structure (number of subjects and relationships between subjects in a group); 2\) points of view; 3\) decision to start a final session (a time when the final session should start), etc. We call the decision regarding a single parameter to be consecutive decision, and decisions regarding distinct parameters to be parallel decisions. Therefore, during the first stage (before the final session) it is possible to make multiple decisions regarding various parameters of the final session. This decisions could be both parallel and consecutive ones. We call such model of decision making to be a multi-stage process of decision making* (Fig.\[multistage\]). ![Multi-stage decision making model.[]{data-label="multistage"}](multistage.png){height="3cm"} Modeling multi-stage decision making processes with RGT ======================================================= Next we consider realization of multi-stage decision making with RGT. Example 2: Change a group structure. Considering the subject from Example 1, we analyze the case when director wants to exclude the 3rd advisor from the group, which will make the final decision. In such a case, there is a single action – 1 – to exclude subject $d$ from the group. Then Boolean algebra of alternatives includes only two elements: 1 and 0. Furthermore, it is enough that director just raise a question to exclude subject $d$ from a group and make influence 1 on each subject: if $с = 1$, then $a=1$, $b=1$ and $d=1$ (Table \[decInt\]). Thus the decision to exclude subject $d$ from the group would be made automatically (Fig.\[polyn2\]). ![Exclusion of a subject $d$ from a group.[]{data-label="polyn2"}](polyn2.png){height="2cm"} Example 3: Realization of a multi-stage decision making. Let the first decision discussed during the first stage is a decision regarding influences (points of view). The next decision was about exclusion of a subject $d$ from the group. Thus, during the first step the formation (setting-up) of points of view has been implemented, then the structure of a group was changed. Therefore the group, which should make a final decision is described by polynomial $ab +c$. The decision equations and their solutions are presented in Table \[decInt3\]. The overall multi-stage decision making process is presented in Fig.\[multiEx\]. [\|c\|c\|c\|]{} &[Decision Equations]{}&[Decision Intervals]{}\ ------------------------------------------------------------------------ a&$a=(b+c)a+c\overline{a}$&$(b+c)\supseteq a \supseteq c$\ ------------------------------------------------------------------------ b&$b=(a+c)b+c\overline{b}$&$(a+c)\supseteq b \supseteq c$\ ------------------------------------------------------------------------ c&$c=c+ab\overline{c}$&$1\supseteq c \supseteq ab$\ \[decInt3\] ![Illustration of multi-stage decision making process. The influences are indicated by the arrow-ends of the ribs. The actual influence is presented near the arrow-end. []{data-label="multiEx"}](multistageEx.png){width="12cm"} We consider that the point of view cannot change without preliminary session regarding the parameter. Therefore we assume that the points of view do not change after the change of group structure. Therefore, during the final session the subjects would make the set-up influences derived from the preliminary session: subjects $a$ and $b$ will make influences $\{\beta\}$ and subject $c$ will have a choice from the interval $1 \supseteq c \supseteq \{\beta\}$. Such process is introduced in Fig. \[multiEx\]. During the 1st stage (first step), the points of view of subjects have been formed. On the 2nd stage (second step), the decision to exclude subject $d$ from a group has been made. Finally, during the 3rd stage the final decision regarding the marketing strategy has been made. Discussion and conclusion ========================= This study introduces the two-stage and multi-stage decision making processed. During the last stage the final decision is made. During the earlier stages the decisions regarding the parameters of a final session are considered. This study shows how before the final decision making the intermediate decision regarding parameters of the final session can be made and how to overall process of decision making could be represented as a sequence of decision making sessions. This approach enables complex decision making, which involves numerous parameters. The important feature of the multi-stage decision making is that during the preliminary sessions subjects can convince other subjects to accept their own point of view. Therefore other subjects can be convince to make decisions beneficial for a particular one. Such approach also allows to distribute the responsibility between all the members of the group, who make the final decision. The results presented in this study allow to extend the scope of applications of RGT to modeling of multi-stage decision making processes. Therefore it becomes possible to perform scenario analysis of various variants of future trends and apply reflexive control to the management of projects. [5]{} Lefebvre, V.A.: Lectures on Reflexive Game Theory. Cogito-Centre, Moscow (2009) (in Russian) Lefebvre, V.A.: Lectures on Reflexive Game Theory. Leaf & Oaks, Los Angeles (2010) Lefebvre, V.A.: Algebra of Conscience. D. Reidel, Holland (1982) Lefebvre, V.A.: The basic ideas of reflexive game’s logic. Problems of systems and structures research. 73–79 (1965) (in Russian)	ArXiv
[A [N]{}ATURAL [S]{}OLUTION TO THE [$\mu$]{} [P]{}ROBLEM]{} J.A. CASAS${}^{,}\;$ and C. MUÑOZ${}^{*}$ ${}^{}$ CERN, CH–1211 Geneva 23, Switzerland ${}^{}$ Instituto de Estructura de la Materia (CSIC),\ Serrano 123, E-28006 Madrid, Spain ${}^{}$ Dept. de Física Teórica C-XI,\ Univ. Autónoma de Madrid, E-28049 Madrid, Spain Abstract* We propose a simple mechanism for solving the $\mu$ problem in the context of minimal low–energy supergravity models. This is based on the appearance of non–renormalizable couplings in the superpotential. In particular, if $H_1H_2$ is an allowed operator by all the symmetries of the theory, it is natural to promote the usual renormalizable superpotential $W_o$ to $W_o+\lambda W_o H_1H_2$, yielding an effective $\mu$ parameter whose size is directly related to the gravitino mass once supersymmetry is broken (this result is maintained if $H_1H_2$ couples with different strengths to the various terms present in $W_o$). On the other hand, the $\mu$ term must be absent from $W_o$, otherwise the natural scale for $\mu$ would be $M_P$. Remarkably enough, this is entirely justified in the supergravity theories coming from superstrings, where mass terms for light fields are forbidden in the superpotential. We also analyse the $SU(2)\times U(1)$ breaking, finding that it takes place satisfactorily. Finally, we give a realistic example in which supersymmetry is broken by gaugino condensation, where the mechanism proposed for solving the $\mu$ problem can be gracefully implemented. [CERN–TH.6764/92]{}\ [IEM–FT–66/92]{}\ [FTUAM 92/45]{}\ [December 1992]{} Introduction ============ One of the interesting features of low–energy supergravity (SUGRA) models is that the electroweak symmetry breaking can be a direct consequence of supersymmetry (SUSY) breaking \[1\]. In the ordinary SUGRA models, SUSY breaking takes place in a hidden sector of the theory, so that the gravitino mass $m_{3/2}$ becomes of the electroweak scale order. Below the Planck mass, $M_P$, one is left with a global SUSY Lagrangian plus some terms (characterized by the $m_{3/2}$ scale) breaking explicitly, but softly, global SUSY. As we will briefly review below, the breakdown of $SU(2)\times U(1)_Y$ appears as an automatic consequence of the radiative corrections to these terms. The so–called $\mu$ problem \[2\] arises in this context. Let us consider a SUGRA theory with superpotential $W(\phi_i)$ and canonical kinetic terms for the $\phi_i$ fields[^1]. Then, the scalar potential takes the form \[3\] $$\begin{aligned} V = e^K \left[ \sum_i \left\| \frac{\partial W}{\partial \phi_i} + \bar{\phi_i}W \right\|^2 - 3\|W\|^2 \right]\;+\;\mathrm{D}\;\mathrm{terms} \;\;, \label{V}\end{aligned}$$ where $K=\sum_i\|\phi_i\|^2$ is the Kähler potential. It is customary to consider $W$ as a sum of two terms corresponding to the observable sector $W^{obs}(\phi_i^{obs})$ and a hidden sector $W^{hid}(\phi_i^{hid})$ $$\begin{aligned} W(\phi_i^{obs},\phi_i^{hid})= W^{obs}(\phi_i^{obs}) + W^{hid}(\phi_i^{hid}) \;\;. \label{W}\end{aligned}$$ $W^{hid}(\phi_i^{hid})$ is assumed to be responsible for the SUSY breaking, which implies that some of the $\phi_i^{hid}$ fields acquire non–vanishing vacuum expectation values (VEVs) in the process. Then, the form of the effective observable scalar potential obtained from eq.(\[V\]), assuming vanishing cosmological constant, is \[4\] $$\begin{aligned} V^{obs}_{eff} = \sum_i \left\| \frac{\partial \hat{W}^{obs}} {\partial \phi_i^{obs}}\right\|^2 &+& m_{3/2}^2 \sum_i \| \phi_i^{obs}\|^2 + \left( Am_{3/2}\hat W^{obs}_t + Bm_{3/2}\hat W^{obs}_b\; +\; \mathrm{h.c.} \right) \nonumber \\ \;&+&\;\mathrm{D}\;\mathrm{terms} \label{Vobs}\end{aligned}$$ with $$\begin{aligned} m_{3/2}^2 = e^{K^{hid}}\|W^{hid}\|^2 \label{m32}\end{aligned}$$ $$\begin{aligned} B = A-1=\sum_i \left( \| \phi_i^{hid}\|^2 + \frac{\bar{\phi}^{hid}} {\bar{W}^{hid}}\frac{\partial \bar{W}^{hid}} {\partial \bar{\phi}_i^{hid}}\right)\;-\;1\;\;, \label{AB}\end{aligned}$$ where $K^{hid}=\sum_i \|\phi_i^{hid}\|^2$, $\hat W^{obs}$ is the rescaled observable superpotential $\hat W^{obs}=e^{K^{hid}/2} W^{obs}$, the subindex $t$($b$) denotes the trilinear (bilinear) part of the superpotential, and $A$, $B$ are dimensionless numbers of $O(1)$, which depend on the VEVs of the hidden fields. Since we are assuming that SUSY breaking takes place at a right scale, the gravitino mass given by eq.(\[m32\]) is hierarchically smaller than the Planck mass (i.e. of order the electroweak scale). In the minimal supersymmetric standard model (MSSM) the matter content consists of three generations of quark and lepton superfields plus two Higgs doublets, $H_1$ and $H_2$, of opposite hypercharge. Under these conditions the most general effective observable superpotential has the form $$\begin{aligned} W^{obs}=\sum_{generations}(h_uQ_LH_2u_R + h_dQ_LH_1d_R + h_eL_LH_1e_R )+ \mu H_1H_2\;\;. \label{Wobs}\end{aligned}$$ This includes the usual Yukawa couplings (in a self–explanatory notation) plus a possible mass term for the Higgses, where $\mu$ is a free parameter. From eq.(\[Vobs\]) the relevant Higgs scalar potential along the neutral direction for the electroweak breaking is readily obtained $$\begin{aligned} V(H_1,H_2)=\frac{1}{8}(g^2+g'^2)\left(\|H_1\|^2-\|H_2\|^2\right)^2 + \mu_1^2\|H_1\|^2 + \mu_2^2\|H_2\|^2 -\mu_3^2(H_1H_2+\mathrm{h.c.}) \;\;, \label{Vhiggs}\end{aligned}$$ where $$\begin{aligned} \mu_{1,2}^2 &=& m_{3/2}^2 + \hat\mu^2 \nonumber \\ \mu_{3}^2 &=& -Bm_{3/2}\hat\mu \nonumber \\ \hat\mu &\equiv & e^{K^{hid}/2}\mu\;\;. \label{mus}\end{aligned}$$ This is the SUSY version of the usual Higgs potential in the standard model. In order for the potential to be bounded from below, the condition $$\begin{aligned} \mu_{1}^2+\mu_2^2-2\|\mu_3^2\|>0 \label{condmus}\end{aligned}$$ must be imposed all over the energy range $[M_Z,M_P]$. This implies in particular $\langle H_{1,2}\rangle = 0$ at the Planck scale. Below the Planck scale, one has to consider the radiative corrections to the scalar potential. Then the boundary conditions of eq.(\[mus\]) are substantially modified in such a way that the determinant of the Higgs mass–squared matrix becomes negative, triggering $\langle H_{1,2}\rangle \neq 0$ and $SU(2)\times U(1)_Y$ symmetry breaking \[1\]. For this scheme to work, the presence of the last term in eq.(\[Wobs\]) is crucial. If $\mu=0$, then the form of the renormalization group equations (RGEs) implies that such a term is not generated at any $Q$ scale since $\mu(Q)\propto \mu$. The same occurs for $\mu_3$, i.e. $\mu_3(Q)\propto \mu$. Then, the minimum of the potential of eq.(\[Vhiggs\]) occurs for $H_1=0$ and, therefore, $d$–type quarks and $e$–type leptons remain massless. Besides, the superpotential of eq.(\[Wobs\]) with $\mu=0$ possesses a spontaneously broken Peccei–Quinn symmetry \[5\] leading to the appearance of an unacceptable Weinberg–Wilczek axion \[6\]. Once it is accepted that the presence of the $\mu$ term in the superpotential is essential, there arises an inmediate question: Is there any dynamical reason why $\mu$ should be small, of the order of the electroweak scale? Note that, to this respect, the $\mu$ term is different from the SUSY soft–breaking terms, which are characterized by the small scale $m_{3/2}$ once we assume correct SUSY breaking. In principle the natural scale of $\mu$ would be $M_P$, but this would reintroduce the hierarchy problem since the Higgs scalars get a contribution $\mu^2$ to their squared mass \[see eq.(\[mus\])\]. Thus, any complete explanation of the electroweak breaking scale must justify the origin of $\mu$. This is the so–called $\mu$ problem \[2\]. This problem has been considered by several authors and different possible solutions have been proposed \[2,7,8\]. In this letter we suggest a scenario in which $\mu$ is generated by non–renormalizable terms and its size is directly related to the gravitino mass. A comparison with the scenarios of refs.\[2,7,8\] is also made. A natural solution to the $\mu$ problem ======================================= Let us start with a simple scenario with superpotential $$\begin{aligned} W=W_o + \lambda W_oH_1H_2 \;\;. \label{WWo}\end{aligned}$$ where $W_o$ is the usual superpotential (including both observable and hidden sectors) [without]{} a $\mu H_1H_2$ term. We have allowed in (\[WWo\]) a non–renormalizable term, characterized by the coupling $\lambda=O(1)$ (in Planck units), which mixes the observable sector with the hidden sector (other higher–order terms of this kind could also be included, but they are not relevant for the present analysis). The $\mu H_1H_2$ term must be absent from $W_o$ since, as was mentioned above, the natural scale for $\mu$ would otherwise be $M_P$. Certainly, this is technically possible in a supersymmetric theory, since the non–renormalization theorems assure that this term cannot be generated radiatively if initially $\mu=0$. One may wonder, however, whether there is a theoretical reason for the absence of the $\mu H_1H_2$ term from $W_o$ in eq.(\[WWo\]), since it is not forbidden by any symmetry of the theory[^2]. It is quite remarkable here that this is provided in the low–energy SUSY theory obtained from superstrings. In this case mass terms (like $\mu H_1H_2$) are forbidden in the superpotential. We will see in section 4 an explicit example in this context. Finally, non-renormalizable terms (like $\lambda W_oH_1H_2$) are in principle allowed in a generic SUGRA theory. Next, we show that the $\lambda W_oH_1H_2$ term yields dynamically a $\mu$ parameter. Using the general expression of eq.(\[V\]), the scalar potential $V$ generated by $W$ has the form $$\begin{aligned} V = &e^K& \left\{ \sum_i \left\| \frac{\partial [W_o(1+\lambda H_1H_2)]}{\partial \phi_i} + \bar{\phi_i}W_o (1+\lambda H_1H_2)\right\|^2 - 3\|W_o(1+\lambda H_1H_2)\|^2 \right\} \nonumber\\ \;&+&\;\mathrm{D}\;\mathrm{terms} \;\;, \label{V2}\end{aligned}$$ which can be written as $$\begin{aligned} V = V^{(1)}\|1+\lambda H_1H_2\|^2 &+& e^K \left\{ \left\| \frac{\partial [W_o(1+\lambda H_1H_2)]}{\partial H_1} + \bar{H_1}W_o (1+\lambda H_1H_2)\right\|^2 +(H_1\leftrightarrow H_2) \right\} \nonumber\\ \;&+&\;\mathrm{D}\;\mathrm{terms} \;\;, \label{V3}\end{aligned}$$ where $$\begin{aligned} V^{(1)}\equiv e^K \left( \sum_i \left\| \frac{\partial W_o} {\partial \phi_i} + \bar{\phi_i}W_o \right\|^2 - 3\|W_o\|^2 \right)\; ;\;\;\phi_i\neq H_{1,2}\;\;. \label{V1}\end{aligned}$$ Since $H_{1,2}$ enter in $W_o$ only through the ordinary Yukawa couplings and we are assuming vanishing VEVs for the observable scalar fields, it is clear (recall that $W_o$ does not contain a $\mu H_1H_2$ coupling) that $\left. \frac{\partial W_o} {\partial H_{1,2}}\right\|_{min}=0$. Besides, the vanishing of the cosmological constant implies $V^{(1)}=0$ at the minimum of the potential. So, we can extract from the second term in eq.(\[V3\]) the soft terms associated with $H_{1,2}$: $$\begin{aligned} V(H_1,H_2)=&\frac{1}{8}&(g^2+g'^2)\left(\|H_1\|^2-\|H_2\|^2\right)^2 + m_{3/2}^2(1+\lambda^2)\|H_1\|^2 + m_{3/2}^2(1+\lambda^2)\|H_2\|^2 \nonumber\\ &+&2m_{3/2}^2\lambda(H_1H_2+\mathrm{h.c.}) \;\;. \label{Vhiggs2}\end{aligned}$$ Comparing eqs.(\[Wobs\]–\[mus\]) with eqs.(\[WWo\],\[Vhiggs2\]) it is clear that $\lambda W_oH_1H_2$ behaves like a $\mu$ term when $W_o$ acquires a non–vanishing VEV dynamically. Defining $\lambda\langle W_o\rangle\equiv\mu$ we can write eq.(\[Vhiggs2\]) as eqs.(\[Vhiggs\],\[mus\]) where now the value of $B$ is $$\begin{aligned} B=2 \;\;. \label{B}\end{aligned}$$ The value of $A$ is still given by eq.(\[AB\]), but the relation $B=A-1$ is no longer true. The fact that the new “$\mu$ parameter” is of the electroweak–scale order is a consequence of our assumption of a correct SUSY–breaking scale $m_{3/2}=e^{K/2}W=O(M_Z)$. Finally, note that the usual condition for the potential to be bounded from below (\[condmus\]) is automatically satisfied by (\[Vhiggs2\]) for any value of $\lambda$. One may wonder how general is the simple scenario of eq.(\[WWo\]). First of all, let us note that the fact that $H_1H_2$ is not forbidden by any symmetry of the theory is a key ingredient for this scenario to work. An obvious generalization of (\[WWo\]) arises when $W_o$ consists of several terms $W_o=W_o^{(1)}+W_o^{(2)}+...$ and $H_1H_2$ couples with a different strength to each term, i.e. $(\lambda_1W_o^{(1)}+ \lambda_2W_o^{(2)}+...)H_1H_2$. However, provided that the hierarchical small value for $\langle W_o\rangle$ is not achieved by a fine–tuning between the VEVs of the various terms $W_o^{(1)},W_o^{(2)},...$, it is clear that the order of magnitude of $\mu$ continues being $m_{3/2}$. Apart from this, it should be noticed that $\lambda_i=O(1)$ (in Planck units) is only natural if $W_o^{(i)}$ is not an operator with a extremely small coupling constant. However, this would be a naturalness problem by itself. This would happen, for instance, for $W_o^{(i)}=m\Phi^2$ with $m<<M_P$. (These terms are forbidden in string theories.) To conclude this section, it is worth noticing that in the context of supergravity theories there is another possible solution to the $\mu$ problem. Since the Kähler potential $K$ is an arbitrary real–analytic function of the scalar fields, we can study for example a theory with the following $K$ $$\begin{aligned} K=\sum_i\|\phi_i\|^2 + f(g(\phi_j, \bar \phi_j)H_1H_2\ +\ \mathrm{h.c.}) \;\;, \label{K}\end{aligned}$$ where $\phi_j\neq H_{1,2}$ and $f$ and $g$ are generic functions ($\langle g(\phi_j, \bar\phi_j)\rangle= O(1)$). Then, although $W_o$ does not contain a $\mu$ term, this is generated in the scalar potential. This is trivial to see for the simplest case (i.e. $f(x)=x$, $g=$ const. $\equiv\lambda$). Then the theory is equivalent to one with Kähler potential $\sum_i\|\phi_i\|^2$ and superpotential $W_oe^{\lambda H_1H_2}$, since the function ${\cal G}=K+\log\|W\|^2$ that defines the SUGRA theory is the same for both. Expanding the exponential, the first two terms coincide with eq.(\[WWo\]) and hence we obtain the same $\mu$ term as in eq.(\[Vhiggs2\]). The possibility (\[K\]) was examined in ref.\[7\] for $f(x)=x$ and when $g$ is a non–trivial function of the hidden fields, in particular for the simplest case $g(\phi_j, \bar\phi_j)=\bar\xi$, where $\bar\xi$ is a hidden field. It remains to be explored whether a Kähler potential similar to that of eq.(\[K\]) can arise in the context of superstring theories. Expectation values for the Higgses ================================== In the above analysed solution to the $\mu$ problem it is assumed that the observable scalar fields have vanishing VEVs at the Planck scale. Since the non–renormalizable term $\lambda W_oH_1H_2$ mixes observable and hidden fields, one may wonder whether that assumption is still true for the Higgses. We will show now that this is in fact the case. We assume here that the initial superpotential $W_o$ gives a correct SUSY breaking, i.e. small gravitino mass and vanishing cosmological constant. This means that $V_o$, i.e. the scalar potential derived from $W_o$, is vanishing at the minimum $\left. V_{o}\right\|_{min}=0$ and thus positive–definite. Using the general expression of eq.(\[V\]), $V_o$ can be decomposed in three pieces $$\begin{aligned} V_o = V^{(1)}\;+\;e^K \left\{ \left\| \frac{\partial W_o}{\partial H_1} + \bar{H_1}W_o \right\|^2 + (H_1\rightarrow H_2)\right\}\;+\; \mathrm{D}\;\mathrm{terms} \;\;, \label{Vo}\end{aligned}$$ where $V^{(1)}$ is defined in eq.(\[V1\]). Recalling that we are assuming that $W_o$ does not contain a $\mu H_1H_2$ term and that $\left. \frac{\partial W_o}{\partial H_{1,2}}\right\|_{min}=0$ (since squarks and sleptons are supposed to have vanishing VEVs), it is clear that $V^{(1)}$ is flat in $H_{1,2}$. So, the minimum of the second piece of (\[Vo\]) is zero and occurs at $H_{1,2}=0$ (for any value of $W_o$). Therefore, necessarily $\left. V^{(1)}\right\|_{min}=0$, i.e. $V^{(1)}$ is also positive–definite. All this is very ordinary: it simply means that the hidden sector is entirely responsible for the breaking. (Note that the $H_{1,2}$ F–terms are vanishing, while some of the hidden fields F–terms must be different from zero.) Notice also that from (\[Vo\]) one obtains $e^{K}\|W_o\|^2(\|H_1\|^2+\|H_2\|^2)= m_{3/2}^2\|H_1\|^2+m_{3/2}^2\|H_2\|^2$ but, because of the absence of a $\mu H_1H_2$ term in $W_o$, there is no $Bm_{3/2}\hat\mu H_1H_2$ term in the scalar potential. Let us now study the impact of doing, according to our approach, $W_o \rightarrow W = W_o + \lambda W_oH_1H_2$. The corresponding scalar potential, $V$, has already been written in eq.(\[V3\]). Now, since $V^{(1)}$ is positive–definite, so is $V$. In fact, the minimum of $V$ is for $V=0$ and occurs when the three pieces of (\[V3\]) are vanishing. Clearly, the minimum of the first and third pieces of (\[V3\]) coincides with that of eq.(\[Vo\]) above, implying $\left. V^{(1)}\right\|_{min}=0$,[^3] and thus the VEV of $W_o$ is the same as when we started with just $W_o$. Finally, recalling that $\left. \frac{\partial W_o}{\partial H_{1,2}} \right\|_{min}=0$, it is clear that the second piece of $V$ in eq.(\[V3\]) has two possible minima $$\begin{aligned} H_1, H_2=0 \;\;, \label{Hs1}\end{aligned}$$ $$\begin{aligned} \lambda H_2+(1+\lambda H_1H_2)\bar H_1&=&0 \nonumber\\ (H_1\leftrightarrow H_2)&=&0 \label{Hs2}\end{aligned}$$ As was explained in section 1, the solution (\[Hs1\]) is the phenomenologically interesting one, whereas the solution (\[Hs2\]) leads to $H_{1,2}\sim M_P$, so it is not phenomenologically viable. We can ignore this solution since if $H_{1,2}$ are initially located at $H_{1,2}=0$ (e.g. by thermal effects) they will remain there as long as (\[Hs1\]) continues to be a minimum solution. Of course, radiative corrections will trigger non–zero VEVs of the correct size for $H_1$, $H_2$. A realistic example =================== As we saw in section 2, the assumption of correct SUSY breaking was crucial for obtaining the $\mu$ parameter of the electroweak–scale order. As a matter of fact, gaugino condensation effects in the hidden sector \[9\] are the most satisfactory mechanism so far explored, able to break SUSY at a scale hierarchically smaller than $M_P$ \[10\]. The reason is that the scale of gaugino condensation corresponds to the scale at which the gauge coupling becomes large, and this is governed by the running of the coupling constant. Since the running is only logarithmically dependent on the scale, the gaugino condensation scale is suppressed relative to the initial one by an exponentially small factor $\sim e^{-1/2\beta g^2}$ ($\beta$ is the one–loop coefficient of the beta function of the hidden sector gauge group $G$). This mechanism has been intensively studied in the context of SUGRA theories coming from superstrings \[11,12\], where the gauge coupling is related to the VEV of the dilaton field $S$ (more specifically Re$S=g^{-2}$). Recall that we have argued in section 2 that superstring theories are precisely a natural context where the solution of the $\mu$ problem presented here can be implemented, since mass terms, such as $\mu H_1 H_2$, appearing in the superpotential are automatically forbidden in superstrings. Besides, non–renormalizable terms like $\lambda W_oH_1H_2$ in eq.(\[WWo\]) are in principle allowed and, in fact, they are usually present \[13\]. In the absence of hidden matter, the condensation process is correctly described by a non–perturbative effective superpotential $$\begin{aligned} W_o\propto e^{-3S/2\beta_o} \;\;, \label{Wcond}\end{aligned}$$ with $\beta_o=3C(G)/16\pi^2$, where $C(G)$ is the Casimir operator in the adjoint representation of $G$. It is difficult to imagine, however, how the mechanism expounded in section 2 could be implemented here. More precisely, it is not clear that we could have something like $W=W_o+\lambda W_oH_1H_2$, due to the effective character of (\[Wcond\]). Fortunately, things are different in the presence of hidden matter, which is precisely the most frequent case in string constructions \[13\]. There is not at present a generally accepted formalism describing the condensation in the presence of massless matter, but the case of massive matter is well understood \[14\]. For example, in the case of $G=SU(N)$ with $M(N+\bar N)$ “quark” representations $Q_\alpha$, $\bar Q_\alpha$, $\alpha=1,...,M$, with a mass term given by $$\begin{aligned} W_o^{pert}=-\sum_{\alpha,\beta}{\cal M}_{\alpha,\beta}Q_{\alpha} \bar Q_\beta \;\;, \label{Wpert}\end{aligned}$$ the complete condensation superpotential can be written as \[12\] $$\begin{aligned} W_o\propto [\mathrm{det}{\cal M}]^{\frac{1}{N}} e^{-3S/2\beta_o} \;\;. \label{Wcond2}\end{aligned}$$ It should be noticed here that, strictly speaking, there are no mass terms like (\[Wpert\]) in the context of string theories. However the matter fields usually have trilinear couplings which play the role of mass terms with a dynamical mass given by the VEV of another matter field. The simplest case occurs when there is an $SU(N)$ singlet field $A$ giving mass to all the quark representations. Then (\[Wpert\]) takes the form $$\begin{aligned} W_o^{pert}=-\sum_{\alpha=1}^M AQ_{\alpha}\bar Q_\alpha \;\;, \label{Wpert2}\end{aligned}$$ and $\mathrm{det}{\cal M}=A^M$. Now, if $H_1H_2$ is an allowed coupling from all the symmetries of the theory, it is natural to promote $W_o^{pert}$ to[^4] $$\begin{aligned} W^{pert}=-\sum_{\alpha}A(1+\lambda' H_1H_2)Q_{\alpha}\bar Q_\alpha \;\;, \label{Wpert3}\end{aligned}$$ so that $\mathrm{det}{\cal M}=[A(1+\lambda' H_1H_2)]^M$, and (\[Wcond2\]) takes the form $$\begin{aligned} W_o\rightarrow W\propto [A(1+\lambda' H_1H_2)]^{\frac{M}{N}} e^{-3S/2\beta_o} \simeq A^{\frac{M}{N}}(1+\frac{M}{N}\lambda' H_1H_2) e^{-3S/2\beta_o} \;\;. \label{Wcond3}\end{aligned}$$ Thus $$\begin{aligned} W=W_o+\lambda W_oH_1H_2 \;\;, \label{WWocond2}\end{aligned}$$ where we have defined $\lambda\equiv\frac{M}{N}\lambda'$. This is precisely the kind of superpotential we wanted (see eq.(\[WWo\])) in order to generate the $\mu$ term dynamically. In ref.\[8\] an interesting solution to the $\mu$ problem was proposed in a similar context with a PQ symmetry, using the presence of a term $H_1H_2Q\bar Q$ in the superpotential and assuming that the scalar components of $Q$ and $\bar Q$ condense at a scale $\Lambda\simeq 10^{11}$ GeV. As mentioned above, the only accepted formalism describing the condensation is in the presence of massive matter. Thus the previous term behaves as a dynamical mass term for the squarks and the complete superpotential (\[Wcond2\]) becomes $W\propto (H_1H_2)^{\frac{1}{N}} e^{-3S/2\beta_o}$. This is phenomenologically unviable since the Higgses must have vanishing VEVs at $M_P$ for a correct phenomenology, which would imply $\langle W\rangle=0$ and thus no SUSY breaking. We can improve this model by including a mass term for $Q\bar Q$. However, a genuine mass term for $Q\bar Q$ would break the PQ symmetry, so one should consider something similar to (\[Wpert2\]). Then the perturbative superpotential is $$\begin{aligned} W^{pert}\sim AQ\bar Q + H_1H_2 Q\bar Q \;\;, \label{Wpert4}\end{aligned}$$ and the scenario becomes much more similar to that given by eq.(\[Wpert3\]). However, there still is an important difference. In eq.(\[Wpert3\]) $H_1H_2$ couples to $AQ\bar Q$ (which is the natural thing if $H_1H_2$ is invariant under all the symmetries of the theory) instead of $Q\bar Q$; thus there is no PQ symmetry. Moreover, (\[Wpert3\]) leads to (\[WWocond2\]) in which the $\mu$ scale is directly given by the $m_{3/2}$ scale ($\mu=O(m_{3/2})$). However from (\[Wpert4\]) the $\mu$ scale is given by the squark condensation scale \[12\] $\langle Q\bar Q\rangle/M_P\simeq m_{3/2}M_P/N\langle A\rangle$, so that the value of $\mu$ in this case tends to be a bit too large. Summary and conclusions ======================= We have proposed a simple mechanism for solving the $\mu$ problem in the context of minimal low–energy SUGRA models. This is based on the appearance of non–renormalizable couplings in the superpotential. In particular, if $H_1H_2$ is an allowed operator by all the symmetries of the theory, it is natural to promote the usual renormalizable superpotential $W_o$ to $W_o+\lambda W_o H_1H_2$, yielding an effective $\mu$ parameter whose size is directly related to the gravitino mass once SUSY is broken (this result is essentially maintained if $H_1H_2$ couples with different strengths to the various terms present in $W_o$). On the other hand, the $\mu$ term must be absent in $W_o$, otherwise the natural scale for $\mu$ would be $M_P$. Certainly this is technically possible in a supersymmetric theory since the non–renormalization theorems assure that this term cannot be generated radiatively if initially $\mu=0$. Remarkably enough, however, a theoretical reason for the absence of the $\mu H_1H_2$ term from $W_o$ is provided in the low–energy SUSY theory obtained from superstrings. In this case mass terms (such as $\mu H_1H_2$) are forbidden in the superpotential (however, non–renormalizable terms like $\lambda W_oH_1H_2$ are in principle allowed and, in fact, they are usually present). We have also addressed other alternative solutions, comparing them with the one proposed here. On the other hand, we have analysed the $SU(2)\times U(1)$ breaking, finding that it takes place satisfactorily. Finally, we have given a realistic example in which SUSY is broken by gaugino condensation in the presence of hidden matter (which is the usual situation in strings), and where the mechanism proposed for solving the $\mu$ problem can be gracefully implemented. [ACKNOWLEDGEMENTS]{} We gratefully acknowledge J. Louis for extremely useful discussions. [99]{} For a recent review, see: L.E. Ibañez and G.G. Ross, CERN–TH.6412/92 (1992), to appear in Perspectives in Higgs Physics, ed. G. Kane, and references therein J.E. Kim and H.P. Nilles, Phys. Lett. B138 (1984) 150 E. 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Nilles and F. Quevedo, Phys. Lett. B210 (1988) 101; I. Antoniadis, J. Ellis, J.S. Hagelin and D.V. Nanopoulos, Phys. Lett. B205 (1988) 459, B213 (1988) 56; A. H. Chamseddine and M. Quirós, Nucl. Phys. B316 (1989) 101 T.R. Taylor, G. Veneziano and S. Yankielowicz, Nucl. Phys. B218 (1983) 493; I. Affleck, M. Dine and N. Seiberg, Nucl. Phys. B241 (1984) 493; D. Amati, K. Konishi, Y. Meurice, G.C. Rossi and G. Veneziano, Phys. Rep. 162 (1988) 169 [^1]: We will consider this case throughout the paper for simplicity. Our general conclusions will not be modified by taking a more general case. [^2]: The $\mu H_1H_2$ term can be forbidden by invoking a Peccei–Quinn (PQ) symmetry \[2,8\]. This is not possible here since (\[WWo\]) does not possess any PQ symmetry. [^3]: The only exception occurs if $\lambda H_1H_2=-1$, but then the second piece of (\[V3\]), which is also positive–definite, is different from zero, so this is not a solution for the minimization of the whole potential. [^4]: We neglect here higher–order non–renormalizable couplings since they do not contribute to the $\mu$ term.	ArXiv
--- abstract: 'We use models of thermal evolution and XUV-driven mass loss to explore the composition and history of low-mass low-density transiting planets. We investigate the Kepler-11 system in detail and provide estimates of both the current and past planetary compositions. We find that a H/He envelope on Kepler-11b is highly vulnerable to mass loss. By comparing to formation models, we show that in situ formation of the system is extremely difficult. Instead we propose that it is a water-rich system of sub-Neptunes that migrated from beyond the snow line. For the broader population of observed planets, we show that there is a threshold in bulk planet density and incident flux above which no low-mass transiting planets have been observed. We suggest that this threshold is due to the instability of H/He envelopes to XUV-driven mass loss. Importantly, we find that this mass loss threshold is well reproduced by our thermal evolution/contraction models that incorporate a standard mass loss prescription. Treating the planets’ contraction history is essential because the planets have significantly larger radii during the early era of high XUV fluxes. Over time low mass planets with H/He envelopes can be transformed into water-dominated worlds with steam envelopes or rocky super-Earths. Finally, we use this threshold to provide likely minimum masses and radial velocity amplitudes for the general population of Kepler candidates. Likewise, we use this threshold to provide constraints on the maximum radii of low-mass planets found by radial velocity surveys.' author: - 'Eric D. Lopez' - 'Jonathan J. Fortney$^1$' - Neil Miller bibliography: - 'myreferences.bib' title: 'How Thermal Evolution and Mass Loss Sculpt Populations of Super-Earths and Sub-Neptunes: Application to the Kepler-11 System and Beyond' --- Introduction ============ In recent years, the frontier of the search for extrasolar planets has pushed towards ever smaller and more Earth-like worlds. We now know of dozens of Neptune mass planets and have even found the first definitively rocky extrasolar planets [Batalha2011,Leger2009]{}. In between, transit searches have begun finding a population of low-mass low-density “super-Earths”. Beginning with the discovery of GJ1214b [Charbonneau2009]{}, these planets represent a new class of exoplanets that do not have any analog in our Solar System. Basic questions about their composition, structure, and formation are still unknown. Are these, in fact, scaled up versions of the Earth that simply have thick hydrogen/helium envelopes atop of rock/iron cores? Or are they instead scaled down versions of Neptune that are rich in water and other volatile ices? The distinction between water-poor super-Earths or water-rich sub-Neptunes has fundamental implications for how these planets formed. So far these low-mass low-density (hereafter LMLD) planets have only been found well inside the snow-line. If these planets only contain rock, iron, and hydrogen/helium, then it is possible they formed close to their current orbits [Hansen2011]{}. However, if a significant fraction of their mass is in water, then they must have formed beyond the snow-line and migrated in to their current locations [Alibert2011,Ida2010,Rogers2011]{}. The Kepler-11 system [Lissauer2011a]{} is an extremely powerful tool for exploring the features of LMLD planets. With six transiting planets orbiting a close solar analog, it is the richest extrasolar system currently known. Moreover, five of the planets have masses from Transit Timing Variations (TTVs), and all five of these fall into the low-mass low-density regime in between Earth and Neptune. These five planets are all interior to Mercury’s orbit, with periods from 10 to 47 days. This provides a unique laboratory to test the possible composition, formation, and evolution of LMLD planets and how these vary as a function of both period and planet mass. Transiting planets with measured masses, like those in Kepler-11, are particularly valuable because we can determine their mean density. All the planets in Kepler-11 have densities too low for pure rock, and therefore must have some sort of thick envelope of volatiles. Likewise, all the planets except Kepler-11b are less dense than pure water and so must have at least some hydrogen/helium. Unfortunately, mass and radius alone cannot uniquely determine a planet’s composition. In general, there is a large degeneracy between the relative amounts of rock, iron, water, and hydrogen/helium [Rogers2010a]{}. This problem is particularly acute for planets with radii $\approx 2-4 \: R_{\mathrm{\oplus}}$, since in this range any of these four constituents can be important. Indeed these sorts of degeneracies have long been a focus of studies of Uranus and Neptune [Hubbard1991,Fortney2011a]{}. One possible solution to the composition problem is to obtain multi-wavelength transmission spectra, as has been done for GJ1214b [Bean2011, Desert2011, Croll2011]{}. Since hydrogen-rich atmospheres have much larger scale heights at a given temperature, near infrared water and methane absorption features will be much more prominent for planets with hydrogen/helium envelopes [Kempton2009,Kempton2010]{}. Unfortunately, these observations are extremely time intensive and even then the possible presence of clouds can make their interpretation difficult. Even worse, nearly all the systems found by Kepler are too faint for these observations with current telescopes. An alternative is to develop models of the formation and evolution of low-mass planets to try and predict what compositions can form and how those compositions change as a planet evolves. In particular, hydrodynamic mass loss from extreme ultra-violet (XUV) heating can remove large amounts of hydrogen/helium from highly irradiated LMLD planets. Models of XUV driven mass loss were first developed to study water loss from early Venus [Hunten1982, Kasting1983]{}, and hydrogen loss from the early Earth [Sekiya1980, Watson1981]{}. These kinds of models have since been developed to study mass loss from hot Jupiters \[e.g.,\]\[\][Lammer2003, Yelle2004, Murray-Clay2009, Ehrenreich2011, Owen2012]{}, where there is strong evidence that atmospheric escape is an important physical process [Vidal-Madjar2004, Davis2009, Lecavelier2010, Lecavelier2012]{}. In Sections \[masslosssec\], \[watersec\], and \[formationsec\] we show that energy-limited hydrodynamic mass loss models, coupled with models of thermal evolution and contraction, can distinguish between water-poor super-Earth and water-rich sub-Neptune scenarios in Kepler-11. Moreover, these models make powerful predictions for the density distribution of the entire population of LMLD transiting planets. In particular, observations show that there is threshold in the bulk density - incident flux distribution above which there are no LMLD planets. In Section \[fdsec\] we examine this threshold and show how it can by reproduced using our thermal evolution models coupled with standard hydrodynamic mass loss prescriptions. Finally, in Section \[constraintsec\] we explore how this threshold can be used to obtain important constraints on planets without measured densities: We constrain the maximum radii of non-transiting radial velocity planets, and the minimum masses of Kepler candidates. Our Model ========= Planet Structure {#intsec} ---------------- We have built on previous work in [Fortney2007]{} and [Nettelmann2011]{} to develop models of the thermal evolution of LMLD planets. To simplify what is undoubtedly a complex interior structure for real planets, we construct model planets with well-defined layers. Low-mass planets are likely to have a significant fraction of their mass in iron and silicate rocks. For simplicity, we assume that these materials are contained in a isothermal rocky core with Earth-like proportions of 2/3 silicate rock and 1/3 iron. For the rock, we use the ANEOS [Thompson1990]{} olivine equation of state (EOS); while for the iron, we use the SESAME 2140 Fe EOS [Lyon1992]{}. On top of this rock/iron core we then attach an interior adiabat. The composition of this adiabat depends on the planet model being considered. For this work, we consider three classes of LMLD planets: rocky super-Earths with H/He envelopes, water-worlds that have pure water envelopes, and sub-Neptunes with a water layer in between the core and the upper H/He layer. For the water-rich sub-Neptune models we assume that this intermediate water-layer has the same mass as the rock/iron core. We choose this value because it is comparable to the water to rock ratio need to fit Kepler-11b as a water-world. This allows us to explore the proposition that all five Kepler-11 planets started out with similar compositions, but that mass loss has subsequently distinguished them. For hydrogen/helium we use the [Saumon1995]{} EOS. Meanwhile for water we use the ab-initio H2O-REOS EOS developed by [Nettelmann2008]{} and [French2009]{}, which was recently confirmed up to 7 Mbar in laboratory experiments [Knudson2012]{}. In the Kepler-11 system, our models predict that water will be in the vapor, molecular fluid, and the ionic fluid phases. The interiors are too hot for high pressure ice phases. Finally, we model the radiative upper atmosphere by assuming that the planet becomes isothermal at pressures where the adiabat is cooler than the planet’s equilibrium temperature, assuming 20% Bond albedo and uniform re-radiation. We then calculate the radius at 10 mbar which we take to be the transiting radius. We connect the different layers of our models models by requiring that pressure and temperature are continuous across boundaries. We then solve for the interior structure assuming hydrostatic equilibrium. A given model is defined by its mass, composition (i.e., the relative proportions in H/He, water, and the rock/iron core) and the entropy of its interior H/He adiabat. By tracking changes in composition and entropy we can then connect these models in time and study the thermal and structural evolution of a given planet. Thermal Evolution {#tesec} ----------------- In order to obtain precise constraints on composition, it is important to fully model how a planet cools and contracts due to thermal evolution. Models that only compute an instantaneous structure [Rogers2010b]{} by necessity must vary the intrinsic luminosity of the planet over several orders of magnitude, which can introduce large uncertainties in the current composition. Obtaining precise constraints from thermal evolution is essential when considering mass loss, since mass loss histories are highly sensitive to uncertainties in the current composition. Moreover, since mass loss depends strongly on planetary radius (to the third power), the mass loss and thermal histories are inextricably linked. Modeling this contraction requires a detailed understanding of a planet’s energy budget. By tracking the net luminosity of a planet, we know how the specific entropy $S$, (i.e., the entropy per unit mass) of the interior adiabat changes with time. For a given mass and composition, this adiabat then defines the planet’s structure and so we can track the planet’s total radius as the model cools and contracts with time. Equation (\[thermaleq\]) shows the energy budget for our models and how this relates to the change in entropy $dS/dt$. $$\label{thermaleq} \int_{M_{\mathrm{core}}}^{M_{\mathrm{p}}} dm \frac{T dS}{dt} = - L_{\mathrm{int}} + L_{\mathrm{radio}} - c_{\mathrm{v}} M_{\mathrm{core}} \frac{dT_{\mathrm{core}}}{dt}$$ The left hand side shows the rate of change of the thermal energy of the interior adiabat. Positive terms on the right hand side represent energy sources that heat and inflate a planet, while negative terms represent energy losses that allow a planet to cool and contract. The term $L_{\mathrm{int}} = L_{\mathrm{eff}}-L_{\mathrm{eq}}$ describes the intrinsic luminosity due to radiation from the planet, where $L_{\mathrm{eq}}$ is the planet’s luminosity due only to absorbed stellar radiation. The $L_{\mathrm{radio}}$ term describes heating due to radioactive decay. The important isotopes are $^{235}$U, $^{40}$K, $^{238}$U, and $^{232}$Th. These have half lives of 0.704, 1.27, 4.47, and 14.1 Gyr, respectively. We assume meteoritic abundances given by [Anders1989]{}. We do not consider the early decay of $^{26}Al$, since we only consider models that are at least 10 Myr after planet formation. The $L_{\mathrm{radio}}$ term has only a minor effect on our models since it is typically an order of magnitude smaller than the other terms in equation (\[thermaleq\]). Lastly, there is the $dT_{\mathrm{core}}/dt$ term, which represents the delay in cooling due to the thermal inertia of the rocky core. As the interior adiabat cools, the core isotherm must also cool, as $T_{\mathrm{core}}$ equals the temperature at the bottom of the adiabat. When the core makes up a large fraction of the planet’s mass, this can significantly slow down the planet’s rate of contraction. We assume a core heat capacity of $c_v = 0.5-1.0$ $\mathrm{J \, K^{-1} \, g^{-1}}$ [Alfe2002, Guillot1995, Valencia2010]{} as in [Nettelmann2011]{}. This range covers values appropriate for both the cores of the Earth and Jupiter. For our three layer sub-Neptune models we still us the mass in rock and iron for $M_{\mathrm{core}}$, since the water layer is generally too hot for ice phases and so it is assumed to be fully convective. For a given interior structure, we determine the intrinsic flux from the interior, at given $S$ of the adiabat, via interpolation in a grid of model atmospheres. The values of $T_{\mathrm{int}}$ (a parametrization of the interior flux), $T_{\mathrm{eq}}$, and $T_{\mathrm{eff}}$ are tabulated on a grid of surface gravity, interior specific entropy, and incident flux for 50$\times$ solar metallicity H/He atmospheres (similar to Neptune). This corresponds to a metal mass fraction of $Z \approx 0.35$ and a mean molecular weight of $\mu \approx 3.5$ $\mathrm{g \, mol^{-1} }$. The grid is the same as that described in [Nettelmann2011]{} for LMLD planet GJ 1214b, where a more detailed description can be found. Here we do expand on that grid to now include a range of incident fluxes, as was done for giant planets in [Fortney2007]{}. In choosing the initial entropy for our evolution model, we assume a “hot start” for model; i.e., we start the models out with a large initial entropy. When then allow the models to cool and contract until either 10 Myr or 100 Myr which is when we begin the coupled thermal and mass loss evolution. This is a common but important assumption. However, in general our thermal evolution models are insensitive to the initial entropy choice by $\sim100$ Myr as in [Marley2007]{}. As a result, we present results at both 10 and 100 Myr. Moreover, to gain confidence in our 10 Myr models we examined the effect of starting those models with a lower initial entropy. Specifically, we ran models in which we started the 10 Myr with the entropies found at 100 Myr. This allowed us to separate the effect of the stellar XUV evolution, from any “hot start” vs. “cold start” uncertainties. Future progress in modeling the formation of water-rich sub-Neptune planets \[e.g.,\]\[\][Rogers2011]{} may allow for an assessment of the most realistic initial specific entropies. XUV-Driven Mass Loss {#masslosssec} -------------------- Close-in planets like those in Kepler-11 are highly irradiated by extreme ultraviolet (EUV) and x-ray photons. These photons photoionize atomic hydrogen high in a planet’s atmosphere, which in turn produces significant heating [Hunten1982]{}. If this heating is large enough, it can generate a hydrodynamic wind that is capable of removing significant mass, potentially including heavier elements as well [Kasting1983]{}. We couple this XUV driven mass loss to our thermal evolution models following the approach of [Jackson2010]{} and [Valencia2010]{}, which explored possible mass loss histories for CoRoT-7b [Leger2009,Queloz2009]{}. Similar approaches have also been used to study the coupled evolution of hot Jupiters \[e.g.,\]\[\][Baraffe2004, Baraffe2005, Hubbard2007a, Hubbard2007b]{} and hot Neptunes [Baraffe2006]{}. A common approach to estimate the mass loss rate is to assume that some fixed fraction of the XUV energy incident on a planet is converted into heat that does work on the atmosphere to remove mass. This is known as the energy-limited approximation [Watson1981]{} and allows a relatively simple analytic description of mass loss rates. $$\label{masslosseq} \dot{M}_{\mathrm{e-lim}} \approx \frac{\epsilon \pi F_{\mathrm{XUV}} R_{\mathrm{XUV}}^3}{G M_{\mathrm{p}} K_{\mathrm{tide}}}$$ $$K_{\mathrm{tide}} = (1 - \frac{3}{2 \xi} + \frac{1}{2 \xi^3})$$ $$\xi = \frac{R_{\mathrm{Hill}}}{R_{\mathrm{XUV}}}$$ Equation (\[masslosseq\]) describes our estimate of the mass loss rate based on the formulation from [Erkaev2007]{}. $F_{\mathrm{XUV}}$ is the total flux between $1-1200$ Å, which is given by Ribas(2005) for Sun-like stars. For stars older than 100 Myr, Ribas found that at 1 AU $F_{\mathrm{XUV}} = 29.7 \tau^{-1.23} \: \mathrm{erg \, s^{-1} \, cm^{-2}}$, where $\tau$ is the age of the star in Gyr. Using this power law, we scale the XUV flux to the appropriate age and semi-major axis for each planet in our models. Although Ribas only targeted Sun-like stars, [Sanz-Forcada2010]{} found similar results for a wide range of stellar types from M3 to F7. Hereafter, we will simply refer to the entire $1-1200$ Å$;$ spectrum as XUV. $R_{\mathrm{XUV}}$ is the planetary radius at which the atmosphere becomes optically thick to XUV photons, which [Murray-Clay2009]{} find occurs at pressures around a nanobar, in the hot Jupiter context. For our work, we assume that the atmosphere is isothermal between the optical and XUV photospheres. This neglects heating from photo-disassociation, which should occur around a $\mathrm{\mu bar}$ [Kempton2012a]{}. However, this effect should be relatively small and if anything will lead to slight underestimate of the mass loss rate. We vary pressure of the XUV photosphere from 0.1 nbar to 10 nbar to include the uncertainty in the structure of the XUV photosphere. For H/He atmospheres on LMLD planets, the nbar radius is typically 10-20% larger than the optical photosphere. $K_{\mathrm{tide}}$ is a correction factor that accounts for the fact that mass only needs to reach the Hill radius to escape [Erkaev2007]{}. For planets like Kepler-11b today this correction factor increases the mass loss rate by $\sim$ 10%, however at early times it can increase the rate by as much as a factor of 2. Finally, $\epsilon$ is an efficiency factor that parametrizes the fraction of the incident XUV flux that is converted into usable work. This efficiency is set by radiative cooling, especially via Lyman $\alpha$, and can depend on the level of incident flux [Murray-Clay2009]{}. Kepler-11 is a $8\pm2$ Gyr old Sun-like star. Using the power law from Ribas et al., this implies that current XUV flux at Kepler-11f is $\approx 37 \: \mathrm{erg \, s^{-1} \, cm^{-2}}$. Similarly, when Kepler-11 was 100 Myr old, the flux at Kepler-11b was $\approx 6\times10^4 \: \mathrm{erg \, s^{-1} \, cm^{-2}}$. [Murray-Clay2009]{} found that at XUV fluxes over $10^5 \: \mathrm{erg \, s^{-1} \, cm^{-2}}$, relevant for many hot Jupiters, mass loss becomes radiation/recombination-limited and highly inefficient. However, at the lower XUV fluxes relevant for the Kepler-11 system mass loss is roughly linear with $F_{\mathrm{XUV}}$ and has efficiencies $\sim 0.1-0.3$. For this work, we assume a default efficiency of $\epsilon=0.1\pm^{0.1}_{0.05}$, although we do examine the effects of lower efficiencies. While we predominantly investigate the loss of H/He envelopes, in some limited cases for Kepler-11b, we also assume this holds for steam envelope loss. [ccccccccc]{}\[h!\] Kepler-11b & $4.3\pm^{2.2}_{2.0}$ & $0.3\pm^{1.1}_{0.25}\%$ & $40\pm^{41}_{29}\%$ & n/a\ \ Kepler-11c & $13.5\pm^{4.8}_{6.1}$ & $4.6\pm^{2.7}_{2.3}\%$ & n/a & $0.3\pm^{1.0}_{0.1}\%$\ \ Kepler-11d & $6.1\pm^{3.1}_{1.7}$ & $8.2\pm^{2.7}_{2.4}\%$ & n/a & $1.3\pm^{0.9}_{0.8}\%$\ \ Kepler-11e & $8.4\pm^{2.5}_{1.9}$ & $17.2\pm^{4.1}_{4.2}\%$ & n/a & $5.5\pm^{2.3}_{3.0}\%$\ \ Kepler-11f & $2.3\pm^{2.2}_{1.2}$ & $4.1\pm^{1.8}_{1.5}\%$ & n/a & $0.4\pm^{0.6}_{0.2}\%$\ \ \[currenttab\] One important implication from equation (\[masslosseq\]) is that mass loss rates are much higher when planets are young. This is due to two reasons. Planetary radii are considerably larger due to residual heat from formation. Moreover, at 100 Myr $F_{\mathrm{XUV}}$ was $\approx 500$ times higher than it is currently [Ribas2005]{}. As a result, most of the mass loss happens in a planet’s first Gyr. Thus although a planet’s envelope may be stable today, its composition may have changed significantly since formation. Likewise, a considerable amount of mass will be lost between the end of planet formation at $\sim10$ Myr [Calvet2002]{} and 100 Myr. Following the x-ray observations of [Jackson2012]{}, we assume that at ages younger than 100 Myr the stellar XUV flux saturates and is constant at the 100 Myr value. Unfortunately, the observations for 10-100 Myr do not cover the EUV (100-1200 Å) part of the spectrum, so there is some uncertainty as to whether this saturation age is uniform across the entire XUV spectrum. Nonetheless, our constraints on the formation of Kepler-11 in sections \[sesec\] and \[formationsec\] come from the lower limits we are able to place on the initial compositions. Assuming that the EUV saturates along with the x-rays is conservative assumption in terms of the amount of mass that is lost. In general, models of LMLD planets that assume H/He envelopes today will predict much larger mass loss histories then models that assume steam envelopes. Partly, this is because the lower mean molecular weight of hydrogen. Mostly, however, it is because when we integrate the compositions back in time from the present, the addition of a small amount of H/He has much larger impact on a planet’s radius than a small amount of water. A larger radius in the past in turn means a higher mass loss rate; and so the integrated mass loss history becomes much more substantial for H/He envelopes. Application to Kepler-11 ======================== Current Compositions from Thermal Evolution ------------------------------------------- The first step in trying to understand the formation and history of a planetary system is to identify the possible current compositions for each of the planets in the absence of any mass loss. This then gives us estimates for the current masses of each planet’s core, which we then use as the starting point for all of our calculations with mass loss. Figure \[mrfig\] shows the Kepler-11 planets in a mass-radius diagram along with curves for different possible compositions. For all planets, we color-code by the incident bolometric flux they receive. The Kepler-11 planets are shown by filled circles with identifying letters next to each one. The other known transiting exoplanets in this mass and radius range are shown by the open squares. In order of increasing radius, these are Kepler-10b [Batalha2011]{}, Kepler-36b [Carter2012]{}, CoRoT-7b [Leger2009,Queloz2009,Hatzes2011]{}, Kepler-20b [Fressin2011,Gautier2011]{}, Kepler-18b [Cochran2011]{}, 55 Cancri e [Winn2011,Demory2011]{}, GJ 1214b [Charbonneau2009]{}, Kepler-36c [Carter2012]{}, Kepler-30b [Fabrycky2012, Sanchis-Ojeda2012]{}, and GJ 3470b [Bonfils2012]{}. Lastly, the open triangles show the four planets in our own solar system that fall in this range: Venus, Earth, Uranus, and Neptune. The curves show various possible compositions. The solid black curve shows a standard Earth-like composition with 2/3 rock and 1/3 iron as described in Section \[intsec\]. The other curves show compositions with thick water or H/He envelopes atop an Earth-like core. These curves include thermal evolution without mass loss to 8 Gyr, the age of Kepler-11. The blue dashed curves show the results for 50% and 100% water-worlds computed at $T_{\mathrm{eq}}=700$ K, approximately the average temperature of the five inaner planets. Likewise, the dotted orange curves show the results for H/He envelopes; however, here each curve is tailored to match a specific Kepler-11 planet and is computed at the flux of that planet. These fits are listed in greater detail in table \[currenttab\]. Here we list the mass of each planet taken from [Lissauer2011a]{}; the H/He fractions needed to match each planet’s current radius for a water-poor super-Earth model; the water fraction needed to match Kepler-11b as a water-world; and the H/He fractions needed to fit Kepler-11c, d, e, and f as sub-Neptunes with an intermediate water layer, as described in Section \[intsec\]. As described in section \[sesec\], we varied the planetary albedo, the heat capacity of the rocky core, and the observed mass, radius, current age, and incident flux. Figure \[mrfig\] and table \[currenttab\] clearly show the degeneracy between various compositions that we are attempting to untangle. There are now four planets including Kepler-11b that can easily be fit either as water-worlds or as water-rich sub-Neptunes with $<2\%$ of their mass in H/He. However, it is worth looking closer at Kepler-11b in particular. It is this the only planet in the system which does not require any hydrogen or helium to match its current radius, although it must have some sort of volatile envelope. Moreover, it is also the most irradiated and it is fairly low gravity. As a result, adding a small amount of hydrogen to its current composition has a large impact on the bulk density, which in turn makes the planet more vulnerable to mass loss, as seen in Eq. (\[masslosseq\]). A clearer picture for this planet emerges when including XUV driven mass-loss and relatively strong constraints from formation models discussed in Section \[formationsec\]. Thus, if there is hope of using mass loss to constrain the composition and formation of the system, it likely lies with Kepler-11b. Mass Loss for a Super-Earth Scenario {#sesec} ------------------------------------ [cccccccccc]{}\[h!\] Kepler-11b & $34.6\pm^{6.5}_{28.2}$ & $87.6\pm^{6.6}_{85.4}\%$ & $44.8\pm^{9.7}_{10.1}$ & $90.4\pm^{5.1}_{8.2}\%$\ \ Kepler-11c & $13.7\pm^{4.7}_{5.8}$ & $6.0\pm^{5.0}_{3.2}\%$ & $14.2\pm^{4.3}_{3.1}$ & $9.1\pm^{28}_{7.3}\%$\ \ Kepler-11d & $6.7\pm^{2.8}_{0.6}$ & $16.5\pm^{22}_{8.5}\%$ & $7.8\pm^{12.8}_{0.8}$ & $28\pm^{56}_{17}\%$\ \ Kepler-11e & $8.8\pm^{2.3}_{1.6}$ & $21.2\pm^{6.0}_{3.2}\%$ & $9.7\pm^{2.5}_{1.9}$ & $28.1\pm^{10.5}_{7.7}\%$\ \ Kepler-11f & $3.1\pm^{5.2}_{0.2}$ & $29\pm^{58}_{24}\%$ & $3.4\pm^{6.6}_{0.4}$ & $35\pm^{57}_{25}\%$\ \ \[masslosstab\] Now that we have estimates for the present day compositions, we will begin considering the effects of mass loss. We will compute mass loss histories that when evolved to the present day, match the current mass and composition. This then tells us what the mass would have to be in the past to result in the current mass and composition. As discussed in Section \[masslosssec\], there is uncertainty in stellar XUV fluxes ages younger than 100 Myr; as a result, we will present results both at 10 Myr and 100 Myr after planet formation. First we will consider water-poor super-Earth models for each planet, which have H/He envelopes atop Earth-like rocky cores. As discussed in Section \[masslosssec\], H/He envelopes are particularly susceptible to mass loss. As an example, Figure \[k11bfig\] shows four possible cooling histories for Kepler-11b. The solid lines show thermal evolution without any mass loss while the dashed lines include mass loss. The orange curves are for water-poor super-Earth models, while the blue curves show water-world models. The red cross shows the current radius and age of Kepler-11b. These curves illustrate the impacts of both thermal evolution and mass loss on the radius of a low-mass planet. The water-world models require that 40% of the current mass must be in water to match the current radius. Assuming our standard efficiency $\epsilon=0.1$, implies an initial composition of 43% water at 10 Myr. This illustrates the relative stability of water envelopes. On the other hand, the dashed orange curve shows the vulnerability of H/He layers. Here we have assumed a efficiency 5$\times$ lower $\epsilon=0.02$ and yet more mass is lost than in the water-world scenario. Even at this low efficiency, Kepler-11b would have to initially be 11% H/He and 4.8 $M_{\mathrm{\oplus}}$ to retain the 0.3% needed to match the current radius. This also shows the large increase in radius that can result from even a relatively modest increase in the H/He mass. Table \[masslosstab\] summarizes the results for Kepler-11 b-f for the water-poor super-Earth scenario. We list the masses predicted by our models when the planets were 10 and 100 Myr old. In addition, we list the fraction of the planets’ masses in the H/He envelope at each age. These results are further illustrated in Figure \[lossfig\]a. Here we have plotted the mass and H/He fraction for each planet at 10 Myr, 100 Myr, and today. Each color corresponds to a particular planet with the squares indicating the current masses and compositions, the circles the results at 100 Myr, and the triangles the results at 10 Myr. In order to calculate the uncertainty on these results, we varied the mass loss efficiency $\epsilon$ from 5-20% and varied the XUV photosphere from 0.1-10 nbar. Likewise, we varied the planetary Bond albedo from 0-0.80 and varied the heat capacity of the rocky core from 0.5-1.0 $\mathrm{J \, g^{-1} \, K^{-1}}$. Also, as discussed in section \[tesec\], we varied the initial entropy for the 10 Myr models, to account for undertainties in “hot-start” vs. “cold-start”. Finally, we factored in the observed uncertainties in mass, radius, and incident flux. Clearly, Kepler-11b is vulnerable to extreme mass loss if it has a H/He envelope atop a rock/iron core. Although less than 1% H/He today, if it is a water-poor super-Earth it could have been have over $\sim90\%$ H/He in the past. At 10 Myr, its mass would have been $45\pm10$ $M_{\mathrm{\oplus}}$, an order of magnitude higher than the current value. Kepler-11b is able to undergo such extreme mass loss because its high XUV flux and the low mass of its rocky core put it in a regime where it is possible to enter a type of runaway mass loss. This happens when the mass loss timescale is significantly shorter then the cooling timescale. After the planet initially loses mass it has an interior adiabat and rocky core that are significantly hotter than would otherwise be expected for a planet of its mass and age. This is because the interior still remembers when the planet was more massive and has not had sufficient time to cool. As a result, the planet will stay inflated for some time and the density stays roughly constant and can actually decrease. A similar effect was seen by [Baraffe2004]{} when they studied coupled thermal evolution and mass loss models for core-less hot Jupiters. We find that this process generally shuts off once the composition drops below $\sim$20% H/He. At that point the presence of the core forces the total radius to shrink even if the planet is unable to cool efficiently. Figure \[runawayfig\] shows this process as Kepler-11b loses mass for three different values of its current mass and therefore its core mass. The curves correspond to the best fit mass from transit-timing as well as the 1$\sigma$ error bars. This shows that the timing of this runaway loss event depends strongly on the mass of the rock/iron core. Super-Earth models of Kepler-11b are unusual in that they are subject to tremendous mass loss and yet they retain a small amount of H/He today. Typically models that start out $\sim90\%$ H/He either experience runaway mass loss and lose their H/He envelopes completely, or they never enter the runaway regime and remain over 50% H/He. The uncertainty in the initial composition of Kepler-11b is due to uncertainty in its TTV mass. At a given current mass, the range of Kepler-11b models that will retain an envelope that is $<1\%$ H/He is extremely narrow. In this sense, the current composition of Kepler-11b requires a rare set of initial conditions if it is a water-poor super-Earth. [cccccccccc]{}\[h!\] Kepler-11b & $4.4\pm^{2.2}_{2.0}$ & $41\pm^{39}_{28}\%$ & $4.5\pm^{2.1}_{1.8}$ & $43\pm^{38}_{29}\%$\ \ Kepler-11c & $13.7\pm^{4.8}_{4.6}$ & $1.8\pm^{18}_{1.4}\%$ & $15.2\pm^{25}_{1.2}$ & $12\pm^{70}_{10}\%$\ \ Kepler-11d & $6.8\pm^{2.8}_{1.2}$ & $11.6\pm^{17}_{8.7}\%$ & $7.6\pm^{7.6}_{0.9}$ & $21\pm^{49}_{10}\%$\ \ Kepler-11e & $9.1\pm^{2.1}_{2.0}$ & $12.8\pm^{4.6}_{6.7}\%$ & $9.7\pm^{2.6}_{2.0}$ & $18\pm^{12}_{10}\%$\ \ Kepler-11f & $2.9\pm^{3.8}_{0.5}$ & $21\pm^{62}_{17}\%$ & $4.0\pm^{4.9}_{1.5}$ & $43\pm^{44}_{36}\%$\ \ \[watertab\] Counterintuitively, if Kepler-11b is more massive today then its implied mass in the past is actually lower. This is because a higher mass today would imply a more massive core, which would increase the planet’s density and decrease its mass loss rate. As a result, a more massive model for Kepler-11b today is less vulnerable to mass loss and so less H/He is needed in the past in order to retain 0.3% today. At 100 Myr, there is a very large uncertainty in the composition due to the uncertainty in the core mass. However, even if we assume the 1$\sigma$ error bar 6.5 $M_{\mathrm{\oplus}}$, Kepler-11b would still be at least 37 $M_{\mathrm{\oplus}}$ and at least 83% H/He at 10 Myr. In section \[formationsec\], we will compare this to models of in situ formation and show that such a scenario is unlikely. On the other hand, Kepler-11c is not particularly vulnerable to mass loss, at least using the best fit mass from transit timing, despite having the second highest flux in system. This is because of the relatively large mass of its rocky core; the high gravity means additional H/He has a more modest effect of the planet’s radius and therefore on the mass loss rate. In fact, along with the incident XUV flux the mass of the rocky core is the single largest factor that determines whether a given planet will be vulnerable to mass loss. As a result, the dominant sources of uncertainty in our mass loss models are the uncertainties in the masses from TTV. These dominate over all the theoretical uncertainties in the thermal evolution and mass loss models. The uncertainty in planet mass from transit timing is particular large for Kepler-11c. If its mass is close to the 1$\sigma$ low value, then it is possible Kepler-11c has undergone more substantial mass loss similar to Kepler-11d-f. Fortunately, as more quarters of data are processed the mass estimates from TTV will become more precise [Agol2005,Holman2005]{}. Finally, Kepler-11d, e, and f are modestly vulnerable to mass loss and are consistent with having originated with $\sim 20\%$ H/He at 100 Myr and $\sim 30\%$ H/He at 10 Myr. In Section \[hillsec\] we will discuss these results in terms of orbital stability. The Water-Rich Scenario {#watersec} ----------------------- Next we consider a water-rich scenario where the entire system formed beyond the snow line. We assume that Kepler-11c-f are water-rich sub-Neptunes as described in Section \[intsec\], while Kepler-11b is currently a water-world. Otherwise the thermal mass loss histories are calculated in the same manner as the water-poor super-Earth scenario. For Kepler-11c-f we calculate the planet mass H/He fraction at 10 and 100 Myr, assuming that only H/He is lost. For Kepler-11b, we examine the vulnerability of both H/He and steam envelopes atop water-rich interiors. The results are summarized in Table \[watertab\] which list the water fraction for a water-world model of Kepler-11b and the H/He fraction for water-rich sub-Neptune models of Kepler-11c-f. Likewise, the results for c-f are shown in Figure \[lossfig\]b. In general, these three layer models are slightly more vulnerable to mass loss than the water-poor super-Earth models presented in section \[sesec\]. Mostly this is because models with a water layer have hotter interiors that cool more slowly. Since models For example, for Kepler-11c without mass loss the models presented in Table \[currenttab\], at 8 Gyr the final entropy in the H/He layer is 6.6 $\mathrm{k_b}$/baryon for the water-rich sub-Neptune model versus 5.8 for the water-poor super-Earth model. The second reason is that counter-intuitively the water-rich sub-Neptune models are slightly more vulnerable to mass loss precisely because they have less of the planet’s mass in H/He today. For a planet that has less H/He today, adding a small amount of H/He at the margin has a larger impact on the planet’s radius and therefore on the mass loss rate. For Kepler-11c-f the results are broadly similar to the those for the water-poor super-Earth scenario. Kepler-11c is again the least vulnerable to mass loss; while Kepler-11d is again the most vulnerable of the four planets that we model as water-rich sub-Neptunes. However, all four of these planets are consistent with having been $\sim10-20\%$ H/He at 100 Myr and $\sim20-30\%$ H/He at 10 Myr. If Kepler-11b was always a water-world, then mass loss was never important for it. Between 10 Myr and the present, it only drops from 43% to 40% water. Moreover, if Kepler-11b was initially a water-rich sub-Neptune similar to the other planets in the system, it could have easily stripped its H/He outer envelope. If we start Kepler-11b at 100 Myr as a water-rich sub-Neptune similar to the other planets with $30\%$ H/He atop 4.3 $M_{\mathrm{\oplus}}$ of rock and water, then assuming $\epsilon=0.1$ the entire H/He envelope will be stripped by 300 Myr. We can set upper limits on the initial mass and H/He fraction of 70 $M_{\mathrm{\oplus}}$ and 94% if Kepler-11b was originally a water-rich sub-Neptune. These are however strictly upper limits, a H/He layer could have been lost at any time between formation and now. Therefore, all five planets are consistent with a scenario is which they formed as water-rich sub-Neptunes with $\sim10\%$ H/He at 100 Myr and $\sim20\%$ H/He at 10 Myr. A Mass Loss Threshold for Low-Mass Low-Density Planets {#fdsec} ====================================================== Although Kepler-11 provides a unique case-study, it is essential to explore how mass loss impacts the larger population of LMLD transiting planets. Figure \[shorelinefig\] shows the bolometric flux these planets receive at the top of their atmospheres vs. their bulk densities. As in Figure \[mrfig\], filled circles show the Kepler-11 planets with the letters indicating each planet. Likewise, the open squares show the other transiting exoplanets that are less than 15 Earth masses. For reference, we have also plotted all other transiting planets between 15 and 100 $M_{\mathrm{\oplus}}$ as gray crosses [Wright2011]{}. The colors indicate possible compositions. All planets with a best-fit mass and radius that lies below a pure rock curve are colored red. These include Kepler-10b, Kepler-36b, CoRoT-7b, and just barely Kepler-20b. Planets that are less dense than pure rock but more dense than pure water, indicating that the could potentially be water-worlds, are colored blue. These include Kepler-11b, Kepler-18b, and 55 Cancri e. Meanwhile those planets that must have a H/He envelope to match their radius are colored orange. These include Kepler-11c, d, e, and f, Kepler-30b, Kepler-36c, GJ 1214b, and GJ 3470b. The dashed black lines show curves of constant mass loss rate according to equation (\[masslosseq\]), assuming $\epsilon=0.1$ and $K_{\mathrm{tide}}=1$. These curves are linear in this plot since the instantaneous mass loss rate goes as the flux over the density. Although Figure \[shorelinefig\] plots the bolometric flux today, we can relate this to an XUV flux at a given time using the [Ribas2005]{} power law for sun-like stars described in Section \[masslosssec\]. The curves show the flux today required to lose mass at 1 $M_{\mathrm{\oplus}} \, \mathrm{Gyr^{-1}}$ when the planets were 1 Gyr old and 100 Myr old, along with another curve showing 0.1 $M_{\mathrm{\oplus}} \, \mathrm{Gyr^{-1}}$ at 100 Myr. Since most of the mass loss happens in the first few hundred Myrs, the bottom two curves can roughly be considered as the respective thresholds for mass loss being important and being unimportant for LMLD planets. One possible explanation of this mass loss threshold is that it caused by XUV driven mass loss from H/He envelopes on low-mass planets. LMLD planets that form above the 100 Myr 1 $M_{\mathrm{\oplus}} \, \mathrm{Gyr^{-1}}$ curve lose mass, increase in density and move to the right until they lie below this threshold. The planets that are left above this line are mostly rocky or at the very least probably do not have H/He envelopes. Planets more massive than $\sim 15$ $M_{\mathrm{\oplus}}$ are not affected since they have a larger reservoir of mass and the loss of a few earth masses of volatiles isn’t sufficient to significantly change their bulk density. To illustrate this, we have plotted our predictions for the bulk densities of each of the Kepler-11 planets at 100 Myr, including the effects of both mass loss and thermal evolution. These are indicated by the shadowed letters at the left of Figure \[shorelinefig\]. The situation becomes even clearer if we instead we plot flux against mass times density as in Figure \[improvedfig\]. The timescale for XUV mass loss goes like $\rho M_{\mathrm{p}}/F_{\mathrm{XUV}}$, so lines in this diagram are constant mass loss timescales. Now the threshold is much clearer and applies to all planets up to all planets with H/He envelopes. This also removes any effects from the somewhat arbitrary 15 $M_{\mathrm{\oplus}}$ cut.The sparsity of planets at low flux and high density is almost certainly a selection effect, since these are likely to be planets with long periods and small radii. However, the interesting result is that there is appears to be a critical mass loss timescale above which we do not find any planets with H/He envelopes. In particular, all five of the inner Kepler-11 planets lie nicely along this threshold. Moreover, of the three planets that lie above the critical mass loss timescale, two are likely rocky. $$\label{tmleq} t_{\mathrm{loss}} = \frac{M_{\mathrm{p}}}{\dot{M}} = \frac{G M_{\mathrm{p}}^2}{\pi \epsilon R_{\mathrm{p}}^3 F_{\mathrm{XUV,E100}}} \frac{F_{\mathrm{\oplus}}}{F_{\mathrm{p}}}$$ The dashed black line in Figure \[improvedfig\] shows our best fit for this critical mass loss timescale. Equation \[tmleq\] defines this mass loss timescale. Here $\epsilon=0.1$ is the mass loss efficiency, $F_{\mathrm{XUV,E100}} = 504$ $\mathrm{erg \, s^{-1} \, cm^{-2}}$ is the XUV flux at the Earth when it was 100 Myr old, and $F_{\mathrm{p}}$ is the current incident bolometric flux at a planet. We find a best fit with $t_{\mathrm{loss,crit}}\approx 12$ Gyr. However, while equation \[tmleq\] accounts for the higher XUV fluxes at earlier times, it does not include the effects of larger radii at formation. The will reduce $t_{\mathrm{loss}}$ by at least another order of magnitude. A similar mass loss threshold was proposed by [Lecavelier2007]{}. Unfortunately, at that time there were relatively few transiting planets and no known transiting super-Earths. As a result, the authors we mostly limited to hot Jupiters from radial velocity surveys and were forced to use a scaling law to estimate radii. Here we are able to confirm the existence of a mass loss threshold and extend it all the way down to $\sim 2$ $M_{\mathrm{\oplus}}$. This mass loss threshold could also help explain features in occurrence rate of planets found by Kepler. [Howard2011a]{} found that the frequency of 2-4 $R_{\mathrm{\oplus}}$ Kepler planet candidates dropped off exponentially for periods within 7 days. This 7 day cutoff corresponds to an incident bolometric flux of 200 $F_{\mathrm{\oplus}}$. There are five planets with measured densities in figure \[shorelinefig\] that lie above 200 $F_{\mathrm{\oplus}}$. Of these five, three planets are consistent with being rocky and two with being water-worlds; none of the five requires a H/He atmosphere to match its observed mass and radius. If all low mass planets orbiting within 7 days lose their H/He atmospheres, then their radii will shrink from 2-4 $R_{\mathrm{\oplus}}$ to $<$2 $R_{\mathrm{\oplus}}$. This could naturally explain the drop off in 2-4 $R_{\mathrm{\oplus}}$ candidates at short periods. Reproducing the Mass Loss Threshold {#reproducesec} ----------------------------------- In order to fully examine whether the mass loss threshold in Figure \[shorelinefig\] can be explained by atmospheric mass loss, we performed a small parameter study with $\sim$800 mass loss models across a wide range of initial masses, compositions, and incident fluxes. For each model we ran thermal evolution and mass loss starting at 10 Myr around a Sun-like star. We ran models with initial masses of 2, 4, 8, 16, 32, and 64 $M_{\mathrm{\oplus}}$. We assumed water-poor super-Earth compositions, meaning H/He envelopes on top Earth-like cores, with initial compositions of 1, 2, 5, 10, 20, and 40% H/He. Finally we varied the incident bolometric flux from 10 to 1000 $F_{\mathrm{\oplus}}$, in order to cover the range of observed planets in Figures \[shorelinefig\] and \[improvedfig\]. We then recorded the resulting masses, densities, and compositions at various ages. The results are shown in Figure \[predictfig\]. As in Figure \[improvedfig\], each panel plots the total incident flux at the top of the atmosphere vs. the planet mass times density assuming different mass loss histories for our full suite of models. The size of each point indicates the mass of the planet, while the color indicates the fraction of its mass in the H/He envelope. The top left panel shows the initial distribution at 10 Myr before we start any mass loss. The other two top panels show the results at 100 Myr and 10 Gyr for our standard mass loss efficiency $\epsilon=0.1$. Meanwhile, the bottom panels show the results at 1 Gyr for a range of different efficiencies. These range from highly inefficient mass loss $\epsilon=0.01$, to our standard efficiency $\epsilon = 0.1$, and finally extremely efficient mass loss $\epsilon=1$. In each panel, as planets cool and lose mass the points move to the right, shrink, and become bluer (less H/He). For reference, we have re-plotted our critical mass loss timescale from Figure \[improvedfig\] in each of the result panels. As we can see, models with mass loss do in general result in a threshold roughly corresponding to a critical mass loss timescale. Moreover, the mass loss threshold observed in Figure \[shorelinefig\] is well reproduced by mass loss models with $\epsilon \approx 0.1$. This is similar to the efficiencies found by detailed models of mass loss from hot Jupiters in the energy-limited regime [Murray-Clay2009]{}. This suggests that our assumption of comparable mass loss efficiencies for LMLD planets is reasonable. It is also apparent that the threshold already in place by 100 Myr, and subsequent evolution has a relatively minor effect. We also examined the effect of beginning our parameter study at 100 rather than 10 Myr; however, this did not significantly affect the location of the threshold. Previous mass loss evolution models \[e.g.,\]\[\][Hubbard2007a, Hubbard2007b, Jackson2012, Owen2012]{} have also predicted mass loss thresholds. However, our models are the first to fully include the effects of coupled mass loss and thermal evolution for LMLD planets. We are able confirm and explain the observed threshold seen in Figure \[improvedfig\] in a region of parameter space where most of the [Kepler]{} planets are being found. Constraints On Mass and Radius for the General Population {#constraintsec} --------------------------------------------------------- If we use the critical mass loss timescale curve from Figure \[improvedfig\] as a approximation for the observed mass loss threshold, then we can write down a simple expression for the threshold. This is shown in equation (\[fdeq\]), which is valid for planets around Sun-like stars with $F_{\mathrm{p}}< 500$ $F_{\mathrm{\oplus}}$. The 500 $F_{\mathrm{\oplus}}$ cut excludes highly irradiated rocky planets like Kepler-10b and CoRoT-7b. These planets may have once had volatiles in the past, but they are likely rocky today and so H/He mass loss is no longer relevant. This cut also excludes the region where energy-limited escape breaks down and mass loss becomes radiation and recombination limited [Murray-Clay2009]{} $$\label{fdeq} \rho M_{\mathrm{p}} \ge \frac{ 3\epsilon F_{\mathrm{XUV,E100}} } { 4 G } \frac{ F_{\mathrm{p}} } { F_{\mathrm{\oplus}} } t_{\mathrm{loss,crit}}$$ The exciting implication of equation (\[fdeq\]) is that we can use it to obtain lower limits on mass for the much larger population of Kepler super-Earths and sub-Neptunes for which we do not have measured densities. This will help identify promising targets for follow-up work with radial velocity observations. This is shown in equation (\[masslimeq\]). $$\label{masslimeq} M_{\mathrm{p}} \ge \sqrt{ \frac{\pi\epsilon F_{\mathrm{XUV,E100}}}{G} \frac{F_{\mathrm{p}}}{F_{\mathrm{\oplus}}} t_{\mathrm{loss,crit}}} \, R_{\mathrm{p}}^{3/2}$$ Table \[masslimtab\] applies equation (\[masslimeq\]) to a list of Kepler candidates smaller than 4 $R_{\mathrm{\oplus}}$ that are well suited to radial-velocity follow-up. We excluded any planets with $F_{\mathrm{p}}> 500$ $F_{\mathrm{\oplus}}$, since equation (\[masslimeq\]) is not valid in that regime. Also, we limited the sample to only those planets with minimum radial velocity semi-amplitudes $K_{\mathrm{min}}>1.0 \: \mathrm{m \, s^{-1}}$ around stars with Kepler magnitude brighter than 13, since these will be the most promising for RV follow-up. In the end, this leaves us with a list of 38 likely detectable targets, eight of which (KOIs 104.01, 107.01, 123.01, 246.01, 262.02, 288.01, 984.01, and 1241.02) have $K_{\mathrm{min}}>2.0 \: \mathrm{m \, s^{-1}}$. Finally, we can also use the mass loss threshold to find an upper limit on the radii of non-transiting planets from radial velocity surveys with $F_{\mathrm{p}} < 500$ $F_{\mathrm{\oplus}}$. This is done in equation (\[radiuslimeq\]). $$\label{radiuslimeq} R_{\mathrm{p}} \le ( \frac{G}{\pi\epsilon F_{\mathrm{XUV,E100}} t_{\mathrm{loss,crit}}}\frac{F_{\mathrm{\oplus}}}{F_{\mathrm{p}}} )^{1/3} M_{\mathrm{p}}^{2/3}$$ Discussion ========== Kepler-11: Comparison to Formation Models, Implications for Migration {#formationsec} --------------------------------------------------------------------- By itself, the constraints from mass loss do not tell us whether Kepler-11 is a system of water-poor super-Earths or water-rich sub-Neptunes. Instead we need to compare our estimates of the initial compositions to models of planet formation. By doing so we can examine whether our estimates of the original compositions for a water-poor super-Earth scenario are consistent with the maximum H/He fraction that can be accreted during in situ formation. [Ikoma2012]{} examine the accretion of H/He atmospheres onto the rocky cores of hot water-poor super-Earths. In particular, they examine the in situ formation of the Kepler-11 system. In addition to a planet’s core mass and temperature, the amount of H/He accreted will depend strongly on the lifetime and dust grain opacity of the accretion disk. As with thermal evolution, the need to cool the rocky core can slow the contraction of the accreting atmosphere and limit the final H/He fraction. They are able to set hard upper limits on the initial compositions for in situ formation by assuming a grain-free, long-lived ($\sim 1$ Myr in the inner 0.2 AU) accretion disk and ignoring the delay in accretion due to cooling the core. In particular, [Ikoma2012]{} find that Kepler-11b could not have accreted more than 10% of its mass in H/He if it formed in situ. Moreover, using a more typical disk lifetime of $10^5$ yr [Gorti2009]{} and including the effect of cooling the core implies that Kepler-11b was $<1\%$ H/He at formation. On the other hand, in Table \[masslosstab\] we showed that thermal evolution and mass loss models predict that if Kepler-11b is a water-poor super-Earth then it was $87\pm^{7}_{85}\%$ and at least $82\%$ at 10 Myr. Combined with the results of [Ikoma2012]{}, this disfavors in situ formation of Kepler-11b. This result appears robust to any uncertainties in thermal evolution or mass loss models. Even if we only look after the period of run-away mass loss, at 3 Gyr Kepler-11b was still 10$\%$ H/He, the maximum allowed by [Ikoma2012]{}. Likewise, we find that Kepler-11f was at least 10$\%$ H/He at 10 Myr, even though the [Ikoma2012]{} models predict that it cannot have accreted its current composition of 4$\%$ H/He if it formed in situ. Furthermore, the co-planar, tightly packed, circular orbits in the system strongly suggest that it could have undergone type 1 migration [Ida2010]{}. As a result, we disfavor in situ formation of the system. If the Kepler-11 system did not form at its current location, then one possibility is that it formed at or beyond the snow-line and then Type 1 migrated to it is current location [Rogers2011]{}. If this is the case, then it is likely a system of water-rich sub-Neptunes and water-worlds as discussed in Section \[watersec\]. As we showed in Section \[watersec\], Kepler-11b is very stable to mass-loss if it is a water-world. If it was initially a water-rich sub-Neptune, it could have easily lost its H/He layer in the first few 100 Myr. Likewise, Kepler-11c-f are all consistent having formed as water-rich sub-Neptunes with $\sim20\%$ of their mass in H/He. The other possibility is that Kepler-11 is a system of water-poor super-Earths that has nonetheless undergone significant migration. For a grain-free accretion disk that lasts $10^6$ yr at 550 K, the critical mass for run-away accretion drops to 5 $M_{\mathrm{\oplus}}$ [Ikoma2012]{}. This implies that Kepler-11b could possibly have formed as a water-poor super-Earth at or beyond the current orbit of 11f. Nonetheless, this assumes a completely grain-free long-lived disk, which may not be realistic. Furthermore, this scenario still requires that Kepler-11b was $\sim$90% H/He when it formed, while all the other planets in the system are consistent with more modest initial compositions. As a result, we favor the water-rich sub-Neptune scenario. Kepler 11: Mass Loss and Orbital Stability {#hillsec} ------------------------------------------ One possible result of significant mass loss is that it could impact the orbital stability of closely packed multi-planet systems like Kepler-11. Although this system is stable in its current configuration, it might not be with the initial masses determined by our models. One relatively simple stability check is to calculate the separation between pairs of planets in terms of their mutual Hill spheres. Figure \[hillfig\] plots the separation in mutual Hill spheres ($\Delta$) between adjacent pairs of planets at both 10 Myr and the present, assuming a water-poor super-Earth composition. [Smith2009]{} found that systems with five or more planets tended to de-stabilize when $\Delta<9$. This threshold is shown as dashed gray lines in figure \[hillfig\]. Although Kepler-11b-c currently lies well below this threshold, [Lissauer2011a]{} showed that the system is nevertheless stable today because planets b and c are dynamically decoupled from the other four planets and so act more like a two planet system. For two planet systems the absolute minimum stable separation is $\Delta=2\sqrt{3}=3.46$ [Gladman1993]{}. This second stability threshold is shown by the dotted lines in figure \[hillfig\]. The Hill radius goes as $M_p^{1/3}$, as a result the change in $\Delta$ from mass loss is relatively modest; nonetheless, the stability of the system is in danger. At 10 Myr, Planets d-e do lie below the approximate $\Delta>9$ stability threshold; however, both pairs on either side of d-e are still relatively stable which may help stabilize the system. More importantly, the separation of planets b-c at 10 Myr skirts dangerously close, $\Delta=3.8\pm^{0.5}_{0.4}$, to the critical $\Delta>2\sqrt{3}$ stability threshold. More detailed modeling needs to be done to assess the impact of mass loss on orbital stability; nonetheless, the $\Delta>2\sqrt{3}$ stability threshold provides another strong reason to be skeptical of a water-poor super-Earth scenario for Kepler-11b. The major caveat to this stability analysis is that we assume that all of the orbits are stationary even as the planets lose mass. This is motivated by [Adams2011]{}, which showed that in the presence of a modest planetary magnetic field XUV driven mass loss from hot Jupiters tends to come out along the magnetic poles. Assuming that the magnetic field is sufficiently strong, dipolar, and perpendicular to the plane of the orbit, then mass loss won’t have any impact on the orbit. In general however, the directionality of mass loss will be an extremely complicated problem determined by the interaction of the ionized hydrodynamic wind, the planetary magnetic field, and the stellar wind. [Boue2012]{} showed that if the mass loss is directed in the plane of the orbit, then it can have a significant impact on both semi-major axis and eccentricity. Future Work ----------- Further confirmation of the mass loss threshold will depend on getting reliable mass estimates for more LMLD planets. Fortunately, there is a large population of super-Earth sized planets in multi-planet systems found by Kepler [Lissauer2011b]{}. For some of these systems TTV can be used to determine masses [Agol2005]{}. Moreover, this will become possible for more systems as more quarters of data are collected. Likewise, as more quarters of transit data are analyzed, previous mass constraints from TTV will become more precise. For Kepler-11 this will allow tighter constraints on both the current and past compositions. In order to better understand mass loss, there is also a strong need to acquire more XUV observations of young ($\sim10$ Myr) stars. Currently the best estimates of EUV fluxes are for G and K stars older than 100 [Ribas2005,Sanz-Forcada2010]{}. Meanwhile, planet formation ends and mass loss becomes important after a few Myr [Calvet2002, Alexander2006]{}. A large amount of planetary mass will be lost in the first 100 Myr and this will depend strongly on the stellar XUV flux. [Jackson2012]{} recently found that x-ray fluxes saturate for stars younger than 100 Myr; however, this needs to be investigated at other wavelengths. Likewise, more UV observations of transiting exospheres are needed. Currently, we only have observations for a handful of hot Jupiters [Vidal-Madjar2004, Lecavelier2010, Lecavelier2012]{}. More observations are needed, especially for Neptune and super-Earth sized planets. Likewise, it is also important get more observations of XUV fluxes and flares from M dwarfs, which are known to be highly active [Reiners2012]{}. This is particularly important for super-Earths in the habitable zones of late M dwarfs where the stability of habitable atmospheres could depend on XUV driven mass loss. The three planets orbiting M-stars in figure \[improvedfig\] all lie an order of magnitude below the threshold for Sun-like stars. This could be due to mass loss from X-ray flares, but without a larger sample size it is impossible to say. The Recently [France2012]{} obtained the first FUV spectrum for GJ876; however, more data are needed. Likewise, non-equilibrium mass loss models need to be developed to understand the impact of flares. In the area of modeling mass loss, we have utilized a relatively simple model that appears to work well for hot Jupiters, although mass loss rates have only been constrained for HD 209458b [Vidal-Madjar2004]{} and HD 189733b [Lecavelier2010, Lecavelier2012]{}. On a broader scale, there may be evidence for a lost population of hot Jupiter planets at very high levels of XUV irradiation \[e.g.,\]\[\][Davis2009]{}. Since we have suggested that H/He mass loss may be more important for more modestly irradiated LMLD planets than for average hot Jupiters (since the smaller planets have much smaller H/He masses, mass loss for LMLD planets can actually transform the very structure of the planets), we encourage further detailed models to quantitatively access mass loss from these atmospheres. Conclusions =========== In order to better understand the structure, history, and formation of low-mass planets, we constructed coupled thermal evolution and mass loss models of water-poor super-Earths, water-worlds, and water-rich sub-Neptunes. The Kepler-11 system represents a new class of low-mass low-density planets that offers a unique test-bed for such models and gives us powerful insights on planet formation and evolution. Applying this understanding more broadly, we find a relation between a planet’s mass, density, and its incident flux that matches the observed population. Moreover, this threshold can help constrain the properties of hundreds of planets. Our primary conclusions are: - XUV-driven hydrogen mass loss coupled with planetary thermal evolution is a powerful tool in understanding the composition and formation of low-mass low-density planets. - A coupled model is essential for this work, due to the much larger planetary radii in the past, when XUV fluxes were significantly higher. - In situ formation of the Kepler-11 system is disfavored, instead it could be a system of water-rich sub-Neptunes that formed beyond the snow line. - If Kepler-11b is a water-poor super-Earth then it likely formed with $\sim 90\%$ H/He beyond 0.25 AU. We believe this is unlikely and instead show that Kepler-11 b-f all could have originated as water-rich sub-Neptunes with $\sim 20\%$ H/He initially. If this is the case, Kepler-11b could have lost its H/He envelope and become a water-world today for a wide range of initial masses and compositions. - There is a sharp observed threshold in incident flux vs. planet density times mass above which we do not find planets with H/He envelopes. To date, low-density planets have not been found above this threshold. - This mass loss threshold is well reproduced by our coupled thermal evolution and mass loss models. - This threshold can be used to provide limits on planet mass or radius for the large population of low-mass low-density planets without measured densities. - In particular, we have identified promising Kepler targets for RV follow-up. [ccccccccc]{}\[h!\] $70.01$ & $12.50$ & $10.85$ & $80.3$ & $3.09$ & $0.73$ & $3.9$ & $1.21$\ \ $70.02$ & $12.50$ & $3.69$ & $343.5$ & $1.92$ & $3.0$ & $3.9$ & $1.75$\ \ $85.01$ & $11.02$ & $5.85$ & $403.4$ & $2.35$ & $2.4$ & $5.8$ & $1.83$\ \ $94.02$ & $12.21$ & $10.42$ & $209.9$ & $3.43$ & $1.0$ & $7.4$ & $1.92$\ \ $104.01$ & $12.90$ & $2.50$ & $233.4$ & $3.36$ & $1.1$ & $7.5$ & $4.08$\ \ $105.01$ & $12.87$ & $8.98$ & $130.3$ & $3.35$ & $0.82$ & $5.6$ & $1.94$\ \ $107.01$ & $12.70$ & $7.25$ & $301.7$ & $3.09$ & $1.4$ & $7.6$ & $2.29$\ \ $110.01$ & $12.66$ & $9.94$ & $220.5$ & $2.92$ & $1.3$ & $5.9$ & $1.62$\ \ $115.02$ & $12.79$ & $7.12$ & $409.1$ & $1.88$ & $3.4$ & $4.2$ & $1.29$\ \ $117.02$ & $12.49$ & $4.90$ & $436.9$ & $1.70$ & $4.1$ & $3.7$ & $1.25$\ \ $122.01$ & $12.35$ & $11.52$ & $108.8$ & $2.78$ & $1.0$ & $3.9$ & $1.06$\ \ $123.01$ & $12.37$ & $6.48$ & $461.4$ & $2.64$ & $2.2$ & $7.4$ & $2.46$\ \ $124.01$ & $12.94$ & $12.69$ & $227.8$ & $3.00$ & $1.2$ & $6.3$ & $1.65$\ \ $246.01$ & $10.00$ & $5.39$ & $404.8$ & $2.53$ & $2.2$ & $6.5$ & $2.27$\ \ $257.01$ & $10.87$ & $6.88$ & $308.6$ & $2.61$ & $1.8$ & $5.9$ & $1.80$\ \ $262.02$ & $10.42$ & $9.37$ & $491.8$ & $2.79$ & $2.1$ & $8.3$ & $2.20$\ \ $277.01$ & $11.87$ & $16.23$ & $177.4$ & $3.82$ & $0.79$ & $8.0$ & $1.91$\ \ $280.01$ & $11.07$ & $11.87$ & $154.9$ & $2.52$ & $1.3$ & $4.0$ & $1.16$\ \ $281.01$ & $11.95$ & $19.55$ & $192.3$ & $3.46$ & $0.95$ & $7.2$ & $1.99$\ \ $285.01$ & $11.57$ & $13.74$ & $180.5$ & $3.38$ & $0.96$ & $6.7$ & $1.61$\ \ $288.01$ & $11.02$ & $10.27$ & $433.9$ & $3.11$ & $1.6$ & $9.2$ & $2.14$\ \ $291.02$ & $12.85$ & $8.12$ & $247.8$ & $2.14$ & $2.2$ & $3.9$ & $1.25$\ \ $295.01$ & $12.32$ & $5.31$ & $339.7$ & $1.77$ & $3.4$ & $3.5$ & $1.22$\ \ $297.01$ & $12.18$ & $5.65$ & $482.1$ & $1.65$ & $4.6$ & $3.7$ & $1.25$\ \ $301.01$ & $12.73$ & $6.00$ & $399.2$ & $1.75$ & $3.8$ & $3.7$ & $1.17$\ \ $323.01$ & $12.47$ & $5.83$ & $166.2$ & $2.17$ & $1.7$ & $3.3$ & $1.21$\ \ $984.01$ & $11.63$ & $4.28$ & $259.7$ & $3.19$ & $1.2$ & $7.4$ & $2.91$\ \ $987.01$ & $12.55$ & $3.17$ & $404.8$ & $1.28$ & $6.1$ & $2.3$ & $1.03$\ \ $1117.01$ & $12.81$ & $11.08$ & $327.5$ & $2.20$ & $2.4$ & $4.7$ & $1.13$\ \ $1220.01$ & $12.99$ & $6.40$ & $441.4$ & $1.95$ & $3.4$ & $4.6$ & $1.52$\ \ $1241.02$ & $12.44$ & $10.50$ & $485.3$ & $3.84$ & $1.3$ & $13.3$ & $3.17$\ \ $1597.01$ & $12.68$ & $7.79$ & $423.6$ & $2.67$ & $2.0$ & $7.2$ & $1.86$\ \ $1692.01$ & $12.56$ & $5.96$ & $175.9$ & $2.65$ & $1.3$ & $4.6$ & $1.61$\ \ $1781.01$ & $12.23$ & $7.83$ & $61.3$ & $3.29$ & $0.58$ & $3.7$ & $1.38$\ \ $1781.02$ & $12.23$ & $3.00$ & $219.6$ & $1.94$ & $2.4$ & $3.2$ & $1.63$\ \ $1921.01$ & $12.82$ & $16.00$ & $172.9$ & $3.09$ & $1.0$ & $5.7$ & $1.28$\ \ $1929.01$ & $12.73$ & $9.69$ & $251.7$ & $2.00$ & $2.4$ & $3.6$ & $1.11$\ \ $2067.01$ & $12.58$ & $13.24$ & $347.2$ & $2.97$ & $1.6$ & $7.6$ & $1.69$\ \ \[masslimtab\] ![image](kepler11_MvR_9_4_2012.ps){width="6.0in" height="4.3in"} ![image](run_k11b_cooling.ps){width="5.in" height="3.57in"} ![image](run_k11b_runaway_mass.ps){width="5.35in" height="3.57in"} ![image](kepler11_form_final.ps){width="5.in" height="6.43in"} ![image](fluxdensity_kepler11.ps){width="6.in" height="4.3in"} ![image](improved_threshold.ps){width="6.in" height="4.3in"} ![image](predict_threshold.ps){width="6.in" height="3.4in"} ![image](kepler11_stability.ps){width="6.in" height="4.3in"}	ArXiv
--- abstract: 'A low energy effective field theory model for QCD with a scalar color octet field is discussed. The model relates the gluon mass, the constituent quark masses and the quark condensate. The gluon mass comes about $\sqrt{N_c}\, \Lambda_{QCD}$ with the quark condensate being proportional to the gluon mass squared. The model suggests that the restoration of chiral symmetry and the deconfinement transition occur at the same temperature and that, near the transition, the critical exponent for the condensate is twice the gluon mass one. The model also favors the decoupling like solution for the gluon propagator.' address: - 'Departamento de Física, Instituto Tecnológico de Aeronáutica, 12228-900 São José dos Campos, SP, Brazil.' - 'Departamento de Física, Universidade de Coimbra, 3004-516 Coimbra, Portugal' author: - 'O. Oliveira' - 'W. de Paula' - 'T. Frederico' title: Linking Dynamical Gluon Mass to Chiral Symmetry Breaking via a QCD Low Energy Effective Field Theory --- , and QCD, gluon mass, chiral symmetry breaking The use of effective field theories is a common practice in Physics (see e.g. [@EffectiveTheory]). There are many examples in the literature of such type of approach for low energy hadronic physics from chiral perturbation theory (see e.g. [@ChiralPertTheory]), to quark models with contact interactions and quark models including pions and meson fields (see e.g.[@QuarkModels]). In this work, we discuss an effective model, invariant under local $SU(3)$ color, for the non-perturbative regime of QCD which uses as degrees of freedom the gluon, the quark and a color octet scalar field $\phi^a$. The scalar field can be viewed as multi-gluon and multi-quark excitation of the vacuum, i.e. it should be interpreted as a collective excitation of the QCD vacuum. Given that the scalar field is not a color singlet, $\phi^a$ does not contribute directly to the S-matrix. The scalar field resumes the non-perturbative properties of QCD and there is no need to specify in detail its dynamics. The $\phi^a$ contribution to the quark and gluon dynamics comes via the color singlet condensate $\langle \phi^a \phi^a \rangle$ which, according to our estimates, turns out to be of the order of $\Lambda_{QCD}$. The scalar field generate a gluon mass $m_g$, shifts the current quark mass and contributes to the quark condensate $\langle \overline q q\rangle$. It turns out that the shift in the quark mass and the quark condensate are all proportional to the effective gluon mass. In this sense, the model links the presence of an effective gluon mass with chiral symmetry breaking ($\chi S B$). The studies of the phase diagram of QCD [@Costa2009] suggest that the restoration of chiral symmetry and the deconfinement phase transition occur at similar values for the temperature and chemical potential, at least for small enough values of the chemical potential $\mu \lesssim 300$ MeV. In the model under discussion either the gluon gets a dynamically generated mass, quarks are confined and chiral symmetry is broken or the gluon mass vanish, quarks become deconfined and chiral symmetry is restored. The model predicts that the restoration of chiral symmetry and deconfinement transition occur simultaneously at low baryon density, in close agreement with the study of the QCD phase-diagram. Some of the results obtained here can be tested comparing the predictions with non-perturbative solutions of QCD, namely with the results from Schwinger-Dyson equations (SDE) and lattice simulations. For pure gauge, the recent solutions of the SDE can be classified into two classes. The scaling solution [@SDEscaling] which predicts a vanishing gluon propagator at zero momenta and has no connection with a dynamical generated gluon mass. The decoupling solution [@Cornwall1982; @Cornwall2009; @Aguilar2011] which describes the gluon as a massive particle in the low energy regime, with the non-vanishing mass being related with gluon confinement. Other authors have also contributed to the understanding of the solutions of the SDE, see [@Maris2003]. The two solutions agree in the ultraviolet region but differ significantly in the infrared region, with the decoupling solution being closer to the results of lattice QCD simulations. Lattice QCD simulations [@Oliveira2011] also suggest that the gluon behaves as a massive particle in the low energy regime and becomes massless in the high energy regime. A non-vanishing gluon mass is welcome to regularize infrared divergences and to solve unitarity problems. Furthermore, a dynamically generated gluon mass is related with the presence of a $\langle A^2 \rangle$ gluon condensate. A non-vanishing gluon condensate is a clear sign of the non-perturbative sector of QCD - see, for example, [@Dudal; @2004] and references therein. Besides the theoretical support for a gluon mass, diffractive phenomena [@Forshaw1999] and inclusive radiative decays of $J/\psi$ and $\Upsilon$ [@Field2002] also suggest a massive gluon. The precise value for the gluon mass depends on the definition used to compute it. Lattice QCD simulations and Schwinger-Dyson equations suggest an infrared gluon mass of $m_g \sim 600$ MeV [@Cornwall1982; @Oliveira2011; @Oliveira2009]. Phenomenology points towards a gluon mass between $ 0.5$ GeV and $ 1.2$ GeV, depending on how the mass is defined; see table 15 in [@Arriola2004]. The effective model predicts a constituent light quark mass $M \propto m^2_g$. This result can be tested looking at the ratio $M / m^2_g$, as a function of momenta, using the decoupling solution of the SDE. The comparison was performed using the results of [@Aguilar2011], which rely on quenched lattice gluon and ghost propagators to solve the quark gap equation in the chiral limit. The decoupling solution depends on the ansatz used for the quark-gluon vertex. In [@Aguilar2011] the authors explore two different definitions for the quark-gluon vertex. It turns out that, in both cases, the ratio of the constituent quark mass over the effective gluon mass squared is compatible with an essentially constant $M / m^2_g$ in the infrared region. In this sense, the predictions of our model get some support from the non-perturbative solution of the QCD Schwinger-Dyson equations. In the formulation of the effective model we assume flavor independence to reduce the number of independent parameters. Flavor independence of color interaction has phenomenological implications that have been checked experimentally at SLAC [@Abe1998], measuring the strong coupling constant $\alpha_s(M_Z)$ in $Z^0 \rightarrow \overline b \, b (g)$, $Z^0 \rightarrow \overline c \, c (g)$ and $Z^0 \rightarrow \overline q_l \, q_l (g)$, where $l = u, d, s$. It was found that, within errors, $\alpha_s(M_Z)$ is flavor independent. Furthermore, flavor independence of strong interactions was also explored on the theoretical side to understand hadronic phenomena [@Beveren1999]. The effective model adds to the usual QCD Lagrangian new operators which are, from the point of view of the quark operators, either color singlet or octet. There is a unique quark singlet operator, which mimics the so-called $^3P_0$ model [@3P0] used to describe OZI-allowed mesonic strong decays. Recall that the $^3P_0$ model assumes that a pair of quark–antiquark is created out from the vacuum, with vacuum quantum numbers, to create new mesons by the recombination of the quarks in the initial state and the “vacuum quark–antiquark” pair. [Effective Degrees of Freedom.]{} Quantum Chromodynamics can accommodate multiquark and pure gluon states. From the point of view of an effective theory, these multiparticle states can be viewed as composite fields. The success of lattice QCD in the quenched approximation can be seen as an indication that the non-perturbative QCD dynamics is mainly in the gluon sector. At the level of the Lagrangian, it is precisely the gluon sector which contains terms of third and fourth order in the gluon field $A^a_\mu$. We will assume that the non-perturbative physics is resumed in the simplest tensor structures, i.e. scalar or pseudoscalar fields, whose main contribution comes from gluonic operators. For global color transformations $A^a_\mu$ belongs to the octet representation. Given that $8 \otimes 8 = 1 \oplus 8 \oplus 8 \oplus 10 \oplus \overline{10} \oplus 27$, the simplest two gluon color operator is a color singlet. Lattice QCD simulations in the quenched approximation [@Cheng2006] predicts the mass of the low-lying glueballs to be 1710 MeV for quantum numbers $J^{PC} = 0^{++}$, 2560 MeV for $J^{PC} = 0^{-+}$ and 4780 for $J^{PC} = 0^{+-}$. We aim to build an effective theory describing hadronic phenomena below the 1 GeV energy scale. From the values for the glueball masses, we expect that the scalar and pseudoscalar glueballs to play a secondary role and we will not include color singlet scalar fields in the effective theory. From the gluon field itself one cannot build a color octet under local transformations. Instead, one can use the non-abelian Maxwell tensor $F^a_{\mu\nu}$ which, for local color transformations, belongs to the adjoint representation of $SU(3)$. The two possible color octet fields with different Lorentz indices are $$\mathcal{A}_{\mu\nu\eta\zeta} = \frac{1}{\Lambda^3} \, f_{abc} \, F^b_{\mu\nu} F^c_{\eta\zeta} \qquad\mbox{ and }\qquad \mathcal{S}_{\mu\nu\eta\zeta} = \frac{1}{\Lambda^3} \, d_{abc} \, F^b_{\mu\nu} F^c_{\eta\zeta} \, , \label{O_def0}$$ where $f_{abc}$ ($d_{abc}$) is the antisymmetric (symmetry) structure constants of $SU(3)$; see, for example, [@Leader96] for definitions and properties. The color transformation properties are derived from $f_{abc}$ and $d_{abc}$ Jacobi identities. Note that in (\[O\_def0\]) one can replace the Maxwell tensor by its dual without changing its transformations properties. The composite fields $\mathcal{A}$ and $\mathcal{S}$ were defined to have the mass dimensions usually associated with a bosonic field. This requires the introduction of the mass scale $\Lambda$. The composite operators $\mathcal{A}$ and $\mathcal{S}$ generate multi-gluon states from the QCD vacuum. The contraction of Lorentz indices for the product of two Maxwell tensors gives rise either to a scalar field $F^a_{\mu\nu} F^{b \, \mu\nu}$ or a pseudoscalar field $F^a_{\mu\nu} \widetilde{F}^{b \, \mu\nu}$. Of course, besides the scalar and pseudoscalar composite fields, other Lorentz tensors can be built from (\[O\_def0\]). However, we are assuming that the scalars or pseudoscalars capture the essential of the non-perturbative glue physics and more complex configurations represented with higher rank tensor operators will be disregarded. The spin zero composite fields belonging to the color adjoint representation built from equation (\[O\_def0\]) are $$\phi^a = \left\{ \begin{array}{lcl} \frac{1}{\Lambda^3}\, d_{abc} \, F^b_{\mu\nu} F^{c \, \mu\nu} \, , & & \mbox{if } \phi^a \mbox{ is a scalar,} \\ \frac{1}{\Lambda^3}\, d_{abc} \, F^b_{\mu\nu} \widetilde{F}^{c \, \mu\nu} \, ,& & \mbox{if } \phi^a \mbox{ is a pseudoscalar field.} \end{array} \right.$$ An operator with the same quantum numbers can be built using only quark operators. Therefore, one can write a scalar color octet composite field as $$\phi^a = \frac{a}{\Lambda^3} \, d_{abc} \, F^b_{\mu\nu} F^{c \, \mu\nu} ~ + ~ \frac{a^\prime}{\Lambda^2} \big[ \overline q \, t^a q \big] ~ + ~ \cdots \, , \label{general_phi_a}$$ where $a$ and $a^\prime$ are constants. The dots in (\[general\_phi\_a\]) stand for other possible operators including multi-gluon, quarks and ghost fields. If $\phi^a$ is a pseudoscalar field, in the first term of (\[general\_phi\_a\]) one should replace one of the non-abelian Maxwell tensors by its dual and in the fermion a $\gamma_5$ should be included. The definition (\[general\_phi\_a\]) allows a naïve estimation of the relative contribution of gluon and quarks to $\phi^a$ via the ratio $ \mathcal{R} = \langle F^2 \rangle / \Lambda \langle \overline q \, q \rangle$, where $\langle F^2 \rangle$ is the gluon condensate and $\langle \overline q \, q \rangle$ the light quark condensate. Taking the values $\alpha_s \langle F^2 \rangle \simeq 0.04$ GeV$^4$ [@Narisson2010], $\langle \overline q \, q \rangle = ( - 270 \mbox{ MeV} )^3$ [@Jamin2002; @Bordes2010] and $\Lambda \sim 0.3$ GeV its follows that $\mathcal{R} \simeq 6.8$. Therefore, one expects $\phi^a$ to be dominated by gluons, in agreement with our initial assumptions. [The Effective Lagrangian .]{} The effective theory includes the gluon field $A^a_\mu$, the quark fields $q_f$, $f$ is a flavor index, an effective scalar field $\phi^a$ belonging to the adjoint representation of $SU(3)$ color group. The effective Lagrangian reads $$\begin{aligned} \mathcal{L} & = & - \frac{1}{4} F^a_{\mu\nu} F^{a \, \mu\nu} ~ + ~ \sum_f \overline q_f \left\{ i \gamma^\mu D_\mu - m \right\} q_f \nonumber \\ & & + ~ \frac{1}{2} \left( D^\mu \phi^a \right) \left( D_\mu \phi^a \right) - V_{oct}( \phi^a \phi^a ) ~ + ~ ~ \mathcal{L}_{GF} ~ + ~ \mathcal{L}_{gh} ~ + ~ \mathcal{L}_{new} \, \label{new_L}\end{aligned}$$ where $D_\mu = \partial_\mu + i g T^a A^a_\mu$ is the covariant derivative, $T^a$ stands for the generators of $SU(3)$ color group, $m_f$ the current quark mass associated with flavor $f$, $V_{oct}$ is the effective potential associated with $\phi^a$. $\mathcal{L}_{GF}$ is the gauge fixing part of the Lagrangian, $\mathcal{L}_{gh}$ contains the ghost terms and $\mathcal{L}_{new}$ includes the new interactions. All terms in $\mathcal{L}$ are gauge invariant. The exception being $\mathcal{L}_{GF}$. The coupling constant $g$ is not the strong coupling constant. The effective model assumes that the non-perturbative physics is summarized in the color octet scalar field $\phi^a$. Therefore, it is expected that the dominant contribution to any process to be associated with $\phi^a$. The effective gauge coupling constant $g$ parameterizes residual interactions and it should be a small number, allowing for a perturbative expansion in powers of $g$. The operators in $\mathcal{L}_{new}$ can be classified accordingly to its mass dimensions. Given that $\phi^a$ has dimensions of mass, the only gauge invariant dimension four operator that can be in $\mathcal{L}_{new}$ without breaking parity is $$G_4 \sum_f \Big[ \overline q \, t^a \, q \Big] ~ \phi^a \, , \label{O4}$$ for a color octet scalar field $\phi^a$. For a pseudoscalar $\phi^a $, $t^a$ should be replaced by $t^a \gamma_5$. Note that we have considered a single coupling constant $G_4$ to preserve flavor symmetry. The allowed dimension five operators in $\mathcal{L}_{new}$ are $$\begin{aligned} & & G_5 \sum_f \Big[ \overline q \, q \Big] ~ \phi^a \phi^a + F_1 \sum_f \Big[ \overline q \, t^a q \Big] ~ d_{abc} \phi^b \phi^c \nonumber \\ & & + ~ F_2 \sum_f \Big[ \overline q \, t^a \gamma^\mu q \Big] ~ D_\mu \phi^a + F_3 \sum_f \Big[ \overline q \, t^a \gamma^\mu D_\mu q \Big] ~ \phi^a \, , \label{O5}\end{aligned}$$ with $G_5 = g_5 / \Lambda$ and $F_i = f_i / \Lambda$ for $i = 1, 2, 3$ and where $g_5$ and $f_i$ are dimensionless coupling constants. For a pseudoscalar color octet field only the operators associated with coupling constant $F_1$ and $F_2$ should be modified by the introduction of a $\gamma_5$ between the quark fields. The effective field theory is summarized in the Lagrangian (\[new\_L\]), with $\mathcal{L}_{new}$ given by the sum of (\[O4\]) and (\[O5\]). $\mathcal{L}$ includes the QCD Lagrangian and verifies the usual soft-pion theorems of chiral symmetry at low energy. The new interactions in $\mathcal{L}_{new}$ introduce new vertices not present in the original QCD Lagrangian , namely $q \overline q \phi^a$, $q \overline q \phi^a \phi^b$ and $q \overline q \phi^a A^b_\mu$, which contribute to quark processes. Note that the only quark color singlet operator mimics the $^3P_0$ model used to describe OZI-allowed mesonic strong decays. So far we have not yet specified the dynamics of the color octet field, i.e. have not defined its potential energy $V_{oct}$. However, given that $\phi^a$ cannot give rise to asymptotic $S$-matrix states, for physical processes one has to integrate over $\phi^a$ and it is unnecessary to detailed its dynamics. As discussed bellow, it will be enough to specify its vacuum expectation values. [Dynamical Gluon and Quark masses.]{} The kinetic term associated with $\phi^a$ gives rise to a gluon mass term through the operator $$\frac{1}{2}�\, g^2 \, \phi^c (T^a T^b)_{cd} \phi^d A^a_\mu A^{b \, \mu} \, .$$ If $\phi^a$ acquire a vacuum expectation value without breaking color symmetry, i.e. $$\langle \phi ^a \rangle = 0 \qquad \mbox{ and } \qquad \langle \phi^a \phi^b \rangle = v^2 \delta^{ab} \, ,$$ and given that for the adjoint representation $\mbox{tr} \left( T^a T^b \right)= N_c \, \delta^{ab}$, then one can write the square of the gluon mass as $$m^2_g = N_c \, g^2 v^2 \, , \label{massa_gluao}$$ where $N_c = 3$. Note that the condensate $\langle \phi^a \phi^b \rangle$, i.e. $v^2$, and therefore the gluon mass is gauge invariant. The proof of gauge invariance follows directly from the transformations properties of $\phi^a$. The gluon mass (\[massa\_gluao\]) is proportional to the effective gauge coupling $g$. As discussed previously, $g$ should be a small number. However, this does not implies necessarily that $m_g$ is also small. The precise value of $m_g$ depends on the relative values of $v$ and $g$. The same mechanism shifts the quark masses due to the operator $$G_5 \, \Big[ \overline q \, q \Big] ~ \phi^a \phi^a \label{anomalous_mass_L}$$ giving rise to a constituent quark mass $$M_f ~ = ~ m_f ~ - ~ \left( N^2_c - 1 \right) \, G_5 \, v^2 ~ = ~ m_f ~ - ~ \, \frac{ N^2_c - 1}{N_c} \, \frac{G_5}{g^2} \, m^2_g \, . \label{M_f}$$ For light quarks, the constituent mass is given by the quark self energy which, in the model, is linked with the gluon mass, the second term in equation (\[M\_f\]); note, for our definitions, $G_5$ is a negative number. If the constituent mass for the light quarks vanishes, chiral symmetry is recovered, and the model predicts that the gluon mass should also vanish. In this way, the model links chiral symmetry breaking with an effective gluon mass. ![Dressed quark propagator. The dashed lines are associated with the color octet scalar field $\phi^a$.[]{data-label="fig:quark_prop"}](fig1.eps) [Light Quark Condensate.]{} The ligth quark condensate $\langle \overline q \, q \rangle$ is an order parameter for chiral symmetry breaking, whose computation requires the knowledge of the quark propagator. In lowest order, the quark propagator is given by the Feynman graph of figure \[fig:quark\_prop\], which includes the contribution from the condensate $\langle \phi^a \phi^b \rangle$. The dressed quark propagator reads $$S_0(k) = i \, \frac{\gamma^\mu k_\mu + M}{ ~ k^2 - M^2 ~} \, ,$$ where $M$ is the constituent quark mass, and the quark condensate is given by $$\begin{aligned} \langle \overline q \, q \rangle = - \, \frac{4 \, N_c \, N_f}{16 \pi^2} \, M \, m^2_g ~ \mathcal{F}\left( \frac{M}{m_g}, \frac{\overline\omega}{m_g} \right), \label{light_condensate}\end{aligned}$$ where $N_f$ is the number of degenerate light flavors, $\overline\omega$ is an energy cut-off and $$\mathcal{F}( x, y) = x^2 \ln \frac{x^2}{x^2 + y^2} \, + \, y^2 \, .$$ For small $M$ or for $M \ll \overline\omega$, equation (\[light\_condensate\]) becomes $$\langle \overline q \, q \rangle ~ = ~ - \, \frac{4 \, N_c \, N_f}{16 \pi^2} \, M \, \overline\omega^2 \, . \label{light_condensate_M0}$$ The quark condensate is proportional to the constituent quark mass, i.e. to the gluon mass squared, and to the square of the energy cut-off. A non-vanishing gluon mass means also a non-vanishing quark condensate and, therefore, chiral symmetry is broken. Equations (\[light\_condensate\_M0\]) and $M$ when combined with the results of lattice QCD simulations [@Oliveira2011] allow us to estimate of the various parameters of the theory. Indeed, taking $\langle \overline q \, q \rangle = \left( - 270 \mbox{ MeV} \right)^3$, $ M = 330$ MeV it follows from (\[light\_condensate\_M0\]) that $\overline\omega = 626$ MeV for $N_f = 2$. For example, this is a typical value used for the cut-off when working with Nambu-Jona-Lasinio type of models. Taking the value for the gluon mass from Landau gauge lattice simulations $m_g = 648$ MeV, then $G_5/g^2 = -0.295$ GeV$^{-1}$ and equation (\[massa\_gluao\]) gives $g \, v = 374$ MeV. Note that $g \, v$ is, apart the color factor $\sqrt{N_c}$, the effective gluon mass and is of order $\Lambda_{QCD}$. [Testing the Gluon and Quark Mass relation.]{} The effective model relates the constituent quark mass $M$ and the gluon mass $m_g$ through equation (\[M\_f\]). For a vanishing current mass, equation (\[M\_f\]) predicts a constant value for the ratio $M / m^2_g$. This prediction can be tested looking at the solutions of the Schwinger-Dyson equations. In the following we will use the results published in [@Aguilar2011]. For the gluon and ghost propagators, the authors use the results of lattice QCD simulations and solved the gap equation for a massless fermion. The calculation does not take into account fermion loops and can be viewed as a quenched approximation. For the gluon propagator, the inverse lattice propagator was fitted to $$\Delta^{-1} (q^2) = m^2(q^2) + q^2 \Big[ 1 + \frac{13 \, C_A \, g^2_1}{96 \, \pi^2} \ln \left( \frac{q^2 + \rho_1 m^2(q^2)}{\mu^2} \right) \Big] \, ,$$ where $$m^2 (q^2) = \frac{m^4}{q^2 + \rho_2 m^2}$$ and $m = 520$ MeV, $g^2_1 = 5.68$, $\rho_1 = 8.55$ and $\rho_2 = 1.91$ are the fitted parameters, $\mu = 4.3$ GeV is the renormalization point and $C_A = 3$ is a $SU(3)$ Casimir invariant. The gluon mass is given by the pole in the propagator and we will use the following definition $$m^2_g (q^2) = \frac{ m^2(q^2) } { 1 + \frac{13 \, C_A \, g^2_1}{96 \, \pi^2} \ln \left( \frac{q^2 + \rho_1 m^2(q^2)}{\mu^2} \right) } \, .$$ The fermionic gap equation was solved for two different ansatz for the quark-gluon vertex, namely a non-Abelian improved version of the Ball-Chiu vertex and an improved version of the Curtis-Pennigton vertex. The choice of vertex leads to slightly different quark mass. In order to distinguish, the results of the Ball-Chiu vertex will be referred as “BC”, while the results from using the Curtis-Pennington vertex will be referred as “CP”. ![On the left hand side, the plot shows the quark masses from solving the fermionic SDE gap equation, using different ansatz for the quark-gluon vertex, and the squared gluon mass computed from quenched lattice simulations. Note that $M(q^2)$ depends slightly on the definition of the quark-gluon vertex. On the right hand side, the plot shows the ratio $M/m^2_g$.[]{data-label="fig:all_masses"}](masses_q_g.eps "fig:") ![On the left hand side, the plot shows the quark masses from solving the fermionic SDE gap equation, using different ansatz for the quark-gluon vertex, and the squared gluon mass computed from quenched lattice simulations. Note that $M(q^2)$ depends slightly on the definition of the quark-gluon vertex. On the right hand side, the plot shows the ratio $M/m^2_g$.[]{data-label="fig:all_masses"}](mq_over_m2glue_mod.eps "fig:") The quark masses computed using the different vertex ansatz and the squared gluon mass are reported in figure \[fig:all\_masses\]. For a cut-off of $\overline\omega = 0.626$ GeV, i.e. up to momenta $q^2 = 0.39$ GeV$^2$, the $M/m^2_g$ ratio increases slightly. Indeed, $M/m^2_g$ changes by less than 15%, relative to its zero momentum value, for the BC vertex, and just below 19% for the CP vertex. We call the reader attention that the gluon and ghost propagators are quenched propagators and the quark mass was computed solving the fermionic gap equation using the quenched propagators. A full dynamical calculation can change the above picture. Lattice QCD simulations [@Bowman2005; @Silva2010] show a decreasing constituent quark mass $M(q^2)$ with the current mass in the infrared region. On the other hand, for dynamical simulations, the gluon propagator is also suppressed relative to the quenched propagator [@Silva2010; @Kamleh2007]. This suggests that $m_g$ is enhanced when we approach the chiral limit. However, the dynamical ghost propagator is enhanced relative to the quenched propagator [@Silva2010] in the infrared region. An enhancement in the ghost propagator implies an enhancement of the quark-gluon vertex [@Aguilar2011] whose effects remain to be investigated. Despite the uncertainties related with the simulation of full QCD, hopefully, the scenario observed in figure \[fig:all\_masses\], where the mass ratio changes by less than 19%, will remain essentially unchanged. Therefore, one can claim that the results summarized in figure \[fig:all\_masses\] suggests that $M/m^2_g$ is essentially constant in the non-perturbative regime. Certainly, given the uncertainties on the calculation of $M$ and $m_g$, figure \[fig:all\_masses\] will not exclude such a possibility. [Summary.]{} We have explored the construction of an effective field theory describing the low energy properties of QCD using scalar like fields as composite gluon operators. We found that color singlet scalar-like fields are unlikely to play a dominant role for the infrared properties. On the other hand, color octet scalar fields allow the building of $SU(3)$ color gauge invariant effective interaction with interesting properties. The effective model is gauge invariant, flavor symmetric and satisfies the usual current algebra relations given that it includes the QCD Lagrangian . It assumes that the non-perturbative physics is resumed in the new condensate $\langle \phi^a \phi^b \rangle = v^2 \delta^{ab}$, with $v \ne 0$ in the confined phase and $v = 0$ in the deconfined phase. Gauge invariance and the scalar condensate are enough to generate a gauge invariant gluon mass, which we associated with the transverse mass measured in Landau gauge Lattice QCD simulations. $\langle \phi^a \phi^b \rangle$ shifts the quark masses, justifying the difference between current and constituent masses, and generates a light quark condensate proportional to the constituent quark mass. Further, the model predicts a constant $M/m^2_g$ ratio. A massive infrared gluon propagator implies a non-vanishing constituent quark mass and a non-vanishing light quark condensate, connecting $m_g$ with the breaking of chiral symmetry. Further, given the relation between $m_g$ and chiral symmetry, the model predicts that the deconfinement transition and the restoration of chiral symmetry occur simultaneously. This is in good agreement with the predictions from the studies of the QCD phase-diagram. If the mass relations are unchanged by temperature and baryonic matter, the model provides a relation between the critical exponents for the gluon mass $\eta_g$ and the light quark condensate $\eta_q$ near the chiral phase transition, namely $\eta_q = 2 \, \eta_g$, that can be tested, for example, with lattice QCD simulations. The mass relation between $M$ and $m_g$ was tested against the decoupling solution of the SDE. The model includes a number of new interactions. The relations discussed above are the result of exploring two operators, the kinetic term for the scalar field and the only new color singlet operator at the quark level $\overline q q \, \phi^a\phi^a$. This last operator mimics the $^3$P$_0$ model used to describe OZI-allowed mesonic strong decays. All the other operators seem to give marginal contributions to the strong decays because either $\langle \phi^a \rangle = 0$, for the operators multiplying $G_4$, $F_2$ and $F_3$, or $d_{abb} = 0$, for the operator multiplying $F_1$. The new vertices can contribute the $S$-matrix at the quark level and can give rise to new quark interactions. In QCD, the quark-gluon vertex is orthogonal to the gluon momenta. The vertices associated with the new interactions do not verify this property and can be viewed as mimicking the longitudinal part of the quark-gluon vertex, which is not constraint by Slavnov-Taylor identities. The effective model has phenomenological implications which have not been explored here and will be the subject of further publications. For example, the dressing of the quark-photon vertex due to $\phi^a$ can give rise to quark anomalous magnetic moments of the type discussed in [@Chang2011]. The model building blocks are the color octet field $\phi^a$, the quark and the gluon fields. The hadronic matter is essentially quark matter. The experimental situation for pure glue states is not so clear. One has to understand why there are no physical states than can be associated with $\phi^a$. An explanation could be its low mass and the weak couplings to the quarks. Looking at the Lagrangian, one can estimate the contribution of the new terms to the mass of $\phi^a$ as $\delta m^2_\phi \approx \| G_5 \, \langle \overline q \, q \rangle \|$. For $G_5 = - 0.295 \, g^2$ GeV$^{-1}$ and $\langle \overline q \, q \rangle = ( -270 \mbox{ MeV})^3$, it follows that $\delta m_\phi \lesssim 76$ MeV. Curiously, recently the authors of [@Beveren2011] have analyzed the BABAR bottomonium decay data and found an excess of signal which could be explained by a light scalar particle with a mass of 38 MeV. 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--- abstract: 'In this article, I present the questions that I seek to answer in my PhD research. I posit to analyze natural language text with the help of semantic annotations and mine important events for navigating large text corpora. Semantic annotations such as named entities, geographic locations, and temporal expressions can help us mine events from the given corpora. These events thus provide us with useful means to discover the locked knowledge in them. I pose three problems that can help unlock this knowledge vault in semantically annotated text corpora: i. identifying important events; ii. semantic search; and iii. event analytics.' author: - \| Dhruv Gupta\ \ \ subtitle: 'Detecting Events in Semantically Annotated Corpora for Search & Analytics' title: Event Search and Analytics --- Introduction ============ Information retrieval systems have largely relied on word statistics in text corpora to satisfy information needs of users by retrieving documents with high relevance for a given keyword query. In my PhD research I hypothesize that information needs of users can be satisfied to a greater extent by using events as a means of navigating text corpora. Events in our context would be an act performed by certain actor(s) at a specific location during a specific time interval. An example would be : Usain Bolt won the gold medal at the 2008 summer Olympics in Bejing. With the availability of annotators that can provide us with accurate semantic annotations in form of named entities, geographic locations, and temporal expressions; we can leverage the growing number of knowledge resources such as Wikipedia [@wiki] and ontologies such as Freebase [@freebase] to understand natural language text and mine important events. Formally the central hypothesis can be stated as follows: As a toy example consider the following text snippet [^1] with demonstrative semantic annotations in Figure \[fig:text\] : In the text snippet (Figure \[fig:text\]), we obtain the named entity whose mention has been identified and disambiguated to point to an external knowledge source. Also identified is a geographical location - , which is disambiguated and resolved to its geographical coordinates. Likewise the temporal expression has also been resolved to time range. Having these semantic annotations we can now devise algorithms that can deduce that the event is that of Usain Bolt winning Olympic competition in Beijing, China. The goal of the proposed research is to leverage the semantic annotations for mining important events and use them to navigate text corpora. The research will find application in many domains of research such as digital humanities, in which social scientists are interested in computational history in large digital-born text collections. Anthropologists are interested in cultural and linguistic shifts that occur in such collections. Collectively we can allow computational culturomics [@culturomics] on corpora to study cultural trends. Events can also be used to link information in multiple and diverse text collections. In short, important events provide a way to create a gist from semantically annotated corpora, which otherwise is not possible through manual human effort. Outline. The article consists of: - a literature survey (Section \[sec:background\]); - an overview of the research problems (Section \[sec:problem\]); - available corpora, test sources and evaluation measures for research (Section \[sec:evaluation\]); - discussion of few open technical problems (Section \[sec:discussion\]). Related Work ============ \[sec:background\] In this section I discuss the progress already made in the area of analyzing different semantic annotations in isolation as well as in conjunction for some of the problems proposed. Temporal Information Retrieval and Extraction. Researchers have considered only temporal annotations in text corpora to improve retrieval effectiveness by analyzing the time sensitivity of keyword queries and incorporating the time dimension in retrieval models. Some methods of analysis of time-sensitive queries rely on publication dates of documents [@diaz_profile; @nattiya_2010], while others also look at the temporal expressions in document contents [@dhruv_2014]. Several works also take into account the time dimension for re-ranking documents [@klaus_2010] and diversifying them along time [@klaus_2013; @nattiya_2014]. One of the seminal works in extracting temporal events was by Ling and Weld [@ling_tie]. They outline a probabilistic model to solve the problem of extracting relations from text with temporal constraints. Important Events in Annotated Corpora. One of the most important seminal works in identifying existing and emerging events were the various tasks in Topic Detection and Tracking (TDT) [@tdt_book]. The TDT program aimed to “search, organize and structure” broadcast news media from multiple sources. The five tasks laid within the ambit of TDT where topic tracking, link detection, topic detection, first story detection, and story segmentation. The task of topic tracking required to build a system to detect on topic stories from an evaluation corpus after being trained on a set on topic stories. The link detection task involved answering a boolean query to whether two given stories are related by a common topic. The topic detection task comprised of declaring new topics from incoming stories which had not been presented to the system. First story detection was another boolean decision task of determining whether the given story is a seed story (first-story) to create a new topic cluster. Story segmentation task required segmentation of an incoming stream of text into stories. Focusing specifically on extracting and summarizing events in the future, Jatowt and Yeung [@adam_cikm11] present a model-based clustering algorithm. The clustering considers both textual and temporal similarities. For computing temporal similarity, the authors model time as a probability distribution by utilizing different family of distributions based on whether its is singular time point, starting date or and ending date. The similarity is then computed using KL-divergence. Radinsky et al. [@kira_jair] present an algorithm Pundit, which based on the past events in text is able to predict a future event given a query to the system. The events are represented as multidimensional attributes such as time, geographic location and participating entities. The algorithm derives these events from external text collection and builds an abstraction tree, which is the result from hierarchical agglomerative clustering. In order to predict the future Pundit is trained to select the most similar cluster from the abstraction tree and produce an event representation. The work by Yeung and Jatowt [@yeung_cikm11] tackles the problem of analysis of historical events in multiple large document collections. They utilize latent Dirichlet allocation to identify topic distributions along time. Thereafter they perform analytics to answer questions such as i. significant years and topics, ii. triggers that caused remembrance of the past and iii. historical similarity of countries. Most recently, Abujabal and Berberich [@mppf] present a system which identifies important events in text collections by counting frequent itemsets of sentences containing named entities and temporal expressions. For evaluation they resort to Wikipedia’s event directory as a ground truth. Semantic Search. Summarizing text collections in a timeline visualization is a natural choice. Swan and Allan [@swan_timeline] present an approach for producing a timeline that depicts most important topics and events closely modeled on the Topic Detection and Tracking task. The algorithms analyzes features based on named entities and noun phrases. The analysis involves construction of $2 \times 2$ contingency table on presence or absence of features, and subsequent measurement of $\chi^2$ statistic for measuring significance of co-occurrence of a pair of features. The seminal work by Baeza-Yates [@yates_future] proposed a future retrieval (FR) system. The FR system considers both text and temporal expressions to identify future events that might be relevant to an input query. Baeza-Yates outlined the components of a FR system to be composed of an information extraction (IE) module, information retrieval (IR) module, and a text mining (TM) module. The IE module would act as a temporal annotator; identify temporal expressions and normalize them. The IR system is designed to incorporate the time dimension in an index; thus retrieving documents with text and time similarity. The TM module would identify the most relevant topics given a time period. He presented a retrieval model, in which each document consists of a multiple temporal events. A temporal event consists of a time segment and its associated likelihood of occurring. The score of the document is thus obtained by its textual similarity and the maximum likelihood of all the temporal events in that document. Bast and Buchhold [@bast_index] outline a joint index structure over ontologies and text. Which allows for fast semantic search and provide context sensitive auto-complete suggestions. Events as a means of search document collections has also been explored by Strötgen and Gertz [@jannik_event]. Events were modeled by the geographic location and time of their occurrence. For temporal queries expressed in simple natural language they outline an extended Backus-Naur form (EBNF) language that incorporates time intervals with standard boolean operations. Geographical queries are also modeled as EBNF language, however the input for them is a minimum bounding rectangle (MBR). Using this multidimensional querying model the user is able to visualize search results in form of events; which are additionally represented on a map. Giving special attention to geographical information retrieval, Samet et al. [@samet_news] present a system NewsStand, that is able to resolve and pinpoint a news article based on the geographic information present in its content. They discuss various methods for toponymn resolution, which is in essence disambiguating the geographic location based on its surface form in the news content. The system involves a streaming clustering algorithm that can keep track of emerging news in new locations and present them in a map-based interface. [l\|p[4cm]{}p[4cm]{}p[4cm]{}]{} \ Event & $c_1$ & $c_2$ & $c_3$\ Words & micheal, phelps, bejing, china, tibet & london, usain, bolt, england, badminton & rio, brazil,copacabana, deodoro, maracanã\ Time &$[08-08-2008, 24-08-2008]$ & $[27-07-2012, 12-08-2012]$ & $[05-08-2016, 21-08-2016]$\ Location & $\langle Beijing, China \rangle$ & $\langle London, England \rangle$ & $\langle Rio de Janeiro, Brazil \rangle$\ Entities & $\langle China \rangle$, $\langle Micheal\_Phelps \rangle$ & $\langle England \rangle$, $\langle Badminton \rangle$ & $\langle Brazil \rangle$, $\langle Copacabana \rangle$\ Event Analytics. By disambiguating and linking named entities to ontologies, Hoffart et al. [@aesthetics; @stics] provide a framework for semantic search and performing analytics on them. They provide features for giving auto-complete suggestions in the form of similar entities for the input named-entity. In [@aesthetics] they provide analytics that leverage accurate entity counts and provide entity co-occurrence statistics which is helpful in analyzing semantically similar named-entities. Research Objectives =================== \[sec:problem\] Given the text corpora with semantic annotations, I describe three important research problems in this section: i. identifying important events; ii. using identified events for improving retrieval effectiveness; and iii. using identified events for analytics. Notation -------- Let us consider multiple corpora for the purpose of analysis. This allows us to capture frequently occurring events as well as link similar events across corpus. Given corpora $$D = \bigcup_{k=1}^{N} D_k,$$ where each document $d \in D$ consists of word sequence $x$ at appropriate granularity (e.g. paragraph or sentence): $$d = \bigcup_{i=1}^{n} x_i.$$ Further each $x \in d$ contains semantic annotations in form of i. named entities ($e$), ii. geographical location ($g$), and iii. temporal expressions ($t$). Additionally $x$ also consists of the a bag of words $\mathcal{W}$ drawn from a vocabulary $\mathcal{V}$. Formally represented as: $$x = \langle \mathcal{E}, g, t, \mathcal{W} \rangle$$ Problem Definition ------------------ The objective is to design a family of algorithms: $$\textsc{Event}(X,Q,\alpha)$$ where $X = \bigcup x$, $Q$ represents an input query and $\alpha \in \mathds{R}^m$, where $\alpha$ is set of parameters. The input query $Q$ can be a combination of following input components: i.* keyword query $q$, ii. time $q_{time}$, iii. geographical location $q_{geo}$, and iv. named entity $q_{entity}$. Given the input, we need to design the algorithms $\textsc{Event}$ according to the different problems. We discuss the design objectives for the three different purposes in this section. Identifying Important Events. Events* are the proposed building blocks for further text analysis. An event in our context is defined to be an activity or an act involving named entities that happens in a specific geographical location anchored to a specific time interval. Mathematically, given a multidimensional query : $$Q = \langle q, q_{time}, q_{geo}, g_{entity} \rangle,$$ and a subset of highly relevant documents $R \subseteq D$, the algorithm for this purpose $\textsc{EventDetect}$ should produce a set of ordered events : $$\mathcal{C} = \{ c_1, c_2, \ldots, c_k \},$$ where, $c = \langle \mathcal{E}, g, t, \mathcal{W} \rangle$. The event $c$ is hence described by the participating named entities $\mathcal{E}$, its location $g$, its time of occurrence $t$, and frequently occurring contextual terms around these semantic annotations $\mathcal{W}$. This requires proposing a probability mass function, $P(\mathcal{C}, R)$, using which we can impose a total order on $\mathcal{C}$. As an example consider the keyword-only query `summer olympics` to the processed corpora of news articles. The designed algorithm shall then identify the important events as in Figure \[fig:event\]. Diversifying and Summarizing Search Results are retrieval tasks that try to address the information need underlying an ambiguous query at different levels of textual granularity. Each task tries to maximize the coverage of different information needs underlying the given ambiguous query. As information intents, we propose to use the mined set of events. Accomplishing these tasks would allow for automatic creation of event timelines or entity biographies. We briefly discuss an intuition of achieving the same. When diversifying search results we would like to present users with documents such that the user finds at least one document that satisfies her information intent. For this we need to devise an algorithm $\textsc{EventDiverse}$ which considers as an input $Q$ and $R \subseteq D$. As an output it returns a set of documents $S \subset R$ which cover all events in $\mathcal{C}$. Summarizing search results would require us to construct an algorithm $\textsc{EventSummary}$ to piece together, sentences $\hat{S} = \bigcup x$, such that the text summary covers all events in $\mathcal{C}$. Semantic Search and Analytics. The mined set of events can further be utilized for search and analytics. For this purpose we can utilize inherent hierarchy in the semantic annotations. For example a given year can be broken down to different months and subsequently days in those months. Similarly, we can utilize the type hierarchies in named entities. Such as and are subtypes of . This can jointly be modeled by using the concept of a data cube [@han_dm] as shown in Figure \[fig:data\_cube\]. Formally, given a query $Q$, the objective would to first model the mined set of events as a data cube and subsequently provide data cube operations [@han_dm]: - roll ($\bigcirc$), - slice ($\ominus$), - dice ($\oplus$), - drill up ($\bigtriangleup$), - drill down ($\bigtriangledown$). ![Example data cube based on set of events $\mathcal{C}$[]{data-label="fig:data_cube"}](cube.pdf) ![Example data cube operations for the query `all races won during 2008 by usain bolt in china` []{data-label="fig:cube_opr"}](cubeopr.pdf) As a concrete example consider the query `all races won during 2008 by usain bolt in china`. To produce an appropriate result the sequence of operations would be: first a slice on the entity ; second dice on ; and finally drill up to year (see Figure \[fig:cube\_opr\]). Data ==== Corpora. There are several readily available massive data sets. They are available from news corporation such as the New York Times [@nyt], English Gigaword [@gigaword]. These corpora have the benefit of being available with reliable publication dates and grammatically well-formed text. On larger scale are Web collections such as ClueWeb’09 [@clueweb09]/’12 [@clueweb12], which are not always accompanied by reliable creation dates and many are ill-formed documents. Semantic Annotations. The text corpora next need to be annotated for text mining. I explain how to obtain the different semantic annotations in the following paragraphs. Named Entities. For disambiguating and linking named entities in text to an external knowledge source such as Wikipedia [@wiki] or an ontology such as YAGO [@yago] or Freebase [@freebase]; I use the AIDA system [@aida]. The AIDA system does named entity disambiguation and linking by leveraging contexts extracted from ontologies such as YAGO. For Web collections such ClueWeb’09/’12 the entity disambiguation and linking has been released as facc1 : Freebase annotation of ClueWeb Corpora [@facc]. Geographical Locations can be obtained by utilizing geographic named entities such as those known to be cities, countries, or continents. Geographical relations stored in an ontology can be used to resolve these locations to its geographical coordinates. Having obtained a set of coordinates, we can subsequently construct a geographical representation such as a minimum bounding rectangle over the coordinates. Temporal Expressions, both implicit and explicit, can be extracted and normalized from text by using temporal taggers such as HeidelTime [@heideltime] or SUTime [@sutime]. Evaluation ========== \[sec:evaluation\] To test our approach we need to construct query sets that contain an event description associated with the query; along with participating named entities, geographical locations where the event took place and relevant time interval associated with it. I describe a tentative approach to achieve this here. Test Data. To evaluate the correctness of the various algorithms, I plan to use reliable encyclopedic resources on the Web such as Wikipedia [@wiki] or other curated knowledge sources. For an objective evaluation, I propose the following different sources depending on the algorithm under evaluation. - Identify important events - Events in a particular year/decade etc. pages available on Wikipedia [@wiki_year]. - Testing of past events can be done by extracting important topics from Category pages on various historical topics on Wikipedia [@wiki_past]. - Events in the future can be evaluated by using important infrastructure projects, engineering projects etc. These can be extracted from Wikipedia and other sources on the Internet. - Current events extracted from Wikipedia [@wiki_current]. - Alternatively, we can manually construct a list of prominent events and extract relevant information such as named entities, geographical location, and time from ontologies such as: YAGO [@yago], Freebase [@freebase], etc. - Diversifying and summarizing search events - Biographies of eminent personalities, for example United States presidents [@wiki_potus]. - Historical timelines of various countries, for example for India [@wiki_timeline]. Structure. Each event in our test bed is then composed of a fact with an accompanying query. Formally, a fact in our testbed is a 4-tuple extracted from one of the aforementioned sources: $$\langle q,\mathcal{E}, g, t, \mathcal{W} \rangle$$ where $q$ consists of keyword query describing the event, $\mathcal{E}$ is a bag of participating entities, $g$ is the geographic location, $t$ is the time of its occurrence, and $\mathcal{W}$ are important terms describing the event. Metrics. Based on the structure of the testbed of events, metrics such as precision, recall and F$_1$ can be utilized to measure the effectiveness of the algorithms for detecting important events in semantically annotated corpora. How effectively the algorithm diversifies documents along multiple dimensions can be evaluated by metrics such as $\alpha$-nDCG [@diverse]. Quality of summaries can be measured by an automatic evaluation metric called Rouge [@rouge]. Discussion ========== \[sec:discussion\] I briefly present some open technical challenges that I will address along with the research objectives in my PhD dissertation. Mathematical Models. One key aspect that occurs in the design of the algorithms is that of computational models for named entities, geographical locations and temporal expressions. What would be the most descriptive mathematical models for each of these semantic annotations? Similarity Functions. Given a pair of named entities, geographical locations or temporal expressions; how can we efficiently compute the similarity between the same type of annotations? Efficiency & Scalability. Identifying data structures for indexing corpora along with their semantic annotations, such that their asymptotic run times scale linearly with the size of the corpora. Evaluation. Since evaluation of the solutions outlined are very subjective in nature; what are other reliable sources of objective ground truth ? What other metrics can be employed to test the effectiveness of our methods ? Conclusion ========== \[sec:conclusion\] In this article I laid out an outline of the research work that I envisage to carry out for my PhD dissertation. The research would in its culmination provide us methods to computationally extract world history as sequence of temporally ordered events and portray future events to take place from semantically annotated corpora. The research would also provide ways to perform semantic search and large scale event analytics on these annotated corpora. I further described already available resources that can be utilized for carrying out the research; test cases that can be built from encyclopedic resources on the Internet; and the metrics that can be utilized for evaluation. [10]{} Clueweb’09. <http://www.lemurproject.org/clueweb09.php/>. Clueweb’12. <http://www.lemurproject.org/clueweb12.php/>. English gigaword. <https://catalog.ldc.upenn.edu/LDC2003T05>. New york times anotated corpus. <https://catalog.ldc.upenn.edu/LDC2008T19>. Abujabal A. and Berberich K. Important events in the past, present, and future. WWW’15 Companion Volume. Agrawal R. et al. Diversifying search results. WSDM’09. Allan J., editor. . Kluwer Academic Publishers, Norwell, MA, USA, 2002. Baeza-Yates R. Searching the future. SIGIR’05 Workshop MF/IR. Bast H. and Buchhold B. An index for efficient semantic full-text search. CIKM’13. Berberich K. et al. A language modeling approach for temporal information needs. ECIR’10. Berberich K. and Bedathur S. Temporal diversification of search results. TAIA’13. Bollacker K. et al. Freebase: A collaboratively created graph database for structuring human knowledge. SIGMOD’08. Chang A. X. and Manning C. D. : A library for recognizing and normalizing time expressions. LREC’12. Ringgaard M. et al. Facc1: Freebase annotation of clueweb corpora, version 1 (release date 2013-06-26, format version 1, correction level 0), June 2013. <http://lemurproject.org/clueweb12/>. Gupta D. and Berberich K. Temporal query classification at different granularities. SPIRE’15. Han J. et al. . Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 3rd edition, 2011. Hoffart J. et al. analytics with strings, things, and cats. CIKM’14. Hoffart J. et al. searching with strings, things, and cats. SIGIR’14. Hoffart J. et al. Robust disambiguation of named entities in text. EMNLP’11. Jatowt A. and Yeung C. A. Extracting collective expectations about the future from large text collections. CIKM’11. Jones R. and Diaz F. Temporal profiles of queries. , 25(3), July 2007. Kanhabua N. and N[ø]{}rv[å]{}g K. Determining time of queries for re-ranking search results. ECDL’10. Ling X. and Weld D. S. Temporal information extraction. AAAI’10. Michel J. B. et al. Quantitative analysis of culture using millions of digitized books. Science’10. Nguyen T. N. and Kanhabua N. Leveraging dynamic query subtopics for time-aware search result diversification. ECIR’14. Radinsky K. et al. Learning to predict from textual data. , 45:641–684, 2012. Samet H. et al. Reading news with maps by exploiting spatial synonyms. , 57(10):64–77, 2014. Str[ö]{}tgen J. and Gertz M. Event-centric search and exploration in document collections. JCDL’12. Strötgen J. and Gertz M. Multilingual and cross-domain temporal tagging. , 47(2):269–298, 2013. Suchanek F. M. et al. Yago: A large ontology from wikipedia and wordnet. , 6(3):203–217, September 2008. Swan R. C. and Allan J. Automatic generation of overview timelines. SIGIR’00. Wikipedia. List of presidents of the united states — [W]{}ikipedia[,]{} the free encyclopedia, 2015. <https://en.wikipedia.org/wiki/List_of_Presidents_of_the_United_States>. Wikipedia. List of years — [W]{}ikipedia[,]{} the free encyclopedia, 2015. <https://en.wikipedia.org/wiki/List_of_years>. Wikipedia. List of years in india — [W]{}ikipedia[,]{} the free encyclopedia, 2015. <https://en.wikipedia.org/wiki/List_of_years_in_India>. Wikipedia. Portal:contents/history and events — [W]{}ikipedia[,]{} the free encyclopedia, 2015. <https://en.wikipedia.org/wiki/Portal:Contents/History_and_events>. Wikipedia. Portal:current\_events — [W]{}ikipedia[,]{} the free encyclopedia, 2015. <https://en.wikipedia.org/wiki/Portal:Contents/History_and_events>. Wikipedia. ikipedia[,]{} the free encyclopedia, 2015. <http://en.wikipedia.org/>. Yeung C. A. and Jatowt A. Studying how the past is remembered: towards computational history through large scale text mining. CIKM’11. Lin C. Y. Rouge: a package for automatic evaluation of summaries. ACL-04 workshop. [^1]: <http://www.bbc.com/sport/0/athletics/34032366>	ArXiv
--- abstract: \| The goal of branch length estimation in phylogenetic inference is to estimate the divergence time between a set of sequences based on compositional differences between them. A number of software is currently available facilitating branch lengths estimation for homogeneous and stationary evolutionary models. Homogeneity of the evolutionary process imposes fixed rates of evolution throughout the tree. In complex data problems this assumption is likely to put the results of the analyses in question. In this work we propose an algorithm for parameter and branch lengths inference in the discrete-time Markov processes on trees. This broad class of nonhomogeneous models comprises the general Markov model and all its submodels, including both stationary and nonstationary models. Here, we adapted the well-known Expectation-Maximization algorithm and present a detailed performance study of this approach for a selection of nonhomogeneous evolutionary models. We conducted an extensive performance assessment on multiple sequence alignments simulated under a variety of settings. We demonstrated high accuracy of the tool in parameter estimation and branch lengths recovery, proving the method to be a valuable tool for phylogenetic inference in real life problems. ${\texttt{Empar}}$ is an open-source C++ implementation of the methods introduced in this paper and is the first tool designed to handle nonhomogeneous data. [: nucleotide substitution models; branch lengths; maximum-likelihood; expectation-maximization algorithm.]{} author: - 'A. M. Kedzierska$^{1,2}$' - 'M. Casanellas$^{2,*}$' title: '${\texttt{Empar}}$: EM-based algorithm for parameter estimation of Markov models on trees.' --- {#section .unnumbered} Assuming that an evolutionary process can be represented in a phylogenetic tree, the tips of the tree are assigned operational taxonomic units (OTUs) whose composition is known. Here, the OTUs are thought of as the DNA sequences of either a single or distinct taxa. Internal vertices represent ancestral sequences and inferring the branch lengths of the tree provides information about the speciation time. Choice of the evolutionary model and the method of inference have a direct impact on the accuracy and consistency of the results [@SulSwo97; @Fel78; @BruHal99; @Pen94; @HueHil93; @Schwartz2010]. We assume that the sites of a multiple sequence alignment (MSA) are independent and identically distributed (i.i.d. hypothesis of all sites undergoing the same process without an effect on each other), the evolution of a set of OTUs along a phylogenetic tree ${\tau}$ can be modeled by the evolution of a single character under a hidden Markov process on ${\tau}$. Markovian evolutionary processes assign a conditional substitution (transition) matrix to every edge of ${\tau}$. Most current software packages are based on the continuous-time Markov processes where the transition matrix associated to an edge $e$ is given in the form $\exp(Q^e t_e)$, where $Q^ee$ is an instantaneous mutation rate matrix. Although in some cases the rate matrices are allowed to vary between different lineages (cf. [@Galtier1998],[@YY99]), it is not uncommon to equate them to a homogeneous* rate matrix $Q$, which is constant for every lineage in ${\tau}$. Relaxing the homogeneity assumption is an important step towards increased reliability of inference (see [@mitoch]). In this work, we consider a class of processes more general than the homogeneous ones: the discrete-time Markov processes. If ${\tau}$ is rooted, these models are given by a root distribution $\pi,$ and a set of transition matrices $A^e$ (e.g. chap. 8 of @Semple2003). The transition matrices $A^e$ can freely vary for distinct edges and are not assumed to be of exponential form, thus are highly applicable in the analyses of non-homogeneous data. Among these models we find the general Markov model (${\mathtt{GMM}}$) and all its submodels, e.g. discrete-time versions of the Jukes-Cantor model (denoted as ${\mathtt{JC69}^{\ast}}$), Kimura two-parameters (${\mathtt{K80}^{\ast}}$) and Kimura 3-parameters models (${\mathtt{K81}^{\ast}}$), and the strand symmetric model ${\mathtt{SSM}}$. Though the discrete-time models provide a more realistic fit to the data [@YY99; @Ripplinger2008; @Ripplinger2010], their complexity requires a solid inferential framework for accurate parameter estimation. In continuous-time models, maximum-likelihood estimation (MLE) was found to outperform Bayesian methods [@Schwartz2010]. The most popular programs of phylogenetic inference (PAML [@Yang1997], PHYLIP [@Felsenstein1989], PAUP\* [@PAUP]) are restricted to the homogeneous models. Though more realistic, the use of nonhomogeneous models in phylogenetic inference is not yet an established practice. Recently, [@Jayaswal2011] proposed two new non-homogeneous models. With the objective of testing stationarity, homogeneity and inferring the proportion of invariable sites, the authors propose an iterative procedure based on the Expectation Maximization (${\mathtt{EM}}$) algorithm to estimate parameters of the non-homogeneous models (cf. [@barryhartigan87]). The ${\mathtt{EM}}$ algorithm was formally introduced by [@Dempster1977] (cf. @Hartley1958). It is a popular tool to handle incomplete data problems or problems that can be posed as such (e.g. missing data problems, models with latent variables, mixture or cluster learning). This iterative procedure globally optimizes all the parameters conditional on the estimates of the hidden data and computes the maximum likelihood estimate in the scenarios, where, unlike in the fully-observed model, the analytic solution to the likelihood equations are rendered intractable. An exhaustive list of references and applications can be found in [@Tanner1996], and more recently in [@Ambroise1998]. Here, we extend on the work of [@Jayaswal2011] and present ${\texttt{Empar}}$, a MLE method based on the ${\mathtt{EM}}$ algorithm which allows for estimating the parameters of the (discrete-time) Markov evolutionary models. ${\texttt{Empar}}$ is an implementation suitable for phylogenetic trees on any number of leaves and currently includes the following evolutionary models: ${\mathtt{JC69}^{\ast}}$, ${\mathtt{K80}^{\ast}}$, ${\mathtt{K81}^{\ast}}$, ${\mathtt{SSM}}$ and ${\mathtt{GMM}}.$ We test the proposed method on simulated data and analyze the accuracy of the parameter and branch length recovery. The tests are conducted in a settings analogue to that of [@Schwartz2010] and evaluate the performance of ${\texttt{Empar}}$ on the four and six-taxon trees with several sets of branch lengths, ${\mathtt{JC69}^{\ast}}$ and ${\mathtt{K81}^{\ast}}$ models under varying alignment lengths. We present an in-depth theoretical study, investigating the dependence of the performance on factors such as model complexity, size of the tree, positioning of the branches, data and total tree lengths. Our findings suggest that the method is a reliable tool for parameter inference of small sets of taxa, best results obtained for shorter branches. The algorithm underlying ${\texttt{Empar}}$ was implemented in C++ and is freely available to download at <http://genome.crg.es/cgi-bin/phylo_mod_sel/AlgEmpar.pl>. METHODS {#methods .unnumbered} ======= Models {#models .unnumbered} ------ We fix a set of $n$ taxa labeling the leaves of a rooted tree ${\tau}$. We denote by $N({\tau})$ the set of all nodes of ${\tau}$, the set of leaves as $L({\tau})$, the set of interior nodes as $Int({\tau}),$ and the set of edges as $E({\tau}).$ We are given a DNA multiple sequence alignment (MSA) associated to the taxa in ${\tau}$ and a discrete-time Markov process on ${\tau}$ associated to an evolutionary by a model ${\mathcal{M}}$, where the nodes in${\tau}$ are discrete random variables with values in the set of nucleotides $\{{\mathtt{A}},{\mathtt{C}},{\mathtt{G}},{\mathtt{T}}\}$. We assume that all sites in the alignment are i.i.d. and model evolution per site as follows: for each edge $e$ of ${\tau}$ we collect the conditional probabilities $P(y\|x,e)$ (nucleotide $x$ being replaced by $y$ at the descendant node of $e$) in a transition matrix $A^e=(P(y\|x,e))_{x,y}$; $\pi=(\pi_{{\mathtt{A}}},\pi_{{\mathtt{C}}},\pi_{{\mathtt{G}}},\pi_{{\mathtt{T}}})$ is the distribution of nucleotides at the root $r$ of ${\tau}$ and $\xi=\{\pi, (A^e)_{e}\}$ the set of continuous parameters of ${\mathcal{M}}$ on ${\tau}$. We denote by $X$ the set of $4^n$ possible patterns at the leaves and $Y$ the set of $4^{\|Int({\tau})\|}$ possible patterns at the interior nodes of ${\tau}.$ In what follows, the joint probability of observing $\textbf{x}=(x_l)_{l \in L({\tau})} \in X$ at the leaves and nucleotides $\textbf{y}=(y_v)_{v\in Int({\tau})} \in Y$ at the interior nodes in ${\tau}$ is calculated as $$p_{\textbf{x},\textbf{y}}(\xi)=\pi_{y_r}\prod_{v \in N({\tau})\setminus\{r\} }A^{e_{an(v),v}}_{y_{an(v)},y_{v}}$$ where $an(v)$ denotes a parent node of node $v,$ $e_{an(v),v}$ is an edge from $an(v)$ to $v$ (note: if $v$ is a leaf, then $y_v=x_v$). In the complete model the states at the interior nodes are observed and the joint distribution is computed as above. On the other hand, the observed model assumes the variables at the interior nodes to be latent. In the latter case, the probability of observing $\textbf{x}=(x_l)_{l \in L({\tau})}$ at the leaves of ${\tau}$ can be expressed as $$p_{\textbf{x}}(\xi)=\sum_{\textbf{y}=(y_v)_{v\in Int({\tau})} \in Y}p_{\textbf{x},\textbf{y}}(\xi).$$ Restricting the shape of the transition matrices $A^e$ leads to different evolutionary models such as ${\mathtt{JC69}^{\ast}}$, ${\mathtt{K80}^{\ast}}$, ${\mathtt{K81}^{\ast}}$, ${\mathtt{SSM}}$ (see @CFK, @arBook, and @AllmanRhodes_chapter4 for references and background on the discrete-time models). The first three are the discrete-time versions of the widely used continuous-time JC69 [@Jukes1969], K80 [@Kimura1980] and K81 [@Kimura1981] models. The Strand Symmetric model ${\mathtt{SSM}}$ (@CS) is a discrete-time generalization to the `HKY` model [@Hasegawa1985] with equal distribution of the pairs of bases (`A,T`) and (`C,G`) at each node of the tree. It reflects the double-stranded nature of DNA and was found to be well-suited for long stretches of data [@Yap2004]. Lastly, the general Markov model ${\mathtt{GMM}}$ (@Allman2003 [@Steel1994a]) is free of restrictions on the entries of $A^e$, non-stationary, and can be thought as a non-homogeneous version of the general time reversible model ([@GTR]). Expectation-Maximization algorithm {#expectation-maximization-algorithm .unnumbered} ---------------------------------- An algebraic approach to the Expectation-Maximization (${\mathtt{EM}}$) algorithm was first introduced in [@Pachter2005]. In this work, we adapted this approach to the context of phylogenetic trees. Let $D$ denote a MSA recorded into a vector of $4^{\|{{L({\tau})}}\|}$ counts of patterns $u_D=(u_\textbf{x})_{\textbf{x}\in X},$ where each $u_\textbf{x}$ stands for the counts of a particular configuration of nucleotides x at the leaves, observed as columns in the alignment. We are interested in maximizing the likelihood function: $$\mathcal{L}_{obs}(\xi ; u_D)=\prod_{\textbf{x}\in X}p_{\textbf{x}}(\xi)^{u_{\textbf{x}}}$$ (up to a constant). Let $U_{cD}=(u_{\textbf{x},\textbf{y}})_{\textbf{x}\in X,\textbf{y}\in Y}$ be an array of counts for the complete model, where $u_{\textbf{x},\textbf{y}}$ is the number of times $\textbf{x}$ was observed at the leaves and $\textbf{y}$ at the interior nodes. The likelihood for the complete model has a multinomial form $$\label{eq:likelEM} \mathcal{L}_c(\xi ; U_{cD})=\prod_{\textbf{x}\in X,\textbf{y}\in Y}p_{\textbf{x},\textbf{y}}(\xi)^{u_{\textbf{x},\textbf{y}}}=\prod_{\textbf{x}\in X,\textbf{y}\in Y}(\pi_{y_r}\prod_{v \in N({\tau})\setminus\{r\} }A^{e_{an(v),v}}_{y_{an(v)},y_{v}})^{u_{\textbf{x},\textbf{y}}}$$ (up to a constant), which is guaranteed to have a global maximum given by a model-specific explicit formula (see ${\textit{Supp. mat.}}$ A). ${\mathtt{EM}}$ algorithm iteratively alternates between the expectation (${\emph{E-step}}$) and maximization step (${\emph{M-step}}$). In the ${\emph{E-step}}$ the algorithm uses the tree topology, current estimates of parameters and the observed data $u_D$ to assign a posterior probability to each of the possible $4^{\|{{L({\tau})}}\|}$ patterns in $X$ and the expected counts of the complete model, $u_{cD}$. This step can be efficiently performed using the peeling algorithm of [@Felsenstein2004]. In the ${\emph{M-step}}$ the updated MLE of the parameters are obtained by maximizing the likelihood of the complete model . The procedure is depicted in Fig. 1. ![Expectation-maximization algorithm.[]{data-label="fig:em_procedure"}](plots/code-crop){width="90.00000%"} The likelihood is guaranteed to increase at each iteration of this process (e.g. [@Wu], [@Husmeier2005]). Moreover, for a compact set of parameters the algorithm converges to a critical point of the likelihood function. Although the output of the algorithm is not guaranteed to be a global maximum, multiple starting points are used for optimal solution. Statistical tests {#statistical-tests .unnumbered} ----------------- The substitution matrices are assumed stochastic. The number $d$ of free parameters for transition matrices in ${\mathtt{JC69}^{\ast}}$, ${\mathtt{K80}^{\ast}}$, ${\mathtt{K81}^{\ast}}$, ${\mathtt{SSM}}$ and ${\mathtt{GMM}}$ models is 1, 2, 3, 6, and 12 respectively. The root distribution under the ${\mathtt{SSM}}$ (respectively ${\mathtt{GMM}}$) model has 2 (resp. 3) free parameters, and is uniform for the remaining models considered here. For clarity of exposition, hereon the reference to the root distribution will be omitted; however, the formulas can be easily modified to include the root. We let $\xi^e$ be the vector of free parameters defined as the off-diagonal elements of a transition matrix $A$ associated to a edge $e$ ($\xi^e_1=A_{1,2}$, $\xi^e_2$ is the next –from left to right, top down– off-diagonal entry distinct from $\xi^e_1$, etc). The procedure is repeated until $\xi^e_d$ is reached. Let $\xi=(\xi_i^e)_{i=1,\ldots,d; e\in{E({\tau})}}$ denote the vector of free parameters for an evolutionary model ${\mathcal{M}}$ as above and let $\hat{\xi}$ be its MLE. Under certain regularity conditions [@Zacks1971 Chap. 5], $\hat{\xi}$ exists, is consistent, efficient and asymptotically normal with mean $\xi$ and the covariance matrix given by the inverse of the Fisher information matrix [@Rao1973; @Efron1978]. The entries of the $d\|{E({\tau})}\|\times d\|{E({\tau})}\|$ Fisher information matrix ${\textbf{I}}$ over free parameters are given by: $${\textbf{I}}(\xi^e_k, \xi^{e'}_{k'}) = -\mathbf{E}\left(\frac{\partial^2 \log{\mathcal{L}}_{obs}(\xi; u_D)}{\partial \xi^{e}_{k}\partial \xi^{e'}_{k'}}\right)$$ (see ${\textit{Supp. mat.}}$ B for details). The Wald statistics for testing the null hypothesis $\xi_i^e=\hat{\xi}_i^e$, $e\in{E({\tau})}, i=1,\ldots,d$, is $$\label{eq:null_distr} (\hat{\xi^e}-\xi^e)^T {\textbf{I}}^e(\hat{\xi^e}-\xi^e)\sim\chi^2_{d},$$ where ${\textbf{I}}^e$ denotes the $d\times d$ slice of ${\textbf{I}}$ corresponding to the free parameters of $e\in{E({\tau})}$. The $p-value$ can thus be easily calculated by looking at the tails of the corresponding $\chi^2$ distribution. We tested the validity of the test statistics in our data by simulating a variety of MSAs under the complete model and compared it to the theoretical distribution . Figure I in the ${\textit{Supp. mat.}}$ C shows high fit and proves that the setting is appropriate. Variances of the free parameters of the model and the full (observed) covariance matrix are saved in the output of ${\texttt{Empar}}$. These in turn can be used as the plug-in estimators in to calculate the $p-values$ and normal confidence intervals for the parameters. We denote by $V^{e}_{i,i}$ the $i^{th}$ diagonal entry of the matrix $({\textbf{I}}^e)^{-1}$ corresponding to the variance of the free parameter $\xi^e_i$, $i=1,\ldots,d$. For the models with $d>1$ (i.e. all but ${\mathtt{JC69}^{\ast}}$), the variances of the free parameters can be summarized in a combined form $cV^e$ for each edge $e$: $$\label{eq:comb_var} cV^{e}(\xi^e)=\frac{\sum^d_{j=1} \left(V^e_{j,j}+\left(\xi^e_j-\frac{\sum^d_{j=1}\xi^e_j}{d}\right)^2\right)}{d}.$$ Branch lengths {#branch-lengths .unnumbered} -------------- The evolutionary distance between two nodes in ${\tau}$ joined by an edge $e$ with substitution matrix $A^e$ is defined as the total number of substitutions per site along $e$. This quantity is referred to as the branch length of edge $e$ (or of matrix $A^e$) and, following on [@barryhartigan87trans], can be approximated by: $$l({{A}^{e}})=-\frac{1}{4}\log\det({{A}^{e}}).\label{eq:brlength}$$ We denote the total length of the tree ${\tau}$ by ${\mathsf{L}_{{\tau}}}$, ${\mathsf{L}_{{\tau}}}=\sum_{e\in\|{E({\tau})}\|}l({{A}^{e}}).$ Now, let $A$ and $A'$ be two invertible $4\times 4$ matrices such that the entries of $(A')^{-1}(A-A')$ are small. From , we get $$\begin{aligned} \label{eq:br_bound} \|l(A)-l(A')\|&=&\frac{1}{4}\|\log\frac{\det(A)}{\det(A')}\|=\frac{1}{4}\|\log\det((A')^{-1}A)\| \nonumber\\ &=&\frac{1}{4}\|\log\det(\textbf{Id} + (A')^{-1}(A-A'))\|\nonumber\\&\approx& \frac{1}{4}\|\log(1+Tr((A')^{-1}(A-A')))\|\nonumber\\ &\approx&\frac{1}{4}\|Tr((A')^{-1}(A-A'))\|\leq \frac{1}{4}4\|\|(A')^{-1}(A'-A))\|\|_{1} \nonumber\\ &\leq& \|\|(A')^{-1} \|\|_{1} \|\|A-A'\|\|_{1},\end{aligned}$$ where $\|\|.\|\|_1$ is the maximum absolute column sum of the matrix. Therefore if $A'$ is a good approximation to $A$, then $l(A')$ is a good approximation to $l(A)$. In what follows, we use the statistical test above to show the accurate recovery of the parameters. By the above argument, we can conclude that the estimates of the branch lengths will also be accurate (see also Results section). Simulated data {#simulated-data .unnumbered} ============== Performance assessment of ${\texttt{Empar}}$ was conducted on the MSAs simulated on four and six-taxon trees following [@Schwartz2010]. In the case of four taxon trees we fixed an inner node as the root and considered three types of topologies: ${{\tau}_{\mathrm{balanced}}}^4$ corresponds to the “balanced” trees with all five branches equal; the inner branch in ${{\tau}_{1:2}}$ is half the length of the exterior branches; and ${{\tau}_{2:1}}$ denotes a topology with the inner branch double the length of the external ones (see Fig. 2). In ${{\tau}_{\mathrm{balanced}}}^4$ and ${{\tau}_{2:1}}$ we let the length $l_0$ of the inner branch vary from 0.01 to 1.4, where starting from 0.05 it increases in steps of 0.05; in ${{\tau}_{1:2}}$ we let $l_0$ vary in $(0,0.7).$ For 6-taxon trees we used only balanced trees ${{\tau}_{\mathrm{balanced}}}^6$ (see Fig. 2) with $l\in (0,0.7)$. ![Unrooted trees used for simulations: ${{\tau}_{\mathrm{balanced}}}^4$, ${{\tau}_{1:2}}$, ${{\tau}_{2:1}}$ and ${{\tau}_{\mathrm{balanced}}}^6$ (from left to right).[]{data-label="fig_trees"}](plots/tree-eq "fig:"){width="20.00000%"} ![Unrooted trees used for simulations: ${{\tau}_{\mathrm{balanced}}}^4$, ${{\tau}_{1:2}}$, ${{\tau}_{2:1}}$ and ${{\tau}_{\mathrm{balanced}}}^6$ (from left to right).[]{data-label="fig_trees"}](plots/tree-short "fig:"){width="20.00000%"} ![Unrooted trees used for simulations: ${{\tau}_{\mathrm{balanced}}}^4$, ${{\tau}_{1:2}}$, ${{\tau}_{2:1}}$ and ${{\tau}_{\mathrm{balanced}}}^6$ (from left to right).[]{data-label="fig_trees"}](plots/tree-long "fig:"){width="20.00000%"} ![Unrooted trees used for simulations: ${{\tau}_{\mathrm{balanced}}}^4$, ${{\tau}_{1:2}}$, ${{\tau}_{2:1}}$ and ${{\tau}_{\mathrm{balanced}}}^6$ (from left to right).[]{data-label="fig_trees"}](plots/tree-6t "fig:"){width="20.00000%"} We simulated multiple sequence alignments on trees with 4 and 6 leaves under ${\mathtt{JC69}^{\ast}}$ and ${\mathtt{K81}^{\ast}}$ models. We used the $\texttt{GenNon-H}$ package available from <http://genome.crg.es/cgi-bin/phylo_mod_sel/AlgGenNonH.pl>. In brief, based on an input phylogenetic tree with given branch lengths, $\texttt{GenNon-H}$ samples the substitution matrices corresponding to these lengths for all edges and uses them to generate the DNA MSAs following discrete-time Markov process on the tree. The output of this software is the alignment, the substitution matrices, root distribution (whenever non-stationary) and the variances of the continuous free parameters. We note that for the ${\mathtt{JC69}^{\ast}}$ model, there is a $1-1$ correspondence between the branch length and the free parameters of the substitution matrix. This does not hold for other models, were different substiution matrices may give the same branch length. We set the alignment length $L$ to $300nt, 500nt, 1,000nt$ and $10,000nt$ for 4-taxa and to $1,000nt$ or $10,000nt$ for 6-taxa. For the ${\mathtt{JC69}^{\ast}}$ and ${\mathtt{K81}^{\ast}}$ evolutionary models, a phylogenetic tree ${\tau}$ (with branch lengths), and a given alignment length, we run each analysis $1,000$ times. and estimated the parameters using ${\texttt{Empar}}$. All MSAs used for the tests are accessible at the ${\texttt{Empar}}$ webpage. Identifiability {#identifiability .unnumbered} --------------- It is known that in certain cases the substitution parameters are not identifiable (e.g. parameters at the edges adjacent to the root of valency 2). As shown in [@Allman2003], the ${\mathtt{GMM}}$ model and its submodels, are identifiable up to a permutation of rows. [@BHident] showed that incorrect order of rows in the matrices can lead to a negative determinant of the substitution matrix from which the branch lengths cannot be calculated. We expected this problem to arise in short data sets and large branch length, as those correspond to the substitution matrices with smaller diagonal value. For all the data sets used for tests, we calculated the percentage of cases among the 1,000 simulations for which the parameters estimated by the EM algorithm were permuted. This phenomenon was only observed in the data sets of $300nt$ and $1,000nt$. In the first case, the estimated matrices were permuted when the initial branch length was 0.55 or longer and corresponded to 0.005-0.023$\%$ of the cases; in the latter, for the branches of 0.6 or longer with at most 0.001$\%$ permuted matrices. Shorter branch lengths and longer alignments did not suffer from the above problem and recovered the underlying order in all of the cases. As shown by [@Chang96], the entries of the Diagonal Largest in Column (DLC) substitution matrices are identifiable. Namely, there exists a unique set of substitution matrices satisfying the DLC condition and a unique root distribution that leads to a given joint distribution at the leaves. In order to ensure the reliability of the results we designed a procedure that scans the tree in the search of the permutations that maximize the number of substitution matrices with larger diagonal entries. It is not possible to maximize it for all edges, thus the goal is to find the permutations giving more weights to the lower parts of the tree, starting with the nodes corresponding to the outer branches. Given a tree ${\tau}$, we choose an interior node to be the root, directing all edges outwards. For each interior node $x$, we apply a permutation $S(x)$ of $\{{\mathtt{A}},{\mathtt{C}},{\mathtt{G}},{\mathtt{T}}\}$ that maximizes the sum of diagonal entries of the matrices assigned to the outgoing edges of $x$. Permutations $S(x)$ are applied recursively to the subtrees of ${\tau}$, moving $x$ from the outer nodes towards the root. RESULTS AND DISCUSSION {#results-and-discussion .unnumbered} ====================== We present the results on the simulated data sets and discuss their dependence on the length of the alignments, the length of the branches and the depth of the branches in the tree–1 for the external branches and 2 for the internal branches [@Schwartz2010]. In cases with multiple branches of equal depth, we chose one of them at random. Results of statistical tests {#results-of-statistical-tests .unnumbered} ---------------------------- Each sample gave rise to a $p-value$ based on the $\chi^2_d$ test given by . The $p-values$ are a measure of strength of evidence against the null hypothesis: for both exceptionally small or large $p-values$ one can reject the null hypothesis. We recorded the proportion of samples for which the $p-value$ lied in the interval $(0.05,0.95)$. The results are shown Table \[tab:pval\_jc\_short\] for the ${\mathtt{JC69}^{\ast}}$ model on the ${{\tau}_{1:2}}$ tree (also see Tab. I-V in the ${\textit{Supp. mat.}}$ C). We observe that even for short alignments of 300nt the null hypothesis cannot be rejected in approximately 95% of the tests. [p[0.8cm]{}p[0.8cm]{}p[0.8cm]{}p[0.8cm]{}p[0.8cm]{}\|p[0.8cm]{}p[0.8cm]{}p[0.8cm]{}p[0.8cm]{}]{} & &\ $\mathbf{l}$ $\mid$ L & & & & & & & &\ 0.01 & 0.971 & 0.972 & 0.968 & 0.946 & 0.972 & 0.949 & 0.868 & 0.958\ 0.05 & 0.947 & 0.951 & 0.947 & 0.948 & 0.974 & 0.943 & 0.953 & 0.952\ 0.10 & 0.949 & 0.953 & 0.964 & 0.952 & 0.952 & 0.948 & 0.948 & 0.955\ 0.15 & 0.952 & 0.954 & 0.958 & 0.938 & 0.946 & 0.953 & 0.940 & 0.947\ 0.20 & 0.957 & 0.944 & 0.944 & 0.954 & 0.949 & 0.965 & 0.944 & 0.954\ 0.25 & 0.957 & 0.955 & 0.955 & 0.956 & 0.945 & 0.939 & 0.955 & 0.936\ 0.30 & 0.957 & 0.943 & 0.945 & 0.955 & 0.943 & 0.946 & 0.941 & 0.948\ 0.35 & 0.952 & 0.943 & 0.958 & 0.958 & 0.948 & 0.943 & 0.950 & 0.960\ 0.40 & 0.955 & 0.946 & 0.947 & 0.957 & 0.951 & 0.951 & 0.936 & 0.944\ 0.45 & 0.949 & 0.944 & 0.944 & 0.947 & 0.948 & 0.955 & 0.958 & 0.958\ 0.50 & 0.948 & 0.935 & 0.942 & 0.941 & 0.929 & 0.949 & 0.954 & 0.946\ 0.55 & 0.954 & 0.949 & 0.946 & 0.957 & 0.936 & 0.944 & 0.944 & 0.952\ 0.60 & 0.940 & 0.942 & 0.937 & 0.953 & 0.944 & 0.934 & 0.948 & 0.955\ 0.65 & 0.940 & 0.934 & 0.955 & 0.952 & 0.938 & 0.938 & 0.945 & 0.948\ 0.70 & 0.944 & 0.936 & 0.942 & 0.946 & 0.917 & 0.940 & 0.944 & 0.948\ 0.75 & 0.922 & 0.932 & 0.947 & 0.934 & 0.922 & 0.932 & 0.943 & 0.950\ 0.80 & 0.909 & 0.932 & 0.926 & 0.957 & 0.957 & 0.928 & 0.943 & 0.941\ 0.85 & 0.912 & 0.912 & 0.932 & 0.948 & 0.968 & 0.930 & 0.936 & 0.947\ 0.90 & 0.870 & 0.885 & 0.919 & 0.951 & 0.980 & 0.918 & 0.929 & 0.953\ 0.95 & 0.852 & 0.888 & 0.939 & 0.951 & 0.981 & 0.965 & 0.908 & 0.944\ 1,00 & 0.824 & 0.866 & 0.893 & 0.935 & 0.982 & 0.981 & 0.896 & 0.933\ 1,05 & 0.816 & 0.853 & 0.889 & 0.930 & 0.980 & 0.981 & 0.898 & 0.937\ 1.10 & 0.806 & 0.852 & 0.891 & 0.921 & 0.990 & 0.995 & 0.925 & 0.945\ 1.15 & 0.784 & 0.812 & 0.867 & 0.938 & 0.980 & 0.987 & 0.982 & 0.951\ 1.20 & 0.797 & 0.785 & 0.823 & 0.923 & 0.986 & 0.986 & 0.984 & 0.942\ 1.25 & 0.786 & 0.803 & 0.824 & 0.938 & 0.983 & 0.981 & 0.984 & 0.941\ 1.30 & 0.789 & 0.793 & 0.800 & 0.894 & 0.981 & 0.976 & 0.992 & 0.925\ 1.35 & 0.755 & 0.787 & 0.786 & 0.893 & 0.973 & 0.991 & 0.989 & 0.912\ 1.40 & 0.761 & 0.789 & 0.785 & 0.864 & 0.970 & 0.974 & 0.994 & 0.879\ \[tab:pval\_jc\_short\] Error in transition matrices {#error-in-transition-matrices .unnumbered} ---------------------------- For a given branch, we quantified the divergence $D$ between the original and estimated parameters of its transition matrix $A$ using the induced $L_1$ norm: $\|\|A-\hat{A}\|\|_{1}$ (see ). The columns in the transition matrices of ${\mathtt{JC69}^{\ast}}$, $\mathtt{K80}^{\ast}$, and the ${\mathtt{K81}^{\ast}}$ are equal and the norm becomes: $$\label{eq:L2norm} D=\sum_{i=1}^4\mid A_{i,1}-\hat{A}_{i,1}\mid. $$ Figure 3 depicts the results for ${\mathtt{JC69}^{\ast}}$ and ${\mathtt{K81}^{\ast}}$ on the three 4-taxon phylogenies, different alignment lengths and depths of the branches. The shapes of the distribution of $D$ for both models are very similar. As expected, the performance is weaker for long branches and short alignments. A great improvement is observed with the increase in the alignment length, e.g. $10,000nt$ depicts very accurate estimates. The performance under the ${\mathtt{JC69}^{\ast}}$ model (Fig. 3a )is better than that of ${\mathtt{K81}^{\ast}}$ (Fig.3b) for shorter branch lengths. Parameter dispersion {#parameter-dispersion .unnumbered} -------------------- Figure 4 shows the variances of the estimated parameters for depth 1 and 2 branches on the ${{\tau}_{\mathrm{balanced}}}$, ${{\tau}_{1:2}}$, ${{\tau}_{2:1}}$ trees under the ${\mathtt{JC69}^{\ast}}$ model. The variances show an exponential increase, which is most significant in the ${{\tau}_{\mathrm{balanced}}}^4$ tree for both depths, and the ${{\tau}_{1:2}}$ for depth 2. The results for the depth 1 branch in ${{\tau}_{\mathrm{balanced}}}$ and ${{\tau}_{1:2}}$ are very similar. The smallest variance was observed for the depth 2 of ${{\tau}_{2:1}}$. For alignments of length $10,000nt$ on four taxa we can say that the method is quite accurate (see also Tab.VI-VIII in the ${\textit{Supp. mat.}}$ C). For the ${\mathtt{K81}^{\ast}}$ model we summarized the results on variances for each edge as the mean of combined variances of all samples (see formula ). The results are analogous to those of the ${\mathtt{JC69}^{\ast}}$ model, see Figure II in ${\textit{Supp. mat.}}$ C. As expected, the parameter estimates are less dispersed for shorter branches and longer alignments (see Tab. IX-XI in the ${\textit{Supp. mat.}}$ C). Error in the branch lengths {#error-in-the-branch-lengths .unnumbered} --------------------------- Using the formula we calculated the actual difference $l_0-\hat{l}$ between the branch length $l_0$ computed from the original parameters $\xi$ and the branch length $\hat{l}$ computed using their MLEs $\hat{\xi^e}$. Negative values of this score imply overestimation of the branch length, while positive values indicate underestimation. The results are shown in Figures 5 and 6. In the case of ${\mathtt{JC69}^{\ast}}$ we observe that the method presented here does not tend to underestimate or overestimate the lengths for the depth 1 branches in all the 4-taxon trees ($l_0-\hat{l}$ is centered at 0 (see Fig. 5). The depth 2 branches have a tendency towards overestimation of the length for branches longer than (approximately) $0.45$ for ${{\tau}_{1:2}}$, $0.9$ for ${{\tau}_{2:1}}$, and $0.8$ for the ${{\tau}_{\mathrm{balanced}}}^4$ trees. In the latter case, lengths longer than $1.2$ for alignments up to $1,000$nt show opposite trend of underestimating the true lengths. The values were accurate when the alignment lengths were increased in the case of ${{\tau}_{1:2}}$ and ${{\tau}_{2:1}}$. On the other hand, for ${{\tau}_{\mathrm{balanced}}}^4$ the alignments of $10,000$nt resulted in overestimation. In the ${\mathtt{K81}^{\ast}}$ model the results are significantly more accurate (see Fig. 6). There is a trend of underestimation for branches longer than (approximately) $0.9$ for shorter alignments. That is especially noticeable for ${{\tau}_{\mathrm{balanced}}}^4$ and depth 1 branches of ${{\tau}_{1:2}}$. This trend diminishes with an increase in the alignment length. Overall, in the case of both models, the variance of the estimate is smaller for shorter lengths and both depth 1 and 2 branches of the ${{\tau}_{2:1}}$ tree. In addition, we calculated the tree length ${\mathsf{L}_{{\tau}}}$ (i.e. the sum of its branch lengths) from the estimated parameters and compared it to the theoretical result on the original branch length $l_0$: $4.5l_0$ for ${{\tau}_{1:2}}$ (where $l_0$ is a depth 1 branch), $3l_0$ for ${{\tau}_{2:1}}$ ($l_0$ for depth 2 branch) and $5l_0$ for ${{\tau}_{\mathrm{balanced}}}^4$. The rightmost columns of Figures 5 and 6 show the results for 4-taxon trees for the ${\mathtt{JC69}^{\ast}}$ and ${\mathtt{K81}^{\ast}}$ models respectively. The length of the tree is estimated accurately for all trees, the estimates being best for ${{\tau}_{2:1}}$. The variance is small and decreasing with an increase in the data length. As the sequences get longer, the distribution is centered around the true value. This is especially visible for the ${\mathtt{K81}^{\ast}}$ model (see Fig. 6). Results for larger trees {#results-for-larger-trees .unnumbered} ------------------------ We run the analysis on the 6-taxon balanced tree, ${{\tau}_{\mathrm{balanced}}}^6$, under the ${\mathtt{K81}^{\ast}}$ model, for alignment lengths of $1,000$nt and $10,000$nt and branch length ${\mathbf{l}}\in\{0.01, 0.1, 0.3, 0.5, 0.7, 0.9, 1.1, 1.3, 1.4\}$. The $p-values$ of the corresponding tests confirm that the performance of the algorithm is very satisfactory (see Tab. XII in the ${\textit{Supp. mat.}}$ C). We have seen in the 4-taxon study that the tree with equal branch lengths gave worse results than the unbalanced trees. Thus, we expect the results of the depth 2 branches to be similarly challenged in this case. Figure 7 depicts the estimated tree lengths. It can be seen that the estimates are accurate and the results improve for the alignments of $10,000$nt. As expected, the variance of the estimates increases with the increase in the length of the branch. By formula , long branches correspond to small values of the determinant of the transition matrix. Thus, statistical fluctuations in the parameter estimates have greater impact on the resulting length of the tree. Next, we calculated the difference between the original and estimated branch lengths. In Figure 8a we see that the depth 1 branches show some degree of underestimation of the length for lengths $1.1-1.4$ and alignments of $1,000$nt. In the case of $10,000$nt, the results improve and can be expected to show little bias for even longer data sets. Branches of depth 2 show higher degree of underestimation with improvement for longer data sets. The divergence of the original and estimated parameters for transition matrices given by formula is shown in Figure 8b. For branches of depth 1 and data of length $10,000$, the error is about 0.2. In the case of branches of depth 2, it is almost doubled for both alignment lengths. In both cases, branch lengths up to 0.5 give satisfactory results. The error of the estimates for longer branches seems to be approaching a plateau. Combined variance of the estimated parameters is much decreased for the $10,000$nt data sets in comparison with the $1,000$nt, and is smaller for the depth 1 branch (see Fig. 8c ). Again, the exponential shape of the plot can be attributed to the logarithm appearing in the formula . CONCLUSION {#conclusion .unnumbered} ========== In order to evaluate the performance of the method proposed here under various circumstances, we conducted many tests on simulated data sets. We observed that the performance of ${\texttt{Empar}}$ is most optimal for long alignments and short branch lengths. It is worth noting that even for short alignments of 300nt or 500nt on 4 taxa, the estimated parameters approximate closely the original parameters in $\approx 95\%$ of the cases as proved by the normality test of the MLE. Moreover, the branch lengths calculated based on the parameters estimated by ${\texttt{Empar}}$ were found very accurate already for short alignments. Though the measure of divergence $D$ for the parameters of transition matrices proposed here accumulates all errors in the entries of the transition matrix, alignment length of 10,000nt showed divergence values smaller than 0.1. In this paper, we provide the first implementation of a tool for inferring continuous parameters under the discrete-time models. The method allows for accurate estimation of branch lengths in non-homogeneous data. There are two limitations to applicability of the method. Firstly, the algorithm has an exponential computational time increase with the number of taxa. This is a restriction due to the fact that the algorithm computes large matrices of dimension that is exponential in the total number of nodes of a tree. Running time of ${\texttt{Empar}}$ on star trees with 3-8 nodes and equal branches of $0.5$ on Ubuntu 11.10, Intel Core i7 920 at 2.67 GHz with 6 Gb is given in Table \[tab:gennong\_times\]. Secondly, the memory usage of ${\texttt{Empar}}$ is approx. $84^{\|N({\tau})\|}$ and corresponds to the memory footprint of the matrix in the ${\mathtt{EM}}$ algorithm, e.g. for this matrix to fit in the memory of a 6Gb machine the bound on the number of nodes is $\|N({\tau})\|$ $\leq 14$. length $\mid$ n 3 4 5 6 7 8 ----------------- ------- ------- ------- ------- ------- ------ 1,000 0.004 0.02 0.033 0.222 1,049 7.14 10,000 0 0.011 0.043 0.171 1,044 6.95 : ${\texttt{Empar}}$ performance time– estimating the parameters of ${\mathtt{K81}^{\ast}}$ on star trees with equal branch lengths of $0.5$, varying number of leaves, $L({\tau})$, for the MSAs of $1,000$ and $10,000nt$. \[tab:gennong\_times\] We conclude that ${\texttt{Empar}}$ is a highly reliable method for estimating branch lengths of relatively small number of taxa and trees with short branch lengths (e.g. closely related species), and achieves high accuracy even when the results are based on short sequences. In particular, ${\texttt{Empar}}$ is a reliable method to compute quartets and to be used with quartet-based methods (see [@Berry99] and [@Berry2000]) on nonhomogeneous data. FUNDING {#funding .unnumbered} ======= Both authors were partially supported by Generalitat de Catalunya, 2009 SGR 1284. MC is partially supported by Ministerio de Educación y Ciencia MTM2009-14163-C02-02.We thank Roderic Guigó for generously providing funding for this project under grant BIO2011-26205 from the Ministerio de Educación y Ciencia (Spain). [45]{} natexlab\#1[\#1]{}url \#1[[\#1]{}]{} E S Allman and J A Rhodes. , 1860 (2):0 113–144, 2003. E S Allman and J A Rhodes. . Cambridge University Press, January 2004. ISBN 0-521-52586-1. E S Allman and J A Rhodes. Phylogenetic invariants. In O Gascuel and MA Steel, editors, [Reconstructing Evolution]{}. Oxford University Press, 2007. C Ambroise. . , 150 (1):0 154–156, January 1998. ISSN 0176-4268. D Barry and J A Hartigan. 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--- abstract: \| We study modes trapped in a rotating ring carrying the self-focusing (SF) or defocusing (SDF) cubic nonlinearity and double-well potential $\cos ^{2}\theta $, where $\theta $ is the angular coordinate. The model, based on the nonlinear Schrödinger (NLS) equation in the rotating reference frame, describes the light propagation in a twisted pipe waveguide, as well as in other optical settings, and also a Bose-Einstein condensate (BEC) trapped in a torus and dragged by the rotating potential. In the SF and SDF regimes, five and four trapped modes of different symmetries are found, respectively. The shapes and stability of the modes, and transitions between them are studied in the first rotational Brillouin zone. In the SF regime, two symmetry-breaking transitions are found, of subcritical and supercritical types. In the SDF regime, an antisymmetry-breaking transition occurs. Ground-states are identified in both the SF and SDF systems. author: - 'Yongyao Li$^{1,2}$, Wei Pang$^{3}$, and Boris A. Malomed$^{1}$' bibliography: - 'apssamp.bib' title: 'Nonlinear modes and symmetry breaking in rotating double-well potentials' --- Introduction ============ The concept of the spontaneous symmetry breaking (SSB) in nonlinear systems was introduced in Ref. [@Chris]. Its significance has been later recognized in various physical settings, including numerous ones originating in nonlinear optics [@Snyder]-[@photo], Bose-Einstein condensates (BECs) [@Milburn]-[@Arik], and degenerate fermionic gases [Padua]{}. A general analysis of the SSB phenomenology was developed too [misc]{}, which is closely related to the theory of bifurcations in nonlinear systems [@Bif]. Fundamental manifestations of the SSB occur in nonlinear systems based on symmetric double-well potentials (DWPs) or dual-core configurations. A paradigmatic example of the latter in nonlinear optics is the twin-core nonlinear fiber, which may serve as a basis for the power-controlled optical switching [@Snyder]. DWP settings in optics were analyzed theoretically and implemented experimentally in photorefractive crystals [@photo]. In the realm of matter waves, main effects predicted in DWPs are Josephson oscillations [@Zapata], the asymmetric self-trapping of localized modes [@Warsaw], and similar effects in binary mixtures [@Mazzarella; @Ng]. Both the Josephson and self-trapping regimes were implemented in the atomic condensate with contact repulsive interactions [@Markus]. The SSB was also analyzed in one- and two-dimensional (1D and 2D) models of BEC trapped in dual-core configurations [@Arik]. Another dynamical setting which has produced a number of interesting effects in media with the intrinsic nonlinearity, especially in BEC, is provided by rotating potentials. It is well known that stirring the self-repulsive condensate typically leads to the formation of vortex lattices [vort-latt]{}, although it was found experimentally [@Cornell] and demonstrated theoretically [@giant] that giant vortices, rather than lattices, may also be formed under special conditions (when the centrifugal force nearly compensates the trapping harmonic-oscillator potential). On the other hand, the rotation of self-attractive condensates gives rise to several varieties of stable localized modes, such as vortices, “crescents" (mixed-vorticity states), and the so-called center-of-mass modes (quasi-solitons) [@rotating-trap]. Further development of this topic was achieved by the consideration of rotating lattice potentials, which can be implemented experimentally as an optical lattice induced in BEC by a broad laser beam transmitted through a revolving sieve [@sieve], or using twisted photonic-crystal fibers in optics [@twistedPC]. In these systems, quantum BEC states and vortex lattices have been studied [@in-sieve], as well as solitons and solitary vortices depinning from the lattice when its rotation velocity exceeds a critical value [@HS]. A specific implementation of the latter settings is provided by the quasi-1D lattice, or a single quasi-1D potential well, revolving about its center in the 2D geometry [@Barcelona]. In particular, the rotation makes it possible to create fundamental and vortical soliton in the self-repulsive medium, where, obviously, nonrotating quasi-1D potentials cannot maintain bright solitons [@Kon]. Furthermore, the rotation of a DWP gives rise to azimuthal Bloch bands [@Ueda; @Stringari]. As mentioned above, the static DWP and its limit form reducing to dual-core systems are fundamental settings for the onset of the SSB [@Snyder]-[Arik]{}. A natural problem, which is the subject of the present work, is the SSB and related phenomenology, i.e., the existence and stability of symmetric, antisymmetric, and asymmetric modes, in rotating DWPs (recently, a revolving DWP configuration was considered in a different context in Ref. [@WenLuo], as a stirrer generating vortex lattices). To analyze basic features of the phenomenology, we here concentrate on the one-dimensional DWP placed onto a rotating ring. As shown in Fig. \[fig\_1\], in optics this setting may be realized as a hollow pipe waveguide twisted with pitch $2\pi /\omega $, while the azimuthal modulation of the refractive index, that gives rise to the effective potential $V(\theta )$, is written into the material of the pipe. Alternatively, a helical potential structure can be created in a straight sheath waveguide by means of optical-induction techniques, using pump waves with the ordinary polarization in a photorefractive material (while the probe wave is to be launched in the extraordinary polarization [@Moti]), or the method of the electromagnetically-induced transparency (EIT) [@Fleischhauer], including its version recently proposed for supporting spatial solitons [Yongyao]{}. In the latter case, one can make the pipe out of Y$_{2}$SiO$_{5}$ crystal doped by Pr$^{3+}$ (Pr:YSO) ions [@Kuznetsova]. In either case of the use of the photorefractive material or EIT, the helical structure may be induced by a superposition of a pair of co-propagating vortical pump waves, with equal amplitudes, a small mismatch of the propagation constants $k_{1,2}$ ($\Delta k\equiv k_{1}-k_{2}\ll k_{1}$), and opposite vorticities $\left( \pm S\right) $, which will give rise to an effective potential profile, $$V(\theta ,z)\sim r^{S}\cos \left( \Delta k\cdot z+2S\theta \right) , \label{V}$$where $z\ $is the propagation distance, while $r$ and $\theta $ are the polar coordinates in the transverse plane. In terms of the BEC, a similar setting may be based on ring-shaped (toroidal) traps, which have been created in experiments [@torus] and investigated in various contexts theoretically [@Salasnich]. In that case, the rotating periodic potential can be added to the toroidal trap [@sieve] , which is equivalent to the consideration of the rotating ring [Stringari,rotating-ring]{}. In this work, we study basic types of trapped modes and their SSB phenomenology in the 1D rotating ring, in both cases of the self-focusing and self-defocusing (SF and SDF) cubic nonlinearities. In Sec. II we formulate the model and present analytical results, which predict a boundary between the symmetric and asymmetric modes, the analysis being possible for the small-amplitude potential and the rotation rate close to $\omega =1/2$. Numerical results are reported in a systematic form, and are compared to the analytical predictions, in Secs. III and IV for the SF and SDF nonlinearities, respectively. The paper is concluded by Sec. IV. The model and analytical considerations ======================================= As said above, we consider the limit of a thin helical shell, which implies a fixed value of the radius in Eq. (\[V\]), $r=r_{0}$, that we normalize to be $r_{0}=1$. Taking the harmonic periodic potential in the form of Eq. (\[V\]) with $S=1$, $V(\theta ,z)=2A\cos ^{2}(\theta -\omega z)$, the corresponding scaled nonlinear Schrödinger equation is $$i{\frac{\partial }{\partial z}}\psi =\left[ -{\frac{1}{2}}{\frac{\partial ^{2}}{\partial \theta ^{2}}}+V(\theta ,z)-\sigma \|\psi \|^{2}\right] \psi , \label{Eq5}$$where $\sigma =+1$ and $-1$ refer to SF and SDF nonlinearities, respectively. Then, we rewrite Eq. (\[Eq5\]) in the helical coordinate system, with $\theta ^{\prime }\equiv \theta -\omega z$: $$i{\frac{\partial }{\partial z}}\psi =\left[ -{\frac{1}{2}}{\frac{\partial ^{2}}{\partial \theta ^{\prime }{}^{2}}}+i\omega {\frac{\partial }{\partial \theta ^{\prime }}}+2A\cos ^{2}(\theta ^{\prime })-\sigma \|\psi \|^{2}\right] \psi , \label{Eq5p}$$where the solution domain is defined at $-\pi \leq \theta ^{\prime }\leq +\pi $. For the narrow toroidal BEC trap with the rotating potential, the respective Gross-Pitaevskii equation, written in the co-rotating reference frame (cf. Refs. [@Ueda; @Stringari]), differs by replacing the propagation distance, $z$, with time $t$ [@review]. Stationary modes with real propagation constant $-\mu $ (in terms of the BEC, $\mu $ is the chemical potential) are sought for as $\psi \left( \theta ^{\prime },z\right) =\exp \left( -i\mu z\right) \phi (\theta ^{\prime })$, with complex function $\phi \left( \theta ^{\prime }\right) $ obeying equation $$\mu \phi =\left[ -{\frac{1}{2}}{\frac{d^{2}}{d\theta ^{\prime }{}^{2}}}% +i\omega {\frac{d}{d\theta ^{\prime }}}+2A\cos ^{2}(\theta ^{\prime })-\sigma \|\phi \|^{2}\right] \phi . \label{phi}$$Equation (\[Eq5p\]) conserves the total power (norm) of the field and its Hamiltonian (energy), $$P=\int_{-\pi }^{+\pi }\left\vert \psi (\theta ^{\prime })\right\vert ^{2}d\theta ^{\prime }, \label{Power}$$$$H=\int_{-\pi }^{+\pi }\left[ {\frac{1}{2}}\left\vert {\frac{\partial \psi }{% \partial \theta ^{\prime }}}\right\vert ^{2}+{\frac{i}{2}}\omega \left( \psi ^{\ast }{\frac{\partial \psi }{\partial \theta ^{\prime }}}-\psi {\frac{% \partial \psi ^{\ast }}{\partial \theta ^{\prime }}}\right) +V(\theta ^{\prime })\|\psi \|^{2}-{\frac{\sigma }{2}}\|\psi \|^{4}\right] d\theta ^{\prime }, \label{Ham}$$with the asterisk stands for the complex conjugate. The periodic boundary conditions, $V(\theta ^{\prime }+2\pi )=V(\theta ^{\prime })$ and $\psi (\theta ^{\prime }+2\pi )=\psi (\theta ^{\prime })$, make Eq. (\[Eq5p\]) invariant with respect to the boost transformation, which allows one to change the rotation speed from $\omega $ to $\omega -N$ with arbitrary integer $N$:$$\psi \left( \theta ^{\prime },z;\omega -N\right) =\psi \left( \theta ^{\prime },z;\omega \right) \exp \left[ -iN\theta ^{\prime }+i\left( \frac{1% }{2}{N}^{2}-N\omega \right) z\right] , \label{boost}$$hence the speed may be restricted to interval $0\leq \omega <1$. Furthermore, Eq. (\[Eq5p\]) admits an additional invariance, relating solutions for opposite signs of the rotation speed: $\psi (\theta ^{\prime },z;\omega )=\psi ^{\ast }(\theta ^{\prime },-z;-\omega )$. If combined with shift $\omega \rightarrow \omega +1$, the latter transformation implies that the solutions for the rotation speeds $\omega $ and $1-\omega $ (with $% 0<\omega <1$) are mutually tantamount, therefore the rotation speed may be eventually restricted to interval $$0\leq \omega \leq 1/2, \label{zone}$$which plays the role similar to that of the fist Brillouin zone in solid-state media [@Kittel]. An analytical approach for small-amplitude modes can be developed if the amplitude of the potential in Eq. (\[Eq5p\]) is small too, $\|A\|~\ll 1$, and $\omega $ is close to the right edge of zone (\[zone\]), $\delta \equiv 1/2-\omega ~\ll 1/2$. In the simplest approximation, the corresponding stationary solutions are looked for as a combination of the zeroth and first angular harmonics, $$\phi \left( \theta ^{\prime }\right) =a_{0}+a_{1}\exp \left( i\theta ^{\prime }\right) , \label{ansatz}$$where $a_{0}$ is fixed to be real, while amplitude $a_{1}$ may be complex. Indeed, setting $\omega =1/2$, $A=0$, and dropping the cubic term, it is obvious that Eq. (\[ansatz\]) is an exact solution to Eq. (\[phi\]). Then, under the aforementioned conditions, the substitution of ansatz ([ansatz]{}) into Eq. (\[phi\]) yields, in the first approximation (taking care of the balance of the zeroth and first harmonics in the equation), the following algebraic equations: $$\begin{aligned} &&\mu a_{0}=Aa_{0}-\sigma a_{0}^{3}-2\sigma \|a_{1}\|^{2}a_{0}, \notag \\ &&\mu a_{1}=\delta a_{1}-\sigma \|a_{1}\|^{2}a_{1}-2\sigma a_{0}^{2}a_{1}+Aa_{1}. \label{Ea0a1}\end{aligned}$$ First, setting $a_{1}=0$ or $a_{0}=0$ in ansatz (\[ansatz\]) corresponds, respectively, to the CW (continuous-wave) mode, and the uniform vortical one, with $\left\vert \psi \left( \theta ^{\prime }\right) \right\vert =% \mathrm{const}$:$$\begin{aligned} a_{1} &=&0,~a_{0}^{2}=\sigma \left( A-\mu \right) , \label{0} \\ a_{0} &=&0,~\left\vert a_{1}\right\vert ^{2}=\sigma \left( A-\mu +\delta \right) \label{1}\end{aligned}$$(recall that $\sigma ^{2}=1$). In terms of full equation (\[Eq5p\]) with $% A\neq 0$, these two solutions extend into non-uniform ones \[with $\left\vert \psi (\theta ^{\prime })\right\vert \neq \mathrm{const}$\], which remain (and are categorized as) symmetric with respect to point $\theta ^{\prime }=0$. The respective vortical mode, obtained as the extension of solution (\[1\]), is investigated in a numerical form in the next section, under the name of FHS (fundamental-harmonic symmetric) mode. Solutions to Eqs. (\[Ea0a1\]) with $a_{0}a_{1}\neq 0$ give rise to the species of asymmetric modes (categorized as FHA, i.e., fundamental-harmonic asymmetric mode, in the next section), with $$a_{0}^{2}={\frac{\sigma }{3}}(A-\mu +2\delta ),~\|a_{1}\|^{2}={\frac{\sigma }{3% }}(A-\mu -\delta ). \label{a0a1}$$According to Eqs. (\[Power\]) and (\[ansatz\]), the propagation constant is related to total power (\[Power\]) of soliton (\[a0a1\]), $P=2\pi (a_{0}^{2}+\|a_{1}\|^{2})$, as follows: $$\mu =A+{\frac{\delta }{2}}-{\frac{3\sigma }{4\pi }}P. \label{E}$$In particular, in the case of the SF nonlinearity, with $\sigma =+1$, Eq. (\[E\]) demonstrates that the asymmetric mode meets the Vakhitov-Kolokolov (VK) criterion, $d\mu /dP<0$, which is a necessary stability condition for modes supported by the SF terms [@VK]. On the other hand, in the case of the SDF sign of the nonlinearity, $\sigma =-1$, the asymmetric mode satisfies the “anti-VK" criterion, $d\mu /dP>0$, which, as argued in Ref. [@anti], may also play the role of a necessary stability condition, in the respective setting. Further, Eq. (\[a0a1\]) predicts a transition between the symmetric (FHS) and asymmetric (FHA) vortical modes at $a_{0}^{2}=0$, i.e., at $\mu =A+2\delta $. Then, Eq. (\[E\]) yields the location of this boundary in terms of the total power, $P_{\mathrm{\min }}=2\pi \sigma \delta $. In view of the above-mentioned equivalence between the modes pertaining to rotation speeds $\omega =1/2\pm \delta $, the latter result predicts the coexistence of the FHS and FHA modes at $$P\geq P_{\mathrm{\min }}=2\pi \|\delta \|. \label{P}$$Comparison of this prediction with numerical findings is presented below. Numerical results for the self-focusing nonlinearity ($% \protect\sigma =+1$) ============================================================ Symmetric, asymmetric, and antisymmetric modes ---------------------------------------------- Solutions to stationary equation (\[phi\]) were constructed by means of numerical code “PCSOM" elaborated in Ref. [@YJK]. In the SF case, the use of different input waveforms makes it possible to identify five distinct species of stationary modes, as listed in Table [tab:table1]{}. Inputs Types of modes ------------------------------- --------------------------------------- -- -- $\cos\theta'$ Fundamental-harmonic symmetric (FHS) $1+\sin\theta'$ Fundamental-harmonic asymmetric (FHA) $\sin^{2}\theta'$ Second-harmonic symmetric (2HS) $1+\sin\theta'$ Second-harmonic asymmetric (2HA) $\sin\theta'$ Anti-symmetric (AnS) $b + \sin\theta',\,0<b\leq 1$ Broken-antisymmetry (BAnS) : Different species of stationary modes in the case of the SF nonlinearity ($\protect\sigma =+1$), labeled by input waveforms which generate them. \[tab:table1\] The symmetry, asymmetry and antisymmetry of the modes designated in Table 1 is realized with respect to point $\theta ^{\prime }=0$, while the solution is considered, as defined above, in the region of $-\pi \leq \theta ^{\prime }\leq +\pi $. Further, the fundamental or second harmonic (“FH" or “2H", respectively) in the nomenclature adopted in the table refers to a dominant term in the Fourier decomposition of the stationary solution (which is made obvious by their shapes, see below). In particular, the stationary patterns of the FHA and 2HA types are generated by the same input in Table 1, $1+\sin \theta ^{\prime }$, but in non-overlapping regions of the parameter space, $\left( \omega ,A,P\right) $, as shown below. Note also that all the inputs displayed in the table are real functions, but the numerical modes found for $\omega \neq 0$ have a complex structure. The stability of the stationary solutions has been identified through the numerical computation of eigenvalues for infinitesimal perturbation modes. To this end, the perturbed solution was taken in the usual form, $$\psi =e^{-i\mu z}[\phi (\theta ^{\prime })+u(\theta ^{\prime })e^{i\lambda z}+v^{\ast }(\theta ^{\prime })e^{-i\lambda ^{\ast }z}], \label{pert}$$where $u(\theta ^{\prime })$ and $v(\theta ^{\prime })$ are perturbation eigenmodes, and $\lambda $ is the corresponding eigenfrequency. The substitution of expression (\[pert\]) into Eq. (\[Eq5p\]) and linearization leads to the eigenvalue problem in the following form: $$\left( \begin{array}{cc} \mu {-}\mathrm{\hat{h}}-i\omega \frac{\partial }{\partial \theta ^{\prime }} & \sigma \phi ^{2} \\ -\sigma \left( \phi ^{\ast }\right) ^{2} & -\mu +\mathrm{\hat{h}}-i\omega \frac{\partial }{\partial \theta ^{\prime }}% \end{array}% \right) \left( \begin{array}{c} u \\ v% \end{array}% \right) =\lambda \left( \begin{array}{c} u \\ v% \end{array}% \right) , \label{lambda}$$where $\mathrm{\hat{h}}=-(1/2)\partial ^{2}/\partial \theta ^{\prime 2}+2A\cos ^{2}(\theta ^{\prime })-2\sigma \|\phi \|^{2}$ is the respective single-particle Hamiltonian. The underlying solution $\phi $ is stable if all the eigenvalues are real. In the case of the SF nonlinearity, the numerical solutions reveal the existence of all the species of the modes indicated in Table 1, except for BAnS, which exist under the SDF nonlinearity, see below. Typical examples of the five species of the stationary modes which are supported by the SDF cubic term are displayed in Fig. \[1stSF\]-\[AntiSF\]. The analysis demonstrates that all these examples are stable. Moreover, the symmetric and asymmetric modes dominated by the fundamental harmonic, FHS and FHA, demonstrate mutual bistability in Fig. \[1stSF\], as these stable modes coexist at common values of the parameters. A distinctive feature of the FHS mode, which is evident in Fig. \[1stSF\], is that maxima and minima of its intensity coincide with local maxima and the minima of the harmonic potential, while its FHA counterpart features an intensity maximum in one potential well, and a minimum in the other. This structure of the FHS suggests that it may correspond to a maximum, rather than minimum, of the energy (see below), but, nevertheless, this mode has a region of stability against small perturbations. On the other hand, Figs. \[2ndSF\] and \[AntiSF\] demonstrates that both the 2HS and AnS modes have two symmetric intensity peaks trapped in the two potential wells (while local minima of the intensity coincide, quite naturally, with potential humps), hence these mode have a chance to realize a minimum of the energy (in the best case, it may be the system’s ground state). The 2HA mode features a similar property in Fig. \[2ndSF\], but with unequal density peaks trapped in the two potential minima. In the limit of the uniform ring (without the potential, $A=0$) the FHS and AnS modes go over into the above-mentioned uniform vortical state (\[1\]), the 2HS pattern degenerates into the CW state (\[0\]), while the FHA mode takes the form of a $2\pi $-periodic cnoidal-wave solution of the nonlinear Schrödinger equation equation (the 2HA pattern does not exist at all in the limit of $A=0$). The evolution of the shape of the modes with the increase of $A$ is shown in Fig. \[Amod\] \[The evolution figure of AnS mode, which is not shown here, is similar to the 2HS type displayed in Fig. \[fig\_5\_c\]\]. Existence and stability diagrams for the different modes -------------------------------------------------------- Results of the systematic numerical analysis are summarized in Fig. [1stPomega]{}, in the form of diagrams for the existence of the stable FHS and FHA modes in the plane of $\left( P,\omega \right) $ at several fixed values of amplitude $A$ of the harmonic potential. In the blank areas, only unstable modes of the FHS type are found \[in direct simulations they feature an oscillatory instability, which is accounted for by a quartet of complex eigenvalues generated by Eq. (\[lambda\])\], while the FHA modes are completely stable in their existence region. The dashed lines in panels (a) and (b) of Fig. \[1stPomega\] demonstrate that Eq. (\[P\]) correctly predicts the bistability boundary at small values of $\delta \equiv 1/2-\omega $ and $P$, provided that $A$ is small enough too \[recall it is the condition under which Eq. (\[P\]) was derived\]. Further, the fact that vertical cuts of the panels in Fig. [1stPomega]{}, that can be drawn through $\omega =\mathrm{const}$, go, with the increase of the total power ($P$), from the region of the monostability of the FHS mode through the FHS/FHA bistability area into the monostability region of the FHA mode (at least, for $\omega $ sufficiently close to $1/2$), clearly suggests that the symmetry-breaking bifurcation of the subcritical type, which features the bistability [@Bif], [@Snyder], occurs along this route. This conclusion is confirmed by a direct consideration of the numerical data (not shown here in detail). As concerns the AnS solitons, in the case of the SF nonlinearity they do not undergo any bifurcation, and are stable in the entire region of their existence, which is shown in the $(\omega ,P)$ plane for fixed values of the lattice’s strength, $A$, in Fig. \[antiSFPomega\] \[as mentioned above, at $% A=0$ the AnS mode coincides with the FHS one, hence the respective existence area is the same as the red (bottom) one in Fig. \[1stPomega\](a)\]. From here we conclude that the stability region of the AnS mode quickly expands with the increase of $A$, and the FHS/FHA bistability areas in panels (b) and (c) of Fig. \[1stPomega\] actually support the tristability, as the AnS mode is stable there too. The multistability between the three species of the modes based on the fundamental harmonic—symmetric, asymmetric, and antisymmetric ones—makes it necessary to compare their energies, which are defined as per Eq. ([Ham]{}). The comparison along the horizontal dotted line, which is drawn in Fig. \[fig\_7\_c\], is presented in Fig. \[fig\_9\_a\]. The figure clearly shows the following relation between the energies: $$H_{\mathrm{FHA}}<H_{\mathrm{AnS}}<H_{\mathrm{FHS}}. \label{HHH}$$ \[1stHam\] For the 2HS and 2HA patterns, which are based on the second angular harmonic, Fig. \[2ndPomega\] displays the stability areas in the $\left( P,\omega \right) $ plane, at different values of the lattice’s strength, $A$ \[cf. Fig. \[1stPomega\]\], together with the stability area for the FHA mode, which may stably coexist with the 2HS pattern. Figure \[2ndPomega\] also demonstrates that, as mentioned above, the 2HA mode, which exists and is stable in the blue (intermediate) area in Fig. \[2ndPomega\], does not exist at $A=0$, emerging and expanding with the increase of $A$. Note that, on the contrary to the situation for the patterns dominated by the fundamental angular harmonic (FHS and FHA) described above, there is no overlap between the stability areas of the symmetric and asymmetric second-harmonic-based modes, 2HS and 2HA, hence (as confirmed by an additional consideration of the numerical results) the symmetry-breaking 2HS $\rightarrow $ 2HA transition, following the increase of $P$ at $A\neq 0$, amounts to a supercritical bifurcation, which features no bistability [@Bif], [@Snyder]. Another peculiarity of the symmetric mode based on the second harmonic (2HS) is that it is stable in two disjoint stability areas—the red and yellow ones in Fig. \[2ndPomega\] (the bottom and rightmost regions), in the latter one the 2HS featuring the bistability with the FHA mode, although they are not linked by any bifurcation. On the other hand, there is no overlap between the stability areas of the asymmetric modes based on the different angular harmonics, FHA and 2HA, therefore these modes may be generated by the same input waveform ($1+\sin \theta ^{\prime }$, in Table 1), in different parameter regions. The effective bistability occurring in Fig. \[2ndPomega\] makes it necessary to compare the energy of the coexisting stable patterns, which is done in Fig. \[fig\_9\_b\], along the horizontal dotted line drawn in Fig. \[2ndPomega\](c). The energy of the coexisting stable AnS mode is included too. In particular, the short segment corresponding to the 2HS mode appears, above the one pertaining to the FHA mode, when the horizontal dotted line in Fig. \[2ndPomega\](c) enters the yellow (rightmost) region of the 2HS-FHA bistability. Thus, adding the results from Fig. \[fig\_9\_b\], we extend energy relations (\[HHH\]) as follows:$$H_{\mathrm{FHA}}~\mathrm{or}~H_{\mathrm{2HA}}<H_{\mathrm{2HS}}<H_{\mathrm{AnS% }}<H_{\mathrm{FHS}}. \label{HHHHH}$$The conclusion is that the asymmetric modes, either FHA or 2HA, represent the ground state of the rotating ring carrying the harmonic potential and SF nonlinearity. Numerical results for the self-defocusing nonlinearity ($% \protect\sigma =-1$) ============================================================== In the case of the SDF nonlinearity, symmetric modes of the FHS and 2HS types have been found too, although, on the contrary to the system with the SF sign, they do not undergo any bifurcations, hence the FHA and 2HA species do not exist in this case. On the other hand, the AnS mode features an antisymmetry-breaking bifurcation, which gives rise to a broken-antisymmetry (BAnS) mode (which does not exist under the SF nonlinearity). A typical example of the stable BAnS mode is displayed in Fig. \[Antiasy\]. In fact, antisymmetry-breaking bifurcations are characteristic to the double-well systems with the SDF nonlinearity [Warsaw]{}. The stationary solutions of the BAnS type can be numerically generated by the initial guess of the form of $(b+\sin \theta ^{\prime })$, with $0<b\leq 1$, see Table 1. Proceeding to the systematic description of the modes supported by the SDF nonlinearity acting in the combination with the rotating harmonic potential, we note, first, that the numerical analysis demonstrates the existence and stability of the mode of the 2HS type at all values of $(\omega ,P,A)$, therefore this solution is not included in the stability diagrams, which are displayed for the FHS mode in Fig. \[1stPAomegaSDF\], and for the AnS and newly found BAnS ones in Fig. \[AntiPAomegaSDF\]. In the blank area of Fig. \[1stPAomegaSDF\], the FHS solutions exist too, featuring an instability accounted for by a quartet of complex eigenvalues \[see Eq. (\[lambda\])\]. In direct simulations, this instability converts the stationary mode into an oscillatory one (not shown here). Further, Fig. \[1stPAomegaSDF\] shows that the antisymmetry-breaking transition, which follows the increase of the total power, $P$, is of a supercritical type. Actually, the supercritical character of the antisymmetry-breaking bifurcation is a characteristic feature of systems with the SDF sign of the nonlinearity [@Warsaw]. Finally, the energy comparison for the model with the SDF nonlinearity is presented in Fig. \[fig\_9\_c\], along the boundary separating the existence (and stability) regions of the AnS and BAnS modes in Fig. \[1stPAomegaSDF\](b). The figure shows the energies of these modes, which coincide along the boundary, along with the results for the FHS and 2HS symmetric modes. From here, it is concluded that the energies are related as follows:$$H_{\mathrm{2HS}}<H_{\mathrm{BAnS}}=H_{\mathrm{AnS}}<H_{\mathrm{FHS}},$$i.e., the 2HS mode represents the ground state in the case of the SDF nonlinearity, cf. Eqs. (\[HHH\]) and (\[HHHHH\]). CONCLUSION ========== The objective of this work is to study the trapped modes of the symmetric, asymmetric, and antisymmetric types, and the symmetry- or antisymmetry-breaking transitions between them, in the rotating ring carrying the DWP (double-well potential) and the cubic nonlinearity with the SF or SDF (self-focusing or defocusing) sign. The analysis has been performed in the first rotational Brillouin zone. In the SF case, five types of different modes, and their stability, have been identified, three dominated by the fundamental angular harmonic (FHS, FHA, and AnS), and two others based on the second harmonic, 2HS and 2HA modes. The SSB (spontaneous symmetry-breaking) transition between the FHS and FHA modes is of the subcritical type, featuring a conspicuous bistability area, while the transition between the between the 2HS and 2HA states is supercritical. There is no overlap between the existence regions of the asymmetric modes of the FHA and 2HA types, one of which realizes the ground state of the SF system. The SDF model supports four distinct species of the trapped modes, viz., FHS, 2HS, AnS, and, in addition, the BAnS (broken-antisymmetry) mode. The latter one appears from the AnS state as a result of supercritical antisymmetry-breaking transition. The 2HS mode represents the ground state of the SDF system. The present work can be naturally extended in other directions. First, it is possible to consider the rotating potential in the form of $A\cos \left( n\theta ^{\prime }\right) $ with $n\geq 3$, while the present analysis corresponds to the DWP with $n=2$ \[note that Eq. (\[V\]) suggests a possibility to create the helical photonic lattice with $n=2S$ and $S>1$\]. This generalization will give rise to many new trapped modes; in particular, it may be possible to build one with a small width, $\Delta \theta ^{\prime }\ll 2\pi $, hence it may be considered as a soliton trapped in the lattice and rotating along with it, cf. Ref. [@HS]. It is relevant too to consider effects of the two-dimensionality (the entire rotating plane, rather than a narrow ring). Our preliminary analysis, based on simulations of the 2D counterpart of Eq. (\[Eq5p\]), demonstrates essentially the same types of trapped modes in rotating annuli of a finite radial size, as reported above in the one-dimensional limit. Another interesting extension is to replace the linear rotating potential by its nonlinear counterpart, generated by modulation of the local nonlinearity along the angular coordinate (as was done in Ref. [@Pu] in the absence of the rotation). 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--- author: - 'V. M. Passegger$^{1}$, A. Reiners$^{1}$, S. V. Jeffers$^{1}$, S. Wende$^{1}$, P. Schöfer$^{1}$, P. J. Amado$^{2}$, J. A. Caballero$^{3}$, D. Montes$^{4}$, R. Mundt$^{5}$, I. Ribas$^{6}$, A. Quirrenbach$^{3}$, and the CARMENES Consortium' bibliography: - 'passegger.bib' title: 'Spectroscopic characterisation of CARMENES target candidates from FEROS, CAFE and HRS high-resolution spectra' --- Introduction ============ The new CARMENES instrument is mounted at the 3.5 m telescope at Calar Alto Observatory, located in the Sierra de los Filabres in southern Spain. It consists of two fibre-fed high-resolution spectrographs, operating in the visible wavelength range from 0.52 to 0.96 $\mu$m and in the near-infrared from 0.96 to 1.71 $\mu$m, having a spectral resolution of R > 80,000. [@Quirrenbach2010; @Quirrenbach2012; @Quirrenbach2014] Both spectrographs will simultaneously perform high-accuracy radial-velocity measurements of about 300 M dwarfs during three years of guaranteed observing time. The aim is to detect low-mass planets within the habitable zones of these stars.\ For science preparation over 1500 high-resolution spectra have been observed with FEROS, CAFE and HRS to determine effective temperature, surface gravity and metallicity. These parameters are fundamental for characterising star-planet systems. The spectra of M dwarfs are very complex, with molecular lines forming due to the low temperatures. This makes it difficult to use a line-by-line approach and requires a full spectral synthesis, which in turn necessitates for accurate models that take into account the formation of molecules. We use the latest generation PHOENIX model grid, the PHOENIX ACES models [@Husser2013]. These models are especially designed for low temperature stellar atmospheres and use a new equation of state to accurately reproduce molecular lines. Methods and Data ================ Name Resolution Coverage \[nm\] No. Spectra No. Stars Observing Period ------- ------------ ----------------- ------------- ----------- -------------------------- CAFE \~65,000 396-950 623 236 2013-01-21 to 2014-09-26 FEROS 48,000 350-920 455 217 2012-12-31 to 2014-07-11 HRS 60,000 420-1100 93 29 2011-09-29 to 2013-06-18 Table \[tab:obs\] summarizes the properties of the spectrographs used for observation and the data taken. Some observed spectra could not be used for analysis because of different issues, e.g. very low signal-to-noise, observation of wrong target, polluting light from close companions. The method we use was described in detail in [@Passegger2016a]. We fit PHOENIX ACES model spectra to our observed spectra. This is done for different spectral ranges, including the $\gamma$- and $\epsilon$-TiO bands (sensitive to temperature and metallicity), the K- and Na-doublets around 768 nm and 819 nm (sensitive to surface gravity and metallicity) and two CaII-lines. Rotational velocities determined by [@Jeffers2016] are included to account for line broadening due to stellar rotation. Other than [@Passegger2016a] a downhill simplex is implemented for linear interpolation between the model grid points and a $\chi2$ -minimization determines the best fit to the data. Figure \[fig:fit\] shows an example fit to CARMENES data. ![image](fit.jpg){width="0.85\linewidth"} Results and Discussion ====================== We obtained stellar parameters for 351 stars from 977 spectra. We find that most stars lie within 3200-3900 K, corresponding to spectral types M1V-M5V, as shown in the upper left panel of Figure \[fig:results\]. The higher the metallicity the higher the temperature for each spectral type (Figure \[fig:results\], lower left panel). This is consistent with results by [@Mann2015]. They showed that with increasing metallicity the radius increases, for fixed temperature. The spectral types have been calculated using spectral indices [@Schoefer2015]. The green squares correspond to a literature computation by [@PecautMamajek2013] for solar metallicity. A literature comparison with [@RojasAyala2012], [@GaidosMann2014] and [@Maldonado2015] shows that our values for metallicity turn out to be higher than published ones. (Figure \[fig:results\], upper right). One possible explanation for this is that PHOENIX ACES models still cannot reproduce the full depths of some lines (see Figure \[fig:fit\], 4th wavelength range), which might cause the algorithm to choose higher metallicity models to fit the lines. On the other hand it seems that the signal-to-noise ratio is also very important for parameter determination. 75 percent of the stars with \[Fe/H\] higher than 0.6 have SNRs lower than 50. We find good agreement with expected \[Fe/H\] values for SNR>50 (Figure \[fig:results\], lower right). For the first four months of CARMENES data we find that the parameters show better agreement with literature, having better SNRs. ![image](results.jpg){width="0.85\linewidth"} Acknowledgments {#acknowledgments .unnumbered} =============== [CARMENES is an instrument for the Centro Astronómico Hispano-Alemán de Calar Alto (CAHA). CARMENES was funded by the German Max-Planck-Gesellschaft (MPG), the Spanish Consejo Superior de Investigaciones Cientícas (CSIC), the European Union through European Regional Fund (FEDER/ERF), Spanish Ministry of Economy and Competitiveness, the state of Baden-Württemberg, the German Science Foundation (DFG), the Junta de Andalucía, and by the Klaus Tschira Stiftung, with additional contributions by the members of the CARMENES Consortium (Max-Planck-Institut für Astronomie, Instituto de Astrofísica de Andalucía, Landessternwarte Königstuhl, Institut de Ciències de l’Espai, Institut für Astrophysik Göttingen, Universidad Complutense de Madrid, Thüringer Landessternwarte Tautenburg, Instituto de Astrofísica de Canarias, Hamburger Sternwarte, Centro de Astrobiología, and the Centro Astronómico Hispano-Alemán).]{}	ArXiv
--- abstract: 'The classical Szegő polynomial approximation theorem states that the polynomials are dense in the space $L^2(\rho)$, where $\rho$ is a measure on the unit circle, if and only if the logarithmic integral of the measure $\rho$ diverges. In this note we give a quantitative version of Szegő’s theorem in the special case when the divergence of the logarithmic integral is caused by deep zeroes of the measure $\rho$ on a sufficiently rare subset of the circle.' author: - 'Alexander Borichev [^1]\' - 'Anna Kononova [^2]\' - 'Mikhail Sodin [^3]' title: \| Notes on the Szegő minimum problem.\ I. Measures with deep zeroes --- Introduction ============ Denote by ${\mathcal P}$ the linear space of algebraic polynomials, and by ${\mathcal P}_n$ its subspace of polynomials of degree $n$. Given a finite positive measure $\rho$ on the unit circle ${\mathbb T}$, put $$e_n(\rho) = \min_{q_0, \ldots , q_{n-1}}\, \sqrt{\int_{{\mathbb T}} \bigl\| q_0+q_1t + \ldots +q_{n-1}t^{n-1} +t^n \bigr\|^2\, {\rm d}\rho(t)} = \text{dist}_{L^2(\rho)}(t^n, {\mathcal P}_{n-1}).$$ Then $$\lim_{n\to\infty} e_n(\rho) = \exp\Bigl( \frac12\int_{\mathbb T}\log\rho'\, {\rm d}m \Bigr),$$ where $m$ is the Lebesgue measure on ${\mathbb T}$ normalized by condition $m({\mathbb T})=1$, and $\rho' = {\rm d}\rho/{\rm d}m$ is the Radon–Nikodym derivative. This is a classical result, first, proven by Szegő for absolutely continuous measures $\rho$, and then, independently, by Verblunsky and Kolmogorov in the general case [@GS Section 3.1] and [@Simon Chapters 1 and 2]. Noting that for $j{\geqslant}0$, $e_{n+j}(\rho)$ coincides with the distance in $L^2(\rho)$ from $ t^{-j}$ to the linear span of $\{ t^k\colon -j+1 {\leqslant}k {\leqslant}n \}$, and recalling that the trigonometric polynomials are dense in $L^2(\rho)$, one sees that the density of algebraic polynomials ${\mathcal P}$ in $L^2(\rho)$ is equivalent to the condition $ \displaystyle\lim_{n\to\infty} e_n(\rho) = 0$, and therefore, to the divergence of the logarithmic integral $$\int_{\mathbb T}\log\rho'\, {\rm d}m = -\infty\,.$$ In these notes we will be occupied by the following question: \[quest\] Suppose $\rho$ is a measure on ${\mathbb T}$ with divergent logarithmic integral. Estimate the rate of decay of the sequence $e_n(\rho)$. Our interest to this question came from the linear prediction for stationary processes. If $\xi\colon {\mathbb Z}\to{\mathbb C}$ is a stationary random sequence with spectral measure $\rho$, then, according to Kolmogorov and Wiener, $e_n(\rho)$ is the error of the best mean-quadratic linear prediction of $\xi(n)$ by $\xi(0), \ldots , \xi(n-1)$; i.e., $$e_n^2(\rho) = \min_{q_0, \ldots , q_{n-1}} {\mathbb E}\Bigl[ \Bigl\| \xi(n) - \sum_{0{\leqslant}j {\leqslant}n-1} q_j \xi(j) \Bigr\|^2 \Bigr].$$ In the case when the logarithmic integral converges, $e_n(\rho)$ has a positive limit $e_\infty(\rho)$, and dependence of the rate of convergence on the smoothness of the density of $\rho$ is well-understood [@Gol; @Ibr]. In the case of divergent logarithmic integral the situation is quite different and not much is known. If the closed support of $\rho$ is not the whole circle, then it is not difficult to show that $e_n(\rho)$ tends to zero at least exponentially. In the other direction, a version of the classical result of Erdős and Turán says if $\rho' >0$ $m$-a.e. on ${\mathbb T}$, then the measure $\rho$ is regular, i.e., $e_n(\rho)^{1/n}\to 1$. Later, stronger criteria for regularity of $\rho$ were found by Widom, Ullman, and Stahl and Totik, see [@ST Chapter 4]. In these notes we show that in several special but interesting situations it is not difficult to estimate decay of the sequence $e_n(\rho)$ using only simple classical tools. Here, we consider the case when the divergence of the logarithmic integral is caused by deep zeroes of the measure $\rho$ on a sufficiently rare subset of ${\mathbb T}$. The results presented in this note extend Theorems 8 and 9 from [@BSW]. Our main idea is that in the case when the measure ${\rm d}\rho = \Phi\, {\rm d}m$ has divergent logarithmic integral (i.e., $\displaystyle \int_{\mathbb T}\log\Phi\, {\rm d}m = -\infty $), the value $\|\log e_n(\rho)\|$ can be controlled by the integral $$\int_{\mathbb T}\min\Bigl\{ \log\Bigl(\frac1{\Phi} \Bigr), A \Bigr\}\, {\rm d}m$$ of the cut-off of $\log\Phi^{-1}$ on an appropriate large level $A$ depending on $n$. This can be viewed as a quantitative version of the regularization of the weight $\Phi$ by $\Phi_{\varepsilon}= \Phi+{\varepsilon}$ with $0<{\varepsilon}\ll 1$ used by Szegő in the proof of his theorem. We succeeded to make this work only under additional regularity assumptions on $\Phi$. The toy example is the absolutely continuous measure ${\rm d}\rho=e^{-H}\, {\rm d}m$, where $H(e^{2\pi{\rm i}\theta})=h(\theta)$, $h\colon {\mathbb R}\to [0, +\infty]$ is a $1$-periodic even function, continuous and decreasing on $(0, \frac12]$, and such that $\displaystyle \int_0 h(\theta)\,{\rm d}\theta = +\infty$. Then, under mild assumptions on $h$, we obtain $$\| \log e_n(\rho) \| \simeq \int_{\mathbb T}h_A\, {\rm d}m\,,$$ where $h_A=\max(h, A)$, and $A=A(n)$ is a solution to the equation $n h^{-1}(a)=a$, $h^{-1}$ is the inverse to the restriction of $h$ on $(0, \frac12]$. Throughout the paper we use the following notation: for positive $A$ and $B$, $A \lesssim B $ means that there is a positive numerical constant $C$ such that $A {\leqslant}CB $, while $A \gtrsim B $ means that $B \lesssim A $, and $A \simeq B $ means that both $A \lesssim B $ and $B \lesssim A $. In the forthcoming second note, we will consider the opposite case when the bulk of the measure $\rho$ is concentrated on a rare subset of ${\mathbb T}$. ### Acknowledgements {#acknowledgements .unnumbered} We thank Sergei Denisov, Fedor Nazarov, and Eero Saksman for several enlightening discussions. Preliminaries ============= Here and elsewhere, $H\colon {\mathbb T}\to [0, +\infty]$ is a measurable function with $$\int_{\mathbb T}H\, {\rm d}m = +\infty.$$ By ${\lambda}_H (a) = m\{H>a\}$ we denote the distribution function of $H$. For $A{\geqslant}1$, we put $H_A(t)=\min(H(t), A)$. To estimate from below and above $\log e_n$, we will use the integrals $$\int_{\mathbb T}H_A\, {\rm d}m = \int_0^A {\lambda}_H (a)\, {\rm d}a = A{\lambda}_H(A) + \int_{\{H{\leqslant}A\}} H\, {\rm d}m$$ with some $A=A(n)$. We record several simple observations, which we frequently use throughout the paper. First, we note that under mild regularity assumptions one of the two terms on the RHS can be discarded. If ${\lambda}_H (a)$ satisfies $$\limsup_{a\to\infty} \frac{{\lambda}_H(a)}{{\lambda}_H(2a)} < 2$$ (i.e., decays not faster than $a^{-p}$ with some $p<1$), then $$\int_{\mathbb T}H_A\, {\rm d}m \simeq A{\lambda}_H(A).$$ On the other hand, if the function $a\mapsto a{\lambda}_H^2(a)$ does not increase (i.e., ${\lambda}_H(a)$ decays as $1/\sqrt a$, or faster), then $$\int_{\mathbb T}H_A\, {\rm d}m \simeq \int_{\{H{\leqslant}A\}} H\, {\rm d}m,$$ provided that ${\lambda}_H(A)$ is separated from $1$ (i.e., $A$ is sufficiently large). To see this, denote by $H^\colon [0, 1]\to [0, +\infty]$ the decreasing rearrangement of $H$, that is, the function inverse to ${\lambda}_H$. Then, the function $s\mapsto s^2H^(s)$ does not decrease. Letting $\alpha={\lambda}_H(A)$ (i.e., $A=H^(\alpha)$), we obtain $$\begin{gathered} A{\lambda}_H(A) = \alpha H^(\alpha) \lesssim \alpha^2 H^(\alpha)\, \int_\alpha^1 \frac{{\rm d}s}{s^2} \\ {\leqslant}\int_\alpha^1 \frac{s^2 H^(s)}{s^2}\, {\rm d}s = \int_\alpha^1 H^(s)\, {\rm d}s = \int_{\{H{\leqslant}A\}} H\, {\rm d}m.\end{gathered}$$ Furthermore, if $A/{\lambda}_H(A) \simeq B/{\lambda}_H(B)$, then $$\int_{\mathbb T}H_A\, {\rm d}m \simeq \int_{\mathbb T}H_B\, {\rm d}m.$$ Assume, for instance, that $A{\leqslant}B$ and that $B/{\lambda}_H(B) {\leqslant}C\cdot A/{\lambda}_H(A)$. Since ${\lambda}_H$ does not increase, $B{\leqslant}C\cdot A$. Then, $$\begin{gathered} \int_{\{H{\leqslant}B\}} H\, {\rm d}m {\leqslant}\int_{\{H{\leqslant}A\}} H\, {\rm d}m + \int_{\{A < H{\leqslant}C\cdot A\}} H\, {\rm d}m \\ {\leqslant}\int_{\{H{\leqslant}A\}} H\, {\rm d}m + C\cdot A{\lambda}_H(A) {\leqslant}C\, \int_{\mathbb T}H_A\, {\rm d}m\,,\end{gathered}$$ and similarly, $$B{\lambda}_H(B) = \frac{B}{{\lambda}_H(B)}\cdot {\lambda}_H^2(B) {\leqslant}C\cdot \frac{A}{{\lambda}_H(A)}\cdot {\lambda}_H^2(A) = C\cdot A{\lambda}_H(A).$$ Thus, $$\int_{\mathbb T}H_B\, {\rm d}m {\leqslant}C\, \int_{\mathbb T}H_A\, {\rm d}m.$$ Since $B{\geqslant}A$, the opposite estimate is obvious. Our last remark concerns regularity of $H$. Since we will be interested only in rather crude lower and upper bounds for $e_n(\rho)$ under conditions ${\rm d}\rho {\geqslant}e^{-H}\, {\rm d}m$ (for lower bounds) or $\displaystyle \int_{\mathbb T}e^H\, {\rm d}\rho <\infty$ (for upper bounds), our estimates will not distinguish between the sequences $e_n(\rho)$ and $C e_n(\rho)$, and we can always replace the function $H$ by any function $\widetilde{H}$ with $\|\widetilde{H}-H\|{\leqslant}1$ without affecting our estimates. Keeping this in mind, we always assume that, for any positive $C$, the equation $C{\lambda}_H(A)=A$ has a unique solution. In Section \[sec5\] (Theorem \[the9\]) we will be using the same tacit assumption for the equation $C{\lambda}^_H(A)=A$, where ${\lambda}_H^(a)$ is the length of the longest open interval within the set $\{H>a\}$. In the same way, we can always assume that $e_n (\rho)<1/2$. The lower bound for $e_n$ via the Remez-type inequality ======================================================= \[thm:LB\] Suppose ${\rm d}\rho {\geqslant}e^{-H}{\rm d}m$. Then $$\| \log e_n (\rho)\| \lesssim \int_0^A {\lambda}_H(a)\,{\rm d}a,$$ where $A=A(n)$ solves the equation $n {\lambda}_H(A)=A$. \[cor:LB\] Suppose $H$ belongs to the weak $L^1(m)$-space, i.e., ${\lambda}_H(a)\lesssim 1/a$ for $a{\geqslant}1$. Then $e_n(\rho)$ does not decay to zero faster than a negative power of $n$. Similarly, if $H$ belongs to the weak $L^p(m)$-space with $0<p<1$, that is ${\lambda}_H(a) \lesssim a^{-p}$, then $$\| \log e_n(\rho) \| \lesssim n^{\frac{1-p}{1+p}}\,.$$ If ${\lambda}_H(a) \lesssim (\log a)^{-1}$ for $a{\geqslant}e$ (in particular, if $\log H$ is integrable), then $$\| \log e_n(\rho) \| \lesssim \frac{n}{\log^2 n}\,,$$ and so on, until we arrive at the classical Erdős-Turán theorem, which states that $$\|\log e_n(\rho)\|=o(n), \quad n\to\infty,$$ provided that $H<+\infty$ a.e. on ${\mathbb T}$ (that is, ${\lambda}_H(a)\to 0$ as $a\to\infty$). Proof of Theorem \[thm:LB\] --------------------------- Let $P$ be an extremal algebraic polynomial of degree $n$ such that $P(0)=1$ and $$\int_{\mathbb T}\|P\|^2\, {\rm d}\rho = e_n(\rho)^2.$$ Then $$\begin{aligned} 0 &{\leqslant}\int_{\mathbb T}\log\|P\|^2\, {\rm d}m \\ &=\int_{\{H>A\}} \log\|P\|^2\, {\rm d}m + \int_{\{H{\leqslant}A\}} \log\bigl( \|P\|^2 e^{-H} \bigr)\, {\rm d}m + \int_{\{H{\leqslant}A\}} H\, {\rm d}m.\end{aligned}$$ Estimating the first and the second integrals on the RHS we let $n$ be so large that $ \int_{\{H{\leqslant}A\}} H\, {\rm d}m {\geqslant}1$ and $$e^A > \rho({\mathbb T}). \label{ko1}$$ By Jensen’s inequality, $$\begin{aligned} \int_{\{H{\leqslant}A\}} \log\bigl( \|P\|^2 e^{-H}\bigr)\, {\rm d}m &{\leqslant}\frac1e + \log\Bigl( \int_{\{H{\leqslant}A\}} \|P\|^2 e^{-H}\, {\rm d}m \Bigr) \\ &{\leqslant}\frac1e + \log\Bigl( \int_{\mathbb T}\|P\|^2 \, {\rm d}\rho \Bigr) = \frac1e + 2 \log e_n(\rho),\end{aligned}$$ and similarly, $$\begin{aligned} \int_{\{H>A\}} \log\|P\|^2\, {\rm d}m &= {\lambda}_H(A) \Bigl( \frac1{{\lambda}_H(A)}\,\int_{\{H>A\}} \log \|P\|^2\, {\rm d}m \Bigr) \\ &{\leqslant}{\lambda}_H(A) \log\Bigl( \frac1{{\lambda}_H(A)}\,\int_{\{H>A\}} \|P\|^2\, {\rm d}m \Bigr) \\ &{\leqslant}\frac1e + {\lambda}_H(A)\log\Bigl( \int_{\mathbb T}\|P\|^2\, {\rm d}m \Bigr).\end{aligned}$$ Next, applying the $L^2$-version of the classical Remez inequality (which follows, for instance, from a more general Nazarov’s result [@Nazarov]), we obtain $$\begin{aligned} \int_{\mathbb T}\|P\|^2\, {\rm d}m &{\leqslant}e^{Cn{\lambda}_H(A)}\int_{\{H{\leqslant}A\}} \|P\|^2\, {\rm d}m \\ &{\leqslant}e^{Cn{\lambda}_H(A)+A}\int_{\{H{\leqslant}A\}} \|P\|^2e^{-H}\, {\rm d}m \\ &{\leqslant}e^{Cn{\lambda}_H(A)+A}\int_{{\mathbb T}} \|P\|^2\, {\rm d}\rho \\ &{\leqslant}e^{Cn{\lambda}_H(A)+A}\rho({\mathbb T}) \qquad \qquad \bigl( {\rm by\ extremality\ of\ } P, \int_{{\mathbb T}} \|P\|^2\, {\rm d}\rho {\leqslant}\rho({\mathbb T})\, \bigr) \\ &{\leqslant}e^{CA} \quad\qquad\qquad\qquad\qquad ({\rm by\ \eqref{ko1}\ and\ the\ equality\ }n{\lambda}_H(A)=A),\end{aligned}$$ whence, $${\lambda}_H(A)\, \log\Bigl( \int_{\mathbb T}\|P\|^2\, {\rm d}m \Bigr) {\leqslant}C A{\lambda}_H(A).$$ Therefore, $$0 {\leqslant}\frac2e + C A{\lambda}_H(A) + 2\log e_n(\rho) + \int_{\{H{\leqslant}A\}} H\, {\rm d}m,$$ and finally, $$\| \log e_n(\rho)\| \lesssim A{\lambda}_H(A) + \int_{\{H{\leqslant}A\}} H\, {\rm d}m = \int_{\mathbb T}H_A\, {\rm d}m,$$ proving Theorem \[thm:LB\]. $\Box$ The upper bound for $e_n$ via Taylor polynomials of an outer function ===================================================================== We give two upper bounds for $e_n(\rho)$. Both of them are based on the construction of monic polynomials of large degree with a good estimate for the $L^2(\rho)$-norm. The first bound uses Taylor polynomials of an outer function $F$ such that $1/F$ mimics the behaviour of $\rho$. It is better adjusted to the case when the distribution function ${\lambda}_H(a)$ decays relatively fast as $a\to\infty$. The second bound uses classical Chebyshev’s polynomials and starts working only when ${\lambda}_H (a)$ decays at infinity slower than $1/a$. Let ${\varphi}\colon [0, \frac12]\to (0, +\infty]$ be a continuous decreasing function, ${\varphi}(0)=\infty$, ${\varphi}(\tfrac12){\leqslant}\inf_{{\mathbb T}} H$. Given $\tau\in [-\frac12, \frac12]$, denote by $\theta_\tau$ solution to the equation ${\varphi}(\theta_\tau) = H(e^{2\pi{\rm i}\tau})$. We call the function $H$ [subordinated to]{} ${\varphi}$ if, for any $\tau$, $$\begin{aligned} H(e^{2\pi{\rm i}\theta}) &{\leqslant}{\varphi}(\theta+\theta_\tau-\tau), \qquad \tau-\theta_\tau < \theta {\leqslant}\tau, \\ H(e^{2\pi{\rm i}\theta}) &{\leqslant}{\varphi}(\tau+\theta_\tau-\theta), \qquad \tau {\leqslant}\theta < \tau+\theta_\tau.\end{aligned}$$ Note that an equivalent way to express the ${\varphi}$-subordination is to say that the function $({\varphi}^{-1}\circ H)(e^{2\pi{\rm i}\theta})$ is a non-negative Lipschitz function on $[-\frac12, \frac12]$ with the Lipschitz constant at most one. We call the unbounded continuous decreasing function ${\varphi}$ on $(0, \frac12]$ [regular]{} if it satisfies at least one of the following two conditions: $${\rm the\ function\ } \theta\mapsto \theta{\varphi}(\theta) \ {\rm does\ not\ decrease \ and\ } {\varphi}(\theta)\gtrsim \log\frac1{\theta} \eqno({\rm Reg}1)$$ $${\varphi}(\theta/2)\lesssim {\varphi}(\theta) \ {\rm and\ } {\varphi}(\theta) \gtrsim 1/\theta. \eqno({\rm Reg}2)$$ \[thm:UB1\] Suppose that $$\int_{\mathbb T}e^H\, {\rm d}\rho <\infty,$$ with $H$ subordinated to a regular function ${\varphi}$. Then $$\| \log e_n (\rho) \| \gtrsim \int_0^A {\lambda}_H(a)\,{\rm d}a,$$ where $A$ solves the equation $n{\varphi}^{-1}(A)=A$ when ${\varphi}$ satisfies condition [(Reg1)]{}, and $A=\sqrt{n}$ when ${\varphi}$ satisfies condition [(Reg2)]{}. \[cor:UB1\] In the assumptions of Theorem \[thm:UB1\], suppose that $\log {\varphi}(\theta)\gtrsim \log\frac1{\theta}$. If $$\liminf_{a\to\infty} a{\lambda}_H(a) > 0,$$ then $e_n(\rho)$ decay to zero at least as a negative power of $n$. Furthermore, $e_n(\rho)$ decay to zero faster than any negative power of $n$, provided that $$\lim_{a\to\infty} a{\lambda}_H(a) = \infty.$$ Note that we need to impose the additional condition $\log{\varphi}(\theta) \gtrsim \log\tfrac1{\theta}$ in this Corollary only in the case when ${\varphi}$ satisfies the first regularity condition (Reg1). Taylor polynomials ------------------ Denote by ${\mathbb P}_r$ the Poisson kernel for the unit disk evaluated at the point $r\in [0, 1)$. \[lemma\_Taylor\] Let $H$ be a weight such that $$H_A {\mathbb P}_{1-\delta} \lesssim H + 1 \qquad {\rm everywhere\ on\ } {\mathbb T},$$ with $ \log \delta^{-1} \lesssim A $. Suppose that $$\int_{\mathbb T}e^H\, {\rm d}\rho < \infty.$$ Then there exists a positive constant $C$ such that, for $n{\geqslant}CA/\delta$, we have $$\|\log e_n(\rho) \| \gtrsim \int_{\mathbb T}H_A\, {\rm d}m.$$ ### Proof of Lemma \[lemma\_Taylor\] Let $M$ be a positive constant such that $ H_A * {\mathbb P}_{1-\delta} {\leqslant}M(H + 1)$, and let $F_A$ be an outer function in ${\mathbb D}$ with the boundary values $\|F_A\|^2 = e^{H_A/M}$, i.e., $$\log \|F_A(rt)\| = \frac1{2M} \bigl( H_A * {\mathbb P}_r \bigr) (t).$$ We expand $F_A((1-\delta)z)$ into the Taylor series $$F_A((1-\delta)z) = \sum_{k{\geqslant}0} f_k z^k,$$ and consider the Taylor polynomials $$P_A(z) = \sum_{k=0}^n f_k z^k.$$ Then, $$e_n(\rho)^2 {\leqslant}\|P_A(0)\|^{-2}\, \int_{\mathbb T}\|P_A\|^2\, {\rm d}\rho.$$ First, we note that $$\|P_A(0)\| = \|F_A(0)\| = \exp\Bigl( \int_{\mathbb T}\log \|F_A\|\, {\rm d}m \Bigr) = \exp\Bigl( \frac1{2M}\, \int_{\mathbb T}H_A \, {\rm d}m \Bigr),$$ i.e., $$e_n(\rho)^2 {\leqslant}\int_{\mathbb T}\|P_A\|^2\, {\rm d}\rho \cdot \exp\Bigl( - \frac1{M}\, \int_{\mathbb T}H_A \, {\rm d}m \Bigr).$$ Next, $$\int_{\mathbb T}\|F_A((1-\delta)t)\|^2\, {\rm d}\rho(t) = \int_{\mathbb T}\exp \Bigl( \frac1{M} H_A * {\mathbb P}_{1-\delta} \Bigr)\, {\rm d}\rho {\leqslant}\int_{\mathbb T}\exp\bigl( H + 1\bigr)\, {\rm d}\rho \lesssim 1,$$ so it remains to estimate the remainder $$\| F_A((1-\delta)z) - P_A(z) \| {\leqslant}\sum_{k>n} \|f_k\|.$$ By Cauchy’s estimates, $$\|f_k\| {\leqslant}(1-\delta)^{k} \max_{{\mathbb T}} \, \|F_A\| {\leqslant}(1-\delta)^{k} e^{A/(2M)},$$ whence, $$\sum_{k>n} \|f_k\| \lesssim \delta^{-1} e^{A/(2M)-n\delta} \lesssim 1\,,$$ provided that $A/\delta \lesssim n$ (here, we use that $\log \delta^{-1} \lesssim A$). Thus, $$\int_{\mathbb T}\|P_A\|^2\, {\rm d}\rho \lesssim 1,$$ which proves the lemma. $\Box$ Estimates of the Poisson integral --------------------------------- Put $p_\delta (\theta) = {\mathbb P}_{1-\delta}(e^{2\pi{\rm i}\theta})$ and recall that $p_\delta(\theta) \lesssim \min(\delta^{-1}, \delta\theta^{-2})$. \[lemma\_Poisson\_A\] Let ${\varphi}\colon (0, \frac12]\to [0, \infty)$ be an unbounded continuous decreasing function, let $\widetilde{\varphi}$ be its even $1$-periodic extension on ${\mathbb R}$, and $\widetilde{\varphi}_A=\min(\widetilde{\varphi}, A)$. Then $$\widetilde{\varphi}_A * p_\delta \lesssim \widetilde{\varphi}+ 1 \qquad {\rm everywhere\ on\ } {\mathbb R},$$ provided that at least one of the following holds: $\rm (i)$ the function $\theta\mapsto \theta^2{\varphi}(\theta)$ does not decrease, and $\delta\lesssim{\varphi}^{-1}(A)$; $\rm (ii)$ ${\varphi}(\theta/2)\lesssim {\varphi}(\theta)$, ${\varphi}(\theta)\gtrsim 1/\theta$, and $\delta\lesssim 1/A$. Note that condition (i) is weaker than condition (Reg1) in Theorem \[thm:UB1\], i.e., the lemma is a bit stronger than what we will use for the proof of Theorem \[thm:UB1\]. We need this version of Lemma \[lemma\_Poisson\_A\] for the proof of Theorem \[thm:DeepZero\]. We also note that condition (i) yields estimate ${\varphi}(\theta/2)\lesssim{\varphi}(\theta)$ from condition (ii). ### Proof of Lemma \[lemma\_Poisson\_A\] We take a sufficiently small $\tau_0>0$ so that ${\varphi}(\tau_0){\geqslant}1$, fix $\tau\in (0, \tau_0]$, and estimate the convolution $(\widetilde{\varphi}_A p_\delta)(\tau)$. There is nothing to prove if $A{\leqslant}{\varphi}\bigl( \frac12 \tau\bigr)$ since in this case $$(\widetilde{\varphi}_A p_\delta)(\tau) {\leqslant}\max_{[-\frac12, \frac12]} \widetilde{\varphi}_A = A {\leqslant}{\varphi}\bigl( \tfrac12 \tau \bigr) \lesssim {\varphi}(\tau)$$ for any $\delta>0$. Hence, in what follows, we assume that $A{\geqslant}{\varphi}\bigl( \frac12 \tau\bigr)$, i.e., $\tau{\geqslant}2{\varphi}^{-1}(A)$. First, we note that for any $\theta\in [0, \frac12]$, we have $\widetilde{\varphi}_A(\tau+\theta) {\leqslant}\widetilde{\varphi}_A(\tau-\theta)$ (to see this, one needs to consider three cases: $0{\leqslant}\theta{\leqslant}\tau$, $\tau{\leqslant}\theta{\leqslant}\frac12-\tau$, and $\frac12-\tau{\leqslant}\theta{\leqslant}\frac12$). Therefore, $$\begin{gathered} \int_{-\frac12}^{\frac12} \widetilde{\varphi}_A(\tau-\theta) p_\delta(\theta)\, {\rm d}\theta = \int_0^{\frac12} \bigl[ \widetilde{\varphi}_A(\tau-\theta) + \widetilde{\varphi}_A(\tau+\theta) \bigr] p_\delta(\theta) \, {\rm d}\theta \\ {\leqslant}2A\, \int_{\|\tau-\theta\|{\leqslant}{\varphi}^{-1}(A)} p_\delta(\theta)\, {\rm d}\theta + 2\, \int_ {\substack{\|\theta-\tau\|{\geqslant}{\varphi}^{-1}(A), \\ 0{\leqslant}\theta{\leqslant}\frac12}} {\varphi}(\tau-\theta) p_\delta(\theta)\, {\rm d}\theta = I+ II.\end{gathered}$$ Before we start estimating integrals on the RHS, observe that $\delta\lesssim\tau$. In the case (i) it is obvious since $\delta\lesssim {\varphi}^{-1}(A){\leqslant}\tau/2$, in the case (ii) it is also obvious since then $\delta\lesssim 1/A \lesssim {\varphi}^{-1}(A) {\leqslant}\tau/2$. Therefore, in the first integral $\theta {\geqslant}\tau-{\varphi}^{-1}(A) {\geqslant}\tau/2 \gtrsim \delta$. Recalling the standard estimate of the Poisson kernel $p_\delta(\theta) \lesssim \min (\delta^{-1}, \delta \theta^{-2}) $, we get $$I \lesssim A\delta\, \int_{\tau-{\varphi}^{-1}(A)}^{\tau+{\varphi}^{-1}(A)} \frac{{\rm d}\theta}{\theta^2} \lesssim \frac{A\delta{\varphi}^{-1}(A)}{\tau^2}.$$ In both cases (i) and (ii) the RHS is $\lesssim{\varphi}(\tau)$. Indeed, if (i) holds, then it is bounded by $ {\varphi}^{-1}(A)^2 A/\tau^2 {\leqslant}\tau^2{\varphi}(\tau)/\tau^2 = {\varphi}(\tau) $. If (ii) holds, then it is bounded by $ {\varphi}^{-1}(A)/\tau^2 \lesssim 1/\tau \lesssim {\varphi}(\tau) $. We split the second integral into four parts $$\int_{\substack{\|\theta-\tau\|{\geqslant}{\varphi}^{-1}(A), \\ 0{\leqslant}\theta{\leqslant}\frac12}} = \int_0^{\min(\delta, \frac12 \tau)} + \int_{\min(\delta, \frac12 \tau)}^{\frac12 \tau} + \int_{{\varphi}^{-1}(A){\leqslant}\|\theta-\tau\| {\leqslant}\frac12 \tau} + \int_{\frac32 \tau}^{1/2}$$ and estimate them one by one. We have $$\begin{gathered} \int_0^{\min(\delta, \frac12 \tau)} {\varphi}(\tau-\theta) p_\delta(\theta)\, {\rm d}\theta \lesssim \frac1{\delta}\, \int_0^{\min(\delta, \frac12 \tau)} {\varphi}(\tau-\theta)\, {\rm d}\theta \\ {\leqslant}\frac1{\delta}\, \min(\delta, \tfrac12 \tau) \cdot {\varphi}(\tfrac12 \tau) \lesssim {\varphi}(\tau),\end{gathered}$$ and $$\begin{gathered} \int_{\min(\delta, \frac12 \tau)}^{\frac12 \tau} {\varphi}(\tau-\theta) p_\delta(\theta)\, {\rm d}\theta \lesssim \delta\, \int_{\min(\delta, \frac12 \tau)}^{\frac12 \tau} \frac{{\varphi}(\tau-\theta)}{\theta^2} \, {\rm d}\theta \\ \lesssim \delta \cdot {\varphi}(\tfrac12 \tau)\, \int_{\min(\delta, \frac12 \tau)}^\infty \frac{{\rm d}\theta}{\theta^2}\, \stackrel{\delta\lesssim\tau}\lesssim \, {\varphi}(\tau).\end{gathered}$$ Next, $$\begin{gathered} \int_{{\varphi}^{-1}(A){\leqslant}\|\theta-\tau\| {\leqslant}\frac12 \tau} {\varphi}(\tau-\theta) p_\delta(\theta)\, {\rm d}\theta \\ \lesssim \delta\, \int_{{\varphi}^{-1}(A){\leqslant}\|\theta-\tau\| {\leqslant}\frac12 \tau} \frac{{\varphi}(\tau-\theta)}{\theta^2}\, {\rm d}\theta \simeq \frac{\delta}{\tau^2}\, \int_{{\varphi}^{-1}(A)}^{\frac12 \tau} {\varphi}(\xi)\, {\rm d}\xi.\end{gathered}$$ In the case (i), the integral on the RHS equals $$\frac{\delta}{\tau^2}\, \int_{{\varphi}^{-1}(A)}^{\frac12 \tau} \frac{\xi^2 {\varphi}(\xi)}{\xi^2}\, {\rm d}\xi {\leqslant}\frac{\delta}{\tau^2}\, \frac{(\tau/2)^2 {\varphi}(\tau/2)}{{\varphi}^{-1}(A)} \, \stackrel{\delta\lesssim {\varphi}^{-1}(A)}\lesssim\, {\varphi}(\tau),$$ while in the case (ii), it does not exceed $$\frac{\delta}{\tau^2} \cdot A\tau/2 \, \stackrel{\delta\lesssim 1/A}\lesssim \, \frac1{\tau} \, \stackrel{\tau{\varphi}(\tau)\gtrsim 1}\lesssim \, {\varphi}(\tau).$$ At last, $$\int_{\frac32 \tau}^{1/2} {\varphi}(\tau-\theta) p_\delta(\theta)\, {\rm d}\theta \lesssim \delta \int_{\frac32 \tau}^{\frac12} \frac{{\varphi}(\theta - \tau)}{\theta^2}\, {\rm d}\theta \lesssim \delta {\varphi}(\tau/2) \cdot \frac1\tau \, \stackrel{\delta\lesssim\tau}\lesssim\, {\varphi}(\tau),$$ completing the proof of the lemma. $\Box$ ### The Poisson integral of $H_A$ \[lemma:Poisson\] Let $H$ be subordinated to a regular function ${\varphi}$, and $H_A=\min (H, A)$. Then $$H_A * {\mathbb P}_{1-\delta} \lesssim H+1 \quad{\rm everywhere\ on\ } {\mathbb T},$$ provided that $\delta\lesssim{\varphi}^{-1}(A)$ when ${\varphi}$ satisfies condition [(Reg1)]{}, and $\delta\lesssim A^{-1}$ when ${\varphi}$ satisfies condition [(Reg2)]{}. Clearly, Lemma \[lemma\_Taylor\] and Lemma \[lemma:Poisson\] combined together yield Theorem \[thm:UB1\]. #### Proof of Lemma \[lemma:Poisson\] We write $H(e^{2\pi{\rm i}\theta})=h(\theta)$, fix the point $\tau\in [-\frac12, \frac12]$ with $h(\tau)<\infty$ at which we will estimate the convolution $(h_A * p_\delta)(\tau)$, and choose $\theta_\tau$ so that ${\varphi}(\theta_\tau)=h(\tau)$. Similarly to the proof of the previous lemma, we assume that $A {\geqslant}{\varphi}(\frac12 \theta_\tau)$, i.e., that ${\varphi}^{-1}(A) {\leqslant}\frac12 \theta_\tau$; otherwise, $$(h_A * p_\delta)(\tau) {\leqslant}A {\leqslant}{\varphi}(\tfrac12 \theta_\tau) \lesssim {\varphi}(\theta_\tau) = h(\tau),$$ and we are done. Now, $$(h_A * p_\delta)(\tau) = \Bigl(\, \int_{\|\theta-\tau\|{\geqslant}\theta_\tau-{\varphi}^{-1}(A)} + \, \int_{\|\theta-\tau\|{\leqslant}\theta_\tau-{\varphi}^{-1}(A)} \, \Bigr)\, h_A(\theta) p_\delta(\tau-\theta)\, {\rm d}\theta = I + II.$$ To estimate the first integral, we note that, since $\theta_\tau-{\varphi}^{-1}(A){\geqslant}\theta_\tau/2$, we have $$I {\leqslant}A \int_{\|\theta-\tau\|{\geqslant}\theta_\tau/2} \, p_\delta(\tau-\theta)\, {\rm d}\theta {\leqslant}2A \int_{\theta_\tau/2}^{1/2} p_\delta (\theta)\, {\rm d}\theta \lesssim A\delta \int_{\theta_\tau/2}^\infty \frac{{\rm d}\theta}{\theta^2} \lesssim \frac{A\delta}{\theta_\tau}\,.$$ In the first case, the RHS is $$\, \stackrel{\delta\lesssim {\varphi}^{-1}(A)}\lesssim\, \frac{A{\varphi}^{-1}(A)}{\theta_\tau} \, \stackrel{{\varphi}^{-1}(A){\leqslant}\frac12 \theta_\tau}\lesssim \, \frac{\frac12 \theta_\tau {\varphi}(\frac12 \theta_\tau)}{\theta_\tau} \lesssim {\varphi}(\theta_\tau).$$ In the second case, $A\delta/\theta_\tau \lesssim 1/\theta_\tau \lesssim {\varphi}(\theta_\tau)$. Therefore, in both cases, the first integral is $\lesssim {\varphi}(\theta_\tau)=h(\tau)$. To estimate the second integral, we note that, by the subordination to ${\varphi}$, it is bounded by $ 2 (\widetilde{\varphi}_A * p_\delta)(\theta_\tau) $, which, by the previous lemma, is $\lesssim {\varphi}(\theta_\tau)+1 = h(\tau)+1$. $\Box$ The upper bound for $e_n$ via Chebyshev polynomials {#sec5} =================================================== Here we assume that the function $H$ is lower semicontinuous; i.e., the sets $\{H\!>\!a\}$ are open, and denote by ${\lambda}^_H(a)$ the length of the longest open interval within $\{H\!>\!a\}$. \[thm:UB2\]\[the9\] Suppose that $$\int_{\mathbb T}e^H\, {\rm d}\rho <\infty.$$ Then $$\| \log e_n (\rho) \| \gtrsim A {\lambda}_H^(A)\,,$$ where $A=A(n)$ solves the equation $n{\lambda}_H^(A)=A$. The following Corollary combines Theorem \[thm:UB2\] with Theorem \[thm:LB\] (and takes into account Observations 2.1 and 2.3) \[cor:UB2\] Let ${\rm d}\rho = e^{-H}\, {\rm d}m $. Suppose that the set $\{H>a\}$ contains an interval with the length comparable to the total length of $\{H>a\}$ (i.e. ${\lambda}_H^ \gtrsim {\lambda}_H$), and that the function ${\lambda}_H$ satisfies $$\limsup_{a\to\infty} \frac{{\lambda}_H(a)}{{\lambda}_H(2a)} <2.$$ Then $$\| \log e_n (\rho)\| \simeq A\lambda_H(A)\,,$$ where $A=A(n)$ is a solution to the equation $n{\lambda}_H(a)=a$. Proof of Theorem \[thm:UB2\] ---------------------------- We will use the following classical lemma (cf., for instance, [@MR]): \[lemma:Chebyshev-Maergoiz\] For any $\alpha\in (0, \frac{\pi}2 ]$ and for any $n\in 2{\mathbb N}$, there exists a monic polynomial $T_{n,\alpha}$ of degree $n$ such that $$\max\{\|T_{n,\alpha}(e^{i\theta})\|\colon \|\theta\| {\geqslant}\alpha \} = 2\cos^n(\alpha/2)\,.$$ For the reader’s convenience, we recall its proof. Put $$T_{2m,\alpha}(e^{i\theta}) = 2\cos^{2m}(\alpha/2)e^{im\theta} \cos\left( 2m\arccos\left(\frac{\cos(\theta/2)}{\cos(\alpha/2)} \right)\right)\,.$$ We only need to show that this is a monic polynomial of degree $n=2m$. Recall that $\cos(2m\arccos x) = 2^{2m-1}Q_m(x^2)$, where $Q_m$ is a monic polynomial of degree $m$. Then $$\begin{aligned} T_{2m,\alpha}(e^{i\theta}) &= 2^{2m} \cos^{2m}(\alpha/2)e^{im\theta} Q_m\left(\frac{\cos^2\theta/2}{\cos^2\alpha/2}\right) \\ &= 2^{2m}\cos^{2m}(\alpha/2)e^{im\theta} Q_m\left(\frac{e^{i\theta}+e^{-i\theta}+2}{4\cos^2\alpha/2}\right)\end{aligned}$$ and it is easily seen that the RHS is a monic polynomial of degree $2m$, proving the lemma. $\Box$ Now, we turn to the proof of Theorem \[thm:UB2\]. Without loss of generality, we assume that $n$ is an even number. Let $J\subset{\mathbb T}$ be the longest arc in the set $\{H>a\}$. We assume that $J=\{t=e^{{\rm i}\varphi}\colon \|\varphi\|{\leqslant}\alpha \}$, $\alpha = \pi{\lambda}_H^(a)$. Let $T=T_{n, \alpha}$ be a monic polynomial of degree $n$ as in Lemma \[lemma:Chebyshev-Maergoiz\]. Then, by a straightforward computation (or by the classical Remez inequality) $$\max_J \|T\| = \max_{\|\varphi\|{\leqslant}\alpha} \left\| T(e^{i\varphi})\right\| \lesssim e^{Cn\alpha} \, \bigl( \cos\frac{\alpha}{2} \bigr)^n\,.$$ Noting that $ \cos^n\tfrac{\alpha}{2} {\leqslant}e^{-cn\alpha^2}$, we get $$\begin{gathered} \int_{\mathbb T}\|T\|^2\, {\rm d}\rho {\leqslant}\max_{{\mathbb T}} \bigl( \|T\|^2e^{-H} \bigr)\, \int_{\mathbb T}e^H\, {\rm d}\rho \\ \lesssim e^{-a}\, \max_J \|T\|^2 + \max_{{\mathbb T}\setminus J} \|T\|^2 \lesssim \Bigl( e^{Cn{\lambda}_H^(a) - a} + 1 \Bigr) e^{-cn{\lambda}_H^(a)^2}.\end{gathered}$$ Letting $A_C$ be the unique solution to the equation $Cn {\lambda}_H^(A_C)=A_C$, we obtain $$\| \log e_n(\rho) \| \gtrsim n {\lambda}_H^(A_C)^2 \simeq A_C{\lambda}_H^(A_C) \simeq A{\lambda}_H^(A),$$ completing the proof of Theorem \[thm:UB2\]. $\Box$ Examples ======== To illustrate our results, we consider the function $H=h\circ d_K$, where $h\colon (0, \frac12]\to (0, +\infty)$ is a $C^1$-smooth decreasing function, $h(0)=+\infty$, and $d_K(t)=\operatorname{dist}(t, K)$, where $K\subset {\mathbb T}$ is a compact set of zero length (recall that we identify ${\mathbb T}$ with ${\mathbb R}/{\mathbb Z}$). Denote by $K_{+s}=\{t\colon d_K(t)<s \}$ the $s$-neighbourhood of $K$ and by $\psi_K(s) = m(K_{+s})$ its length. Then $${\lambda}_H(a) = m\{H>a\} = \begin{cases} \psi_K(h^{-1}(a)), &a{\geqslant}h(\tfrac12) \\ 1, &a<h(\tfrac12), \end{cases}$$ and $$\int_0^A {\lambda}_H(a)\, {\rm d}a= \Bigl(\int_0^{h(1/2)}+\int_{h(1/2)}^A\Bigr) {\lambda}_H(a)\, {\rm d}a= \int_{h^{-1}(A)}^{1/2} \psi_K \|h'\| + h(\tfrac12),$$ provided that $A>h(\frac12)$. To estimate the function $\psi_K$ it is convenient to use that $ \psi_K(s) \simeq s N_K(s) \simeq sP_K(s) $, where $N_K(s)$ is the covering number of $K$ and $P_K(s)$ is the packing number of $K$, see, for instance, [@Falc Chapter 3]. We call the set $K$ $\gamma$-regular if $\psi_K(s) \simeq s^{1-\gamma}$. For instance, the set $e^{2\pi {\rm i}\mathcal C}$, where $\mathcal C$ is the standard ternary Cantor set is $\gamma$-regular with $\gamma = \frac{\log 2}{\log 3}$, while the set $\{t=\exp(2\pi {\rm i} n^{-\nu})\colon n\in{\mathbb N}\} \cup \{1\}$ is $\gamma$-regular with $\gamma=(\nu+1)^{-1}$. Two corollaries --------------- We get straightforward corollaries to our results taking $h(s)=s^{-p}$. \[cor:d\_K2\] Let $K\subset{\mathbb T}$ be a $\gamma$-regular compact set with some $\gamma\in [0, 1)$. Suppose that ${\rm d}\rho = \exp \bigl( - d_K^{\gamma-1} \bigr)\, {\rm d}m$. Then $ \| \log e_n(\rho)\| \simeq \log n$. The second corollary pertains to the case when the length of the longest interval in the set $\{d_K<s\}$ is comparable with the length of the whole set $\{d_K<s\}$. Then Corollary \[cor:UB2\] applies. \[cor:d\_K3\] Let $\nu>0$, $K=\{t=\exp(2\pi {\rm i}n^{-\nu})\colon n\in{\mathbb N}\} \cup \{1\}$, and ${\rm d}\rho = \exp \bigl( - d_K^{-p} \bigr)\, {\rm d}m$ with $p>\vartheta = \frac{\nu}{\nu+1}$. Then $$\|\log e_n (\rho)\| \simeq n^{\frac{p-\vartheta}{p+\vartheta}}.$$ Measures with deep zero at one point ------------------------------------ The last illustration to our estimates pertains to the simplest case when the measure $\rho$ has a deep zero at one point and is symmetric with respect to this point. In this case, our estimates yield a relatively complete result. \[thm:DeepZero\] Let $h\colon (0, \frac12]\to [0, +\infty)$ be a continuous decreasing function such that $$\int_0 h(a)\,{\rm d}a = +\infty.$$ Suppose that $h$ satisfies at least one of the following two conditions: - $$\theta\mapsto \theta^2 h(\theta) \quad {\rm does\ not\ decrease},$$ and $$\|\log \theta\| = O(h(\theta)),\qquad \theta\to 0;$$ - $$\limsup_{a\to\infty} \frac{h^{-1}(a)}{h^{-1}(2a)} < 2$$ Let $\rho$ be an absolutely continuous measure on ${\mathbb T}$ with density $e^{-h(\|\theta\|)}$. Then $$\|\log e_n (\rho)\| \simeq \int_0^{1/2} h_A(a)\,{\rm d}a$$ where $A$ solves the equation $n h^{-1}(A)=A$ and $h_A=\min(h, A)$. The lower bound for $e_n$ (i.e., the upper bound for $\|\log e_n\|$) follows from Theorem \[thm:LB\] and does not need any regularity assumptions on $h$. Conditions (i) and (ii) are needed for the proof of the upper bound for $e_n$. In the case (i), it is a consequence of Lemma \[lemma\_Taylor\] combined with the first case of Lemma \[lemma\_Poisson\_A\]. In the case (ii), it follows from Theorem \[thm:UB2\]. Note that these two cases overlap, e.g., the function $h(\theta)=\theta^{-p}$ with $1<p{\leqslant}2$ satisfies both of them. The following corollary gives an idea about the rate of decay of $e_n(\rho)$ for several explicitly written functions $h$. Let $\rho$ be an absolutely continuous measure on ${\mathbb T}$ with density $e^{-h(\|\theta\|)}$. Then for $n{\geqslant}4$ we have - If $h(\theta)\simeq \theta^{-1}\log^{-1}(1/\theta)$, then $\|\log e_n (\rho)\| \simeq \log\log n$, - If $p>-1$ and $h(\theta)\simeq \theta^{-1}\log^p(1/\theta)$, then $\|\log e_n (\rho)\| \simeq (\log n)^{p+1}$, - If $p>1$ and $h(\theta)\simeq \theta^{-p}$, then $\|\log e_n (\rho)\| \simeq n^{(p-1)/(p+1)}$, - If $p>0$ and $h(\theta)\simeq \exp(\theta^{-p})$, then $\|\log e_n (\rho)\| \simeq n(\log n)^{-2/p}$. Our last remark is that, plausibly, the technique based on the potential theory in the external field developed by Mhaskar–Saff, Rakhmanov, Levin–Lubinsky, Totik and others should allow one to obtain more precise estimates of $e_n$ in the situation considered in Theorem \[thm:DeepZero\]. See for instance, Theorem 1.22 and Examples 3 and 4 in Section 1.6 in [@LL] which contain similar results for orthogonal polynomials on the real line. On the other hand, likely, this will require much stronger regularity assumptions on the function $h$ and more technical proofs. [A]{} A. Borichev, M. Sodin, B. Weiss, Spectra of stationary processes on ${\mathbb Z}$. In: $50$ years with Hardy spaces, 141–157, Oper. Theory Adv. Appl., [261]{}, Birkhäuser-Springer, 2018. K. Falconer, Fractal geometry. Mathematical foundations and applications. Third edition. John Wiley & Sons, 2014. B. L. Golinskii, The asymptotic behavior of the prediction error. Teor. Verojatnost. i Primenen. [19]{} (1974), 724–739. U. Grenander, G. Szegő, Toeplitz forms and their applications. Univ. California Press, 1958. I. A. Ibragimov, On the asymptotic behaviour of the prediction error. Teor. Verojatnost. i Primenen. [9]{} (1964), 695–703. E. Levin, D. Lubinsky, Orthogonal polynomials for exponential weights. CMS Books in Mathematics, 4, Springer–Verlag, New York, 2001. L. S. Maergoiz, N. N. Rybakova, Chebyshev polynomials with zeros on the circle and related problems. St. Petersburg Math. J. [25]{} (2014), 965–979. F. Nazarov, Complete version of Turán’s lemma for trigonometric polynomials on the unit circumference. In: Complex analysis, operators, and related topics, 239–246, Oper. Theory Adv. Appl., [113*]{}, Birkhäuser, 2000. B. Simon, Orthogonal Polynomials on the Unit Circle. Part 1: Classical Theory. Amer. Math. Soc., 2005. H. Stahl, V. Totik, General Orthogonal Polynomials. Cambridge Univ. Press, 1992. [A.B.: Institut de Mathématiques de Marseille, Aix Marseille Université, CNRS, Centrale Marseille, I2M, Marseille, France A.K.: Department of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, Russia M.S.: School of Mathematics, Tel Aviv University, Tel Aviv, Israel ]{} [^1]: Supported by a joint grant of Russian Foundation for Basic Research and CNRS (projects 17-51-150005-NCNI-a and PRC CNRS/RFBR 2017-2019) and by the project ANR-18-CE40-0035. [^2]: Supported by a joint grant of Russian Foundation for Basic Research and CNRS (projects 17-51-150005-NCNI-a and PRC CNRS/RFBR 2017-2019). [^3]: Supported by ERC Advanced Grant 692616 and ISF Grant 382/15.	ArXiv
--- abstract: 'In the simplified setting of the Schwinger model we present a systematic study on the simulation of dynamical fermions by global accept/reject steps that take into account the fermion determinant. A family of exact algorithms is developed, which combine stochastic estimates of the determinant ratio with the exploitation of some exact extremal eigenvalues of the generalized problem defined by the ‘old’ and the ‘new’ Dirac operator. In this way an acceptable acceptance rate is achieved with large proposed steps and over a wide range of couplings and masses.' author: - \| Francesco Knechtli and Ulli Wolff[^1]\ Institut für Physik, Humboldt Universität\ Newtonstr. 15\ 12489 Berlin, Germany title: Dynamical fermions as a global correction --- 0.5 cm HU-EP-03/12\ SFB/CCP-03-07 [^1]: e-mail: [email protected], [email protected]	ArXiv
--- abstract: \| The results in [@O2] (see [@O1] for the quasistatics regime) consider the Helmholtz equation with fixed frequency $k$ and, in particular imply that, for $k$ outside a discrete set of resonant frequencies and given a source region $D_a\subset \RR^d$ ($d=\overline{2,3}$) and $u_0$, a solution of the homogeneous scalar Helmholtz equation in a set containing the control region $D_c\subset \RR^d$, there exists an infinite class of boundary data on $\partial D_a$ so that the radiating solution to the corresponding exterior scalar Helmholtz problem in $\RR^d\setminus D_a$ will closely approximate $u_0$ in $D_c$. Moreover, it will have vanishingly small values beyond a certain large enough “far-field" radius $R$ (see Figure \[fig:mainsetup\] for a geometric description). In this paper we study the minimal energy solution of the above problem (e.g. the solution obtained by using Tikhonov regularization with the Morozov discrepancy principle) and perform a detailed sensitivity analysis. In this regard we discuss the stability of the the minimal energy solution with respect to measurement errors as well as the feasibility of the active scheme (power budget and accuracy) depending on: the mutual distances between the antenna, control region and far field radius $R$, value of regularization parameter, frequency, location of the source. author: - Mark Hubenthal - Daniel Onofrei bibliography: - 'controlpaper.bib' title: Sensitivity analysis for active control of the Helmholtz equation --- Introduction ============ During recent years, there has been a growing interest in the development of feasible strategies for the control of acoustic and electromagnetic fields with one possible application being the construction of robust schemes for sonar or radar cloaking. One main approach controls fields in the regions of interest by changing the material properties of the medium in certain surrounding regions ([@Chan3; @Chan2; @Cummer; @Green2; @Green1; @Gunther-pr; @Pendry] and references therein). Several alternative techniques are proposed in the literature (other than transformation optics strategies) such as: plasmonic designs (see [@Alu] and references therein), strategies based on anomalous resonance phenomena (see [@Mil1; @Mil3; @Mil2]), conformal mapping techniques (see [@Ulf2; @Ulf1]), and complementary media strategies (see [@Chan]). In the applied community, active designs for the manipulation of fields appear to have occurred initially in the context of low-frequency acoustics (or active noise cancellation). Especially notable are the pioneering works of Lueg [@Lueg] (feed-forward control of sound) and Olson & May [@Olson-May] (feedback control of sound). The reviews [@Elliot; @Fuller; @Tsynkov; @T1; @Peake; @Peterson], provide detailed accounts of past and recent developments in acoustic active control. In the context of cloaking, the [*[interior]{}]{} strategy proposed in [@Miller] employs a continuous active layer on the boundary of the control region while the [[exterior]{}]{} scheme discussed in [@OMV4; @OMV1; @OMV2; @OMV3] (see also [@CTchan-num]), uses a discrete number of active sources located in the exterior of the control region to manipulate the fields. The active exterior strategy for 2D quasistatics cloaking was introduced in [@OMV1], and, based on a priori* information about the incoming field, the authors constructively described how one can create an almost zero field control region with very small effect in the far field. However, the proposed strategy did not work for control regions close to the active source. It “cloaked" large objects only when they are far enough from the source region (see [@OMV4]) and was not adaptable to three space dimensions. The finite frequency case was studied in the last section of [@OMV1] and in [@OMV3] (see also [@OMV4] for a recent review) where three (or four in 3D) active sources were needed to create a zero field region in the interior of their convex hull, while creating a very small scattering effect in the far field. The broadband character of the proposed scheme was numerically observed in [@OMV2]. All the above results were obtained assuming large amplitude and highly oscillatory currents on the active source regions. In this regard, in [@Norris] (see also [@Devaney0; @Miller]) the authors presented theoretical and numerical evidence that increasing the number of sources will decrease the power needed on each source and thus increase the feasibility of the scheme. Experimental designs and testing of active cloaking schemes in various regimes are reported in [@Du; @Ma; @Eleftheriades1; @Eleftheriades2]. In a recent development in [@O1], a general analytical approach based on the theory of boundary layer potentials is proposed for the active control problem in the quasi-static regime. By using the same integral equation approach, in [@O2] we extended the results presented in [@O1] to the active control problem for the exterior scalar Helmholtz equation. In particular, we characterized an infinite class of boundary functions on the source boundary $\partial D_a$ so that we achieve the desired manipulation effects in several mutually disjoint exterior regions. The method is novel in the sense that instead of using microstructures, exterior active sources modeled with the help of the above boundary controls are employed for the desired control effects. Such exterior active sources can represent velocity potential, pressure or currents. In the current paper we study the active control problem in the context of cloaking, where one antenna $D_a$ protects a given control region $D_c$ from far field interrogation on $\partial B_{R}(\mathbf{0})$, with $R \gg 1$ (see Figure \[fig:mainsetup\]). We make use of the results in [@O2] and present a detailed sensitivity and feasibility study for the minimal norm solution of the problem. The paper is organized as follows: In Section \[sec:background\] we recall the general result obtained in [@O2] in the context of exterior active cloaking. In Section \[sec:stability\] we present an $L^2$ conditional stability result for the minimal norm solution with respect to measurement errors of the incoming field. In Section \[sec:numerics\] we present the numerical details of the Tikhonov regularization algorithm with the Morozov discrepancy principle for the computation of the minimal norm solution of the exterior active cloaking problem in two dimensions. We will numerically observe the fact that the scheme requires large antenna powers in the far field and we will provide numerical support for our theoretical stability results. An important part of this section will be focused on the sensitivity analysis, where we will study: the dependence of the control results as a function of mutual distances between the antenna, control region and far field region; and the broadband character of our scheme in the near field region. Finally, in Section \[sec:conclusions\] we highlight the main results of the paper and discuss current and future challenges and extensions of our research. Background {#sec:background} ========== In this section we will recall the main result regarding the active exterior control problem for the Helmholtz equation obtained in [@O2]. We will focus only on the case where one active external source (antenna) $D_a$ protects a control region $D_c$ from an interrogating far field and maintains an overall small signature beyond a disk of large enough radius $R$. The general setup for this question will be as follows. Let $B_{R} \subset \mathbb{R}^{d}$ be the ball of radius $R > 0$. We assume $\B0\in D_{a} \subset B_{R}$ is the region inside a single antenna with sufficiently smooth boundary $\partial D_{a}$. We also let $D_{c} \Subset B_{R}$ be the control region, which is assumed to satisfy $\overline{D_{c}} \cap \overline{D_{a}} = \emptyset$ (see Figure \[fig:mainsetup\]). The numerical simulations in the current work are performed for the two dimensional case but the methods are adaptable to the three dimensional setting as well. Consider the function space $$\Xi = L^{2}(\partial D_{c}) \times L^{2}(\partial B_{R}),$$ endowed with the scalar product $$(\phi,\psi)_{\Xi} = \int_{\partial D_{c}}\phi_{1}(\mathbf{y})\overline{\psi}_{1}(\mathbf{y})\,dS_{\mathbf{y}} + \int_{\partial B_{R}} \phi_{2}(\mathbf{y})\overline{\psi}_{2}(\mathbf{y})\,dS_{\mathbf{y}},$$ which is a Hilbert space. For the remainder of the paper we will assume that every $L^2$ space of complex valued functions will be endowed with the usual inner product. As in [@O2] consider $K: L^{2}(\partial D_{a}) \to \Xi$, the double layer potential operator restricted to $\partial D_{c}$ and $\partial B_{R}$, respectively, defined by $$\label{EQ:K} K\phi(\Bx,\Bz) = (K_{1}\phi(\Bx), K_{2}\phi(\Bz)), \quad \phi \in L^{2}(\partial D_{a}),$$ where $$\begin{aligned} K_1:L^{2}(\partial D_{a})\rightarrow L^{2}(\partial D_{c}), \, K_1\phi(\Bx) & = \int_{\partial D_{a}}\phi(\By)\frac{\partial \Phi(\Bx,\By)}{\partial {{\mbox{\boldmath ${\nu}$}}}_{\By}}ds_{\By}, \mbox{ for }\Bx\in \partial D_c, \notag\\ K_2:L^{2}(\partial D_{a})\rightarrow L^{2}(\partial B_R), \, K_2\phi(\Bz) & = \int_{\partial D_{a}}\phi(\By)\frac{\partial \Phi(\Bz,\By)}{\partial {{\mbox{\boldmath ${\nu}$}}}_{\By}}ds_{\By}, \mbox{ for }\Bz\in \partial B_R(\B0). \label{8}\end{aligned}$$ Here $\Phi(\Bx, \By)$ represents the fundamental solution of the relevant Helmholtz operator, i.e., $$\label{9} \Phi(\Bx,\By)=\vspace{0.15cm}\left\{\begin{array}{ll} \vspace{0.15cm}\displaystyle\frac{e^{ik\|\Bx-\By\|}}{4\pi\|\Bx-\By\|}, \mbox{ for } d=3\vspace{0.15cm}\\ \frac{i}{4}H_0^{(1)}(k\|\Bx-\By\|) , \mbox{ for } d=2\end{array}\right.$$ with $H_{0}^{(1)} = J_{0} + iY_{0}$ representing the Hankel function of first type. Note that in the integrals are to be understood as singular integrals defined through an operator extension from $C(\partial D_a)$. We will also consider $k$ such that $$\begin{aligned} \label{ass-lambda-0} && \mathbf{1)} \, -k^2 \mbox{ is not a Neumann eigenvalue for the Laplace operator in $D_a$ or $B_R(\B0)$},\nonumber\\ && \mathbf{2)} \, -k^2 \mbox{ is not a Dirichlet eigenvalue for the Laplace operator in $D_c$}.\end{aligned}$$ As in [@O2] we introduce the adjoint operator $K^{}: \Xi \to L^{2}(\partial D_{a})$, which can be shown to satisfy $$K^{}\psi(\mathbf{x}) = \int_{\partial D_{c}} \psi_{1}(\mathbf{y}) \overline{\frac{\partial \Phi(\mathbf{y},\mathbf{x})}{\partial \nu_{\mathbf{x}}}}\,dS_{\mathbf{y}} + \int_{\partial B_{R}} \psi_{2}(\mathbf{y}) \overline{\frac{\partial \Phi(\mathbf{y},\mathbf{x})}{\partial \nu_{\mathbf{x}}}}\, dS_{\mathbf{y}}, \quad \mathbf{x} \in \partial D_{a}. \label{eq:dlpotentialadjoint}$$ This paper proposes a sensitivity study for the following problem: Let $V\Subset D_c$ and $R'>R$. For a fixed wave number $k > 0$ and fixed $0 < \mu \ll 1$, find a function $h \in C(\partial D_{a})$ such that there exists $u \in C^{2}(\RR^{n} \setminus \overline{D_{a}}) \cap C^{1}(\RR^{n} \setminus D_{a})$ solving $$\left\{ \begin{array}{rl} (\Delta + k^{2}) u(\mathbf{x}) & = 0 \quad \mathbf{x} \in \RR^{n} \setminus \overline{D_{a}}\\ u & = h \quad \textrm{ on $\partial D_{a}$}\\ \\|u - f_{1}\\|_{C(\overline{V})} & = \mathcal{O}(\mu)\mbox { and }\\|u\\|_{C(\RR^{n} \setminus B_{R'}(\mathbf{0}))} = \mathcal{O}(\mu), \end{array} \right. \label{eq:inverseproblem}$$ where $f_{1}$ is a solution of the Helmholtz equation in a neighborhood of the control region $D_{c}$. In fact, by using the operator $K$ and regularity arguments it is shown in [@O2] that a class of solutions for problem can be obtained by considering the following problem: for a fixed wave number $k>0$ satisfying conditions , a given function $f=(f_1,0) \in \Xi$ and $\mu > 0$, find a density function $\phi \in C(\partial D_{a})$ such that $$\\|K\phi - f\\|_{\Xi} \leq \mu. \label{eq:controlinequality}$$ Problem is in fact a Fredholm integral equation of the first kind, and it was studied in a very general setting in [@O2]. There the authors proved that the bounded and compact operator $K$ is also one-to-one and has a dense (but not closed) range, thus proving the existence of a class of solutions for (and thus for ). However, the fact that $K$ is compact and that its range is not closed also implies that problem is ill-posed. By using regularization, one can approximate a solution to problem with an arbitrary level of accuracy $\mu \ll 1$. There are several methods known in the literature, but we will use in this paper the Tikhonov regularization method [@reg1; @reg2]. This method, when applied to the operator $K:L^2(\partial D_a)\rightarrow \Xi$, proposes a solution $\phi_{\alpha}\in C(\partial D_{a})$ of the form $$\label{Tikhonov} \phi_{\alpha}=(\alpha I+K^K)^{-1}K^f, \textrm{ for } 0<\alpha \ll 1,$$ where $\alpha$ is a suitably chosen regularization parameter. It is known that $\Vert K\phi_\alpha-f\Vert_\Xi\rightarrow 0$ as $\alpha\rightarrow 0$, (see [@Kirsch-Book96], Theorem 2.16), but the optimal choice of $\alpha$ is an essential step in designing a feasible method (e.g., finding a minimal norm solution), and there are various modalities to do this. In this paper we will use the Morozov discrepancy principle associated to the following weighted residual: $$\label{disc-fun} E(\phi,h)= \displaystyle\frac{1}{\\|h_{1}\\|_{L^{2}(\partial D_{c})}^{2}}\\| K_1\phi - h_{1}\\|_{L^{2}(\partial D_{c})}^{2} + \frac{1}{2\pi R} \\|K_2\phi \\|_{L^{2}(\partial B_{R})}^{2},$$ for every given $h=(h_1,0)\in \Xi$. The reasoning behind using the weighted residual discrepancy functional defined at \[disc-fun\] is as follows. Due to the asymptotic behavior of $\frac{\partial \Phi(\mathbf{x},\, \mathbf{y})}{\partial \nu_{\mathbf{y}}} = \mathcal{O}(\|\mathbf{x}-\mathbf{y}\|^{-1/2})$ as $\|\mathbf{x}-\mathbf{y}\| \to \infty$, we have that given a fixed density $\phi$, $\\|K\phi\\|_{L^{2}(\partial B_{R})} = \mathcal{O}(1)$ as $R \to \infty$. In other words, using the space $L^{2}(\partial B_{R})$ with the standard surface measure is not really suited to the decay properties of double layer potential solutions, because the decay of the normal derivative $\partial_{\nu}\Phi$ is too weak. Similarly, we use the relative norm $$\label{eq:F1} \frac{\\|K_1\phi - h_{1}\\|_{L^{2}(\partial D_{c})}}{\\|h_{1}\\|_{L^{2}(\partial D_{c})}}$$ on $\partial D_{c}$ because this is a useful quantity for determining how good the control is, regardless of the norm of $h_{1}$. Thus our procedure for finding an approximate solution for problem is to first make use of the Tikhonov regularization for the operator $K:L^2(\partial D_a)\rightarrow \Xi$ as described in to obtain $\phi_\alpha$ and then apply the Morozov’s discrepancy principle for the unique choice of $\alpha$ ([@Kress-Book99]), i.e. such that $$\label{Morozov0} E(\phi_{\alpha}, f)=\delta^2,$$ with $\displaystyle\delta^2\leq \mu^2\min\left\{\frac{1}{2\\|f_{1}\\|^2_{L^{2}(\partial D_{c})}}, \frac{1}{4\pi R}\right\}$. In what follows, we will account for noise and measurement errors and will consider with $f=(f_1,0)\in \Xi$ replaced by $f_\epsilon=(f_{\epsilon,1}, f_{\epsilon,2})\in \Xi$, given by $$\label{random-f} f_{\epsilon} = (f_{1} + \epsilon s, \, 0)\in \Xi,$$ where $s\in L^{2}(\partial D_{c})$ is a random perturbation with $\\|s\\|_{L^{2}(\partial D_{c})} \leq 2 \\|f_{1}\\|_{L^{2}(\partial D_{c})}$ and $f_1$ is a solution of the Helmholtz equation in a neighborhood of the control region $D_{c}$. We mention that in the numerical experiments of Section \[sec:numerics\], $f_{1}$ denotes the $k$ frequency component of the far field of a far field observer. Note that this assumption about the interrogating signal ensures that $f_1$ is a solution of the Helmholtz equation in $B_R$. In the noisy case (i.e. when $f$ is replaced by $f_\epsilon$) equation becomes $$\label{Morozov} E(\phi_{\alpha}, f_\epsilon)=\delta^2,$$ where $\phi_{\alpha}=(\alpha I+K^K)^{-1}K^f_\epsilon$ is the Tikhonov regularization solution. From the definition of $E$ and classical results, [@Kirsch-Book96; @Kress-Book99], it follows that admits at least a solution $\alpha$. Moreover, as we will discuss in Section \[sec:stability\], motivated by numerical evidence, we formulate the hypothesis that there exists $\epsilon_0>0$ such that for each $\epsilon\in (0,\epsilon_0)$, problem has a unique solution $\alpha(\epsilon)$ which uniquely defines a differentiable function $\epsilon\rightsquigarrow\alpha(\epsilon)$. We will study the minimal norm solution uniquely determined by , discuss its stability for given noisy data in $\Xi$ and, in the case of data corresponding to a point source, analyze its sensitivity with respect to parameters such as: mutual distances between $D_a$, $D_c$ and $B_R(\B0)$; wave number $k$; and the location of the point source. Stability estimate for the Tikhonov regularization {#sec:stability} ================================================== In this section we present analytical and numerical arguments which indicate the stability of the minimum norm solution $\phi_{\alpha}$ with respect to noise level $\epsilon$ for a given fixed discrepancy level $\delta$. Next, we present below Lemma \[lemma:phif1constant\] which will provide bounds for $\\|f_{1}\\|_{L^{2}(\partial D_{C})}$ and $\alpha$ in terms of the operatorial norm of $K_{1}^{}$. Let $0<\delta<\frac{1}{\sqrt{2}}$ and $z=(z_1,0) \in \Xi'$ with $z_1\neq 0$. Consider the Tikhonov regularization solution $\phi_{\alpha}=(\alpha I + K^{}K)^{-1}K^z \in C(\partial D_a)$, with $\alpha$ such that $\\|K\phi_{\alpha} - z\\|_{\Xi'} \leq \delta$. Then we have $$\begin{aligned} \\|z_{1}\\|_{L^2(\partial D_c)} & \leq 4\\|K_1^\\|_{\mathcal O}\\|\phi_{\alpha}\\|_{L^{2}(\partial D_{a})},\label{lb-phi}\\ \alpha & \leq 4\delta \Vert K_1^\Vert^2_{\mathcal O},\label{ub-alpha}\end{aligned}$$ where $K_1^$ is the adjoint operator for $K_1$ defined by and $\\|\cdot\\|_{\mathcal O}$ denotes the operatorial norm. \[lemma:phif1constant\] We will start with the proof of . Note that since $E(\phi_{\alpha}, z)=\delta^{2}$, we have $$\label{5.17} \\|K_{1}\phi_{\alpha} - z_{1}\\|_{L^{2}(\partial D_{c})}^{2} \leq \delta^{2}\\|z_{1}\\|_{L^2(\partial D_c)}^{2}.$$ From we obtain $$\\|K_{1}\phi_{\alpha}\\|_{L^2(\partial D_c)}^{2} - 2 \, {\textrm{Re}}(K_{1}\phi_{\alpha}, \, z_{1})_{L^2(\partial D_c)} + \\|z_{1}\\|_{L^2(\partial D_c)}^{2} \leq \delta^{2}\\|z_{1}\\|_{L^2(\partial D_c)}^{2},$$ and this implies $$\\|z_{1}\\|_{L^2(\partial D_c)}^{2}(1 -\delta^{2}) \leq 2 \, {\textrm{Re}}(\phi_{\alpha}, \, K_{1}^{}z_{1})_{L^2(\partial D_a)} \leq 2 \\|\phi_{\alpha}\\|_{L^2(\partial D_a)}\\|K_{1}^{} z_{1}\\|_{L^2(\partial D_a)}. \label{pas1}$$ Then gives $$\begin{aligned} \\|\phi_{\alpha}\\|_{L^2(\partial D_a)} & \geq \frac{\\|z_{1}\\|_{L^2(\partial D_c)}^{2}(1 - \delta^{2})}{2 \\|K_{1}^{}z_{1}\\|_{L^2(\partial D_a)}} = {\left ( \frac{1 - \delta^{2}}{2}\right )}{\left ( \frac{\\|z_{1}\\|_{L^2(\partial D_c)}}{\\|K_{1}^{}z_{1}\\|_{L^2(\partial D_a)}} \right )}\\|z_{1}\\|_{L^2(\partial D_c)}\notag\\ & \geq \frac{1 - \delta^{2}}{2} {\left ( \frac{\\|z_{1}\\|_{L^2(\partial D_c)}}{\\|K_{1}^{}\\|_{\mathcal O}}\right )}\notag\\ &\geq \frac{\\|z_1\\|_{L^2(\partial D_c)}}{4\\|K_{1}^{}\\|_{\mathcal O}}.\label{margine}\end{aligned}$$ Next we proceed towards proving . From the definition of $\phi_{\alpha}$ we have $$\begin{aligned} \alpha \phi_{\alpha} + K^{}K\phi_{\alpha}&=& K^z=K_1^z_1,\nonumber\\ \alpha \phi_{\alpha}+K_2^{}K_2\phi_{\alpha} &=& K_1^(z_1-K_1\phi_{\alpha}).\label{5.18}\end{aligned}$$ Here we have used from and that $$\begin{aligned} \label{adjunct-rel} K^\psi=K_1^\psi_1+K^_2\psi_2, \mbox{ for all }\psi\in\Xi,\nonumber\\ K^Kv=K_1^K_1v+K_2^K_2v, \mbox{ for all }v\in L^2(\partial D_a).\end{aligned}$$ Multiplying with $\phi_{\alpha}$ in the sense of the usual scalar product in $L^2(\partial D_a)$, we obtain $$\label{rel11} \alpha \\|\phi_{\alpha}\\|^2_{L^2(\partial D_a)} +\\|K_2\phi_{\alpha}\\|^2_{L^2(\partial D_a)} = (K_1^(z_1-K_1\phi_{\alpha}),\phi_{\alpha})_{L^2(\partial D_a)}.$$ Using , and in we then have $$\alpha \\|\phi_{\alpha}\\|_{L^2(\partial D_a)} \leq \delta\\|K_1^\\|_{\mathcal O}\\|z_1\\|_{L^2(\partial D_c)}\Longrightarrow \alpha\leq 4\delta\\|K_1^\\|^2_{\mathcal O} \label{alfa}.$$ Next, before presenting the main stability result of this section, i.e., Proposition \[theorem:stabilityestimate\] below, we must understand the conditions on $\epsilon > 0$ under which admits a unique solution $\alpha(\epsilon)$ with the property that the resulting function $\epsilon\rightsquigarrow \alpha(\epsilon)$ is differentiable. For this, we consider the function $g:(0,\infty)\times(0,\infty)\rightarrow (0,\infty)$ defined by $$g(\alpha, \epsilon) = E(\phi_\alpha, f_\epsilon)\label{eq:galphaeps},$$ where $f_{\epsilon} \in \Xi$ was introduced in , and $\phi_\alpha$ is the Tikhonov regularization solution introduced in . With this notation, can be rewritten as $$\label{Morozov-g} g(\alpha, \epsilon) = \delta^2,$$ where $\delta$ is the desired fixed discrepancy level. By using classical results (e.g., [@Kirsch-Book96; @Kress-Book99]) it can be observed that for every $\epsilon$, admits at least one solution in $(0,\infty)$ and that $g$ defined by is differentiable with respect to positive $\alpha$ and $\epsilon$, respectively. In fact, it follows from classical arguments that a maximum value of $\alpha$ for a given $\epsilon$ exists. This solution of corresponds to the $L^2$ minimal energy solution and we will further refer to it as the Morozov solution. For the remainder of the paper, unless otherwise specified, $C$ will denote a generic constant which depends only on the operator $K$, $d_c=diam(D_c)$ and $d=dist(\partial D_c,\partial D_a)$. The next Proposition states a central stability result concerning the Morozov solution of . We have, Let $0<\delta$ be as above, and $f_\epsilon$ and $f_1$ as defined in . For every $\epsilon \geq 0$ consider $\phi_{\alpha_{\epsilon}}=(\alpha_\epsilon I + K^{}K)^{-1}K^{}f_{\epsilon}\in C(\partial D_a)$ with $\alpha_\epsilon=\alpha(\epsilon)$ the Morozov solution of . Then we have, $$\displaystyle\frac{\\| \phi_{\alpha_\epsilon} - \phi_{\alpha_0}\\|_{L^{2}(\partial D_{a})}}{\\|\phi_{\alpha_{\epsilon}}\\|_{L^{2}(\partial D_{a})}} \leq \displaystyle\frac{ \left\|\displaystyle\frac{\alpha_{\epsilon}}{\alpha_0} - 1\right\| + \sqrt{ \left\|\displaystyle\frac{\alpha_{\epsilon}}{\alpha_0} -1 \right\|^{2} + 16\displaystyle \frac{\epsilon{\left ( 2\delta + C\delta\epsilon +C\epsilon\right )}}{\alpha_0}\\|K^_{1}\\|_{\mathcal O}}}{2}.$$\[theorem:stabilityestimate\] Fix $\epsilon > 0$ and let $f=f_\epsilon\mbox{ for } \epsilon=0$. Let us recall that $\alpha(\epsilon)$ is uniquely implicitly defined by the equation $E((\alpha_{\epsilon}I+K^K)^{-1}K^f_{\epsilon}, f_{\epsilon})= \delta^2$ and by Lemma \[lemma-diff-alpha\] it will be differentiable in some interval $(0,\epsilon_0)$ for all wavenumbers $k$. Next consider $$\begin{aligned} \alpha_{\epsilon}\phi_{\alpha_{\epsilon}} + K^{}K\phi_{\alpha_{\epsilon}} & = K^{}f_{\epsilon},\\ \alpha_{0} \phi_{\alpha_{0}} + K^{}K\phi_{\alpha_0} & = K^{}f.\end{aligned}$$ Subtracting, we obtain $$\begin{aligned} \alpha_{0}\phi_{\alpha_0}-\alpha_{\epsilon}\phi_{\alpha_{\epsilon}} + K^{}K(\phi_{\alpha_0}-\phi_{\alpha_{\epsilon}}) & = K^{}(f - f_{\epsilon}),\notag\\ \alpha_0(\phi_{\alpha_0} - \phi_{\alpha_{{\epsilon}}}) + (\alpha_0 - \alpha_{\epsilon})\phi_{\alpha_{{\epsilon}}} + K^{}K(\phi_{\alpha_0}-\phi_{\alpha_{\epsilon}}) & = K^{}(f - f_{\epsilon}).\label{3.1.1}\end{aligned}$$ Integrating both sides of against $\phi_{\alpha_0} - \phi_{\alpha_{{\epsilon}}}$ yields $$\begin{aligned} & \displaystyle\alpha_0 \\|\phi_{\alpha_0} - \phi_{\alpha_{{\epsilon}}}\\|_{L^2(\partial D_a)}^{2} + (\alpha_0 - \alpha_{{\epsilon}})(\phi_{\alpha_{{\epsilon}}}, \, \phi_{\alpha_0} - \phi_{\alpha_{{\epsilon}}})_{L^2(\partial D_a)} + \\|K(\phi_{\alpha_0} - \phi_{\alpha_{{\epsilon}}})\\|_\Xi^{2}\notag\\ & \quad\quad= \displaystyle(K(\phi_{\alpha_0} - \phi_{\alpha_{{\epsilon}}}), \, f - f_{\epsilon})_{\Xi}= \displaystyle(K_{1}(\phi_{\alpha_0} - \phi_{\alpha_{{\epsilon}}}), \, f_{1} - f_{\epsilon, 1})_{L^2(\partial D_c)}.\label{3.1.2}\end{aligned}$$ where, we have used and the fact that $f_2=f_{\epsilon,2}=0$ in the last equality above. Thus, $$\begin{aligned} \label{ecuatie-1} \!\!\!\!\!\!\!\!\!\alpha_0 \\|\phi_{\alpha_0} - \phi_{\alpha_{{\epsilon}}}\\|_{L^2(\partial D_a)}^{2} \leq & \|\alpha_0 - \alpha_{{\epsilon}}\|\\|\phi_{\alpha_{{\epsilon}}}\\|_{L^2(\partial D_a)}\\|\phi_{\alpha_0} - \phi_{\alpha_{{\epsilon}}}\\|_{L^2(\partial D_a)} \nonumber\\ + & \\|f_{1} - f_{\epsilon, 1}\\|_{L^2(\partial D_c)}\\|K_{1}(\phi_{\alpha_0} - \phi_{\alpha_{{\epsilon}}})\\|_{L^2(\partial D_c)}\end{aligned}$$ Observe that $$\begin{aligned} \\|K_{1}(\phi_{\alpha_0} - \phi_{\alpha_{{\epsilon}}})\\|_{L^2(\partial D_c)} & \leq \\|K_{1} \phi_{\alpha_0} - f_{ 1}\\|_{L^2(\partial D_c)} + \\|K_{1} \phi_{\alpha_{{\epsilon}}} - f_{{\epsilon},1}\\|_{L^2(\partial D_c)} + \\|f_{1} - f_{\epsilon, 1}\\|_{L^2(\partial D_c)}\nonumber\\ &\leq {\delta}\\|f_{1}\\|_{L^2(\partial D_c)}+ {\delta}\\|f_{{\epsilon},1}\\|_{L^2(\partial D_c)} + C\epsilon \\|f_{1}\\|_{L^2(\partial D_c)}\nonumber\\ & = {\left ( 2\delta + C\delta\epsilon +C\epsilon\right )}\\|f_{1}\\|_{L^2(\partial D_c)}. \label{ecuatie-2}\end{aligned}$$ where $f_{\epsilon,1}=f_{1} + \epsilon s$ with $\\|s\\|_{L^{2}(\partial D_{c})} \leq C \\|f_{1}\\|_{L^{2}(\partial D_{c})}$, and we have used the definition of $\phi_{\alpha_{\epsilon}}$ and in the inequalities above. Using in we obtain $$\begin{aligned} \alpha_0 \\|\phi_{\alpha_0} - \phi_{\alpha_{{\epsilon}}}\\|_{L^2(\partial D_a)}^{2} & \leq \|\alpha_0 - \alpha_{{\epsilon}}\| \\|\phi_{\alpha_{{\epsilon}}}\\|_{L^2(\partial D_a)}\\|\phi_{\alpha_0} - \phi_{\alpha_{{\epsilon}}}\\|_{L^2(\partial D_a)} \notag\\ & + \epsilon{\left ( 2\delta + C\delta\epsilon +C\epsilon\right )}\\|f_{1}\\|^2_{L^2(\partial D_c)}. \label{ecuatie-3}\end{aligned}$$ If we define $A := \frac{\\|\phi_{\alpha_{\epsilon}} - \phi_{\alpha_{0}} \\|_{L^{2}(\partial D_{a})}}{\\|\phi_{\alpha_{{\epsilon}}}\\|_{L^{2}(\partial D_{a})}}$, inequality implies that $$\begin{aligned} \alpha_0 A^{2} & \leq \|\alpha_0 - \alpha_{{\epsilon}}\| A + \frac{\epsilon {\left ( 2\delta + C\delta\epsilon +C\epsilon\right )} \\|f_{1}\\|_{L^{2}(\partial D_{c})}^{2}}{\\|\phi_{\alpha_{{\epsilon}}}\\|_{L^{2}(\partial D_{a})}^{2}} \notag\\ & \leq \|\alpha_0 - \alpha_{{\epsilon}}\| A + 16\epsilon {\left ( 2\delta + C\delta\epsilon +C\epsilon\right )}\\|K^_1\\|^2_{\mathcal O}, \label{quadratic}\end{aligned}$$ where we have used of Lemma \[lemma:phif1constant\] in the last inequality above. Next, consider $$h(A) := A^{2} - \left\|\displaystyle\frac{\alpha_{{\epsilon}}}{\alpha_0} - 1 \right\| A - 16\\|K^_1\\|^2_{\mathcal O}\displaystyle\frac{\epsilon {\left ( 2\delta + C\delta\epsilon +C\epsilon\right )}}{\alpha_0}.$$ Then, from we have $h(A) \leq 0$ and this implies that $$A \leq \displaystyle\frac{ \left\|\displaystyle\frac{\alpha_{{\epsilon}}}{\alpha_0} - 1 \right\| + \sqrt{\left\|\displaystyle\frac{\alpha_{{\epsilon}}}{\alpha_0} - 1 \right\|^{2} + 64\\|K^_1\\|^2_{\mathcal O}\displaystyle\frac{\epsilon {\left ( 2\delta + C\delta\epsilon +C\epsilon\right )}}{\alpha_0}}}{2}$$ which completes the proof. Regarding the monotonic character of $g$, we note that, as suggested by the numerics, $g$ is not in general globally monotonic with respect to $\alpha$ as can be seen in Figure \[fig:nonincreasing\_F\], which considers an antenna of radius $a = 0.01$, region of control characterized in polar coordinates by $r_{1} = 0.011$, $r_{2} = 0.015$, $\theta \in [3\pi/4, 5\pi/4]$, wave number $k=10$, $f_\epsilon$ given by with $f_1=\frac{1}{4}H^{(1)}_0(k\|\Bx-\Bx_0\|)$ with $\mathbf{x}_{0} = [10000,0]^{T}$, and noise level $\epsilon=0.005$. But on the other hand, for the same geometry and functional settings as in Figure \[fig:nonincreasing\_F\], we observe in Figure \[fig:plot\_g\_dg\_alpha\] that for each $\epsilon<0.015$, $g(\alpha, \epsilon) = E(\phi_{\alpha_{\epsilon}}, \, f_{\epsilon})$ is strictly increasing with respect to $\alpha$ in the interval $(10^{-4}, 1)$. Moreover, for every $\epsilon<0.015$ the Morozov solution $\alpha(\epsilon)$ is the unique solution of in $(10^{-4}, 1)$. In fact, for the same geometry and functional settings as in Figure \[fig:plot\_g\_dg\_alpha\] and for $k \in [1,100]$ and $\epsilon = 0.005$, Table \[tab:pktable\] summarizes the values of $p_k>0$ for which $g(\alpha, \epsilon)$ is locally strictly monotonic with respect to $\alpha$ in an interval $(10^{-p_k}, 1)$, as well as the value of the Morozov solution for each $k$ (also see Figure \[fig:pk\_plot\]). Together with Figure \[fig:plot\_g\_dg\_alpha\] where $k=10$ and $\epsilon$ is varied in the interval $[0,0.015]$, this suggests that the Morozov solution $\alpha(\epsilon)$ of satisfies $\alpha(\epsilon)\in(10^{-p_k}, 1)$ at least for $\epsilon < 0.015$. This in turn together with the strict monotonicity $g$ implies the existence of a unique solution $\alpha(\epsilon)$ for . Then, uniqueness together with the fact that $\frac{\partial}{\partial \alpha}g(\alpha, \epsilon)\neq 0$ in $ (10^{-p_k}, 1)\times (0,0.015)$ (for $k=10$ shown in Figure \[fig:plot\_g\_dg\_alpha\]) implies the differentiability of the Morozov solution $\alpha(\epsilon)$ by using the implicit function theorem. --------------- ---------------- ------------------ ------ ---------------- ------------------ $k$ $-p_{k}$ Morozov $\alpha$ $k$ $-p_{k}$ Morozov $\alpha$ \[0.5ex\] 1.0 -5.74057337341 0.0021397 51.0 -5.63387601498 0.023987 6.0 -5.15371857022 0.0022445 56.0 -5.5538513243 0.028226 11.0 -4.61487654411 0.0027985 61.0 -5.58052339557 0.032734 16.0 -4.43348001737 0.0037643 66.0 -7.93866063316 0.038053 21.0 -7.22374205154 0.0052228 71.0 -7.78393971408 0.043891 26.0 -6.73292218326 0.0072707 76.0 -7.60786606025 0.048884 31.0 -6.41280555768 0.0098052 81.0 -7.41580451387 0.05425 36.0 -6.18339417827 0.012607 86.0 -7.2077344143 0.060871 41.0 -5.97530913764 0.015782 91.0 -7.00500135956 0.066982 46.0 -5.78858575492 0.01971 96.0 -6.90364624001 0.07278 --------------- ---------------- ------------------ ------ ---------------- ------------------ : Table of values $-p_{k}$ such that $g(\alpha, \epsilon)$ is increasing with respect to $\alpha$ for $\alpha \geq 10^{-p_{k}}$. $\epsilon$ is fixed at $0.005$ in this case.[]{data-label="tab:pktable"} ![Threshold value $p_{k}$ for which $F(\alpha)$ is increasing when $\alpha > 10^{-p_{k}}$. Also shows the value of $\alpha$ for which $F(\alpha) = \delta^{2}$ with the same setting as in Figure \[fig:plot\_g\_dg\_alpha\], where $\delta = 0.02$ and the noise level $\epsilon = 0.005$.[]{data-label="fig:pk_plot"}](pk_plot){width="\textwidth"} For simplicity of notation, in what follows we will write sometimes $\alpha$ instead of $\alpha_\epsilon=\alpha(\epsilon)$ and we will use $\alpha'$ and $f_{\epsilon,i}'$ to denote $\frac{d\alpha}{d\epsilon}$ and $\frac{df_{\epsilon,i}}{d\epsilon}$ respectively. Motivated by the above numerics, we formulate the following more general hypothesis: \[hypothesis-I\] Assume the same geometrical setup as in Section \[sec:background\] and let $f_\epsilon, f_1$ be as in . Then there exists $p_0>0$ and $\epsilon_0>0$ such that $\frac{\partial}{\partial \alpha}g(\alpha, \epsilon)\neq 0$ in $ (10^{-p_0}, 1)\times (0,\epsilon_0)$ for all wave numbers $k$, and the Morozov solution $\alpha(\epsilon)$ is the unique solution of in $(10^{-p_0}, 1)$. For example, as shown in Table \[tab:pktable\] and Figure \[fig:pk\_plot\], for $f_\epsilon, f_1$ as in with $s=\widehat{\nu}\Vert f_1\Vert_{ L^2(\partial D_c)}$ where $\widehat{\nu}\in L^2(\partial D_c)$ is a random perturbation with $\Vert \widehat{\nu}\Vert_{ L^2(\partial D_c)}=1$, and for the same geometry and data as in Figure \[fig:plot\_g\_dg\_alpha\], we have that Hypothesis \[hypothesis-I\] is satisfied for $p_0=10^{-4}$ and $\epsilon_0=0.015$ for all $k=\overline{1,100}$. Thus, whenever Hypothesis \[hypothesis-I\] is satisfied, the definition of $\alpha(\epsilon)$ and the implicit function theorem imply: \[lemma-diff-alpha\] There exists $\epsilon_0>0$ such that for every $\epsilon\in(0,\epsilon_0)$, the function $\alpha:(0,\epsilon_0)\rightarrow(0,\infty)$, where for each $\epsilon\in(0,\epsilon_0)$, $\alpha(\epsilon)$ represents the Morozov solution of , will be differentiable for all wave numbers $k$. The next Lemma is a technical result needed in the stability estimate obtained in Corollary \[corolar-1\]. \[teorema-int\] Let $f_{1}$ be a solution of the Helmholtz equation in a neighborhood of $D_c$ satisfying the following source type condition: $$\label{source-cond} \displaystyle\left\Vert K_1\psi_0-\frac{f_1}{\Vert f_1\Vert_{L^{2}(\partial D_{c})}}\right\Vert_{L^{2}(\partial D_{c})}\!\!\!\!\leq C\delta \mbox{ for some }\psi_0\in L^2(\partial D_a) \mbox{ with } \Vert\psi_0\Vert_{L^2(\partial D_a)}\leq C \delta.$$ Assume $R$ (radius of $B_R(\B0)$) is such that, $$\label{R-bound} \Vert f_1\Vert_{L^{2}(\partial D_{c})} \leq \sqrt{\pi R}.$$ Consider $s_\epsilon=f_{1} + \frac{\epsilon}{2} \widehat{\nu}\\|f_{1}\\|_{L^{2}(\partial D_{c})}$ where $\widehat{\nu}\in L^2(\partial D_c)$ is a random perturbation with $\\|\widehat{\nu}\\| \leq 1$. Assume the same functional framework as in Proposition \[theorem:stabilityestimate\] and that Hypothesis \[hypothesis-I\] holds true in the case when $f_\epsilon$ is given by $$\label{hyp-1} f_{\epsilon} = (f_{\epsilon,1}, f_{\epsilon,2})= (f_{1} + \epsilon s_\epsilon, \, 0).$$ Then, there exists $\epsilon_0>0$ such that the Morozov solution of equation $\alpha=\alpha(\epsilon)$ satisfies $$\alpha\|\alpha'\| \leq C\frac{\delta^2}{\sqrt{\alpha}}, \mbox{ for all } \epsilon<\epsilon_0.$$ Define the weights $$\begin{aligned} w_{1} & := \frac{1}{\\|f_{\epsilon, 1}\\|_{L^{2}(\partial D_{c})}^{2}}\\ w_{2} & := \frac{1}{2 \pi R}.\end{aligned}$$ and denote $T_{\alpha} := (K^{}K + \alpha I)^{-1}$, $R_{\alpha} := T_{\alpha}K^{}$. Then using the Einstein summation convention, we may write $$E(\phi_{\alpha}, f_{\epsilon})= w_{i}\\|K_{i}\phi_{\alpha} - f_{\epsilon,i} \\|_{L^{2}(W_{i})}^{2} = w^{i} {\left ( K_{i}\phi_{\alpha} - f_{\epsilon,i}, \, K_{i}\phi_{\alpha} - f_{\epsilon,i}\right )}_{L^{2}(W_{i})},$$ where $W_{1} = \partial D_{c}$, $W_{2} = \partial B_{R}$. Next, as in Lemma \[lemma-diff-alpha\], we observe that Hypothesis \[hypothesis-I\] together with the implicit function theorem imply the uniqueness and differentiability of $\alpha(\epsilon)$, on the interval $\epsilon \in (0, \epsilon_0)$ for some $\epsilon_0>0$, where $\alpha(\epsilon)$ is uniquely and implicitly defined by the equation $E((\alpha_{\epsilon}I+K^K)^{-1}K^f_{\epsilon}, \, f_{\epsilon})= \delta^2$. Differentiating the equation $E((\alpha_{\epsilon}I+K^K)^{-1}K^f_{\epsilon}, \, f_{\epsilon})= \delta^2$ with respect to $\epsilon$ and noting that $\delta$ is fixed, we obtain $$\begin{aligned} 0 = \partial_{\epsilon}E(\phi_{\alpha}, f_{\epsilon})& = 2w_{i}\,{\textrm{Re}}{ {\left ( K_{i}\phi_{\alpha}' - f_{\epsilon,i}', \, K_{i}\phi_{\alpha} - f_{\epsilon, i}\right )}}_{L^{2}(W_{i})} \notag\\ & \quad - 2(w_{1})^{2}\,{\textrm{Re}}{{\left ( f_{\epsilon,1}', f_{\epsilon, 1}\right )}}_{L^{2}(\partial D_{c})}\\|K_{1}\phi_{\alpha} - f_{\epsilon,1}\\|_{L^{2}(\partial D_{c})}^{2}\label{eq:diffnorm1}.\end{aligned}$$ Next, from $(K^{}K + \alpha I)\phi_{\alpha} = K^{}f_{\epsilon}$ we observe that $$\phi_{\alpha}' = R_{\alpha}f_{\epsilon}' - \alpha' T_{\alpha}\phi_{\alpha}. \label{eq:phialphadiff}$$ Thus, we may write $$K_{i}\phi_{\alpha}' - f_{\epsilon, i}' = -\alpha' K_{i} T_{\alpha}\phi_{\alpha} + K_{i}R_{\alpha}f_{\epsilon}' - f_{\epsilon, i}'.\label{eq:sub1}$$ By using (\[eq:sub1\]) and we obtain that $$\begin{aligned} \label{ec-11} \!\!\!\!\!\!\!\!\! 2{\left ( K_{i}\phi_{\alpha}' - f_{\epsilon,i}', \, K_{i}\phi_{\alpha}\right )}_{L^{2}(W_{i})} &= -2\alpha' {\left ( T_{\alpha}\phi_{\alpha}, \, K_{i}^{}K_{i}\phi_{\alpha}\right )}_{L^{2}(\partial D_{a})}\notag\\ & + 2{\left ( K_{i}R_{\alpha}f_{\epsilon}' - f_{\epsilon, i}', \, K_{i}\phi_{\alpha}\right )}_{L^{2}(W_{i})},\end{aligned}$$ and $$\begin{aligned} \label{ec-12} \!\!\!\!\!\!\!\!\!-2{\left ( f_{\epsilon,i}, \, K_{i}\phi_{\alpha}' - f_{\epsilon, i}'\right )}_{L^{2}(W_{i})} & = 2\alpha'{\left ( f_{\epsilon,i}, \, K_{i}T_{\alpha}\phi_{\alpha}\right )}_{L^{2}(W_{i})} - 2{\left ( f_{\epsilon,i}, \, K_{i}R_{\alpha}f_{\epsilon}' - f_{\epsilon, i}'\right )}_{L^{2}(W_{i})}\notag\\ & = 2\alpha'{\left ( K_{i}^{}f_{\epsilon,i}, \, T_{\alpha}\phi_{\alpha}\right )}_{L^{2}(\partial D_{a})} - 2{\left ( f_{\epsilon,i}, \, K_{i}R_{\alpha}f_{\epsilon}' - f_{\epsilon, i}'\right )}_{L^{2}(W_{i})}.\end{aligned}$$ Let $P,Q$ be defined by $$\begin{aligned} P & = 2w_{i}\left[ {\left ( K_{i}\phi_{\alpha} - f_{\epsilon, i}, \, K_{i}R_{\alpha}f_{\epsilon}' - f_{\epsilon,i}'\right )}_{L^{2}(W_{i})} + \alpha' {\left ( K_{i}^{}f_{\epsilon,i} - K_{i}^{}K_{i}\phi_{\alpha}, \, T_{\alpha}\phi_{\alpha}\right )}_{L^{2}(\partial D_{a})} \right]\label{P}\\ Q & = 2(w_{1})^{2}\,{\left ( f_{\epsilon,1}', f_{\epsilon, 1}\right )}_{L^{2}(\partial D_{c})}\\|K_{1}\phi_{\alpha} - f_{\epsilon,1}\\|^{2}.\label{Q}\end{aligned}$$ Then from , used in we obtain $$\label{ec-15'} 0=\partial_{\epsilon}E(\phi_{\alpha}, f_{\epsilon})= {\textrm{Re}}(P) - {\textrm{Re}}(Q).$$ We focus first on $P$ introduced in . In this regard, let us define $$\label{P1} L_{i} := {\left ( K_{i}\phi_{\alpha} - f_{\epsilon, i}, \, K_{i}R_{\alpha}f_{\epsilon}' - f_{\epsilon, i}'\right )}_{L^{2}(W_{i})}.$$ Observe that implies $$f_\epsilon'=(f'_{\epsilon,1}, f'_{\epsilon,2})=(f_1+\epsilon\widehat{\nu}\Vert f_1\Vert_{L^2(\partial D_c)},0)\label{P2}.$$ First note that by using classical arguments based on the singular value decomposition for $K:L^2(\partial D_a)\rightarrow\Xi$, one can adapt the results in [@Kirsch-Book96] (Theorem 2.7) and obtain, $$\label{regl-1} \\|KR_{\alpha}Kz - Kz \\|_{\Xi} \leq C\sqrt{\alpha} \Vert z\Vert_{L^2(\partial D_a)}, \mbox{ for every } z\in L^2(\partial D_a).$$ Let $\displaystyle f=\left(\frac{f_1}{\Vert f_1\Vert_{L^2(\partial D_c)}},0\right)$ and $\displaystyle v=\frac{f_\epsilon'}{\Vert f_1\Vert_{L^2(\partial D_c)}}-f$. By using the definition of $E$ and $\Xi$, , , , , , Cauchy’s inequality and the triangle inequality in , we obtain $$\begin{aligned} \label{P4} 2\|w_iL_i\| &\leq& C\delta (\sqrt{w_1}\\|K_1R_{\alpha}f_{\epsilon}' - f_{\epsilon,1}' \\|_{L^2(\partial D_c)}+\sqrt{w_2}\\|K_2R_{\alpha}f_{\epsilon}' - f_{\epsilon,2}' \\|_{L^2(\partial B_R(\B0))})\nonumber\\ &\leq&C\delta (\\|K_1R_{\alpha}(v+f) - (v_1+f_1) \\|_{L^2(\partial D_c)}+\\|K_2R_{\alpha}(v+f) - (v_2+f_2) \\|_{L^2(\partial B_R(\B0))})\nonumber\\ &\leq& C\delta \\|KR_{\alpha}(v+f) - (v+f) \\|_{\Xi}\nonumber\\ &\leq& C\delta ( \\|KR_{\alpha}f - f\\|_{\Xi}\!+\!\\|KR_{\alpha}v\! - \!v\\|_{\Xi})\nonumber\\ &\leq& C\delta(\\|KR_{\alpha}K\psi_0 - K\psi_0 \\|_{\Xi}+\\|KR_{\alpha}(K\psi_0 - f) \\|_{\Xi}\!+\!\\|K\psi_0 - f\\|_{\Xi})+C\frac{\delta\epsilon}{\sqrt{\alpha}}\nonumber\\ &\leq& C\delta^2\sqrt{\alpha} +C\frac{\delta^2}{\sqrt{\alpha}}+C\delta^2+C\frac{\delta\epsilon}{\sqrt{\alpha}}\nonumber\\ &\leq& C\frac{\delta^2}{\sqrt{\alpha}},\end{aligned}$$ where Einstein summation convention was used and where, in the second inequality above we make use of to obtain $\sqrt{\frac{w_2}{w_1}}\leq 1$ and $\sqrt{w_1}\Vert f_1\Vert_{L^2(\partial D_c)}<C$ for small enough $\epsilon$, and in the fourth and fifth inequalities above we have used that $\|\|R_\alpha\|\|_{{\cal O}}\leq\frac{1}{2\sqrt{\alpha}}$ (e.g. see [@Kirsch-Book96]), and respectively, that $\epsilon < \delta$ and $\psi_0$ satisfies the source condition . Expanding $P$ defined in and using the fact that $f_{\epsilon,2} = 0$ and $\phi_{\alpha} = T_{\alpha}K^{}f_{\epsilon} = T_{\alpha}K_{1}^{}f_{\epsilon,1}$, we obtain $$\begin{aligned} \label{ec-13} P & = \frac{2\alpha'}{\\|f_{\epsilon,1}\\|_{L^{2}(\partial D_{c})}^{2}} {\left ( K_{1}^{}f_{\epsilon,1},\, T_{\alpha}\phi_{\alpha}\right )}_{L^{2}(\partial D_{a})} - \frac{2\alpha'}{\\|f_{\epsilon,1}\\|_{L^{2}(\partial D_{c})}^{2}}{\left ( K_{1}^{}K_{1}\phi_{\alpha}, \, T_{\alpha}\phi_{\alpha}\right )}_{L^{2}(\partial D_{a})}\notag\\ & \quad - \frac{2 \alpha'}{2\pi R} {\left ( K_{2}^{}K_{2}\phi_{\alpha},\, T_{\alpha}\phi_{\alpha}\right )}_{L^{2}(\partial D_{a})} + 2w_{i}L_{i}\notag\\ & = \frac{2\alpha'}{\\|f_{\epsilon,1}\\|^{2}} {\left ( T_{\alpha}^{-1}\phi_{\alpha},\,T_{\alpha}\phi_{\alpha}\right )}_{L^{2}(\partial D_{a})} - \frac{2\alpha'}{\\|f_{\epsilon,1}\\|^{2}}{\left ( K_{1}^{}K_{1}\phi_{\alpha}, T_{\alpha}\phi_{\alpha}\right )}_{L^{2}(\partial D_{a})}\notag\\ & \quad - \frac{2 \alpha'}{2\pi R} {\left ( K_{2}^{}K_{2}\phi_{\alpha},\, T_{\alpha}\phi_{\alpha}\right )}_{L^{2}(\partial D_{a})} + 2w_{i}L_{i}\notag\\ & = \frac{2\alpha'}{\\|f_{\epsilon,1}\\|^{2}} {\left ( (\alpha I + K_{2}^{}K_{2})\phi_{\alpha},\,T_{\alpha}\phi_{\alpha}\right )}_{L^{2}(\partial D_{a})} - \frac{2 \alpha'}{2\pi R} {\left ( K_{2}^{}K_{2}\phi_{\alpha},\, T_{\alpha}\phi_{\alpha}\right )}_{L^{2}(\partial D_{a})} + 2w_{i}L_{i}\notag\\ & = \frac{2\alpha \alpha'}{\\|f_{\epsilon,1}\\|^{2}} {\left ( \phi_{\alpha},\,T_{\alpha}\phi_{\alpha}\right )}_{L^{2}(\partial D_{a})} + \frac{\alpha'}{\\|f_{\epsilon,1}\\|^{2}}B{\left ( K_{2}^{}K_{2}\phi_{\alpha}, \, T_{\alpha}\phi_{\alpha}\right )}_{L^{2}(\partial D_{a})} + 2w_{i}L_{i},\end{aligned}$$ where $B = 2 - \displaystyle\frac{\\|f_{\epsilon,1}\\|_{L^{2}(\partial D_{c})}^{2}}{\pi R}$ and we have used that $T_\alpha^{-1}=\alpha I+K_1^K_1+K_2^K_2$ in the third equality above. Observe that implies $B\geq 0$. Introduce the notation $\widetilde{K}_{2} := \sqrt{B}K_{2}$, and denote $\displaystyle v_{\alpha} := \frac{\phi_{\alpha}}{\\|f_{\epsilon,1}\\|_{L^{2}(\partial D_{c})}}$. Then becomes $$\begin{aligned} \label{ec-14} P & = 2 \alpha \alpha' {\left ( v_{\alpha}, \, T_{\alpha}v_{\alpha}\right )}_{L^{2}(\partial D_{a})} + \alpha' {\left ( \widetilde{K}_{2}^{}\widetilde{K}_{2}v_{\alpha}, \, T_{\alpha}v_{\alpha}\right )}_{L^{2}(\partial D_{a})} + 2w_{i}L_{i}\notag\\ & = \alpha \alpha' {\left ( v_{\alpha}, \, T_{\alpha}v_{\alpha}\right )}_{L^{2}(\partial D_{a})} + \alpha' {\left ( (\alpha I + \widetilde{K}_{2}^{}\widetilde{K}_{2})v_{\alpha}, \, T_{\alpha}v_{\alpha}\right )}_{L^{2}(\partial D_{a})} + 2w_{i}L_{i}.\end{aligned}$$ From [@kato] (see Section V.3.10), for any self-adjoint linear operator $A:H\rightarrow H$, where $H$ is a given Hilbert space (real or complex), we have that: $$\label{Kato} 0<\gamma = \inf_{\lambda \in \mathrm{Sp}(A)}\lambda \Longrightarrow (Ax, x)_H \geq \gamma(x,x)_H,$$ where $(\cdot,\cdot)$ in denotes the usual Hilbert product and where $\mathrm{Sp}(A)$ denotes the real spectrum of the operator $A$. Then, by using for the operator $T_\alpha$ we obtain $$\begin{aligned} \label{Kato-1} {\left ( v_{\alpha}, \, T_{\alpha}v_{\alpha} \right )}_{L^{2}(\partial D_{a})} &\geq \frac{1}{\alpha + \mu_{1}^{2}}\\|v_{\alpha}\\|_{L^{2}(\partial D_{a})}^{2}\geq \frac{1}{1 + \mu_{1}^{2}}\\|v_{\alpha}\\|_{L^{2}(\partial D_{a})}^{2}\notag\\ &\geq C \\|v_{\alpha}\\|_{L^{2}(\partial D_{a})}^{2},\end{aligned}$$ where we have used that $\displaystyle \frac{1}{\alpha+\mu_1^2} = \inf_{\lambda \in \mathrm{Sp}(T_\alpha)}\lambda$ with $\mu_1$ denoting the largest singular value of $K$. Next consider $\displaystyle D_{\alpha} := \alpha I + \widetilde{K}_{2}^{}\widetilde{K}_{2}$. Then, because $D_\alpha$ and $T_\alpha$ are linear, bounded, self-adjoint, invertible and positive definite operators, we have that $D_\alpha T_{\alpha}$ will also be linear, bounded, self-adjoint, invertible and have strictly positive eigenvalues. Indeed, for any eigenvalue-eigenvector pair $(x,\lambda)$ of $D_\alpha T\alpha$ we have $$D_\alpha T_\alpha x=\lambda x\Rightarrow T_\alpha x=\lambda D_\alpha^{-1}x\Rightarrow \lambda=\frac{( T_\alpha x,x)}{( D^{-1}_\alpha x,x)}\geq 0.$$ Observing that $0\notin \mathrm{Sp}(D_\alpha T_\alpha)$ we have the claim, and the positive definiteness of $D_\alpha T_\alpha$ follows. Thus we have $$\label{ec-15} {\left ( (\alpha I + \widetilde{K}_{2}^{}\widetilde{K}_{2})v_{\alpha}, \, T_{\alpha}v_{\alpha}\right )}_{L^{2}(\partial D_{a})} = {\left ( D_{\alpha}v_{\alpha},\, T_{\alpha}v_{\alpha}\right )}_{L^{2}(\partial D_{a})}\geq 0$$ From , and used in , we obtain $$\begin{aligned} \label{ec-17} \|2w_{i}L_{i}\| + \|Q\| & \geq \|\alpha'\| \left\| \alpha {\left ( v_{\alpha}, \, T_{\alpha} v_{\alpha}\right )}_{L^{2}(\partial D_{a})} + {\left ( D_{\alpha}v_{\alpha}, \, T_{\alpha}v_{\alpha}\right )}_{L^{2}(\partial D_{a})} \right\|\notag\\ & \geq \|\alpha'\| C\alpha\\|v_{\alpha}\\|_{L^{2}(\partial D_{a})}^{2} \notag\\ & \geq C \alpha \|\alpha'\| \\|v_{\alpha}\\|_{L^{2}(\partial D_{a})}^{2}.\end{aligned}$$ From and elementary algebraic manipulations we obtain, $$\label{ec-18} \|Q\| \leq \frac{2 \\|f_{1}\\|_{L^{2}(\partial D_{c})}}{\\|f_{\epsilon, 1}\\|_{L^{2}(\partial D_{c})}^{4}} \\|f_{\epsilon, 1}\\|_{L^{2}(\partial D_{c})}\cdot \delta^{2} \\|f_{\epsilon,1}\\|_{L^{2}(\partial D_{c})}^{2} = \frac{2 \\|f_{1}\\|_{L^{2}(\partial D_{c})} \delta^{2}}{\\|f_{\epsilon, 1}\\|_{L^{2}(\partial D_{c})}} \leq C\delta^{2}$$ Recalling that $\displaystyle v_{\alpha} := \frac{\phi_{\alpha}}{\\|f_{\epsilon,1}\\|_{L^{2}(\partial D_{c})}}$, Lemma \[lemma:phif1constant\] implies $$\label{ec-19} \\|v_{\alpha}\\|_{L^{2}(\partial D_{a})} \geq C.$$ Then from , , and used in we finally obtain the statement of the Lemma: $$\label{ecuatie-4} \alpha \|\alpha'\| \leq C \delta^{2}+ C\frac{\delta^2}{\sqrt{\alpha}}\leq C\frac{\delta^2}{\sqrt{\alpha}}, \mbox{ for } \epsilon<\epsilon_0.$$ Note that $\\|s_\epsilon\\|_{L^{2}(\partial D_{c})}\leq (1+\frac{\epsilon}{2})\\|f_{1}\\|_{L^{2}(\partial D_{c})}$ and thus $f_\epsilon$ as defined above satisfies . The next result is a simple consequence of Proposition \[theorem:stabilityestimate\] and Lemma \[teorema-int\]. We have, \[corolar-1\] Assume the same notation and the same framework as in Proposition \[theorem:stabilityestimate\]. Assume also that for $d_c=diam(D_c)$ small enough there exists $d=dist(\partial D_c,\partial D_a)$ small enough so that $\alpha(0)=\alpha_0\approx\delta$. Then we have $$\displaystyle\frac{\\| \phi_{\alpha_\epsilon} - \phi_{\alpha_0}\\|_{L^{2}(\partial D_{a})}}{\\|\phi_{\alpha_{\epsilon}}\\|_{L^{2}(\partial D_{a})}} \leq C\sqrt{\epsilon}.$$ From Lemma \[teorema-int\] we obtain that $$\label{estimare} \|\alpha_\epsilon'\| \alpha_\epsilon^{\frac{3}{2}}\leq C\delta^2.$$ Estimate and Cauchy’s theorem implies that $$\label{ecuatie-5} \left\|\frac{\alpha_\epsilon^{\frac{5}{2}}}{\alpha_0^{\frac{5}{2}}}-1\right\|= \frac{5}{2}\epsilon\left\|\alpha'_{\epsilon_}\alpha_{\epsilon_}^{\frac{3}{2}}\alpha_0^{-\frac{5}{2}}\right\|\leq C\epsilon\delta^2\alpha_0^{-\frac{5}{2}}\leq C\sqrt{\epsilon},$$ where $\epsilon_\in(0,\epsilon)$ and we used that $\epsilon<\delta$. Next, simple algebraic manipulation and imply $$\left\|\frac{\alpha_\epsilon}{\alpha_0}-1\right\|\leq \left\|\frac{\alpha_\epsilon^5}{\alpha_0^5}-1\right\|\leq C\sqrt{\epsilon}\left(\frac{\alpha_\epsilon^{\frac{5}{2}}}{\alpha_0^{\frac{5}{2}}}+1\right)\leq C\sqrt{\epsilon}(2+ C\sqrt{\epsilon})\leq C\sqrt{\epsilon}.$$ This together with Proposition \[theorem:stabilityestimate\] imply the statement of the Corollary. We note that all the above results can be adapted to three dimensions. The proof follows exactly the same steps by considering the natural extension of the definition of the discrepancy function $E$ in three dimensions. The assumption made in Corollary \[corolar-1\] that $\alpha_0\approx\delta$ for $d_c=diam(D_c)$ and $dist(\partial D_c,\partial D_a)=d$ small enough is based on of Lemma \[lemma:phif1constant\] and on the numerical results presented in Section \[sec:numerics\]. In particular, for the same settings as in Figure \[fig:nonincreasing\_F\], Figure \[fig:besselFS\_alpha0\_vary\_k\_mu\] shows that given small $d_c$ for small enough $d$, we have roughly that $10^{-2.5} \leq \alpha_{0} \leq 10^{-1}$. Since $\delta = 2\cdot10^{-2}$ we have in this case that $\frac{1}{2\sqrt{5}} \delta \lesssim \alpha_{0} \lesssim 5 \delta$. \[constant\] As suggested by our numerical results in Section \[sec:numerics\] we beleive that the constants (denoted by $C$) in Proposition \[theorem:stabilityestimate\], Lemma \[teorema-int\] and Corollary \[corolar-1\] will only have small values for $d_c=diam(D_c)$ and $dist(\partial D_c,\partial D_a)=d$ small enough. Numerics {#sec:numerics} ======== In this section we proceed with the numerical study of the minimal norm solution for obtained through Tikhonov regularization with the Morozov discrepancy principle for the choice of the regularization parameter in two dimensions. First we focus on the general setup of our numerical approach, and then in Section \[subsec:parametersetup\] we discuss more specifically the parameters used in our numerical examples. In Sections \[subsec:nearfieldstability\] and \[subsec:sphericalsource\] we present numerical data which demonstrates how stably $\phi$ depends on $f$ and various control statistics for a spherical point source. All figures generally display their respective parameters in an offset legend. For all of the numerical computations done in this section, we discretize the integral operator $K$ via the method of moment collocation. We refer to ([@Kress-Book99], §17.4) for more details on the method. First we choose an approximate basis of functions for $L^{2}(\partial D_{a})$. To do this we suppose the domain $D_{a}$ can be parametrized in polar coordinates by points $$(s(\tau)\cos{\tau}, \, s(\tau)\sin{\tau})), \quad \tau \in [0,2\pi],$$ where $s:\mathbb{R} \to \mathbb{R}_{+}$ is a $2\pi$-periodic smooth function. Using these coordinates, any function $\phi$ defined on $\partial D_{a}$ can be realized via the pullback as a function of $\tau$: $$\phi(s(\tau)\cos{\tau}, \, s(\tau)\sin{\tau}).$$ For convenience, let us use the notation $\widehat{\tau} = (\cos{\tau},\sin{\tau})$ and $\widehat{\tau}^{\perp} = (-\sin{\tau}, \cos{\tau})$. Now let $n_{a} \in \mathbb{N}$ and let $\tau_{j} = \frac{2\pi j}{n_{a}}$, $0 \leq j \leq n_{a}-1$ be $n_{a}$ equally spaced points on the interval $[0,2\pi)$. We then use the exponential basis functions $\{e^{il\tau}\}_{l=0}^{n_{a}-1}$ for $L^{2}([0,2\pi])$ and approximate a given $\phi \in L^{2}(\partial D_{a})$ via interpolation at the points $\{\tau_{j}\}_{j=0}^{n_{a}-1} \subset [0,2\pi]$. Note that $$\begin{aligned} \int_{\partial D_{a}}\phi(\mathbf{y}) \frac{\partial \Phi}{\partial \nu_{\mathbf{y}}}(\mathbf{x}, \, \mathbf{y})\,dS_{\mathbf{y}} & = \int_{0}^{2\pi}\phi(s(\tau)\cos{\tau}, \, s(\tau)\sin{\tau})\frac{\partial \Phi}{\partial \nu_{y}}(\mathbf{x}, (s(\tau)\cos{\tau}, \, s(\tau)\sin{\tau})) \notag\\ & \quad \cdot \sqrt{s(\tau)^{2} + s'(\tau)^{2}}\,d\theta.\end{aligned}$$ Furthermore, since $\left( s'(\tau)\cos{\tau} - s(\tau)\sin{\tau}, \, s(\tau)\cos{\tau} + s'(\tau)\sin{\tau}\right)$ is a tangent vector to $\partial D_{a}$, we have that $$\begin{aligned} \nu(\mathbf{y}) = \nu(\tau) & = \frac{(s(\tau)\cos{\tau} + s'(\tau)\sin{\tau}, \, s(\tau)\sin{\tau} - s'(\tau)\cos{\tau})}{\sqrt{ s(\tau)^{2} + s'(\tau)^{2}}}\\ & = \frac{s(\tau)\widehat{\tau} - s'(\tau)\widehat{\tau}^{\perp}}{\sqrt{ s(\tau)^{2} + s'(\tau)^{2}}}\end{aligned}$$ is the unit outward normal vector to $\partial D_{a}$. It is then straightforward to compute in the case of the Helmholtz equation in 2D that $$\begin{aligned} & \quad \frac{\partial \Phi}{\partial \nu_{\mathbf{y}}}(\mathbf{x}, \, (s(\tau)\cos{\tau}, \, s(\tau)\sin{\tau}))\\ & = \nabla_{y}\Phi(\mathbf{x}, (s(\tau)\cos{\tau}, \, s(\tau)\sin{\tau})) \cdot \nu(\tau)\\ & = \frac{ik}{4}H_{0}^{(1)'}(k\|\mathbf{x} - s(\tau)\widehat{\tau}\|)\frac{s(\tau)\widehat{\tau} - \mathbf{x}}{\sqrt{s(\tau)^{2} + \|\mathbf{x}\|^{2} - 2s(\tau)\mathbf{x}\cdot \widehat{\tau}}} \cdot \frac{s(\tau)\widehat{\tau} - s'(\tau)\widehat{\tau}^{\perp}}{\sqrt{s(\tau)^{2} + s'(\tau)^{2}}}.\end{aligned}$$ Let $n_{a} \in \mathbb{N}$ be the number of sample points on the antenna, $\partial D_{a}$, and let $n_{c} \in \mathbb{N}$ be the total number of sample points on $\partial D_{c}$. Also let $n_{R}$ be the total number of sample points on $\partial B_{R}$. We write the $2 \times (n_{c} + n_{R})$ matrix of points $$\mathbf{X} := [x_{1}, \ldots, x_{n_{c}+n_{R}}],$$ where each $x_{j}$ is a $2$-vector, $\{x_{j}\}_{j=1}^{n_{c}} \subset \partial D_{c}$ and $\{x_{j}\}_{j=n_{c}+1}^{n_{c} + n_{R}} \subset \partial B_{R}$. Approximations of all the integrals involved are then computed using a standard left endpoint sum with the appropriate quadrature weights. All the numerical examples presented herein take $D_{c}$ to be an annular sector parametrized by $r \in [r_{1}, r_{2}]$ and $\theta \in [\theta_{1}, \theta_{2}]$. See Figure \[fig:numericssetup\] for details. For each $1 \leq j \leq n_{c} + n_{R}$ and each $0 \leq l \leq n_{a}-1$, we compute $K[e^{il\tau}](x_{j})$ via the approximation $$\frac{2\pi}{n_{a}}\sum_{m=0}^{n_{a}-1} \frac{\partial \Phi( x_{j}, [s(\tau_{m})\cos{\left ( \tau_{m}\right )}, \, s(\tau_{m})\sin{\tau_{m} }]^{T})}{\partial \nu_{\mathbf{y}}} e^{il \tau_{m}}\sqrt{s(\tau_{m})^{2} + s'(\tau_{m})^{2}}.$$ If we fix $j$ and vary $l$, we see that the above sum is equivalent to computing the discrete Fourier transform of the $n_{a}$-vector $$\mathbf{v}_{j} := \left[\frac{\partial \Phi( x_{j}, [s(\tau_{m})\cos{\tau_{m}}, \, s(\tau_{m})\sin{\tau_{m} }]^{T})}{\partial \nu_{\mathbf{y}}}\sqrt{s(\tau_{m})^{2} + s'(\tau_{m})^{2}}\right]_{m=0}^{n_{a}-1}, \label{eq:vj}$$ which can be computed efficiently using the Fast Fourier Transform algorithm. In particular, for the vector $\mathbf{v}_{j}$ in $$[K\{e^{il\tau}\}(x_{j})]_{l=0}^{n_{a}-1} \approx 2\pi \mathsf{FFT}(\mathbf{v}_{j}),$$ where $\mathsf{FFT}$ is defined on $n_{a}$-vectors $\mathbf{w} = [w_{1},\ldots, w_{n_{a}}]^{T} \in \mathbb{C}^{n_{a}}$ by $$\mathsf{FFT}(\mathbf{w}) = \left[ \frac{1}{n_{a}}\sum_{j=1}^{n_{a}}w_{j}e^{\frac{2\pi i (j-1)(l-1)}{n_{a}}} \right]_{l=1}^{n_{a}}. \label{eq:fft}$$ So the matrix representation of $K$ is then the $n_{a} \times (n_{c} + n_{R})$ matrix $$A := 2\pi [\mathsf{FFT}(\mathbf{v}_{1}), \cdots, \mathsf{FFT}(\mathbf{v}_{n_{c}+n_{R}})].$$ Now, in order to approximately solve the ill-posed problem $K\phi = (f_{1}, f_{2})$, we attempt to solve the linear system $$\begin{aligned} K_{1}\phi(x_{j}) & = f_{1}(x_{j}), \quad 1 \leq j \leq n_{c}\\ K_{2}\phi(x_{j}) & = f_{2}(x_{j}), \quad n_c + 1 \leq j \leq n_c + n_{R}.\end{aligned}$$ Since $A$ is computed with respect to the functions $e^{il\theta}$, we first consider the approximate coefficients of $\phi$ with respect to the finite basis $\{e^{il\tau}\}_{l=0}^{n_{a}-1}$, given by $$c_{l} := \frac{1}{n_{a}}\sum_{m=0}^{n_{a}-1}e^{-il\tau_{m}}\phi(s(\tau_{m})\cos(\tau_{m}), \, s(\tau_{m})\sin(\tau_{m}))\,d\tau \approx \frac{1}{2\pi}\int_{0}^{2\pi} e^{-il\tau}\phi(s(\tau)\cos(\tau), \, s(\tau)\sin(\tau))\,d\tau.$$ Let $$h = [c_{0}, c_{1}, \ldots, c_{n_{a} - 1}]^{T} \in \mathbb{C}^{n_{a}}.$$ We then numerically compute the Tikhonov regularized solution $$h_{\alpha} := (A^{}A + \alpha I)^{-1}A^{}f,$$ with $\alpha > 0$. The solution vector $h_{\alpha}$ yields the approximate coefficients $c_{l}$ of the desired density $\phi$ with respect to the functions $\{e^{il\tau}\}_{l=0}^{n_{a}-1}$. We obtain the density $\phi_{\alpha}$ corresponding to $h_{\alpha}$ sampled at the angles $\tau_{m}$ on $\partial D_{a}$ by the formula $$\phi_{\alpha}(\tau_{m}) := \sum_{l=0}^{n_{a}-1}[h_{\alpha}]_{l}e^{il\tau_{m}}.$$ After computing the residual $K\phi - f$ (e.g. for $\phi = \phi_{\alpha}$), we will then need to compute $$E(\phi_\alpha, f) = \frac{1}{\\|f_{1}\\|_{L^{2}(\partial D_{c})}^{2}}\\| K_{1}\phi - f_{1}\\|_{L^{2}(\partial D_{c})}^{2} + \frac{1}{2\pi R} \\|K_{2}\phi - f_{2}\\|_{L^{2}(\partial B_{R})}^{2}.$$ Recall that the discrepancy function $F(\alpha)$ was defined by $$F(\alpha) = E(\phi_\alpha, f) - \delta^{2}, \label{eq:F}$$ where $\delta > 0$ is a fixed error parameter. As discussed in Section \[sec:stability\], the mapping $$\alpha \mapsto E(\phi_\alpha, f)$$ is not globally increasing, as can be numerically demonstrated. However, for certain feasible regions of $(\alpha, \epsilon)$, $F$ is increasing. And in this case, there is a unique $\alpha_{\delta}$ such that $F(\alpha_{\delta}) = 0$. To find $\alpha_{\delta}$, we use Newton’s method combined with an initial coarse line search to identify a good starting point. First note that if we split the matrix $A$ into two blocks $A_{near}$ ($n_{c}$ by $n_{a}$) and $A_{far}$ ($n_{R}$ by $n_{a}$) so that $$A = \left[ \begin{array}{c} A_{near}\\ A_{far} \end{array}\right],$$ then $[A\phi]_{1} = A_{near}\phi$, $[A\phi]_{2} = A_{far}\phi$, and $A^{}A = A_{near}^{}A_{near} + A_{far}^{}A_{far}$. In the discretized setting, instead of (\[eq:F1\]) we take $$F(\alpha) =\frac{1}{\\|f_{1}\\|^{2}} \\|A_{near}h_{\alpha} - f_{1}\\|_{L^{2}(\partial D_{c})}^{2} + \frac{1}{2\pi R}\\|A_{far}h_{\alpha} - f_{2}\\|_{L^{2}(\partial B_{R})}^{2} - \delta^{2}$$ with $$h_{\alpha} = (A^{}A + \alpha I)^{-1}A^{}f = (A_{near}^{}A_{near} + A_{far}^{}A_{far} + \alpha I)^{-1}{\left ( A_{near}^{}f_{1} + A_{far}^{}f_{2}\right )}.$$ Then in the same spirit as that presented in [@CoKr-Book98] for Tikhonov regularization with respect to the standard $L^2$ norm, we compute $$\begin{aligned} F'(\alpha) & = \frac{-2\alpha}{\\|f_{1}\\|_{L^{2}(\partial D_{c})}^{2}}{\textrm{Re}}{\left ( \frac{\partial h_{\alpha}}{\partial \alpha}, \, h_{\alpha}\right )} \notag\\ & \quad + {\left ( \frac{1}{\pi R} - \frac{2}{\\|f_{1}\\|_{L^{2}(\partial D_{c})}^{2}}\right )}{\textrm{Re}}{\left ( \frac{\partial h_{\alpha}}{\partial \alpha}, \, A_{far}^{}A_{far}h_{\alpha} - A_{far}^{}f_{2}\right )} \label{eq:dF1}\\ \frac{\partial h_{\alpha}}{\partial \alpha} & = -(A^{*}A + \alpha I)^{-1}h_{\alpha}, \label{eq:dF2}\end{aligned}$$ where $( \cdot, \cdot )$ denotes the $L^{2}$ inner product on $\partial D_{a}$. The function $f_{1}$ defined on $\partial D_{c}$ could be, for example, the trace of a plane wave, or of the fundamental solution to the Helmholtz equation based at some fixed point $\mathbf{x}_{0}$, i.e. a point source. For this paper, we focus on the case where $f_{1}$ is a point source and where $f_{2} \equiv 0$ on $\partial B_{R}$. A spherical point source is represented as $$\frac{i}{4}H_{0}^{(1)}(k\|\mathbf{x} - \mathbf{x}_{0}\|), \label{eq:sphericalsource}$$ where $\mathbf{x}_{0}$ is the source point (typically outside of $B_{R}$). For such an $f_{1}$, there are some quantities in which we will be interested so as to determine the effectiveness of a given density $\phi$ in solving the problem $K\phi =f$. These are: the relative error of $K\phi$ on $\partial D_{c}$; the $L^{2}$ average of $K\phi$ on $\partial B_{R}$; the relative and absolute stability of $\phi$ when applying a small perturbation to $f_{1}$; the norm of $\phi$ on $\partial D_{a}$. In other words, we will measure $$\frac{\\|K_{1} \phi - f_{1}\\|_{L^{2}(\partial D_{c})}}{\\|f_{1}\\|_{L^{2}(\partial D_{c})}}, \quad \frac{1}{\sqrt{2\pi R}} \\|K_{2}\phi\\|_{L^{2}(\partial B_{R})}, \label{eq:computevar1}$$ $$\frac{\\|\phi_{\alpha_{\epsilon}}- \phi_{\alpha_0}\\|_{L^{2}(\partial D_{a})}}{\\|\phi_{\alpha_0}\\|_{L^{2}(\partial D_{a})}}, \quad \\|\phi_{\alpha_\epsilon} - \phi_{\alpha_0}\\|_{L^{2}(\partial D_{a})}, \label{eq:computevar2}$$ and $$\\|\phi\\|_{L^{2}(\partial D_{a})}, \label{eq:phinorm}$$ where $\phi_{\alpha_\epsilon}$ is the Tikhonov regularization solution to $K\phi = (f_{1,\epsilon},0)$ with $\\|f_{1} - f_{1,\epsilon}\\|_{L^{2}(\partial D_{c})} = \epsilon \\|f_{1}\\|_{L^{2}(\partial D_{c})}$, and $\phi_{\alpha_0}$ is the solution with unperturbed $f_{1}$. The Morozov solution $\alpha_0$ and $\alpha_\epsilon$ are computed via Newton’s Method using and such that $$\begin{aligned} E(\phi_{\alpha_0}, f) & = \delta^2 \notag\\ E(\phi_{\alpha_\epsilon}, f_{\epsilon}) & = \delta^2.\end{aligned}$$ See also the discussion following . Recall from (\[random-f\]), that when adding noise to the data $(f_{1}, 0)$, we choose a random perturbation $\eta \in L^{2}(\partial D_{c})$ and set $$f_{1,\epsilon} = f_{1} + \epsilon \widehat{\eta} \\|f_{1}\\|_{L^{2}(\partial D_{c})}, \label{eq:noiseterm}$$ where $\epsilon > 0$ represents the relative percentage of noise added. In the discrete case, the noise is chosen to be a complex $n_c$-vector $\nu$ whose real and imaginary components are pseudorandom numbers (we used uniformly distributed noise, but any distribution would yield similar results) on the interval $(-1,1)$. Furthermore, for reproducibility, whenever generating $\nu$ using a pseudorandom number generator, we always reset the seed to the same value. Parameters Used for Numerical Experiments\[subsec:parametersetup\] ------------------------------------------------------------------ Here we describe some of the parameters used for the various numerical experiments presented. In Section \[subsec:sphericalsource\] we always assume that $\partial D_{a}$ is a circle with radius given by $a = 0.01$, and that $\partial D_{c}$ is a sector of an annulus with $\theta_{1} = 3\pi/4$ and $\theta_{2} = 5\pi/4$. We also take $R = 10$ in all computational examples. We remark that we always restrict the distance from $D_{c}$ to $D_{a}$ to be no smaller than $10^{-3}$ due to the numerical limitations of our approach. This is due to the fact that $K\phi$ is a singular integral when evaluating at points very near to $\partial D_{a}$. Therefore, at points on $\partial D_{c}$ that are near $\partial D_{a}$ the limitations of machine precision become more and more apparent. Numerically, we observed that our direct approach starts to break down near $d = \mathrm{dist}(\partial D_{c}, \partial D_{a}) = 10^{-4}$. However, we stress that one could most likely obtain high accuracy in computing $K\phi$ for $d \leq 10^{-4}$ by using the Nyström method as discussed in [@Kress-Book99]. For the collocation method, we use $n_{a} = 256$ sample points on $\partial D_{a}$, and $n_{\mathrm{arc}_{1}} = 256$ points on the inner arc of $\partial D_{c}$, with the remaining points chosen so as to keep the quadrature weights approximately constant. Thus for a very thin region, $n_{c} \approx 512$. We also take $n_{R} = 256$ (number of sample points on $\partial B_{R}$). Note that increasing $n_{c}$ or $n_{R}$ will put more emphasis on matching $f$ on $\partial D_{c}$ or $\partial B_{R}$, respectively. The discrepancy parameter $\delta$ used for Tikhonov regularization will typically be fixed at $0.02$. The key variables under consideration are $d = r_{1} - a$, $k$, and $\epsilon$ (perturbation parameter for adding noise to $f_{1}$). All of the plots presented in the following sections involve varying two of the aforementioned parameters and plotting different quantities of interest, as stated in (\[eq:computevar1\])-(\[eq:phinorm\]). When evaluating the relative change in $\phi$ given a perturbation of $f_{1}$, denoted by $f_{\epsilon, 1}$, we remark that for the parameter choices we used, a $0.5\%$ change ($\epsilon = 0.005$) in $f_{1}$ yielded a roughly $5\%$ change in $\phi$. However, one must keep in mind that this depends quite a lot on the parameters used. In particular, setting the discrepancy $\delta = 0.02$ in all the simulations had an important effect on the numerical results. If we had used $\delta = 0.05$ instead (which leads to approximately a $5 \%$ mismatch on the region $\partial D_{c}$), then the relative change in $\phi$ given $\epsilon = 0.005$ would be noticeably smaller. So ultimately there is a tradeoff between $\mu$, $R$, $k$, $\delta$, and $\epsilon$ which is not entirely trivial, but this still can be examined experimentally as we have done. Near field stability\[subsec:nearfieldstability\] ------------------------------------------------- We present below Figure \[fig:singularvalues\], which shows how the first $50$ singular values of the operator $K = (K_{1}, K_{2})$ vary with $d$. It is clear that for $d$ small (i.e. for control in the nearfield of the antenna), the rate of decay of the singular values of $K$ is considerably slower than for larger $d$. This in turn provides some experimental evidence for the fact that nearfield control seems to be more feasible in terms of stable dependence of the solution $\phi$ on $f$. We also show Figure \[fig:singularvaluesdiff\], which shows the behavior of the first and sixth singular value of $K$ with respect to $d$ and $k$. ![Plot of first $50$ singular values of $K:L^{2}(\partial D_{a}) \to \Xi$ for $\partial D_{a}$ a circular antenna of radius $a = 0.01$ and $\partial D_{c}$ an annular region of varying distance from $\partial D_{a}$.[]{data-label="fig:singularvalues"}](50_singular_values){width="\textwidth"} ![Plot of first singular value of $K:L^{2}(\partial D_{a}) \to \Xi$ as well as the relative difference of the first and sixth singular values with respect to $d$ and $k$. Again, $\partial D_{a}$ is a circular antenna of radius $a = 0.01$ and $\partial D_{c}$ an annular region.[]{data-label="fig:singularvaluesdiff"}](singular_values_diff){width="\textwidth"} Control for a Spherical Point Source\[subsec:sphericalsource\] -------------------------------------------------------------- We now consider the case that $$f_{1}(\mathbf{x}) = \frac{i}{4}H_{0}^{(1)}(k\|\mathbf{x} - \mathbf{x}_{0}\|),$$ where $\mathbf{x}_{0}$ is the source point. In all examples presented in this section, we have $R = 10$ unless otherwise noted, and $\mathbf{x}_{0} = [20, 0]^{T}$ or $\mathbf{x}_{0} = [10000, 0]^{T}$ (to approximate a source at infinity). First we observe how the frequency $k$ and distance $d$ from $\partial D_{c}$ to $\partial D_{a}$ affects the various control criteria. In figures \[fig:besselFS\_epsilon\_0005\_delta\_002\_vary\_k\_01\_to\_100\_mu\_0001\_to\_003\_data\] and \[fig:besselNS\_epsilon\_0005\_delta\_002\_vary\_k\_01\_to\_100\_mu\_0001\_to\_003\_data\] we vary $k$ from $0.1$ to $100$ and $d$ from $0.001$ to $0.003$ with $a = 0.01$. With the error discrepancy set at $\delta = 0.02$, we have in both figures that relative error on $\partial D_{c}$ is very close to $2\%$ for all data points. Moreover, with $0.5\%$ noise added to $f$, roughly a $5\%$ change in $\phi$ is observed at all frequencies when $d$ is near its lower limit. A bit more sensitivity is observed for frequencies $k < 20$ when $d$ increases beyond $0.01$. Interestingly, for $k > 20$ the optimal parameter $\alpha$ is larger and corresponding power budget smaller in order to achieve discrepancy $\delta$. In figures \[fig:besselFS\_epsilon\_0005\_delta\_002\_vary\_k\_01\_to\_100\_mu\_0001\_to\_003\_data\] and \[fig:besselNS\_epsilon\_0005\_delta\_002\_vary\_k\_01\_to\_100\_mu\_0001\_to\_003\_data\] it is clear that for smaller $d$ values the sensitivity of $\phi$ to $0.5\%$ noise added to $f$ is close to $5\%$. As $d$ increases, sensitivity of $\phi$ to noise increases as expected. Having $\mathbf{x}_{0}$ nearer or farther from $\partial B_{R}$ does not have a very significant effect on the overall shape of each subplot. Figures \[fig:besselFS\_vary\_mu\_epsilon\] and \[fig:besselNS\_vary\_mu\_epsilon\] show how the quantities of interest change with $d$ and the noise factor $\epsilon$, both with $k = 10$. The reason for choosing $k=10$ instead of, e.g., $k=1$ is that from figure \[fig:besselFS\_epsilon\_0005\_delta\_002\_vary\_k\_01\_to\_100\_mu\_0001\_to\_003\_data\] we see a slightly higher sensitivity of $\phi$ to noise for approximately $1 < k < 20$ when $d$ starts to increase. So the goal was to capture the worst case scenario for the control stability. For smaller values of $d$ we see as before that a roughly $0.5\%$ change in $f_{1}$ yields about a $5\%$ change in $\phi$. Moreover, the dependence on $\epsilon$ for fixed $d$ is superlinear, consistent with the illposedness of the problem. Interestingly, sensitivity of $\phi$ at $d \approx 0.015$ is better than at nearby values, but of course such a value depends on the other parameters of the problem setup. Finally, we consider Figure \[fig:besselFS\_vary\_k\_R\], which shows the dependence on $R$ and $k$ for a source at $\mathbf{x}_{0} = [10000,\, 0]^{T}$. Overall, one can see that $R$ can be decreased to around $R = 3$ at any frequency between $0.1$ and $100$ and still achieve the same approximate level of sensitivity for $\phi$ as in the previous plots with $R = 10$. Conclusions and Future Work {#sec:conclusions} =========================== In this paper we studied the feasibility of the active control scheme for the scalar Helmholtz equation. In the $L^2$ setting, we presented analytic conditional stability results as well as detailed numerical sensitivity studies for the minimal energy solution. We provided several analytic and numerical arguments for the scheme’s feasibility and broadband character in the near field when the interrogating field is a far field of a far field observer. We focused our discussion in this paper only on the case of an interrogating far field point source (i.e. similar to a plane wave with corresponding decay) because we believe that this situation is relevant in usual radar or sonar detection problems. In contrast, the case of an interrogating plane wave corresponds to a different problem, where the observer is close to the source and control region and thus the interrogating signal does not have sufficient decay. In fact, we have numerically studied the case when the interrogating field is a plane wave without decay or a given uniform field. We observed the scheme does not behave well for the uniform field and that although the stability and accuracy of the near field scheme are essentially independent of the plane wave direction, the overall performance of the scheme is not as good when compared to the case of an interrogating signal coming from a far field observer presented above. In fact, for the same settings as in Figure \[fig:nonincreasing\_F\] when comparing the case of an interrogating far field point source with an interrogating plane wave, we obtained $5\%$ versus $8\%$ stability error and power budget levels of $\approx 10^{-1}$ versus $\approx 10$. We conclude that the scheme performance depends not only on the location of the control region with respect to the source region but also on the amplitude and oscillatory pattern of the incoming field. Currently we are considering a more localized basis for $L^2(\partial D_a)$ (e.g. delta function basis, or splines) in order to better observe the field characteristics in the control region $D_c$ and around the antenna $D_a$. We also plan to study the active control scheme for linear arrays and for large elongated antennas. Then, as a next step in our research efforts, we will work on the extension of the current numerical sensitivity study to three dimensions and full Maxwell system and on the study of near field control with planar and conformal arrays.	ArXiv
--- abstract: 'For a nonnegative weakly irreducible tensor $\mathcal{A}$, we give some characterizations of the spectral radius of $\mathcal{A}$, by using the digraph of tensors. As applications, some bounds on the spectral radius of the adjacency tensor and the signless Laplacian tensor of the $k$-uniform hypergraphs are shown.' address: - 'School of Science, Harbin Institute of Technology, Harbin 150001, PR China' - 'School of Mathematical Sciences and Shanghai Key Laboratory of Contemporary Applied Mathematics, Fudan University, Shanghai, 200433, PR China' - 'College of Science, Harbin Engineering University, Harbin 150001, PR China' author: - Lizhu Sun - Baodong Zheng - Yimin Wei - Changjiang Bu title: Characterizations of the spectral radius of nonnegative weakly irreducible tensors via digraph --- Nonnegative tensor, Spectral radius, Digraph, $k$-uniform hypergraph\ AMS classification: 15A69, 15A18 Introduction {#Introduction } ============ An order $m$ dimension $n$ tensor $\mathcal{A}$ consist of $n^{m}$ complex entries, $$\mathcal{A}=\left( {a_{i_1i_2\cdots i_m } } \right),~{1 \leqslant i_j\leqslant n}~ \left( {j = 1,2,\ldots ,m} \right).$$ The tensor $A=(a_{i_1i_2\cdots i_m})$ is called symmetric if $a_{i_1i_2\cdots i_m}=a_{\sigma(i_1)\sigma(i_2)\cdots \sigma(i_m)}$, where $\sigma$ is any permutation of the indices. The tensor $\mathcal{A}$ is called nonnegative if all the entries $a_{i_1\ldots i_m } \geqslant 0$. Let $\mathbb{R}^{[m,n]}_+$ be the set of order $m$ dimension $n$ nonnegative tensors. And let $\mathbb{R}^{n}_{++}$ be the set of the dimension $n$ positive vectors (all the entries positive). In 2005, the eigenvalue of tensors is defined by Qi [@Qi2005] and Lim [@Lim; @2005], respectively. For a complex tensor $\mathcal{A}$ of order $m$ dimension $n$, a complex number $\lambda $ and a nonzero complex vector $x=(x_1,x_2,\ldots,x_n)^{\rm T}$ is called an eigenvalue and an eigenvector (corresponding to $\lambda$) of the tensor $\mathcal{A}$, respectively, if they satisfy $$\mathcal{A}x^{m - 1} = \lambda x^{\left[ {m - 1} \right]},$$ where $\mathcal{A}x^{m - 1}$ is a dimension $n$ vector with entry $$\left( {\mathcal{A}x^{m - 1} } \right)_i = \sum\limits_{i_2 ,...,i_m = 1}^n {a_{ii_2 ...i_m } x_{i_2 } ...x_{i_m } }, i=1,2,\ldots,n,$$ and $x^{\left[ {m - 1} \right]} = \left( {x_1^{m - 1} ,x_2^{m - 1},\ldots,x_n^{m - 1} } \right)^\mathrm{T}$ (see [@Qi2005]). Let $\rho(\mathcal{A})=\sup\{\|\lambda\|:\lambda \in {\rm spec}(\mathcal{A})\}$ be the spectral radius of tensor $\mathcal{A}$, where ${\rm spec}(\mathcal{A})$ is the set of all the eigenvalues of $\mathcal{A}$. Recently, the spectral theory of tensors has attracted much attention [@Bu2015; @Chang; @Chang; @2011; @Yang; @2010; @Yang; @2013]. In 2005, Lim [@Lim; @2005] proposes the definition of irreducible tensors. In 2008, Chang et al. [@Chang] give the Perron-Frobenius Theorem for nonnegative irreducible tensors. It is shown that $\rho(\mathcal{A})$ is an eigenvalue of nonnegative irreducible tensors, and $\rho(\mathcal{A})$ is the only eigenvalue with nonnegative eigenvectors [@Chang]. Similarly as the Collatz-Wielandt Theorem of matrices, the Minimax Theorem of the spectral radius of nonnegative irreducible tensors $\mathcal{A}$ is given as follows (see [@Chang]) $$\min \limits_{x \in \mathbb{R}_{++} ^n }\max \limits_{1 \leqslant i \leqslant n} \frac{{\left( {\mathcal{A}x^{m - 1} } \right)_i }} {{x_i^{m - 1} }} = \rho \left( \mathcal{A} \right) = \max \limits_{x \in \mathbb{R}_ {++} ^n} \min \limits_{1 \leqslant i \leqslant n} \frac{{\left( {\mathcal{A}x^{m - 1} } \right)_i }} {{x_i^{m - 1} }} .$$ Scholars pay much attention to find the largest eigenvalue of nonnegative irreducible tensors [@Chen2013; @Ng; @Ni2014; @Zhou; @2013]. The weakly irreducible tensors are defined by associated with tensors a digraph [@SF; @2013; @Pearson2014]. \[def1\] An order $m$ dimension $n$ tensor $\mathcal{A}$ over real field is called weakly irreducible if the digraph $G_{\mathcal{A}}$ is strongly connected. Digraph $G_{\mathcal{A}}$ and the strongly connectivity are introduced in Section 2. If tensor $\mathcal{A}$ is irreducible then it is weakly irreducible [@SF; @2013; @Pearson2014]. For the nonnegative weakly irreducible tensor $\mathcal{A}$, there exists a positive eigenvector corresponding to the eigenvalue $\rho(\mathcal{A})$ (see [@SF; @2013]). A hypergraph $\mathcal{H}=(V(\mathcal{H}), E(\mathcal{H}))$ is called $k$-uniform if each edge of $\mathcal{H}$ contains exactly $k$ distinct vertices [@Cooper2012]. The adjacency tensor of $\mathcal{H}$, denoted by $\mathcal{A}_{\mathcal{H}}=(a_{i_1i_2\cdots i_k})$, is an order $k$ dimension $\|V(\mathcal{H})\|$ tensor with entries $$a_{i_1i_2\cdots i_k}= \left\{ {\begin{array}{{20}{c}} \frac{1}{(k-1)!}, ~ \{i_1i_2\cdots i_k\}\in E(\mathcal{H}), \\ 0 ,~~~~~~~\mbox{otherwises} .~~~~~ \\ \end{array}} \right.$$ The degree tensor* of the $k$-uniform uniform hypergraph $\mathcal{H}$, denoted by $\mathcal{D}_{\mathcal{H}}$, is an order $k$ dimension $\|V(\mathcal{H})\|$ diagonal tensor whose $(i\cdots i)$-diagonal entry is the degree of vertex $i$, $i=1,2,\ldots,\|V(\mathcal{H})\|$. The tensor $\mathcal{Q}_{\mathcal{H}}=\mathcal{D}_{\mathcal{H}}+\mathcal{A}_{\mathcal{H}}$ is called the signless Laplacian tensor of the $\mathcal{H}$. Recently, the spectral theory of hypergraphs developed rapidly [@Cooper2012; @Hu2015; @Qi2014; @Shao2015; @Zhou2014]. It is well-known that the nonnegative irreducible matrices are closely related to the digraphs [@RA1991]. For a nonnegative irreducible matrices $M$ with all diagonal entries zero, Brualdi [@RA1982] gives the characterizations of the spectral radius of $M$ by using the associated digraph. In this paper, we use the digraph of tensors to characterize the spectral radius of the nonnegative weakly irreducible tensors, which generalize the results of matrices to tensors [@RA1982]. By applying the characterizations, some bounds on the spectral radius of the adjacency tensor and signless Laplacian tensor of a $k$-uniform hypergraph are shown. Preliminary =========== For an order $m$ dimension $n$ tensor $A=(a_{i_1i_2\cdots i_m})$, let $G_\mathcal{A} = \left({V(G_\mathcal{A} ),E( G_\mathcal{A} )} \right)$ be the digraph of the tensor $\mathcal{A }$ with vertex set $V (G_\mathcal{A} )= \left\{ {1,2,\ldots,n} \right\}$ and arc set $E( G_\mathcal{A} ) = \{ (i,j)\| a_{ii_2 ...i_m } \neq 0,j \in \{ i_2 ,...,i_m \}\}$ (see [@SF; @2013; @Pearson2014]). If there exist directed paths from $i$ to $j$ and $j$ to $i$ for each $i, ~j\in V(G_\mathcal{A} )$ ($i\neq j$), then $ G_\mathcal{A}$ is called strongly connected. Denote the set of the circuits in $G_\mathcal{A} $ by $C( G_\mathcal{A} )$ (Loops in the circuits are allowed). Let $G_\mathcal{A} ^ + \left( v \right): = \left\{ {u\in V(G_\mathcal{A} ):\left( {v,u} \right) \in E( G_\mathcal{A} )} \right\}$. Define a map $f$ from the vertex set of $G_\mathcal{A} $ to the real field, $f:V(G_\mathcal{A} )\rightarrow \mathbb{R}$, $f$ is called a vertex labelling of $G_\mathcal{A} $. By the Lemma 2.6 of [@RA1982], we can get the following result. [[@RA1982]]{}\[lem1\] Let $G_\mathcal{A} $ be the digraph of $\mathcal{ A}$ with a vertex labelling $f$ on $V(G_\mathcal{A} )$. If $G_\mathcal{A} ^ + \left( v \right)$ is nonempty for each $v \in V(G_\mathcal{A} )$, then there exist circuits $\{v_{i_1 },v_{i_2 } ,...,v_{i_k } ,v_{i_{k + 1} } \\= v_{i_1 }\}$ and $\{v_{t_1 },v_{t_2 } ,...,v_{t_s } ,v_{t_{s + 1} } = v_{t_1} \}$ (Loops in the circuits are allowed) such that $f(v_{i_{j + 1} })=\max\{f(v):v\in G_\mathcal{A} ^ + ( v_{i_j} )\}$ and $f(v_{t_{l + 1} })=\min\{f(v):v\in G_\mathcal{A}^ + ( v_{t_l} )\}$ $j = 1,2,\ldots,k$, $ l = 1,2,\ldots,s$, respectively. [[@SF; @2013]]{}\[lem3\] Let $\mathcal{A}\in \mathbb{R}^{[m,n]}_+$ be a weakly irreducible tensor. Then $\rho(\mathcal{A})$ is an eigenvalue of $\mathcal{A}$, and there exists a unique positive eigenvector corresponding to $\rho(\mathcal{A})$ up to a multiplicative constant. [[@shao; @2013]]{}\[lem5\] For an order $m$ dimension $n$ tensor $\mathcal{A}={(a_{i_1i_2\ldots i_m})}$ and an invertible diagonal matrix $D={\rm diag}(d_{11}, d_{22},\ldots, d_{nn})$, $\mathcal{B}=D^{-(m-1)}\mathcal{A}D$ is an order $m$ dimension $n$ tensor with entries $$b_{i_1i_2\ldots i_m}= d_{i_1 i_1}^{-(m-1)}a_{i_1i_2\ldots i_m}d_{i_2i_2}\cdots d_{i_mi_m}.$$ In this case, $\mathcal{A}$ and $\mathcal{B}$ are called diagonal similar, and $\mathcal{A}$ and $\mathcal{B}$ have the same spectrum. \[lem6\][[@Pearson2014]]{} Let $\mathcal{H}$ be an $k$-uniform hypergraph. Then $\mathcal{A}_{\mathcal{H}}\in \mathbb{R}^{[m,n]}_+$ (and $\mathcal{Q}_{\mathcal{H}}\in \mathbb{R}^{[m,n]}_+$) is weakly irreducible if and only if $\mathcal{H}$ is connected. Main results ============ For a tensor $\mathcal{A}=\left( {a_{i_1 i_2 \cdots i_m } } \right)\in \mathbb{R}^{[m,n]}_+$, we denote the sum of $i$-th slice of $\mathcal{A}$ by $K_i= \sum\limits^n_{i_2,\ldots,i_m =1}{a_{ii_2 \cdots i_m } } $, $i=1,2,\ldots,n$. Let $\|\gamma\|$ be the length of the circuit $\gamma\in C (G_\mathcal{A} )$. We first give two results on the bounds of spectral radius for nonnegative weakly irreducible tensors, which extend Theorem 4.7 and Corollary 4.6, 4.8 of [@RA1982] to tensors. \[thm2\] Let $\mathcal{A} = \left( {a_{i_1 i_2 ...i_m } } \right) \in \mathbb{R}^{[m,n]}_+$ be a weakly irreducible tensor. Then $$\min_{\gamma \in C(G_\mathcal{ A})}\left(\prod\limits_{i\in \gamma } {K_i }\right)^{\frac{1}{\|\gamma\|}}\leqslant \rho (\mathcal{A}) \leqslant \max_{\gamma \in C(G_\mathcal{ A})} \left(\prod\limits_{i\in \gamma } {K_i }\right)^{\frac{1}{\|\gamma\|}}.$$ By Lemma \[lem3\], we suppose that $x = \left( {x_1 ,x_2, \ldots,x_n } \right)^{\rm T}$ is a positive eigenvector corresponding to the eigenvalue $\rho(\mathcal{A})$. Let $f(i)=x_i$ ($i \in V(G_\mathcal{A})$) be the vertex labelling of digraph $G_\mathcal{A}$. From Definition \[def1\], we have $G_\mathcal{A}^+(i)$ is nonempty for each $i\in V(G_\mathcal{A})$. Lemma \[lem1\] gives that there exists at least one circuit $\gamma_1 = \{i_1 ,i_2,\ldots,i_p ,i_{p + 1} = i_1\}$ such that $x_{i_{j + 1} } \geqslant x_k $, for each $ k \in G_\mathcal{A} ^ + \left( {i_j } \right)$, $j = 1,2,\ldots,p$. Hence, $$\begin{aligned} \rho(\mathcal{A}) x_{i_j }^{m - 1} & = \sum\limits_{ k_2 ,\cdots,k_m =1}^n a_{i_j k_2 ...k_m } x_{k_2} \cdots x_{k_m } \hfill \\ &\leqslant \left( {\sum\limits_{ k_2 ,\cdots,k_m =1}^n {a_{i_j k_2 ...k_m } }} \right)x_{i_{j + 1} }^{m - 1} \hfill \\ & = K_{i_j } x_{i_{j + 1} }^{m - 1} \hfill ,\end{aligned}$$ for $j = 1,2,\ldots,p$. Thus $$\left(\rho (\mathcal{A})\right)^p \prod\limits_{j = 1}^p x_{i_j }^{m - 1} \leqslant \prod\limits_{j = 1}^p {K_{i_j } x_{i_{j + 1} }^{m - 1} } .$$ Note that $x = \left( {x_1 ,x_1,\ldots,x_n } \right)^{\rm T}$ is positive, so we can get $$\rho (\mathcal{A}) \leqslant \left(\prod\limits_{j = 1}^p {K_{i_j } }\right)^{\frac{1}{p}} ,$$ that is $$\rho (\mathcal{A}) \leqslant \left(\prod\limits_{i\in \gamma_1 } {K_i }\right)^{\frac{1}{p}}.$$ Lemma \[lem1\] also shows that there exists at least one circuit $\gamma_2=\{v_{t_1 } ,v_{t_2 },\ldots,v_{t_s } ,v_{t_{s + 1} } = v_{t_1 }\}$ such that $x_{t_{l + 1} } \leqslant x_k $ for each $ k \in G_\mathcal{A} ^ + ( t_l )$, $l = 1,2,\ldots,s$. Similarly as the above proof, we can get $$\rho (\mathcal{A}) \geqslant \left(\prod\limits_{i\in \gamma_2 } {K_i }\right)^{\frac{1}{s}}.$$ Thus $$\min_{\gamma \in C(G_\mathcal{ A})}\left(\prod\limits_{i\in \gamma } {K_i }\right)^{\frac{1}{\|\gamma\|}} \leqslant \rho (\mathcal{A}) \leqslant \max_{\gamma \in C(G_\mathcal{ A})} \left(\prod\limits_{i\in \gamma } {K_i }\right)^{\frac{1}{\|\gamma\|}}.$$ Remark. It is easy to see that if $K_1=K_2=\cdots=K_n$, the equalities in the above theorem hold. Lemma \[lem6\] gives that the adjacency tensor (and signless Laplacian tensor) of a connected hypergraph $\mathcal{A}_{\mathcal{H}}$ (and $\mathcal{Q}_{\mathcal{H}}$) is nonnegative weakly irreducible, so $\rho(\mathcal{A}_{\mathcal{H}})$ (and $\rho(\mathcal{Q}_{\mathcal{H}})$) is an eigenvalue of $\mathcal{A}_{\mathcal{H}}$ (and $\mathcal{Q}_{\mathcal{H}}$). In [@Cooper2012], it is shown that the largest eigenvalue $\lambda_{\max}$ of $\mathcal{A}_{\mathcal{H}}$ is between the maximum degree $d_{\max}$ and the average degree $\overline{d}$ of $\mathcal{H}$, $\overline{d} \leqslant\lambda_{\max}\leqslant d_{\max} $. By using Theorem \[thm2\], we also show the bounds on the largest eigenvalue of adjacency tensor and signless Laplacian tensor in terms of the degrees. \[thm5\] Let $\mathcal{H}$ be a connected $k$-uniform hypergraph with $n$ vertices. Suppose that $d_i$ is the degree of vertex $i$, $i=1,2,\ldots,n$. Then $$\min_{\gamma \in C(G_{\mathcal{ A}_{\mathcal{H}}})}\left(\prod\limits_{i\in \gamma } {d_i }\right)^{\frac{1}{\|\gamma\|}}\leqslant \rho(\mathcal{A}_{\mathcal{H}}) \leqslant \max_{\gamma \in C(_{\mathcal{ A}_{\mathcal{H}}})} \left(\prod\limits_{i\in \gamma } {d_i }\right)^{\frac{1}{\|\gamma\|}},$$ and $$\min_{\gamma \in C(G_{\mathcal{ Q}_{\mathcal{H}}})}\left(\prod\limits_{i\in \gamma } {2d_i }\right)^{\frac{1}{\|\gamma\|}}\leqslant \rho(\mathcal{Q}_{\mathcal{H}}) \leqslant \max_{\gamma \in C(_{\mathcal{ Q}_{\mathcal{H}}})} \left(\prod\limits_{i\in \gamma } {2d_i }\right)^{\frac{1}{\|\gamma\|}}.$$ The shortest length of the circuits in $G_\mathcal{ A}$ is called the girth of $G_\mathcal{ A}$. If we bring an order to the slice sums $K_i$ $(i=1,2,\ldots,n)$ of tensor $\mathcal{A}$, the following result can be obtained. \[thm1\] Let $\mathcal{A} = \left( {a_{i_1 i_2 ...i_m } } \right)\in \mathbb{R}^{[m,n]}_+ $ be a weakly irreducible tensor. Suppose that $K_1 \leqslant K_2\leqslant \cdot\cdot\cdot \leqslant K_n$ and the girth of $G_\mathcal{ A}$ is $g$. Then $$\left ({K_1K_2\cdots K_g }\right)^{\frac{1}{g}}\leqslant \rho \left( \mathcal{A} \right) \leqslant \left({K_{n - g + 1} K_{n - g + 2}\cdots K_n }\right)^{\frac{1}{g}} .$$ By Theorem \[thm2\], we have there exist circuits $\gamma_1,\gamma_2\in C(G_\mathcal{ A})$ of length $\|\gamma_1\|\geqslant g,~\|\gamma_2\|\geqslant g$ such that $$\left({\prod\limits_{i \in \gamma_2 } {K_i } }\right)^{\frac{1}{\|\gamma_2\|}}\leqslant \rho \left( \mathcal{A} \right) \leqslant \left({\prod\limits_{i \in \gamma_1 } {K_i } }\right)^{\frac{1}{\|\gamma_1\|}} .$$ Since $$\left({\prod\limits_{i \in \gamma_1 } {K_i } }\right)^{\frac{1}{\|\gamma_1\|}} \leqslant \left({K_{n - \|\gamma_1\| + 1} K_{n - \|\gamma_1\| + 2}\cdots K_n }\right)^{\frac{1}{\|\gamma_1\|}} \leqslant \left({K_{n - g + 1} K_{n - g + 2}\cdots K_n }\right)^{\frac{1}{g}}$$ and $$\left({\prod\limits_{i \in \gamma_2 } {K_i } }\right)^{\frac{1}{\|\gamma_2\|}}\geqslant \left(K_1K_2\cdots K_{\|\gamma_2\|} \right)^{\frac{1}{\|\gamma_2\|}}\geqslant \left(K_1K_2\cdots K_g \right)^{\frac{1}{g}},$$ we obtain the theorem holds. For a connected $k$-uniform hypergraph $\mathcal{H}$, According to the definition of hypergraph, there are no loops in $G_{\mathcal{A}_{\mathcal{H}}}$. And since $\mathcal{A}_{\mathcal{H}}$ is symmetric tensor, we have the girth of $G_{\mathcal{A}_{\mathcal{H}}}$ is $2$. Clearly, the girth of $G_{\mathcal{Q}_{\mathcal{H}}}$ is $1$. Hence, it follows from Theorem \[thm5\] and \[thm1\] that the follow result can be obtained, which are shown in [@Cooper2012] and [@Hu2015+], respectively. For a connected $k$-uniform hypergraph $\mathcal{H}$, if $\mathcal{H}$ is regular with degree $d$, then $\rho(\mathcal{A}_{\mathcal{H}})=d$ and $\rho(\mathcal{Q}_{\mathcal{H}})=2d$. As we introduce in the first section, there are some results to find the spectral radius (largest eigenvalue) of nonnegative irreducible tensors. Here, we also give two theorems on the minimum and the maximum characterizations of the spectral radius of nonnegative weakly irreducible tensors. \[thm3\] Let $\mathcal{A} = \left( {a_{i_1 i_2 ...i_m } } \right) \in \mathbb{R}^{[m,n]}_+$ be a weakly irreducible tensor. Then $$\min_{x\in \mathbb{R}_{++}^n} \max_{\gamma\in C(G_\mathcal{ A})}\left(\prod_{i\in\gamma} \frac{(\mathcal{A}x^{m-1})_i}{x_i^{m-1}}\right)^{\frac{1}{\|\gamma\|}}=\rho(\mathcal{A})=\max_{x\in \mathbb{R}_{++}^n} \min_{\gamma\in C(G_\mathcal{ A})}\left(\prod_{i\in\gamma} \frac{(\mathcal{A}x^{m-1})_i}{x_i^{m-1}}\right)^{\frac{1}{\|\gamma\|}}.$$ Let $\mathcal{B}=X^{-(m-1)}\mathcal{A}X$, where the matrix $X={\rm diag}(x_1,x_2,\ldots,x_n)$, $x_i>0$, $i=1,2,\ldots,n$. It is easy to see $\mathcal{B}$ is nonnegative weakly irreducible, and $G_\mathcal{ A}$ and $G_\mathcal{ B}$ are the same digraph. So by Theorem \[thm2\], we have $$\min_{\gamma\in C(G_\mathcal{ A})}\left(\prod_{i\in\gamma} K_i(\mathcal{B})\right)^{\frac{1}{\|\gamma\|}}\leqslant \rho(\mathcal{B})\leqslant\max_{\gamma\in C(G_\mathcal{ A})}\left(\prod_{i\in\gamma} K_i(\mathcal{B})\right)^{\frac{1}{\|\gamma\|}}.$$ Calculation gives that $$K_i(\mathcal{B})= \frac{1}{{x_i^{m - 1} }} \sum\limits^n_{ i_2,\cdots,i_m =1 } { {a_{ii_2 ...i_m } } x_{i_2 } ...x_{i_m } }=\frac{(\mathcal{A}x^{m-1})_i}{x_i^{m-1}},$$ and since Lemma \[lem5\] shows that $\rho(\mathcal{A})=\rho(\mathcal{B})$, we obtain $$\min_{\gamma\in C(G_\mathcal{ A})}\left(\prod_{i\in\gamma} \frac{(\mathcal{A}x^{m-1})_i}{x_i^{m-1}}\right)^{\frac{1}{\|\gamma\|}}\leqslant \rho(\mathcal{A})\leqslant \max_{\gamma\in C(G_\mathcal{ A})}\left(\prod_{i\in\gamma} \frac{(\mathcal{A}x^{m-1})_i}{x_i^{m-1}}\right)^{\frac{1}{\|\gamma\|}}.$$ When $(x_1,x_2,\ldots, x_n)^{\rm T}$ is a positive eigenvector of $\mathcal{A}$ corresponding to $\rho(\mathcal{A})$, the equalities in the above equation hold, so we can get $$\min_{x\in \mathbb{R}_{++}^n} \max_{\gamma\in C(G_\mathcal{ A})}\left(\prod_{i\in\gamma} \frac{(\mathcal{A}x^{m-1})_i}{x_i^{m-1}}\right)^{\frac{1}{\|\gamma\|}}=\rho(\mathcal{A})=\max_{x\in \mathbb{R}_{++}^n} \min_{\gamma\in C(G_\mathcal{ A})}\left(\prod_{i\in\gamma} \frac{(\mathcal{A}x^{m-1})_i}{x_i^{m-1}}\right)^{\frac{1}{\|\gamma\|}}.$$ By the proof of the above theorem, we get the following result. Let $\mathcal{A} = \left( {a_{i_1 i_2 ...i_m } } \right) \in \mathbb{R}^{[m,n]}_+$ be a weakly irreducible tensor. Then $$\min_{\gamma\in C(G_\mathcal{ A})}\left(\prod_{i\in\gamma} \frac{(\mathcal{A}x^{m-1})_i}{x_i^{m-1}}\right)^{\frac{1}{\|\gamma\|}}\leqslant \rho(\mathcal{A})\leqslant \max_{\gamma\in C(G_\mathcal{ A})}\left(\prod_{i\in\gamma} \frac{(\mathcal{A}x^{m-1})_i}{x_i^{m-1}}\right)^{\frac{1}{\|\gamma\|}},$$ where $x=(x_1,x_2,\ldots,x_n)$ is a positive vector. From Theorem \[thm1\], the following result can be obtained. \[thm4\] Let $\mathcal{A}\in \mathbb{R}^{[m,n]}_+$ be a weakly irreducible tensor. Let the girth of $G_\mathcal{ A}$ is $g$. Then $$\min_{X\in D^n} \left({\prod_{i=n-g+1}^n K_i(X^{-(m-1)}\mathcal{A}X)}\right)^{\frac{1}{g}}=\rho(\mathcal{A}) =\max_{X\in D^n} \left({\prod_{i=1}^gK_i(X^{-(m-1)}\mathcal{A}X)}\right)^{\frac{1}{g}},$$ where the slice sums of tensor $X^{-(m-1)}\mathcal{A}X$ are in the order $K_1 \leqslant K_2 \leqslant\cdot\cdot\cdot \leqslant K_n$ and $D^n$ is the set of all the $n\times n$ positive diagonal matrices. Remark. Brualdi gives the characterizations of the spectral radius of a nonnegative irreducible matrices with all diagonal entries zero ( Corollary 4.10 and 4.11 of [@RA1982]). Theorem \[thm3\] and \[thm4\] generalize them to general nonnegative weakly irreducible tensors without the condition that diagonal entries are zero. References [00]{} R.A. Brualdi, H.J. Ryser, Combinatorial Matrix Theory, Cambridge University Press, Cambridge, 1991. R.A. Brualdi, Matrices, eigenvalues, and directed graphs, Linear and Multilinear Algebra 11(1982) 143-165. C. Bu, Y. Wei, L. Sun, J. 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--- abstract: 'We introduce defects, with internal gauge symmetries, on a knot and Seifert surface to a knot into the combinatorial construction of finite gauge-group Dijkgraaf-Witten theory. The appropriate initial data for the construction are certain three object categories with coefficients satisfying a partially degenerate cocycle condition.' author: - 'I. J. Lee and D. N. Yetter' title: 'Dijkgraaf-Witten Type Invariants of Seifert Surfaces in 3-Manifolds' --- Introduction ============ Throughout we work exclusively in the PL category, all manifolds are compact, and we write composition in diagrammatic order. The goal of this paper is to construct state-sum invariants of triples $K \subset \Sigma \subset M$, in which $M$ is an oriented 3-manifold, $K$ an oriented knot or link in $M$ and $\Sigma$ an oriented surface with boundary embedded in $M$ with $\partial \Sigma = K$, extending and generalizing the state-sum construction of finite-gauge group Dijkgraaf-Witten theory given by Wakui [@W]. To avoid confusion between link in the knot-theoretic sense and the link of a simplex in the sense of combinatorial topology, we will always refer to $K$ as “the knot”, even though it might be a link in the knot-theoretic sense. The arrangement $K \subset \Sigma \subset M$ is a very simple stratified space of the sort Crane and Yetter [@CY] called “starkly stratified ”. However, as was the case in two dimensions (cf. [@DPY]), our constructions will require us to restrict our attention to flag-like triangulations. The first task will, thus, be to show that the instances of extended bistellar moves which preserve flag-likeness suffice to give all PL homeomorphisms between stratified PL spaces of the form $K \subset \Sigma \subset M$. From [@CY] recall A [starkly stratified space]{} is a PL space $X$ equipped with a filtration $$X_0 \subset X_1 \subset \ldots \subset X_{n-1} \subset X_n = X$$ satisfying 1. There is a triangulation $\mathcal T$ of $X$ in which each $X_k$ is a subcomplex. 2. For each $k = 1,\ldots n$ $X_k \setminus X_{k-1}$ is a(n open) $k$-manifold. 3. If $C$ is a connected component of $X_k \setminus X_{k-1}$, then $\mathcal T$ restricted to $\bar{C}$ gives $\bar{C}$ the structure of a combinatorial manifold with boundary. 4. For each combinatorial ball $B_k$ with $\accentset{\circ}B_k \subset X_k \setminus X_{k-1}$, $\accentset{\circ}B_k$ admits a closed neighborhood $N$ given inductively as a cell complex as follows (although we require $B_k$ to be a combinatorial ball, the triangulation is then ignored): $N = N_n$, where $N_m$ for $k \leq m \leq n$ is given inductively by $$N_k = B_k$$ and $$N_{\ell+1} = N_\ell \cup \bigcup_{v \in S_\ell} L(v) \ast v$$ for $S_\ell$ a finite set of points in $X\setminus X_\ell$, andl $L(\cdot)$ a function on $S_\ell$ valued in $$\{ L \| L\; \mbox{\rm is a combinatorial ball and}\; B_k \subset L \subset N_\ell \}$$ We will call such a neighborhood of the interior of a combinatorial ball lying in a stratum of the same dimension a [stark neighborhood]{}. It is quite easy to show that for a knot or link $K$ with a Seifert surface $\Sigma$ in a 3-manifold $M$, the filtered space $\emptyset \subset K \subset \Sigma \subset M$ is a starkly stratified space – the stark neighborhoods of open simplexes $\Sigma$ are simply the joins of their closure with two points of $M \setminus \Sigma$, one on each side of $\Sigma$, $\Sigma$ being, of course, two-sided, while those of open simplexes in $K$ are iterated joins, first with a point in $\Sigma$ and a point in $M \setminus \Sigma$ and then of with two points of $M \setminus \Sigma$, one on either side of $\Sigma$ (or more properly the union of $\Sigma$ with the first join, which is easily seen to also be an oriented surface with boundary). Throughout, despite allowing the possibility that is is actually a link with more than one component, we will refer to $K$ as “the knot” to avoid the ambiguity of the word “link” in the context of PL topology, as we will have cause to mention the link of a simplex in the PL sense fairly often in our exposition. We will also need A triangulation $\mathcal T$ of a stratified PL space $$X_0 \subset X_1 \subset \ldots \subset X_{n-1} \subset X_n = X$$ is [flag-like]{} if each of the $X_i$ is a subcomplex, and moreover for each simplex $\sigma$ of $\mathcal T$, the restriction of the filtration to the simplex, that is the distinct non-empty intersections in the sequence $$X_0\cap \sigma \subset X_1 \cap \sigma \subset \ldots \subset X_{n-1} \cap \sigma \subset \sigma$$ form a (possibly incomplete) simplicial flag. Observe that in any starkly stratified space, it follows from the iterative construction of a stark neighborhood, that the restriction of the filtration to any simplex of a stark neighborhood is a simplical flag. Moreover, in general, flag-like triangulations are plentiful. In particular, it is easy to show If $\mathcal T$ is a triangulation of a stratified PL space $$X_0 \subset X_1 \subset \ldots \subset X_{n-1} \subset X_n = X$$ for which each $X_i$ is a subcomplex, then its barycentric subdivision $\beta X$ is flag-like. As our construction will require a flag-like triangulation, the most direct way to show topological invariance will be show that combinatorial moves which preserve flag-likeness suffice to give all PL homeomorphisms between our stratified spaces $K \subset \Sigma \subset M$. An Alexander move (stellar subdivision or stellar weld), a Pachner move (bistellar move), or an extended Pachner move (extended bistellar move in the sense of [@CY]) on a triangulation of a stratified space is [flag-like]{} if the restriction of the filtration to each simplex of both its initial and its final state is a simplicial flag. From Alexander Moves to Extended Pachner Moves ============================================== As observed in [@DPY], any Alexander subdivision is flag-like, while there exist non-flag-like Alexander welds, but the theorem of [@DPY] that flag-like Alexander moves suffice to characterize PL homeorphism of stratified PL spaced equipped with flag-like triangulations hold in full generality: If $\mathcal T$ and ${\mathcal T}^\prime$ are flag-like triangulations of a stratified PL spaces $$X_0 \subset X_1 \subset \ldots \subset X_{n-1} \subset X_n = X$$ and $$X_0^\prime \subset X_1^\prime \subset \ldots \subset X_{n-1}^\prime \subset X_n^\prime = X^\prime ,$$ respectively, then there is a stratification-preserving PL homeomorphism from $X$ to $X^\prime$ if and only if there is a sequence of flag-like Alexander moves starting with $\mathcal T$ and resulting in a triangulation of $X$ which is combinatorially equivalent to the triangulation ${\mathcal T}^\prime$ of $X^\prime$. The proof is identical to that given in the special case in [@DPY]: Sufficiency is clear, necessity follows from the full force of Alexander’s theorem – that two complexes are PL homoemorphic if and only if there is a common subdivision by Alexander subdivisions – and the observation that subdivisions of flag-like triangulations are always flag-like: starting at $\mathcal T$ and ${\mathcal T}^\prime$, apply Alexander subdivisions (necessarily flag-like) until a common combinatorial type of triangulation is reached. The sequence in the theorem is then the sequence of subdivsions beginning at $\mathscr T$ followed by the welds reversing the subdivisions starting at ${\mathscr T}^\prime$ in reverse order. Working with Alexander moves in the context of state-sum models is difficult, since in dimensions three and greater there are infintely many combinatorial types of Alexander move, and what is more, the number of simplexes in the subdivided triangulation is quite large. We therefore wish to reduce PL homeorphism of our triples $K \subset \Sigma \subset M$, equipped with flag-like triangulations to flag-like instances of the extended Pachner moves of [@CY]. \[sufficient\_moves\] Every flag-like Alexander move relating triangulations of a knot, Seifert surface, 3-manifold triple $K \subset \Sigma \subset M$ can be accomplished by a sequence of flag-like extended Pachner moves. Every flag-like Alexander move relating triangulation of a knot, Seifert surface, 3-manifold triple $K \subset \Sigma \subset M$ can be accomplished by a sequence of flag-like moves of the following types: Pachner moves on tetrahedra, extended Pachner moves on triangles of the Seifert surface, and Alexander subdivision of an edge of the knot whose link has exactly three edges (a 3-6 move). It suffices to show that flag-like Alexander subdivisions can be accomplished by sequences of flag-like extended Pachner moves. For subdivisions of tetrahedra, subdivisions of triangles lying in $\Sigma$ and of edges of $K$ with regular neighborhoods as depicted below, (1.9,-0.5) –(1.5,-1.8) – (1.5,1.8); (0,0) –(1.9,-0.5) – (3,0.5) – (1.5,1.8)– (0,0); (0,0) –(1.5,-1.8)– (3,0.5); (1.9,-0.5)– (1.5,-1.8); (1.5,1.8)– (1.9,-0.5); (0,0) – (1.2,1.0) – (3,0.5); (1.5,1.8) – (1.2,1.0) – (1.5,-1.8); (1.5,1.8) – (1.5,-0.2); (1.5,-0.6) – (1.5,-1.8); (1.5,1.8)–(1.9,-0.5)–(1.5,-1.8); (1.5,1.8) circle (1mm); (1.9,-0.5) circle (1mm); (1.5,-1.8) circle (1mm); the first statement is immediate, since the Alexander move itself is an extended Pachner move. Likewise the second statement is immediate for subdivisions of tetrahedra, triangles lying in $\Sigma$ and edges of $K$ with regular neighborhoods as depicted here: (1.9,-0.5) –(1.5,-1.8) – (1.5,1.8); (0,0) –(1.9,-0.5) – (3,0.5) – (1.5,1.8)– (0,0); (0, 0) – (3,0.5); (0,0) –(1.5,-1.8)– (3,0.5); (1.9,-0.5)– (1.5,-1.8); (1.5,1.8)– (1.9,-0.5); (1.5,1.8) – (1.5,-0.2); (1.5,-0.6) – (1.5,-1.8); (1.5,1.8)–(1.9,-0.5)–(1.5,-1.8); (1.5,1.8) circle(1mm); (1.9,-0.5) circle(1mm); (1.5,-1.8) circle(1mm); For subdivisions of edges in $\Sigma$, the result follows by extending the Pachner moves in the proof of the corresponding result in [@DPY] to a stark neighborhood formed by taking the join with a point on either side of the surface. Subdivisions of triangles with interior in $M \setminus \Sigma$ are easily seen to be accomplished by a sequence of two flag-like Pachner moves, regardless of how the filtration restricts to the pair of tetrahedra sharing the triangular face: first perform a 1-4 Pachner move in one of the tetrahedra, as a subdivision, of flag-like triangulation this move is necessarily flag-like. The resulting complex than has the face to be subdivided shared by the other of the original tetrahedra and one of those from the subdivsion, whose interesection with lower-dimensional strata necessarily lies in the boundary of triangle being subdivided. The 2-3 Pachner move on this pair of tetrahedra is necessarily flag-like since the intersection with the lower dimensional strata lies entirely in the original tetrahedron, and performing it accomplished the desired subdivision. The main difficulty in the proof of each statement comes from handling subdivisions of edges lying on the knot, the link of which has a different number of edges than that included in the set of moves, and subdivisions of edges with interior in the top dimensional stratum. For the latter, we proceed as follows: first, observe that the link of any given edge is a polygon, and the star of the edge consists of tetrahedra including an edge of the link and the given edge (as one of its pairs of disjoint edges) and triangles shared by two of the tetrahedra, having one vertex in the link and the given edge as their opposites side, and the faces of these. As the edge has interior in the top dimensional stratum, and the triangulation is flag-like, the given edge has at most one vertex in a lower dimensional stratum (the knot or the Seifert surface), and the higher Now, perform a 1-4 Pachner move on each tetrahedron in the star, as Alexander subdivisions of a flag-like triangulation, these moves are necessarily flag-like. Having done this, consider the triangles which were shared by the original tetrahedra of the given edge. They are now each shared by two tetrahedra which intersect the lower-dimensional strata only in the intersection of the triangle with the lower-dimensional – by flag-likeness this must be empty, a single vertex (in $\Sigma$ or in $K$) or an edge with its end-points (all in $\Sigma$, all in $K$, or in $\Sigma$, except for a vertex in $K$) – and thus the entire star of each of the triangles in the new triangulation intersects the lower-dimensional strata only in the empty set or the same vertex or edge. It is immediate then that the 2-3 Pachner move on the star of each triangle is flag-like. Now, perform all of the 2-3 Pachner moves on the stars of the triangles shared by the original tetrahedra. The link of the given edge in this new triangulation has as its vertices exactly the new vertices introduced by the 1-4 moves of the first step, and as its edges precisely the new edges introduced by the 2-3 moves. It is easy to see that the intersection of the star of the given edge in this new triangulation with the lower-dimensional strata is exactly the intersection of (the closure of) the given edge, that is, either empty, or a single vertex in either $\Sigma$ or $K$. In any event, any Pachner moves performed within the star of the given edge in the last triangulation are necessarily flag-like. By Cassali’s [@C] improvement of Pachner’s result, we can reach any triangulation of the star (as a manifold with boundary) with the same triangulation of the boundary by a squence of Pachner moves on the interior of the star, in particular we can perform the Alexander subdivision of the edge by a sequence of such moves. Having subdivided the given edge in the modified triangulation, we must now apply more flag-like Pachner moves to obtain the Alexander subdivision of the given edge [in the original triangulation]{}. Now, consider the cells in the most recently constructed triangulation within which we had performed the 2-3 Pachner moves. These cells are still cells of the latest triangulation, and still intersect the lower-dimensional strata in such a way that any moves performed in their interiors are necessarily flag-like, and are still each the star of and edge introduced by one of the 2-3 moves, but this star now consists of four closed tetrahedra all sharing the edge – being in particular the join of a pair of triangles sharing the edge with two vertices. By Cassali [@C] we can perform the join with the two vertices of a 2-dimensional 2-2 Pachner move removing the edge introduced by the 2-3 move. Now, observe that the last constructed triangulation is a proper subdivision the triangulation we had after applying the 1-4 moves (and thus of the original triangulation), and, moreover, in it, the triangle shared by the tetrahedra of the original star have been divided into two triangles, exactly as in the result of the desired Alexander move. Applying Cassali’s main theorem [@C] to the subdivision of each of the original tetrahedra (any moves within which much be flag-like) then obtains the desired triangulation by a sequence of flag-like Pachner moves. Finally, we must reduce all Alexander subdivisions of an edge of the knot to flag-like extended Pachner moves on simplexes not on the knot, together with the subdivion of an edge of the knot with a four (resp. three) edge link to establish the first (resp. second) statement. For the second statement, consider first the case of reducing the Alexander subdivision of the edge in the first picture above to flag-like extended Pachner moves on simplexes not on the knot, together with the subdivion of an edge of the knot with a four three edge link. The star of the edge to be subdivided is show below. (2,0)–(1.7,3)–(1.7,-2); (A) at (0,0); (B) at (2,0); (C) at (3,1.2); (D) at (1,1.2); (E) at (1.7,3); (F) at (1.7,-2); (G) at (1.7,0.3); (A)–(B)–(C); (A)–(D)–(C); (A)–(E)–(B); (E)–(C); (D)–(E); (A)–(F)–(B); (F)–(C); (D)–(F); (E)–(G); (1.7,-0.2)–(F); (E) circle (3pt); (F) circle (3pt); (E)–(B)–(F); First, apply a 2-3 Pachner move to two tetrahedra $AFED$ and $DFCE$, sharing the triangle $DEF$. Observe that this move is flag-like. Giving the triangulation given within the original star below. (2,0)–(1.7,3)–(1.7,-2); (A) at (0,0); (B) at (2,0); (C) at (3,1.2); (D) at (1,1.2); (E) at (1.7,3); (F) at (1.7,-2); (G) at (1.7,0.3); (A)–(B)–(C); (A)–(D)–(C); (A)–(C); (A)–(E)–(B); (E)–(C); (D)–(E); (A)–(F)–(B); (F)–(C); (D)–(F); (E)–(G); (1.7,-0.2)–(F); (E) circle (3pt); (F) circle (3pt); (E)–(B)–(F); Then we can apply 3-6 move to the new star of the edge EF. (2,0)–(1.7,3)–(1.7,-2); (A) at (0,0); (B) at (2,0); (C) at (3,1.2); (D) at (1,1.2); (E) at (1.7,3); (F) at (1.7,-2); (G) at (1.7,0.3); (A)–(B)–(C); (A)–(D)–(C); (A)–(C); (A)–(E)–(B); (E)–(C); (D)–(E); (A)–(F)–(B); (F)–(C); (D)–(F); (E)–(G); (1.7,-0.2)–(F); (A)–(G); (B)–(G); (C)–(G); (E) circle (3pt); (F) circle (3pt); (E)–(B)–(F); Subdividing the edge AC using flag-like Pachner moves as in the argument above, then reversing a like sequence of flag-like Pachner moves to introduce an edge DG then gives the desired final configuration shown below. (2,0)–(1.7,3)–(1.7,-2); (A) at (0,0); (B) at (2,0); (C) at (3,1.2); (D) at (1,1.2); (E) at (1.7,3); (F) at (1.7,-2); (G) at (1.7,0.53); (A)–(B)–(C); (A)–(D)–(C); (A)–(E)–(B); (E)–(C); (D)–(E); (A)–(F)–(B); (F)–(C); (D)–(F); (E)–(G); (1.7,-0.2)–(F); (A)–(G); (B)–(G); (C)–(G); (D)–(G); (E) circle (3pt); (F) circle (3pt); (E)–(B)–(F); Now for edges of the knot with links of greater than four edges, the number of edges in the link can be reduced by a 2-3 move, as was done in the initial step of the construction just completed, and after the application of the 3-6 move, a sequence of pairs of a subdivision of an edge introduced by a 2-3 move, followed by a reverse subdivision to introduce a radial edge from the subdivision point introduced by the 3-6 move, will accomplish the desired Alexander subdivision. Thus we establish the second statement. The same argument will also reduce Alexander subdivisions of edges on the knot with links of more than four edges to sequences of flag-like extended Pachner moves. To establish the first statement, it thus remains only to show that the 3-6 move can be accomplished by a sequence of flag-like extended Pachner moves. Consider an edge of the knot with a three edge link. The link consists of two edges incident with the Seifert surface and one not incident with the Seifert surface. As shown above, the edge not incident with the Seifert surface can be subdivided by a sequence of flag-like Pachner moves. Having done this, the edge of the knot now has a 4 edge link, and can be subdivided by the extended Pachner move (4-8 move). It remains then to weld two edges which had been the edge of the link not incident with the Seifert surface. This, however, is the reverse of a subdivision of an edge not incident with either the knot or the Seifert surface, so as shown above can be done by a sequence of flag-like Pachner moves, thus establishing the result. Initial Data: Untwisted Case ============================ The initial data describing local states in Wakui’s construction [@W] of state-sum for finite-gauge group Dikjkgraaf-Witten theory is a finite group, $G$, together with a 3-cocycle $\alpha:G^3 \rightarrow K^\times$ valued in the multiplicative group of a field $K$. In [@DPY], 2-dimensional Dijgraaf-Witten theory was extended to pairs $C\subset \Sigma$ of a closed curve lying in a surface, using local states given by a triple $(H,X,G)$ of two groups $H$ and $G$ and a set $X$ with commuting actions of the groups on the left and right, respectively, together with $K^\times$-valued coefficients extending a 2-cocycle on $G$ and satisfying 2-cocycle-like conditions. For now we defer consideration of the analogue of a cocycle and describe the initial data sufficient for the “untwisted” construction, which will turn out to be equivalent to the construction involving an analogue of a cocycle which is identically $1$. Brief consideration of the conditions on the triple $(H,X,G)$ show that the data can be regarded as a two-object category in which the elements of $H$ are the endomorphisms of one object, which we denote 1, the elements of $G$ the endomorphisms of the other, which we denote 2, there are no arrows from 2 to 1 and the elements of $X$ are the arrow from 1 to 2 – the commutativity of the actions being part of the associativity of composition in the category. A $\mathcal P$-[parcel]{} is a small category $\mathcal C$ equipped with a surjective conservative functor $U:{\mathcal C}\rightarrow {\mathcal P}$, which is injective on objects, to a poset $\mathcal P$. A finite $\mathsf 3$-parcel, where $\mathsf 3$ is the three-element chain $1 < 2 < 3$, will be the initial data needed for the untwisted construction, and part of the data needed for the general construction. Throughout we will denote an object of a $\mathsf 3$-parcel by the number which is its image under the functor $U$. The State-Sum Construction: Untwisted Case ========================================== We now describe a state-sum invariant of knot, Seifert surface, 3-manifold triples (which by general principles, cf. [@Y], can be seen as arising from a generalized TQFT given by a functor on cobordisms of surfaces marked with curves with boundary), using only a finite $\mathsf 3$-parcel as initial data. Let $\mathcal T$ be a flag-like triangulation of $K \subset \Sigma \subset M$ for a knot (or link) $K$ with a Seifert surface $\Sigma$ in a 3-manifold $M$. The $\Gamma({\mathcal T})$, [directed graph]{} of $\mathcal T$, is the directed graph with ${\mathcal T}_0$, the set of vertices of $\mathcal T$ as vertices, and an edge $(v,w)$ whenever $v$ and $w$ are vertices connected by an edge of $\mathcal T$ and $\dim(v) \leq \dim(w)$, where the dimension of a vertex is the dimension of the stratum in which it lies. ${\mathcal C}({\mathcal T})$, the category of $\mathcal T$, is then the quotient of the path category of $\Gamma({\mathcal T})$ by the relations $(v,w)(w,v) = Id_v$ whenever there are edges in both directions between two vertices, and $(v,w)(w,x) = (v,x)$ whenever there is a 2-simplex of $\mathcal T$ with vertices $v,w,$ and $x$. ${\mathcal C}({\mathcal T})$ then has a surjective conservative functor $D$ to $\mathsf 3$, given on object by mapping each vertex to the dimension of the stratum in which it lies. Given a $\mathsf 3$-parcel, $U:{\mathcal C}\rightarrow {\mathsf 3}$, a $\mathcal C$-coloring of a flag-like triangulation $\mathcal T$ of a knot-Seifert surface-3-manifold triple $K \subset \Sigma \subset M$ is a functor $\sigma:{\mathcal C}({\mathcal T}) \rightarrow {\mathcal C}$ such that $U(\sigma) = D$. \[untwisted\] Let $K \subset \Sigma \subset M$ be a knot, Seifert surface, 3-manifold triple, $\mathcal T$ a flag-like triangulation, and $U:{\mathcal C}\rightarrow {\mathsf 3}$ a $\mathsf 3$-parcel. Let $[\mathcal T, \mathcal C]$ denote the set of $\mathcal C$-clorings of $\mathcal T$, and $G_i = {\mathcal C}(i,i)$ for $i = 1, 2,3$, denote the endomorphism groups of $\mathcal C$. And finally let $\mathcal T_0^i$ for $i = 1,2,3$ denote the set of vertices of $\mathcal T$ lying in the stratum of dimension $i$. Then, using $\#$ to denote cardinality, the quantity $$\frac{ \#[\mathcal T, \mathcal C] } { \prod_{i=1}^3 \#G_i^{\#T_0^i} }$$ is independent of the triangulation and thus gives a topological invariant of the triple. We prove this theorem using Alexander moves. The extended Pachner moves of section 2 will become important for the twisted case. Whenever an Alexander subdivision is performed, a single new vertex is introduced. Chosing an edge from the new vertex to an adjacent vertex in the same stratum induces an equivalence between the category of the old triangulation and the category of the new triangulation, by functors commuting with the underlying functors to $\mathsf 3$. A $\mathcal C$-coloring of the old triangulation determines most of a $\mathcal C$-coloring of the new – all that remains to be specified are the images of the arrows named by edges incident with the new vertex. A brief consideration shows that the choice of an element of $G_i$ (where $i$ is the dimension of the stratum in which the new vertex lies) to be the image of the edge chosen to induce the equivalence of categories completely determines a unique $\mathcal G$-coloring of the new triangulation, and thus whenever a simplex in the $i$-dimensional stratum is subdivided, the number of $\mathcal G$-colorings of the new triangulation is exactly $\#G_i$ times the number of $\mathcal G$-colorings of the old triangulation, thus establishing the result. As with the invariants of $n$-manifolds given by untwisted Dikjgraaf-Witten theory, which for connected $n$-manifolds count group homomorphisms from the fundamental group to the guage group, and more generally count groupoid homomorphisms from any skeleton of the fundamental groupoid to the guage group, the invariants given by Theorem \[untwisted\] admit a counting interpretation. Observe that the relation $x \preceq y$ on points of $M$ given by $x \preceq y$ if $\dim(x) \leq \dim(y)$, where the dimension of a point is the dimension of the stratum in which it lies, is a preorder, whose restriction to each stratum is chaotic. Taking the family of directed paths in $M$ to be the continuous maps $p:[0,1]\rightarrow M$ which are non-decreasing with respect to the usual order on $[0,1]$ and the preorder on $M$, $M$ is given the structure of a directed topological space, or $d$-space in the sense of Grandis [@Gr1; @Gr2]. It is easy to establish The fundamental category $\uparrow\!\!\Pi_1(X)$ with respect to the $d$-space structure of the previous paragraph has an underlying functor to $\mathsf 3$ given on objects by $x \mapsto \dim(x)$, and $\uparrow\!\!\pi_1(X)$ is any skelton of $\uparrow\!\!\Pi_1(X)$, the invariant of Theorem \[untwisted\] counts the number of functors from $\uparrow\!\!\pi_1(X)$ to $\mathcal C$, which commute with the underlying functors to $\mathsf 3$. Partial Cocycles ================ We now turn to the matter of introducing local coefficients which generalize the group 3-cocycle in Wakui’s construction [@W]. The obvious generalization of a $K^\times$-valued group 3-cocycle to a category – a function from composable triples of arrows in $\mathcal C$ to $K^\times$, satisfying the group 3-cocycle condition [mutatis mutandis]{} for every composable quadruple of arrows – will give an instance of the rest of the necessary data. However, a moment’s thought reveals that the values of a cocycle on composable triples for which the target of the composition is the object $1$ or $2$ will be irrelevant to the construction, since they cannot occur as labels of a tetrahedron, and, in fact weaker conditions on the values on triples whose composition has target $3$ suffice. The analogue of Wakui’s 3-cocycles in our construction is given by a function on the composable triples of arrows which can occur as labels of the long oriented path in a tetrahedron, satisfying equations corresponding to the combinatorial moves that suffice to ensure topological invariance: A [$K^\times$-valued partial 3-cocycle]{} on a $\mathsf 3$-parcel $U:{\mathcal C}\rightarrow {\mathsf 3}$ is a function $$\alpha: \{ (f,g,h) \| f,g,h \in Arr({\mathcal C}), \; fgh \; \mbox{\rm is defined,}\; t(h) = 3, \; \mbox{\rm and } s(h) \neq 1\; \} \rightarrow K^\times$$ for some field $K$, satisfying 1. $\alpha_{j,k,3,3}(g,h,k)\cdot\alpha_{i,k,3,3}^{-1}(fg,h,k)\cdot\alpha_{i,j,3,3}(f,gh,k)\cdot$\ $\alpha_{i,j,k,3}^{-1}(f,g,hk)\cdot\alpha_{i,j,k,3}(f,g,h) = 1$, whenever $k \geq 2$, and\ $f,g,h,k$ is composable. 2. $ \alpha_{i , j , 2 , 3} ( a , b c , d ) \cdot \alpha_{j , 2 , 2 , 3} ( b , c , d ) \cdot \alpha^{-1}_{i , j , 2 , 3} ( a , b , c d ) \cdot \alpha^{-1}_{i , 2 , 2 , 3} ( a b , c , d ) = $\ $ \alpha_{i , j , 2 , 3} ( a , b c , e ) \cdot \alpha_{j , 2 , 2 , 3} ( b , c , e ) \cdot \alpha^{-1}_{i , j , 2 , 3} ( a , b , c e ) \cdot \alpha^{-1}_{i , 2 , 2 , 3} ( a b , c , e ), $\ whenever $a,b,c,d$ and $a,b,c,e$ are both composable. 3. $\alpha_{1 , 1 , 2 , 3} ( a , b , c ) \cdot \alpha^{-1}_{1 , 1 , 2 , 3} ( a , b , e ) \cdot \alpha_{1 , 1 , 3 , 3} ( a , b c , d ) = $\ $\alpha_{1 , 1 , 2 , 3} ( f , g b , c ) \cdot \alpha^{-1}_{1 , 1 , 2 , 3} ( f , g b , e ) \cdot \alpha_{1 , 1 , 3 , 3} ( f , g b c , d ) \cdot $\ $\alpha_{1 , 1 , 2 , 3} ( g , b , c ) \cdot \alpha^{-1}_{1 , 1 , 2 , 3} ( g , b , e ) \cdot \alpha_{1 , 1 , 3 , 3} ( g , b c , d ) $, whenever\ $a,b,c,d$ is composable, $e$ is any arrow such that $be = bcd$,\ and $fg = a$. where $\alpha_{i,j,k,3}$ denotes the restriction of $\alpha$ to $\{ (f,g,h) \| s(f) = i, t(f) = j = s(g), t(g) = k = s(h), t(h) = 3 \}$ for $1 \leq i \leq j \leq k \leq3$, and the quadruples of maps in the condition have sources and targets agreeing with those specified by the subscripts on instances of $\alpha$. If $\beta:\{(f,g,h) \| f,g,h \in Arr({\mathcal C}) \; \mbox{\rm and } fgh \;\mbox{\rm is defined} \}\rightarrow K^\times$ is a 3-cocycle on the category $\mathcal C$ of a $\mathsf 3$-parcel, then the restriction of $\beta$ to $\{ (f,g,h) \| f,g,h \in Arr({\mathcal C}), \; fgh \; \mbox{\rm is defined,}\; t(h) = 3 \; \mbox{\rm and } s(h) \neq 1\}$ is a partial 3-cocycle. The first condition in the definition of a partial cocycle follows trivially from the cocycle condition, being simply instances of the cocycle condition. The second follows since in the presence of the cocycle condition each side equals $\beta(a, b, c)$. The third requires a little work. Solving the equation to move all factors to the right hand side, gives the equivalent condition $$\begin{aligned} 1 & = & \alpha^{-1}_{1 , 1 , 2 , 3} ( a , b , c ) \cdot \alpha_{1 , 1 , 2 , 3} ( a , b , e ) \cdot \alpha^{-1}_{1 , 1 , 3 , 3} ( a , b c , d ) t\\ & & \alpha_{1 , 1 , 2 , 3} ( f , g b , c ) \cdot \alpha^{-1}_{1 , 1 , 2 , 3} ( f , g b , e ) \cdot \alpha_{1 , 1 , 3 , 3} ( f , g b c , d ) \cdot \\ & &\alpha_{1 , 1 , 2 , 3} ( g , b , c ) \cdot \alpha^{-1}_{1 , 1 , 2 , 3} ( g , b , e ) \cdot \alpha_{1 , 1 , 3 , 3} ( g , b c , d ). \end{aligned}$$ Now, suppressing the subscripts denoting sources and targets, and substituting the cocycle $\beta$ gives $$\begin{aligned} 1 & = & \beta^{-1} ( a , b , c ) \cdot \beta ( a , b , e ) \cdot \beta^{-1}( a , b c , d ) \cdot\\ & & \beta( f , g b , c ) \cdot \beta^{-1} ( f , g b , e ) \cdot \beta ( f , g b c , d ) \cdot \\ & &\beta( g , b , c ) \cdot \beta^{-1} ( g , b , e ) \cdot \beta ( g , b c , d ). \end{aligned}$$ Now, recalling the $fg = a$, the factors aligned in each column in the preceding expression are three factors of an instance of the coboundary of $\beta$, for the composable quadruples $f,g,b,c$; $f,g,b,e$ and $f,g,bc,d$, respectively. Replacing the factors in each column with the product of two factors equal to them by the cocycle condition gives $$\begin{aligned} 1 & = & \beta( f , g , bc ) \cdot \beta^{-1} ( f , g , be ) \cdot \beta ( f , g , bcd ) \cdot \\ & & \beta^{-1} ( f, g, b ) \cdot \beta ( f, g, b ) \cdot \beta^{-1}(f, g, bc ) \end{aligned}$$ the factors on the right hand side of which cancel in pairs, when the condition that $be = bcd$ is recalled, thus completing the proof. As usual, to twist by coefficients, we need to specify orientations on the edges of the triangulation, and in the proof of invariance show not only invariance under the combinatorial moves giving PL homoemorphism, for us the moves of the second statement of Theorem \[sufficient\_moves\], but under changes to the edge-orientations. The orientation on the knot and the non-invertibility conditions in the definition of the category ${\mathcal C}({\mathcal T})$ remove some of the need to make choices, but not all – the edges of the knot are oriented in agreement with the orientation of the knot, and edges incident with the knot, but not lying in it, are all oriented away from the knot, and edges incident with the Seifert surface, but not lying in it, are all oriented away from the surface. To orient the other edges, as usual chose a linear ordering of the vertices (in this case it suffices to chose separate linear orderings of the vertices in the interior of the Seifert surface and of the vertices in the complement of the closed Seifert surface) and orient the remaing edges from the earlier vertex to the later vertex. Once the edges are oriented, each tetrahedron in the triangulation has a longest oriented path of edges, and a combinatorial orientation induced by the ordering of the vertices along the longest path. We can now state our main theorem: \[main\] Let $U:{\mathcal C} \rightarrow {\mathsf 3}$ be a $\mathsf 3$-parcel, and $\alpha$ be a partial 3-cocycle on it. Then for any knot, Seifert surface, 3-manifold triple $K \subset \Sigma \subset M$, the quantity given by $$\sum_{\lambda:{\mathcal C}({\mathcal T})\rightarrow {\mathcal C}} \frac{\prod_{\sigma \in {\mathcal T}_3} \alpha^{\epsilon(\sigma)}(\lambda(\sigma))} { \prod_{i=1}^3 \#G_i^{\#T_0^i} }$$ where the summation ranges over all $\mathcal C$-colorings of $\mathcal T$, $\lambda(\sigma)$ denotes the composable of arrows $\lambda$ assigns to the longest path of the tetrahedron $\sigma$ and $\epsilon(\sigma)$ is $+1$ if the combinatorial orientation of $\sigma$ agrees with the orientation of $M$, and $-1$ otherwise, for any flag-like triangulation $\mathcal T$ and a choice of ordering of the vertices off the knot, is independent of the triangulation and choice of ordering, and thus a topological invariant of the triple. As in the case of Theorem \[untwisted\], moves which introduce a new vertex involve a choice of a label for one of the edges incident with the new vertex and lying in the same stratum as the new vertex, which suffices, in the presence of labels from the remaining edges not subdivided by the move to determine uniquely a $\mathcal C$-coloring of the new triangulation. The denominator is thus multiplied by the number of summands in the numerator for the new triangulation. It thus suffices to show that for any move which does not introduce a new vertex (the 2-3 Pachner moves and the 4-4 extended Pachner move) for any $\mathcal C$-coloring of the initial triangulation, there is a unique $\mathcal C$-coloring of the triangulation resulting from the move, agreeing with the given coloring on all edges not modified by the move and that the summand in the numerator for these colorings are equal, and similarly for moves which do introduce a new vertex (the 1-4 Pachner move, the 2-6 extended Pachner move and the 3-6 move on an edge of the knot) that given a $\mathcal C$-coloring of the initial triangulation and a choice of label for one of the new edges incident with the new vertex and lying in the same stratum, there is a unique $\mathcal C$-coloring of the new triangulation agreeding with the given coloring on all edges not modified by the move, having the chosen label on the new edge, and, moreover, that the summand in the numerator for the new triangulation for this coloring equals the summand in the numerator for the old triangulation of the given coloring. Condition (3) is exactly the condition needed to ensure that each summand corresponding to labelings of the six-tetrahedon state in the 3-6 move are equal to the summand corresponding to the labeling of the three-tetrahedron state, as illustrated in Figures \[three\_tetrahedra\] and \[six\_tetrahedra\]. --------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------- $\displaystyle $ \left\{ \begin{tikzpicture}%[thick][scale=1.5] \begin{array}{ll} \fill[blue,blue, opacity=.5] (1.5,-1.8)--(1.9,-0.5)--(1.5,1.8); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (1.5,1.8){}; \begin{tikzpicture}%[thick][scale=1.5] \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (0,0){}; \fill[blue,blue, opacity=.5] (1.5,-1.8)--(1.9,-0.5)--(1.5,1.8); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (1.5,1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (0,0){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (1.9,-0.5){}; \begin{scope} \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \draw [dashed,midarrow={>}] (B)--(D) node[font=\tiny, midway, left]{$ d $}; \begin{scope} \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, left]{$ a b c $}; \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, left]{$ a b c $}; \draw [midarrow={>}] (C)--(B) node[font=\tiny, midway, below]{$ c $}; \draw [midarrow={>}] (C)--(B) node[font=\tiny, midway, below]{$ c $}; \draw [midarrow={>}] (C)--(D) node[font=\tiny, midway, left]{$ c d $}; \draw [midarrow={>}] (A)--(C) node[font=\tiny, midway, right]{$ a b $}; \draw [midarrow={>}] (A)--(C) node[font=\tiny, midway, right]{$ a b $}; \draw [midarrow={>}] (E)--(B) node[font=\tiny, midway, left]{$ b c $\,}; \draw [midarrow={>}] (A)--(D) node[font=\tiny, midway, right]{\,$ a b c d $\,}; \draw [midarrow={>}] (E)--(C) node[font=\tiny, midway, right] {$ b $\,}; \draw [midarrow={>}] (E)--(B) node[font=\tiny, midway, left]{$ b c $\,}; \draw [dashed,thick, midarrow={>}] (1.5,1.8) -- (1.5,-0.2) node[font=\tiny, midway, left]{$ a $}; \draw [midarrow={>}] (E)--(C) node[font=\tiny, midway, right] {$ b $\,}; \draw [dashed,thick] (1.5,-0.6) -- (1.5,-1.8); \draw [midarrow={>}] (E)--(D) node[font=\tiny, midway, right]{$ b c d $}; \fill[red, opacity=.5] (A) circle (3pt); \draw [dashed,thick, midarrow={>}] (1.5,1.8) -- (1.5,-0.2) node[font=\tiny, midway, left]{$ a $}; \fill[red, opacity=.5] (E) circle (3pt); \draw [dashed,thick] (1.5,-0.6) -- (1.5,-1.8); \fill[blue, opacity=.5] (C) circle (3pt); \fill[red, opacity=.5] (A) circle (3pt); \draw[thick,red,red, opacity=.5] (A)--(E); \fill[red, opacity=.5] (E) circle (3pt); \draw[thick,blue,blue, opacity=.5] (A)--(C); \fill[blue, opacity=.5] (C) circle (3pt); \draw[thick,blue,blue, opacity=.5] (E)--(C); \draw[thick,red,red, opacity=.5] (A)--(E); \end{scope} \draw[thick,blue,blue, opacity=.5] (A)--(C); \end{tikzpicture} \draw[thick,blue,blue, opacity=.5] (E)--(C); \end{scope} & \alpha_{1 , 1 , 2 , 3} ( a , b , c ) \end{tikzpicture}$ \\ \\ \begin{tikzpicture}%[thick][scale=1.5] \fill[blue,blue, opacity=.5] (1.5,-1.8)--(1.9,-0.5)--(1.5,1.8); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (1.5,1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \begin{scope} \draw [midarrow={>}] (C)--(D) node[font=\tiny, midway, left]{$ c d $}; \draw [midarrow={>}] (A)--(C) node[font=\tiny, midway, right]{$ a b $}; \draw [midarrow={>}] (A)--(D) node[font=\tiny, midway, right]{\,$ a b c d $\,}; \draw [midarrow={>}] (E)--(C) node[font=\tiny, midway, right] {$ b $\,}; \draw [midarrow={>}] (E)--(D) node[font=\tiny, midway, right]{$ b c d $}; \draw [midarrow={>}] (A)--(E) node[font=\tiny, midway, left]{$ a $}; \fill[red, opacity=.5] (A) circle (3pt); \fill[red, opacity=.5] (E) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \draw[thick,red,red, opacity=.5] (A)--(E); \draw[thick,blue,blue, opacity=.5] (A)--(C); \draw[thick,blue,blue, opacity=.5] (E)--(C); \end{scope} \end{tikzpicture} & \displaystyle \alpha^{-1}_{1 , 1 , 2 , 3} ( a , b , c d ) \\ \\ \begin{tikzpicture}%[thick][scale=1.5] \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (1.5,1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (0,0){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \begin{scope} \draw [dashed,midarrow={>}] (B)--(D) node[font=\tiny, midway, left]{$ d $}; \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, left]{$ a b c $}; \draw [midarrow={>}] (A)--(D) node[font=\tiny, midway, right]{\,$ a b c d $\,}; \draw [midarrow={>}] (E)--(B) node[font=\tiny, midway, left]{$ b c $\,}; \draw [midarrow={>}] (E)--(D) node[font=\tiny, midway, right]{$ b c d $}; \draw [midarrow={>}] (A)--(E) node[font=\tiny, midway, right]{$ a $}; \fill[red, opacity=.5] (A) circle (3pt); \fill[red, opacity=.5] (E) circle (3pt); \draw[thick,red,red, opacity=.5] (A)--(E); \end{scope} \end{tikzpicture} & \alpha_{1 , 1 , 3 , 3} ( a , b c , d ) \\ \end{array} \right.$ --------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------- $\displaystyle $ \left\{ \begin{tikzpicture}%[thick][scale=1.5] \begin{array}{ll} \fill[blue,blue, opacity=.5] (1.5,-1.8)--(1.9,-0.5)--(1.5,1.8); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (1.5,1.8){}; \begin{tikzpicture}%[thick][scale=1.5] \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (0,0){}; \fill[blue,blue, opacity=.5] (1.5,0)--(1.9,-0.5)--(1.5,1.8); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (1.5,1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (0,0){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (F) at (1.5,-0.1){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (F) at (1.5,0){}; \begin{scope} \begin{scope} \draw [dashed,midarrow={>}] (B)--(D) node[font=\tiny, midway, left]{$ d $}; \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, left]{$ ( a b c = ) f g b c $}; \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, left]{$ a b c $}; \draw [midarrow={>}] (C)--(B) node[font=\tiny, midway, below]{$ c $}; \draw [midarrow={>}] (C)--(B) node[font=\tiny, midway, below]{$ c $}; \draw [midarrow={>}] (A)--(C) node[font=\tiny, midway, right]{$ a b ( = f g b )$}; \draw [midarrow={>}] (C)--(D) node[font=\tiny, midway, right]{$ c d $}; \draw [dashed,thick, midarrow={>}] (1.5,1.8) -- (1.5,0) node[font=\tiny, midway, left]{$ f $}; \draw [midarrow={>}] (A)--(C) node[font=\tiny, midway, right]{$ a b $}; \fill[red, opacity=.5] (A) circle (3pt); \draw [midarrow={>}] (A)--(D) node[font=\tiny, midway, right]{\,$ a b c d $\,}; \fill[blue, opacity=.5] (C) circle (3pt); \draw [midarrow={>}] (E)--(B) node[font=\tiny, midway, left]{$ b c $\,}; \draw[thick,red,red, opacity=.5] (A)--(1.5,0); \draw [midarrow={>}] (E)--(C) node[font=\tiny, midway, right] {$ b $\,}; \draw[thick,blue,blue, opacity=.5] (A)--(C); \draw [midarrow={>}] (E)--(D) node[font=\tiny, midway, right]{$ b c d $}; \draw [dashed, thick, midarrow={>}] (F)--(B) node[font=\tiny, midway, above]{$ g b c $}; \draw [dashed,thick, midarrow={>}] (1.5,1.8) -- (1.5,-0.2) node[font=\tiny, midway, left]{$ f $}; \draw [dashed, thick, midarrow={>}] (F)--(C) node[font=\tiny, midway, left]{$ g b $}; \draw [dashed,thick, midarrow={>}] (1.5,-0.6) -- (1.5,-1.8) node[font=\tiny, midway, left]{$ g $}; \draw[thick,blue,blue, opacity=.5] (F)--(C); \fill[red, opacity=.5] (A) circle (3pt); \fill[red, opacity=.5] (F) circle (3pt); \fill[red, opacity=.5] (E) circle (3pt); \end{scope} \fill[blue, opacity=.5] (C) circle (3pt); \end{tikzpicture} \draw[thick,red,red, opacity=.5] (A)--(E); \draw[thick,blue,blue, opacity=.5] (A)--(C); & \alpha_{1 , 1 , 2 , 3} ( f , g b , c ) \draw[thick,blue,blue, opacity=.5] (E)--(C); \draw [thick, midarrow={>}] (F)--(B) node[font=\tiny, midway, above]{$ g b c $}; \\ \draw [thick, midarrow={>}] (F)--(C) node[font=\tiny, midway, left]{$ g b $}; \\ \draw [thick, midarrow={>}] (F)--(D) node[font=\tiny, midway, above]{$ g b c d $}; \end{scope} \begin{tikzpicture}%[thick][scale=1.5] \end{tikzpicture}$ \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (1.5,1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (0,0){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (F) at (1.5,-0.1){}; \begin{scope} \draw [dashed,midarrow={>}] (B)--(D) node[font=\tiny, midway, left]{$ d $}; \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, left]{$ ( a b c = ) f g b c $}; \draw [midarrow={>}] (A)--(D) node[font=\tiny, midway, right]{\,$ f g b c d ( = a b c d ) $\,}; \draw [thick, midarrow={>}] (1.5,1.8) -- (F) node[font=\tiny, midway, left]{$ f $}; \fill[red, opacity=.5] (A) circle (3pt); \draw[thick,red,red, opacity=.5] (A)--(F); \draw [thick, midarrow={>}] (F)--(B) node[font=\tiny, midway, below]{$ g b c $}; \draw [thick, midarrow={>}] (F)--(D) node[font=\tiny, midway, below]{$ g b c d $}; \fill[red, opacity=.5] (F) circle (3pt); \end{scope} \end{tikzpicture} & \displaystyle \alpha_{1 , 1 , 3 , 3} ( f , g b c , d ) \\ \\ \begin{tikzpicture}%[thick][scale=1.5] \fill[blue,blue, opacity=.5] (1.5,-0.1)--(1.9,-0.5)--(1.5,1.8); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (1.5,1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (F) at (1.5,-0.1){}; \begin{scope} \draw [midarrow={>}] (C)--(D) node[font=\tiny, midway, below]{$ c d $}; \draw [midarrow={>}] (A)--(C) node[font=\tiny, midway, right]{$ f g b(=\!a b ) $}; \draw [midarrow={>}] (A)--(D) node[font=\tiny, midway, right]{\,$ f g b c d ( = a b c d ) $\,}; \draw [dashed,thick, midarrow={>}] (1.5,1.8) -- (1.5,-0.2) node[font=\tiny, midway, left]{$ f $}; \fill[red, opacity=.5] (A) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \draw[thick,red,red, opacity=.5] (A)--(F); \draw[thick,blue,blue, opacity=.5] (A)--(C); \draw [thick, midarrow={>}] (F)--(C) node[font=\tiny, midway, left]{$ g b $}; \draw [dashed, thick, midarrow={>}] (F)--(D) node[font=\tiny, midway, right]{$ g b c d $}; \draw[thick,blue,blue, opacity=.5] (F)--(C); \fill[red, opacity=.5] (F) circle (3pt); \end{scope} \end{tikzpicture} & \alpha^{-1}_{1 , 1 , 2 , 3} ( f , g b , c d ) \\ \\ \begin{tikzpicture}%[thick][scale=1.5] \fill[blue,blue, opacity=.5] (1.5,-1.8)--(1.9,-0.5)--(1.5,0); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (0,0){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (F) at (1.5,0){}; \begin{scope} \draw [midarrow={>}] (C)--(B) node[font=\tiny, midway, below]{$ c $}; \draw [midarrow={>}] (E)--(B) node[font=\tiny, midway, left]{$ b c $\,}; \draw [midarrow={>}] (E)--(C) node[font=\tiny, midway, right] {$ b $\,}; \draw [dashed,thick, midarrow={>}] (F) -- (1.5,-1.8) node[font=\tiny, midway, left]{$ g $}; \fill[red, opacity=.5] (E) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \draw[thick,blue,blue, opacity=.5] (E)--(C); \draw [thick, midarrow={>}] (F)--(B) node[font=\tiny, midway, above]{$ g b c $}; \draw [thick, midarrow={>}] (F)--(C) node[font=\tiny, midway, above]{\,\,\,\,$ g b $}; \fill[red, opacity=.5] (F) circle (3pt); \draw[thick,red,red, opacity=.5] (F)--(E); \draw[thick,blue,blue, opacity=.5] (F)--(C); \end{scope} \end{tikzpicture} & \alpha_{1 , 1 , 2 , 3} ( g , b , c ) \\ \\ \begin{tikzpicture}%[thick][scale=1.5] \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (0,0){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (F) at (1.5,-0.1){}; \begin{scope} \draw [midarrow={>}] (B)--(D) node[font=\tiny, midway, above]{$ d $}; \draw [midarrow={>}] (E)--(B) node[font=\tiny, midway, left]{$ b c $\,}; \draw [midarrow={>}] (E)--(D) node[font=\tiny, midway, right]{$ b c d $}; \draw [thick, midarrow={>}] (F) -- (1.5,-1.8) node[font=\tiny, midway, left]{$ g $}; \fill[red, opacity=.5] (F) circle (3pt); \fill[red, opacity=.5] (E) circle (3pt); \draw[thick,red,red, opacity=.5] (F)--(E); \draw [thick, midarrow={>}] (F)--(B) node[font=\tiny, midway, below]{$ g b c $}; \draw [thick, midarrow={>}] (F)--(D) node[font=\tiny, midway, below]{$ g b c d $}; \end{scope} \end{tikzpicture} & \alpha_{1 , 1 , 3 , 3} ( g , b c , d ) \\ \\ \begin{tikzpicture}%[thick][scale=1.5] \fill[blue,blue, opacity=.5] (1.5,-1.8)--(1.9,-0.5)--(1.5,-0.1); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (F) at (1.5,-0.1){}; \begin{scope} \draw [midarrow={>}] (C)--(D) node[font=\tiny, midway, right]{$ c d $}; \draw [midarrow={>}] (E)--(C) node[font=\tiny, midway, right] {\!\!$ b $\,}; \draw [midarrow={>}] (E)--(D) node[font=\tiny, midway, right]{$ b c d $}; \draw [thick, midarrow={>}] (F) -- (1.5,-1.8) node[font=\tiny, midway, left]{$ g $}; \fill[red, opacity=.5] (F) circle (3pt); \fill[red, opacity=.5] (E) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \draw[thick,red,red, opacity=.5] (F)--(E); \draw[thick,blue,blue, opacity=.5] (E)--(C); \draw[thick,blue,blue, opacity=.5] (F)--(C); \draw [thick, midarrow={>}] (F)--(C) node[font=\tiny, midway, below]{$ g b $}; \draw [thick, midarrow={>}] (F)--(D) node[font=\tiny, midway, above]{$ g b c d $}; \end{scope} \end{tikzpicture} & \alpha^{-1}_{1 , 1 , 2 , 3} ( g , b , c d ) \\ \end{array} \right.$ ---------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------- Conditions (1) and (2) each ensure invariance under two types of moves – the 1-4 and 2-3 Pachner moves and the 2-6 and 4-4 extended Pachner moves, respectively. In each case, the differently indexed instances of the conditions correspond to the ways in which a flag-like triangulation of the state with fewer tetrahedra can intersect the strata. For instance, the cases of the 4-4 move and the 2-6 move in which the tetrahedra being modified by the moves are not incident with the knot are show in Figures \[four\_before\] and \[four\_after\] and Figures \[two\_extended\] and \[six\_no\_knot\], respectively. The latter depends on the choice of the label $f$, which then induces the labels $g$ and $h$ such that there are factorizations of $c$ and $d$ as $c = fg$ and $d = fh$, respectively. ---------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------- $\displaystyle $ \left\{ \begin{tikzpicture}%[scale=1.5] \begin{array}{ll} \fill[blue,blue, opacity=.5] (0,0)--(2.5,0)--(3.5,1.2)--(1,1.2)--(0,0); \node[circle, fill, inner sep=.8pt, outer sep=0pt] (A) at (0,0){}; \begin{tikzpicture}%[scale=1.5] \node[circle, fill, inner sep=.8pt, outer sep=0pt] (B) at (2.5,0){}; \fill[blue,blue, opacity=.5] (0,0)--(2.5,0)--(1,1.2)--(0,0); \node[circle, fill, inner sep=.8pt, outer sep=0pt] (C) at (3.5,1.2){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (A) at (0,0){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (D) at (1,1.2){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (B) at (2.5,0){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (E) at (1.7,3){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (D) at (1,1.2){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (F) at (1.7,-2){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (E) at (1.7,3){}; \begin{scope} \begin{scope} \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, below]{$ a $\,\,\,}; \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, below]{$ a $\,\,\,}; \draw [midarrow={>}] (B)--(C) node[font=\tiny, midway, left]{$ b $}; \draw [dashed,midarrow={>}] (A)--(D) node[font=\tiny, midway, right]{$ abc $}; \draw [dashed,midarrow={>}] (C)--(D) node[font=\tiny, midway, below]{\,\,\,\,$ c $}; \draw [dashed,thick,midarrow={>}] (B)--(D) node[font=\tiny, midway, right]{$ bc $}; \draw [dashed,midarrow={>}] (A)--(D) node[font=\tiny, midway, right]{$ abc $}; \draw [midarrow={>}] (A)--(E) node[font=\tiny, midway, left]{$ abcd $}; \draw [thick,midarrow={>}] (B)--(D) node[font=\tiny, midway, right]{$ bc $}; \draw [midarrow={>}] (B)--(E) node[font=\tiny, midway, right]{$ bcd $}; \draw [midarrow={>}] (A)--(E) node[font=\tiny, midway, left]{$ abcd $}; \draw [dashed,midarrow={>}] (D)--(E) node[font=\tiny, midway, right]{$ d $}; \draw [midarrow={>}] (B)--(E) node[font=\tiny, midway, right]{$ bcd $}; \draw[thick,blue,blue] (A)--(B); \draw [midarrow={>}] (C)--(E) node[font=\tiny, midway, right]{$ cd $}; \draw[dashed,thick,blue,blue] (D)--(A); \draw [dashed,midarrow={>}] (D)--(E) node[font=\tiny, midway, right]{$ d $}; \draw[dashed,thick,blue,blue] (D)--(B); \draw [midarrow={>}] (A)--(F) node[font=\tiny, midway, left]{$ abce $}; \fill[blue, opacity=.5] (A) circle (3pt); \draw [midarrow={>}] (B)--(F) node[font=\tiny, midway, left]{$ bce$\!\!}; \fill[blue, opacity=.5] (B) circle (3pt); \draw [midarrow={>}] (C)--(F) node[font=\tiny, midway, right]{$ ce $}; \fill[blue, opacity=.5] (D) circle (3pt); \draw [dashed,midarrow={>}] (D)--(F) node[font=\tiny, midway, right]{$ e $}; \end{scope} \draw[thick,blue,blue] (A)--(B)--(C); \end{tikzpicture} \draw[dashed,thick,blue,blue] (C)--(D)--(A); \fill[blue, opacity=.5] (A) circle (3pt); & \alpha_{2 , 2 , 2 , 3} ( a , b c , d ) \fill[blue, opacity=.5] (B) circle (3pt); \\ \fill[blue, opacity=.5] (C) circle (3pt); \\ \fill[blue, opacity=.5] (D) circle (3pt); \end{scope} \begin{tikzpicture}%[scale=1.5] \end{tikzpicture}$ \fill[blue,blue, opacity=.5] (2.5,0)--(3.5,1.2)--(1,1.2); \node[circle, fill, inner sep=.8pt, outer sep=0pt] (B) at (2.5,0){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (C) at (3.5,1.2){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (D) at (1,1.2){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (E) at (1.7,3){}; \begin{scope} \draw [midarrow={>}] (B)--(C) node[font=\tiny, midway, left]{$ b $}; \draw [dashed,midarrow={>}] (C)--(D) node[font=\tiny, midway, below]{\,\,\,\,$ c $}; \draw [thick,midarrow={>}] (B)--(D) node[font=\tiny, midway, right]{$ bc $}; \draw [midarrow={>}] (B)--(E) node[font=\tiny, midway, right]{$ bcd $}; \draw [midarrow={>}] (C)--(E) node[font=\tiny, midway, right]{$ cd $}; \draw [midarrow={>}] (D)--(E) node[font=\tiny, midway, right]{$ d $}; \draw[thick,blue,blue] (B)--(C); \draw[thick,blue,blue] (B)--(D); \draw[dashed,thick,blue,blue] (C)--(D); \fill[blue, opacity=.5] (B) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \fill[blue, opacity=.5] (D) circle (3pt); \end{scope} \end{tikzpicture} & \displaystyle \alpha_{2 , 2 , 2 , 3} ( b , c , d ) \\ \\ \begin{tikzpicture}%[scale=1.5] \fill[blue,blue, opacity=.5] (0,0)--(2.5,0)--(1,1.2)--(0,0); \node[circle, fill, inner sep=.8pt, outer sep=0pt] (A) at (0,0){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (B) at (2.5,0){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (D) at (1,1.2){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (F) at (1.7,-2){}; \begin{scope} \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, below]{$ a $\,\,\,}; \draw [midarrow={>}] (A)--(D) node[font=\tiny, midway, right]{$ abc $}; \draw [thick,midarrow={>}] (B)--(D) node[font=\tiny, midway, right]{$ bc $}; \draw [midarrow={>}] (A)--(F) node[font=\tiny, midway, left]{$ abce $}; \draw [midarrow={>}] (B)--(F) node[font=\tiny, midway, left]{$ bce$\!\!}; \draw [dashed,midarrow={>}] (D)--(F) node[font=\tiny, midway, right]{$ e $}; \draw[thick,blue,blue] (A)--(B); \draw[thick,blue,blue] (D)--(A); \draw[thick,blue,blue] (D)--(B); \fill[blue, opacity=.5] (A) circle (3pt); \fill[blue, opacity=.5] (B) circle (3pt); \fill[blue, opacity=.5] (D) circle (3pt); \end{scope} \end{tikzpicture} & \displaystyle \alpha^{-1}_{2 , 2 , 2 , 3} ( a , b c , e ) \\ \\ \begin{tikzpicture}%[scale=1.5] \fill[blue,blue, opacity=.5] (2.5,0)--(3.5,1.2)--(1,1.2); \node[circle, fill, inner sep=.8pt, outer sep=0pt] (B) at (2.5,0){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (C) at (3.5,1.2){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (D) at (1,1.2){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (F) at (1.7,-2){}; \begin{scope} \draw [midarrow={>}] (B)--(C) node[font=\tiny, midway, left]{$ b $}; \draw [midarrow={>}] (C)--(D) node[font=\tiny, midway, below]{\,\,\,\,$ c $}; \draw [thick,midarrow={>}] (B)--(D) node[font=\tiny, midway, right]{$ bc $}; \draw [midarrow={>}] (B)--(F) node[font=\tiny, midway, left]{$ bce$\!\!}; \draw [midarrow={>}] (C)--(F) node[font=\tiny, midway, right]{$ ce $}; \draw [midarrow={>}] (D)--(F) node[font=\tiny, midway, right]{$ e $}; \draw[thick,blue,blue] (B)--(C); \draw[thick,blue,blue] (C)--(D); \draw[thick,blue,blue] (B)--(D); \fill[blue, opacity=.5] (B) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \fill[blue, opacity=.5] (D) circle (3pt); \end{scope} \end{tikzpicture} & \displaystyle \alpha^{-1}_{2 , 2 , 2 , 3} ( b , c , e ) \\ \end{array} \right.$ ---------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------- $\displaystyle $ \left\{ \begin{tikzpicture}%[scale=1.5] \begin{array}{ll} \fill[blue,blue, opacity=.5] (0,0)--(2.5,0)--(3.5,1.2)--(1,1.2)--(0,0); \node[circle, fill, inner sep=.8pt, outer sep=0pt] (A) at (0,0){}; \begin{tikzpicture}%[scale=1.5] \node[circle, fill, inner sep=.8pt, outer sep=0pt] (B) at (2.5,0){}; \fill[blue,blue, opacity=.5] (0,0)--(2.5,0)--(3.5,1.2)--(0,0); \node[circle, fill, inner sep=.8pt, outer sep=0pt] (C) at (3.5,1.2){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (A) at (0,0){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (D) at (1,1.2){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (B) at (2.5,0){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (E) at (1.7,3){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (C) at (3.5,1.2){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (F) at (1.7,-2){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (E) at (1.7,3){}; \begin{scope} \begin{scope} \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, below]{$ a $\,\,\,}; \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, below]{$ a $\,\,\,}; \draw [midarrow={>}] (B)--(C) node[font=\tiny, midway, left]{$ b $}; \draw [midarrow={>}] (B)--(C) node[font=\tiny, midway, left]{$ b $}; \draw [dashed,midarrow={>}] (C)--(D) node[font=\tiny, midway, below]{\,\,\,\,$ c $}; \draw [dashed,thick,midarrow={>}] (A)--(C) node[font=\tiny, midway, below]{$ ab $}; \draw [dashed,midarrow={>}] (A)--(D) node[font=\tiny, midway, right]{$ abc $}; \draw [midarrow={>}] (A)--(E) node[font=\tiny, midway, left]{$ abcd $}; \draw [thick,midarrow={>}] (A)--(C) node[font=\tiny, midway, below]{$ ab $}; \draw [midarrow={>}] (B)--(E) node[font=\tiny, midway, right]{$ bcd $}; \draw [midarrow={>}] (A)--(E) node[font=\tiny, midway, left]{$ abcd $}; \draw [midarrow={>}] (C)--(E) node[font=\tiny, midway, right]{$ cd $}; \draw [midarrow={>}] (B)--(E) node[font=\tiny, midway, right]{$ bcd $}; \draw[thick,blue,blue] (A)--(B)--(C); \draw [midarrow={>}] (C)--(E) node[font=\tiny, midway, right]{$ cd $}; \draw[dashed,thick,blue,blue] (C)--(A); \draw [dashed,midarrow={>}] (D)--(E) node[font=\tiny, midway, right]{$ d $}; \fill[blue, opacity=.5] (A) circle (3pt); \draw [midarrow={>}] (A)--(F) node[font=\tiny, midway, left]{$ abce $}; \fill[blue, opacity=.5] (B) circle (3pt); \draw [midarrow={>}] (B)--(F) node[font=\tiny, midway, left]{$ bce$\!\!}; \fill[blue, opacity=.5] (C) circle (3pt); \draw [midarrow={>}] (C)--(F) node[font=\tiny, midway, right]{$ ce $}; \end{scope} \draw [dashed,midarrow={>}] (D)--(F) node[font=\tiny, midway, right]{$ e $}; \end{tikzpicture} \draw[thick,blue,blue] (A)--(B)--(C); \draw[dashed,thick,blue,blue] (C)--(D)--(A); & \alpha_{2 , 2 , 2 , 3} ( a , b , c d ) \fill[blue, opacity=.5] (A) circle (3pt); \\ \fill[blue, opacity=.5] (B) circle (3pt); \\ \fill[blue, opacity=.5] (C) circle (3pt); \fill[blue, opacity=.5] (D) circle (3pt); \begin{tikzpicture}%[scale=1.5] \end{scope} \fill[blue,blue, opacity=.5] (0,0)--(3.5,1.2)--(1,1.2)--(0,0); \end{tikzpicture}$ \node[circle, fill, inner sep=.8pt, outer sep=0pt] (A) at (0,0){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (C) at (3.5,1.2){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (D) at (1,1.2){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (E) at (1.7,3){}; \begin{scope} \draw [dashed,midarrow={>}] (C)--(D) node[font=\tiny, midway, below]{\,\,\,\,$ c $}; \draw [dashed,midarrow={>}] (A)--(D) node[font=\tiny, midway, right]{$ abc $}; \draw [thick,midarrow={>}] (A)--(C) node[font=\tiny, midway, below]{$ ab $}; \draw [midarrow={>}] (A)--(E) node[font=\tiny, midway, left]{$ abcd $}; \draw [midarrow={>}] (C)--(E) node[font=\tiny, midway, right]{$ cd $}; \draw [dashed,midarrow={>}] (D)--(E) node[font=\tiny, midway, right]{$ d $}; \draw[thick,blue,blue] (A)--(C); \draw[dashed,thick,blue,blue] (C)--(D)--(A); \fill[blue, opacity=.5] (A) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \fill[blue, opacity=.5] (D) circle (3pt); \end{scope} \end{tikzpicture} & \displaystyle \alpha_{2 , 2 , 2 , 3} ( a b , c , d ) \\ \\ \begin{tikzpicture}%[scale=1.5] \fill[blue,blue, opacity=.5] (0,0)--(2.5,0)--(3.5,1.2)--(0,0); \node[circle, fill, inner sep=.8pt, outer sep=0pt] (A) at (0,0){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (B) at (2.5,0){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (C) at (3.5,1.2){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (F) at (1.7,-2){}; \begin{scope} \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, below]{$ a $\,\,\,}; \draw [midarrow={>}] (B)--(C) node[font=\tiny, midway, left]{$ b $}; \draw [thick,midarrow={>}] (A)--(C) node[font=\tiny, midway, below]{$ ab $}; \draw [midarrow={>}] (A)--(F) node[font=\tiny, midway, left]{$ abce $}; \draw [midarrow={>}] (B)--(F) node[font=\tiny, midway, left]{$ bce$\!\!}; \draw [midarrow={>}] (C)--(F) node[font=\tiny, midway, right]{$ ce $}; \draw[thick,blue,blue] (A)--(B)--(C); \draw[thick,blue,blue] (C)--(A); \fill[blue, opacity=.5] (A) circle (3pt); \fill[blue, opacity=.5] (B) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \end{scope} \end{tikzpicture} & \displaystyle \alpha^{-1}_{2 , 2 , 2 , 3} ( a , b , c e ) \\ \\ \begin{tikzpicture}%[scale=1.5] \fill[blue,blue, opacity=.5] (0,0)--(3.5,1.2)--(1,1.2)--(0,0); \node[circle, fill, inner sep=.8pt, outer sep=0pt] (A) at (0,0){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (C) at (3.5,1.2){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (D) at (1,1.2){}; \node[circle, fill, inner sep=.8pt, outer sep=0pt] (F) at (1.7,-2){}; \begin{scope} \draw [dashed,midarrow={>}] (C)--(D) node[font=\tiny, midway, below]{\,\,\,\,$ c $}; \draw [dashed,midarrow={>}] (A)--(D) node[font=\tiny, midway, right]{$ abc $}; \draw [thick,midarrow={>}] (A)--(C) node[font=\tiny, midway, below]{$ ab $}; \draw [midarrow={>}] (A)--(F) node[font=\tiny, midway, left]{$ abce $}; \draw [midarrow={>}] (C)--(F) node[font=\tiny, midway, right]{$ ce $}; \draw [dashed,midarrow={>}] (D)--(F) node[font=\tiny, midway, right]{$ e $}; \draw[thick,blue,blue] (A)--(C); \draw[thick,blue,blue] (C)--(D)--(A); \fill[blue, opacity=.5] (A) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \fill[blue, opacity=.5] (D) circle (3pt); \end{scope} \end{tikzpicture} & \displaystyle \alpha^{-1}_{2 , 2 , 2 , 3} ( a b , c , e ) \\ \end{array} \right.$ ---------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------ $\displaystyle $ \left\{ \begin{tikzpicture}%[thick][scale=1.5] \begin{array}{ll} \fill[blue,blue, opacity=.5] (0,0)--(1.9,-0.5)--(3,0.5)--(0,0); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (0,0){}; \begin{tikzpicture}%[thick][scale=1.5] \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (1.9,-0.5){}; \fill[blue,blue, opacity=.5] (0,0)--(1.9,-0.5)--(3,0.5)--(0,0); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (0,0){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (1.5,1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (3,0.5){}; \begin{scope} \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (1.5,1.8){}; \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, below]{$ a $}; \begin{scope} \draw [midarrow={>}] (B)--(C) node[font=\tiny, midway, left]{$ b $}; \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, below]{$ a $}; \draw [dashed,midarrow={>}] (A)--(C) node[font=\tiny, midway, below]{$ a b $\,\,\,}; \draw [midarrow={>}] (B)--(C) node[font=\tiny, midway, right]{$ b $}; \draw [midarrow={>}] (A)--(D) node[font=\tiny, midway, left]{$ a b c $}; \draw [dashed,midarrow={>}] (A)--(C) node[font=\tiny, midway, below]{$ a b $\,\,\,}; \draw [midarrow={>}] (B)--(D) node[font=\tiny, midway, right]{$ b c $}; \draw [midarrow={>}] (A)--(D) node[font=\tiny, midway, left]{$ a b c $}; \draw [midarrow={>}] (C)--(D) node[font=\tiny, midway, right]{$ c $}; \draw [midarrow={>}] (B)--(D) node[font=\tiny, midway, right]{$ b c $}; \draw [midarrow={>}] (A)--(E) node[font=\tiny, midway, left]{$ a b d $}; \draw [midarrow={>}] (C)--(D) node[font=\tiny, midway, right]{$ c $}; \draw [midarrow={>}] (B)--(E) node[font=\tiny, midway, left] {$ b d $}; \fill[blue, opacity=.5] (A) circle (3pt); \draw [midarrow={>}] (C)--(E) node[font=\tiny, midway, right]{$ d $}; \fill[blue, opacity=.5] (B) circle (3pt); \fill[blue, opacity=.5] (A) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \fill[blue, opacity=.5] (B) circle (3pt); \draw[thick,blue,blue] (A)--(B); \fill[blue, opacity=.5] (C) circle (3pt); \draw[thick,blue,blue] (B)--(C); \draw[thick,blue,blue] (A)--(B); \draw[dashed,thick,blue,blue] (A)--(C); \draw[thick,blue,blue] (B)--(C); \end{scope} \draw[dashed,thick,blue,blue] (A)--(C); \end{tikzpicture} \end{scope} \end{tikzpicture}$ & \alpha_{2 , 2 , 2 , 3} ( a , b , c ) \\ \\ \begin{tikzpicture}%[thick][scale=1.5] \fill[blue,blue, opacity=.5] (0,0)--(1.9,-0.5)--(3,0.5)--(0,0); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (0,0){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \begin{scope} \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, below]{$ a $}; \draw [midarrow={>}] (B)--(C) node[font=\tiny, midway, left]{$ b $}; \draw [midarrow={>}] (A)--(C) node[font=\tiny, midway, above]{$ a b $\,\,\,}; \draw [midarrow={>}] (A)--(E) node[font=\tiny, midway, left]{$ a b d $}; \draw [midarrow={>}] (B)--(E) node[font=\tiny, midway, left] {$ b d $}; \draw [midarrow={>}] (C)--(E) node[font=\tiny, midway, right]{$ d $}; \fill[blue, opacity=.5] (A) circle (3pt); \fill[blue, opacity=.5] (B) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \draw[thick,blue,blue] (A)--(B); \draw[thick,blue,blue] (B)--(C); \draw[thick,blue,blue] (A)--(C); \end{scope} \end{tikzpicture} & \displaystyle \alpha^{-1}_{2 , 2 , 2 , 3} ( a , b , d ) \\ \end{array} \right.$ ------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------ ----------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------- $\displaystyle $ \left\{ \begin{tikzpicture}%[thick][scale=1.5] \begin{array}{ll} \fill[blue,blue, opacity=.5] (0,0)--(1.9,-0.5)--(3,0.5)--(0,0); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (0,0){}; \begin{tikzpicture}%[thick][scale=1.5] \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (1.9,-0.5){}; \fill[blue,blue, opacity=.5] (0,0)--(1.9,-0.5)--(1.5,0.1)--(0,0); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (0,0){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (1.5,1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (1.5,1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (F) at (1.5,-0.1){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (F) at (1.5,0.1){}; \begin{scope} \begin{scope} \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, below]{$ a $}; \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, below]{$ a $}; \draw [midarrow={>}] (B)--(C) node[font=\tiny, midway, below]{\!$ b $}; \draw [midarrow={>}] (A)--(D) node[font=\tiny, midway, left]{$ abc $}; \draw [dashed,midarrow={>}] (A)--(C) node[font=\tiny, midway, above]{$ a b $\,\,\,\,\,\,\,}; \draw [midarrow={>}] (B)--(D) node[font=\tiny, midway, right]{$ bc $}; \draw [midarrow={>}] (A)--(D) node[font=\tiny, midway, left]{$ abc $}; \draw [dashed,thick, midarrow={>}] (1.5,0.1) -- (1.5,1.8) node[font=\tiny, midway, left]{$ g $}; \draw [midarrow={>}] (B)--(D) node[font=\tiny, midway, right]{$ bc $}; \fill[blue, opacity=.5] (A) circle (3pt); \draw [midarrow={>}] (C)--(D) node[font=\tiny, midway, right]{$ c $}; \fill[blue, opacity=.5] (B) circle (3pt); \draw [midarrow={>}] (A)--(E) node[font=\tiny, midway, left]{$ abd $}; \fill[blue, opacity=.5] (F) circle (3pt); \draw [midarrow={>}] (B)--(E) node[font=\tiny, midway, right] {$ bd $}; \draw[thick,blue,blue] (A)--(B); \draw [midarrow={>}] (C)--(E) node[font=\tiny, midway, right]{$ d $}; \draw [dashed,thick, midarrow={>}] (A)--(F) node[font=\tiny, midway, above]{$ abf $}; \draw [dashed,thick, midarrow={>}] (1.5,-0.1) -- (1.5,1.8) node[font=\tiny, midway, left]{$ g $}; \draw [dashed,thick, midarrow={>}] (B)--(F) node[font=\tiny, midway, right]{\,\,$ bf $}; \draw [dashed,thick, midarrow={>}] (1.5,-0.25) -- (1.5,-1.8) node[font=\tiny, midway, left]{$ h $}; \draw[dashed,thick,blue,blue] (A)--(F); \fill[blue, opacity=.5] (A) circle (3pt); \draw[dashed,thick,blue,blue] (F)--(B); \fill[blue, opacity=.5] (B) circle (3pt); \end{scope} \fill[blue, opacity=.5] (C) circle (3pt); \end{tikzpicture} %\draw[thick,red,red, opacity=.5] (A)--(E); \draw[thick,blue,blue] (A)--(B); & \alpha_{2 , 2 , 2 , 3} ( a , b f , g ) \draw[thick,blue,blue] (B)--(C); \draw[dashed,thick,blue,blue] (A)--(C); \\ \draw [thick, midarrow={>}] (A)--(F) node[font=\tiny, midway, above]{$ abf $}; \\ \draw [thick, midarrow={>}] (B)--(F) node[font=\tiny, midway, below]{$ bf $\,\,}; \draw [thick, midarrow={>}] (C)--(F) node[font=\tiny, midway, below]{\!\!\!\!\!$ f $}; \begin{tikzpicture}%[thick][scale=1.5] \end{scope} \fill[blue,blue, opacity=.5] (0,0)--(1.5,-0.1)--(3,0.5)--(0,0); \end{tikzpicture}$ \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (0,0){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (1.5,1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (F) at (1.5,-0.1){}; \begin{scope} \draw [dashed,midarrow={>}] (A)--(C) node[font=\tiny, midway, above]{$ a b $\,\,\,\,\,\,\,}; \draw [midarrow={>}] (A)--(D) node[font=\tiny, midway, left]{$ abc $}; \draw [midarrow={>}] (C)--(D) node[font=\tiny, midway, right]{$ c $}; \draw [thick, midarrow={>}] (1.5,-0.1) -- (1.5,1.8) node[font=\tiny, midway, right]{$ g $}; \fill[blue, opacity=.5] (A) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \fill[blue, opacity=.5] (F) circle (3pt); \draw[dashed,thick,blue,blue] (A)--(C); \draw [thick, midarrow={>}] (A)--(F) node[font=\tiny, midway, below]{$ abf $}; \draw [thick, midarrow={>}] (C)--(F) node[font=\tiny, midway, below]{$ f $}; \draw[thick,blue,blue] (F)--(A); \draw[thick,blue,blue] (F)--(C); \end{scope} \end{tikzpicture} & \displaystyle \alpha^{-1}_{2 , 2 , 2 , 3} ( ab , f , g ) \\ \\ \begin{tikzpicture}%[thick][scale=1.5] \fill[blue,blue, opacity=.5] (1.5,-0.1)--(1.9,-0.5)--(3,0.5)--(1.5,-0.1); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (1.5,1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (F) at (1.5,-0.1){}; \begin{scope} \draw [midarrow={>}] (B)--(C) node[font=\tiny, midway, below]{\!$ b $}; \draw [midarrow={>}] (B)--(D) node[font=\tiny, midway, right]{$ bc $}; \draw [midarrow={>}] (C)--(D) node[font=\tiny, midway, right]{$ c $}; \draw [thick, midarrow={>}] (1.5,-0.1) -- (1.5,1.8) node[font=\tiny, midway, left]{$ g $}; \fill[blue, opacity=.5] (B) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \fill[blue, opacity=.5] (F) circle (3pt); \draw[thick,blue,blue] (B)--(C); \draw [thick, midarrow={>}] (B)--(F) node[font=\tiny, midway, below]{$ bf $\,\,\,}; \draw [dashed,thick, midarrow={>}] (C)--(F) node[font=\tiny, midway, above]{$ f $}; \draw[thick,blue,blue] (F)--(B); \draw[dashed,thick,blue,blue] (F)--(C); \end{scope} \end{tikzpicture} & \alpha_{2 , 2 , 2 , 3} ( b , f , g ) \\ \\ \begin{tikzpicture}%[thick][scale=1.5] \fill[blue,blue, opacity=.5] (0,0)--(1.9,-0.5)--(1.5,0.1)--(0,0); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (0,0){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (F) at (1.5,0.1){}; \begin{scope} \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, below]{$ a $}; \draw [midarrow={>}] (A)--(E) node[font=\tiny, midway, left]{$ abd $}; \draw [midarrow={>}] (B)--(E) node[font=\tiny, midway, right] {$ bd $}; \draw [dashed,thick, midarrow={>}] (F) -- (1.5,-1.8) node[font=\tiny, midway, left]{$ h $}; \fill[blue, opacity=.5] (A) circle (3pt); \fill[blue, opacity=.5] (B) circle (3pt); \fill[blue, opacity=.5] (F) circle (3pt); \draw[thick,blue,blue] (A)--(B); \draw [thick, midarrow={>}] (A)--(F) node[font=\tiny, midway, above]{$ abf $}; \draw [thick, midarrow={>}] (B)--(F) node[font=\tiny, midway, right]{$ bf $}; \draw[thick,blue,blue] (F)--(B); \draw[thick,blue,blue] (F)--(A); \end{scope} \end{tikzpicture} & \alpha^{-1}_{2 , 2 , 2 , 3} ( a , b f , h ) \\ \\ \begin{tikzpicture}%[thick][scale=1.5] \fill[blue,blue, opacity=.5] (0,0)--(1.5,-0.1)--(3,0.5)--(0,0); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (0,0){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (F) at (1.5,-0.1){}; \begin{scope} \draw [midarrow={>}] (A)--(C) node[font=\tiny, midway, above]{$ a b $\,\,\,\,\,\,\,}; \draw [midarrow={>}] (A)--(E) node[font=\tiny, midway, left]{$ abd $}; \draw [midarrow={>}] (C)--(E) node[font=\tiny, midway, right]{$ d $}; \draw [thick, midarrow={>}] (F) -- (1.5,-1.8) node[font=\tiny, midway, left]{$ h $}; \fill[blue, opacity=.5] (A) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \fill[blue, opacity=.5] (F) circle (3pt); \draw[dashed,thick,blue,blue] (A)--(C); \draw [thick, midarrow={>}] (A)--(F) node[font=\tiny, midway, below]{$ abf $}; \draw [thick, midarrow={>}] (C)--(F) node[font=\tiny, midway, below]{\!\!\!\!\!$ f $}; \draw[thick,blue,blue] (F)--(C); \draw[thick,blue,blue] (F)--(A); \end{scope} \end{tikzpicture} & \alpha_{2 , 2 , 2 , 3} ( a b , f , h ) \\ \\ \begin{tikzpicture}%[thick][scale=1.5] \fill[blue,blue, opacity=.5] (1.5,-0.1)--(1.9,-0.5)--(3,0.5)--(1.5,-0.1); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (F) at (1.5,-0.1){}; \begin{scope} \draw [midarrow={>}] (B)--(C) node[font=\tiny, midway, below]{\!$ b $}; \draw [midarrow={>}] (B)--(E) node[font=\tiny, midway, right] {$ bd $}; \draw [midarrow={>}] (C)--(E) node[font=\tiny, midway, right]{$ d $}; \draw [thick, midarrow={>}] (F) -- (1.5,-1.8) node[font=\tiny, midway, left]{$ h $}; \fill[blue, opacity=.5] (B) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \fill[blue, opacity=.5] (F) circle (3pt); \draw[thick,blue,blue] (B)--(C); \draw [thick, midarrow={>}] (B)--(F) node[font=\tiny, midway, below]{$ bf $\,\,}; \draw [thick, midarrow={>}] (C)--(F) node[font=\tiny, midway, below]{\!\!\!\!\!$ f $}; \draw[thick,blue,blue] (F)--(C); \draw[thick,blue,blue] (F)--(B); \end{scope} \end{tikzpicture} & \alpha^{-1}_{2 , 2 , 2 , 3} ( b , f , h ) \\ \end{array} \right.$ ----------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------- In each case, separating the factors involving $d$ (resp. $g$) from those involving $e$ (resp. $h$) gives an instance of condition (2) in the definition of partial cocycle, which is thus seen to ensure invariance under the illustrated instance of the 4-4 extended Pachner move and the 2-6 extended Pachner move. Invariance under moves of these forms with a vertex (resp. an edge) of the polygon in the Seifert surface being modified by the move lying in the knot is, by an identical argument, given by the instances of condition (2) with $i=1$ and $j =2$ (resp. $i = j = 1$). In the case illustrated of the 2-6 extended Pachner move, the new vertex is added to the ordering after the vertices of the triangle lying in the Seifert surface. However, if it is inserted anywhere else in the ordering, an undefying calculation shows that an equation of the same form ensures invariance under the 2-6 move. The case of 1-4 and 2-3 Pachner moves is similar. Invariance under each is ensured by condition (1). We illustrate this with the instances of the moves in which the boundary contains an edge of the knot bounding a triangle of the Seifert surface. The before and after states of the 2-3 move are shown in Figure \[2-3\_with\_knot\_and\_surface\] [ll]{} $\displaystyle \begin{tikzpicture}%[thick][scale=1.5] \fill[blue,blue, opacity=.5] (1.5,1.8)--(0,0)--(1.9,-0.5); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (1.5,1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (0,0){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \begin{scope} \draw [dashed,midarrow={>}] (B)--(D) node[font=\tiny, midway, below]{$b \cdot c$}; \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, left]{$a$}; \draw [midarrow={>}] (B)--(C) node[font=\tiny, midway, below]{$b$}; \draw [midarrow={>}] (C)--(D) node[font=\tiny, midway, above]{$c$}; \draw [midarrow={>}] (A)--(C) node[font=\tiny, midway, right]{$a \cdot b$}; \draw [midarrow={>}] (A)--(D) node[font=\tiny, midway, right]{\,$a \cdot b \cdot c$\,}; \draw [midarrow={>}] (B)--(E) node[font=\tiny, midway, left]{$b \cdot c \cdot d$\,}; \draw [midarrow={>}] (C)--(E) node[font=\tiny, midway, left] {$c \cdot d$\,}; \draw [midarrow={>}] (D)--(E) node[font=\tiny, midway, right]{$d$}; \fill[red, opacity=.5] (A) circle (3pt); \fill[red, opacity=.5] (B) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \draw[thick,red,red, opacity=.5] (A)--(B); \draw[thick,blue,blue, opacity=.5] (B)--(C)--(A); \end{scope} \end{tikzpicture}$ & $ \left\{ \begin{array}{ll} \begin{tikzpicture}%[thick][scale=1.5] \fill[blue,blue, opacity=.5] (1.5,1.8)--(0,0)--(1.9,-0.5); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (1.5,1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (0,0){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (3,0.5){}; \begin{scope} \draw [dashed,midarrow={>}] (B)--(D) node[font=\tiny, midway, below]{$b \cdot c$}; \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, left]{$a$}; \draw [midarrow={>}] (B)--(C) node[font=\tiny, midway, below]{$b$}; \draw [midarrow={>}] (C)--(D) node[font=\tiny, midway, above]{$c$}; \draw [midarrow={>}] (A)--(C) node[font=\tiny, midway, right]{$a \cdot b$}; \draw [midarrow={>}] (A)--(D) node[font=\tiny, midway, right]{\,$a \cdot b \cdot c$\,}; \fill[red, opacity=.5] (A) circle (3pt); \fill[red, opacity=.5] (B) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \draw[thick,red,red, opacity=.5] (A)--(B); \draw[thick,blue,blue, opacity=.5] (B)--(C)--(A); \end{scope} \end{tikzpicture} & \alpha^{-1}_{1 , 1 , 2 , 3} ( a , b , c ) \\ \\ \begin{tikzpicture}%[thick][scale=1.5] \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (0,0){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \begin{scope} \draw [midarrow={>}] (B)--(D) node[font=\tiny, midway, below]{$b \cdot c$}; \draw [midarrow={>}] (B)--(C) node[font=\tiny, midway, below]{$b$}; \draw [midarrow={>}] (C)--(D) node[font=\tiny, midway, above]{$c$}; \draw [midarrow={>}] (B)--(E) node[font=\tiny, midway, left]{$b \cdot c \cdot d$\,}; \draw [midarrow={>}] (C)--(E) node[font=\tiny, midway, left] {$c \cdot d$\,}; \draw [midarrow={>}] (D)--(E) node[font=\tiny, midway, right]{$d$}; \fill[red, opacity=.5] (B) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \draw[thick,blue,blue, opacity=.5] (B)--(C); \end{scope} \end{tikzpicture} & \displaystyle \alpha^{-1}_{1 , 2 , 3 , 3} ( b , c , d ) \\ \end{array} \right.$ \ \ $\displaystyle \begin{tikzpicture}%[thick][scale=1.5] \fill[blue,blue, opacity=.5] (1.5,1.8)--(0,0)--(1.9,-0.5); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (1.5,1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (0,0){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \begin{scope} \draw [dashed,midarrow={>}] (B)--(D) node[font=\tiny, midway, left]{$bc$}; \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, left]{$a$}; \draw [midarrow={>}] (B)--(C) node[font=\tiny, midway, below]{$b$}; \draw [midarrow={>}] (C)--(D) node[font=\tiny, midway, above]{$c$}; \draw [midarrow={>}] (A)--(C) node[font=\tiny, midway, right]{$ab$}; \draw [midarrow={>}] (A)--(D) node[font=\tiny, midway, right]{\,$abc$\,}; \draw [midarrow={>}] (B)--(E) node[font=\tiny, midway, left]{$bcd$\,}; \draw [midarrow={>}] (C)--(E) node[font=\tiny, midway, right] {$cd$\,}; \draw [midarrow={>}] (D)--(E) node[font=\tiny, midway, right]{$d$}; \draw [dashed,thick, midarrow={>}] (1.5,1.8) -- (1.5,-0.2) node[font=\tiny, midway, left]{$abcd$}; \draw [dashed,thick] (1.5,-0.6) -- (1.5,-1.8); \fill[red, opacity=.5] (A) circle (3pt); \fill[red, opacity=.5] (B) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \draw[thick,red,red, opacity=.5] (A)--(B); \draw[thick,blue,blue, opacity=.5] (B)--(C)--(A); \end{scope} \end{tikzpicture}$ & $ \left\{ \begin{array}{ll} \begin{tikzpicture}%[thick][scale=1.5] \fill[blue,blue, opacity=.5] (1.5,1.8)--(0,0)--(1.9,-0.5); \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (1.5,1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (0,0){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \begin{scope} \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, left]{$a$}; \draw [midarrow={>}] (B)--(C) node[font=\tiny, midway, below]{$b$}; \draw [midarrow={>}] (A)--(C) node[font=\tiny, midway, right]{$ab$}; \draw [midarrow={>}] (B)--(E) node[font=\tiny, midway, left]{$bcd$\,}; \draw [midarrow={>}] (C)--(E) node[font=\tiny, midway, right] {$cd$\,}; \draw [dashed,thick, midarrow={>}] (1.5,1.8) -- (1.5,-0.2) node[font=\tiny, midway, left]{$abcd$}; \draw [dashed,thick] (1.5,-0.6) -- (1.5,-1.8); \fill[red, opacity=.5] (A) circle (3pt); \fill[red, opacity=.5] (B) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \draw[thick,red,red, opacity=.5] (A)--(B); \draw[thick,blue,blue, opacity=.5] (B)--(C)--(A); \end{scope} \end{tikzpicture} & \alpha^{-1}_{1 , 1 , 2 , 3} ( a , b , c d ) \\ \\ \begin{tikzpicture}%[thick][scale=1.5] \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (1.5,1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (C) at (1.9,-0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \begin{scope} \draw [midarrow={>}] (C)--(D) node[font=\tiny, midway, above]{$c$}; \draw [midarrow={>}] (A)--(C) node[font=\tiny, midway, right]{$ab$}; \draw [midarrow={>}] (A)--(D) node[font=\tiny, midway, right]{\,$abc$\,}; \draw [midarrow={>}] (C)--(E) node[font=\tiny, midway, right] {$cd$\,}; \draw [midarrow={>}] (D)--(E) node[font=\tiny, midway, right]{$d$}; \draw [midarrow={>}] (A)--(E) node[font=\tiny, midway, left]{$abcd$}; \fill[red, opacity=.5] (A) circle (3pt); \fill[blue, opacity=.5] (C) circle (3pt); \draw[thick,blue,blue, opacity=.5] (A)--(C); \end{scope} \end{tikzpicture} & \displaystyle \alpha^{-1}_{1 , 2 , 3 , 3} ( a b , c , d ) \\ \\ \begin{tikzpicture}%[thick][scale=1.5] \node[circle, fill, inner sep=.9pt, outer sep=0pt] (A) at (1.5,1.8){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (B) at (0,0){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (D) at (3,0.5){}; \node[circle, fill, inner sep=.9pt, outer sep=0pt] (E) at (1.5,-1.8){}; \begin{scope} \draw [dashed,midarrow={>}] (B)--(D) node[font=\tiny, midway, left]{$bc$}; \draw [midarrow={>}] (A)--(B) node[font=\tiny, midway, left]{$a$}; \draw [midarrow={>}] (A)--(D) node[font=\tiny, midway, right]{\,$abc$\,}; \draw [midarrow={>}] (B)--(E) node[font=\tiny, midway, left]{$bcd$\,}; \draw [midarrow={>}] (D)--(E) node[font=\tiny, midway, right]{$d$}; \draw [midarrow={>}] (A)--(E) node[font=\tiny, midway, right]{$abcd$}; \fill[red, opacity=.5] (A) circle (3pt); \fill[red, opacity=.5] (B) circle (3pt); \draw[thick,red,red, opacity=.5] (A)--(B); \end{scope} \end{tikzpicture} & \alpha_{1 , 1 , 3 , 3} ( a , b c, d ) \\ \end{array} \right.$ Plainly invariance under this move is equivalent to the relation $$\begin{aligned} \lefteqn{ \alpha^{-1}_{1 , 1 , 2 , 3} ( a , b , c ) \, \cdot \, \alpha^{-1}_{1 , 2 , 3 , 3} ( b , c , d ) =} \\ & & \alpha^{-1}_{1 , 1 , 2 , 3} ( a , b , c d ) \, \cdot \, \alpha^{-1}_{1 , 2 , 3 , 3} ( a b , c , d ) \, \cdot \, \alpha_{1 , 1 , 3 , 3} ( a , b c, d ) \end{aligned}$$ which solves to give the instance of condition (1) with indices 1,1,2,3,3. A little effort shows that the same relation suffices to give invariance under the other instance of a 2-3 move with an triangle on the boundary lying in the Seifert surface, with an edge on the knot – in which the edge of the knot lies on the equatorial triangle, and under a 1-4 move in which the tetrahedron subdivided similarly has a triangle of its boundary lying in the Seifert surface with an edge on the knot, regardless of whether the new vertex is inserted into the ordering before or after the original vertex lying in the 3-dimensional stratum. Likewise the other instances of condition (1) with indices 3,3,3,3,3 (resp. 2,3,3,3,3; 1,3,3,3,3; 2,2,3,3,3; 1,2,3,3,3; 1,1,3,3,3; 2,2,2,3,3; 1,2,2,3,3) give invariance under 2-3 and 1-4 moves in which the boundary of the cell modified is not incident with either the knot or Seifert sufrace (resp. intersect the Seifert surface in a single vertex not on the knot; intersect the knot in a single vertex; is not incident with the knot, but intersects the Seifert surface in an edge; intersects the knot in a single vertex and the Seifert surface in an edge ending at that vertex; intersects the knot in a single edge; is not incident with the knot, but intersects the Seifert surface in a triangle; intersect the Seifert surface in a triangle with a single vertex on the knot). Again, inserting the new vertex in 1-4 moves anywhere in the ordering gives rise to different instances of the same relation. Thus the value of the expression is independent of both the triangulation and the ordering(s) of the vertices. As with the untwisted invariant, which is the special case of the preceding in which $\alpha \equiv 1$, by general principles explained in [@Y], this invariant is the restriction to the endomorphisms of the monoidal identity, the empty triple of spaces, of a generalized TQFT, a monoidal functor from (3,2,1)-[COBORD]{}, the category whose objects are triples of an oriented surface containing an oriented curve with boundary, with disjoint union as the monoidal product, and whose arrows are PL-homeomorphism classes of cobordisms between these, to $k$-[v.s.]{}. Examples of Initial Data ======================== Finite $\mathsf 3$-parcels are easy to construct. Given any three finite groups $G_i$ $i = 1, 2,3$ and finite sets $X_{1,2}$ and $X_{2,3}$, with $X_{i,j}$ equipped with a left $G_i$-action and a right $G_j$ action, which commute with each other, there is a $\mathsf 3$-parcel $\mathcal C$ with ${\mathcal C}(i,i) = G_i$, ${\mathcal C}(i,j) = X_{i,j}$ for $(i,j) = (1,2)$ or $(2,3)$ and $${\mathcal C}(1,3) = X_{1,2} \times X_{2,3} / \sim$$ where $\sim$ is the equivalence relation induced by $$(xg,y) \sim (x,gy)$$ whenever $x\in X_{1,2}$, $g\in G_2$ and $y\in X_{2,3}$, where the actions of the group are denoted by the null infix. Composition is given by the group multiplications, group actions, and the map $(x,y) \mapsto [x,y]$ where $[x,y]$ is the equivalence class of the pair under $\sim$ for $x\in X_{1,2}$ and $y\in X_{2,3}$, as appropriate. The identity arrows for the object $i$ is the group identity of $G_i$. It is trivial to verify associativity. If we add to the data a finite set $X_{1,3}$ with a commuting left $G_1$ action and a right $G_3$ action, together with an biequivariant map $\varphi: X_{1,2} \times X_{2,3} / \sim \rightarrow X_{1,3}$, and let the composition arrows $x:1\rightarrow 2$ and $y:2\rightarrow 3$ be given by $\varphi([x,y])$, we obtain a complete description of all finite $\mathsf 3$-parcels. Among these are $\mathsf 3$-parcels constructed from subgroups and subsets of a groups with suitable closure properties after the manner of [@DPY] Example 5.2, on which it is, moreover, easy to find examples of partial 3-cocycles using ordinary group cohomology: Fix a group $G$. Let $G_i$ $i = 1,2,3$ be subgroups of $G$ and $X_{1,2}$ (resp. $X_{2,3}$) be a subset of $G$ closed under left multiplication by elements of $G_1$ (resp. $G_2$) and right multiplication by elements of $G_2$ (resp. $G_3$), and let $X_{1,3} := \{ x\xi \; \|\; x \in X_{1,2}\; \&\; \xi \in X_{2,3} \}$. The disjoint union of the $G_i$ for $i = 1,2,3$ and the $X_{i,j}$ for $(i,j) = (1,2), (2,3), (1,3)$ is easily seen to be a $\mathsf 3$-parcel $\mathcal C$ under a composition induced by the group law on $G$ with the obvious underlying functor to $\mathsf 3$. It is also easy to see that given any $K^\times$-valued 3-cocycle $\alpha$ on $G$, the map induced on $Arr({\mathcal C})$ by composing the canonical maps to $G^3$, induced by the inclusions of the homsets into $G$, with $\alpha$ is a 3-cocycle on $\mathcal C$, and thus a partial 3-cocycle. [1]{} Alexander, J., “The Combinatorial Theory of Complexes,” [Ann. of Math.]{} [31]{} (2) (1930) 292-320. Casali, Maria Rita, “A note about bistellar operations on PL-manifolds with boundary,” [Geometria Dedicata]{} [56]{} (1995) 257-262. Crane, L. and Yetter, D.N., “Moves on Filtered PL Manifolds and Stratified PL Spaces,” \#arXiv:1404.3142. Dougherty, Aria L., Park, Hwajin, and Yetter, D.N., “On 2-dimensional Dikjgraaf-Witten Theory with Defects,” (2014) Grandis, Marco, “Directed Homotopy Theory, I. The fundamental category”, [Cah. Topol. Géom. Diff. Catég.]{} [44]{} (2003), 281-316. Grandis, Marco, [Directed Algebraic Topology: Models of non-reversible worlds]{}, Cambridge University Press, (2009). Wakui, Michihisa, “On Dijkgraaf-Witten Invariant for 3-manifolds,” [Osaka J. Math.]{} [29]{} (4) (1992) 675-696. Yetter, D.N., “Triangulations and TQFTs,” in [Conference Proceedings on Quantum Topology]{}, (Randy Baadhio, ed.), World Scientific (1993) 354-370.	ArXiv
--- abstract: 'We study the spin-orbit coupling induced by the splitting between TE and TM optical modes in a photonic honeycomb lattice. Using a tight-binding approach, we calculate analytically the band structure. Close to the Dirac point,we derive an effective Hamiltonian. We find that the local reduced symmetry ($\mathrm{D_{3h}}$) transforms the TE-TM effective magnetic field into an emergent field with a Dresselhaus symmetry. As a result, particles become massive, but no gap opens. The emergent field symmetry is revealed by the optical spin Hall effect.' author: - 'A. V. Nalitov' - 'G. Malpuech' - 'H. Terças' - 'D. D. Solnyshkov' bibliography: - 'reference.bib' title: 'Spin-orbit coupling and optical spin Hall effect in photonic graphene' --- Spin-orbit coupling in crystals allows to create and control spin currents without applying external magnetic fields. These phenomena have been described in the seventies [@Dyakonov] and are nowadays called the spin Hall effect (SHE) [@Hirsch1999; @reviewSHE]. In 2005, the interplay between the spin-orbit coupling and the specific crystal symmetry of graphene[@Geim2007] has been proposed [@Kane2005] to be at the origin of a new type of spin Hall effect, the quantum spin Hall effect, in which the spin currents are supported by surface states and are topologically protected [@QSHE; @Kane2010]. This result has a special importance, since it defines a new class of $Z_2$-topogical insulator [@Kane2005b], not associated with the quantization of the total conductance, but associated with the quantization of the spin conductance. However, from an experimental point of view, the realization of any kind of SHE is difficult, because spin-orbit coupling does not only lead to the creation of spin current, but also to spin decoherence [@Dyakonov2]. In graphene, the situation is even worse, since the spin-orbit coupling is extremely weak. Deposition of adatoms has been proposed to increase the spin-orbit coupling [@Gmitra2013], and it allowed the recent observation of the SHE [@Balakrishnan2013], but associated with a very short spin relaxation length, of the order of 1 $\mu$m. On the other hand, artificial honeycomb lattices for atomic Bose Einstein Condensates (BEC) [@hon_atom] and photons [@Peleg2007; @Kuhl2010; @Polini2013; @Kalesaki2014; @Jacqmin2014] have been realized. These systems are gaining a lot of attention due to the large possible control over the system parameters, up to complete Hamiltonian engineering[@Hafezi; @Umucalilar]. In BECs, the recent implementation of synthetic magnetic fields [@Lin1] and of non-Abelian, Rashba-Dresselhauss gauge fields [@Lin2] appears promising in the view of the achievement of topological insulator analogs. Photonic systems, and specifically photonic honeycomb lattices appear even more promising. They are based on coupled wave guide arrays [@Rechtsman2013], on photonic crystals with honeycomb symmetry [@Won2011], and on etched planar cavities [@Jacqmin2014]. A photonic Floquet topological insulator has been recently reported [@Chong2013], and some others based on the magnetic response of metamaterials predicted [@Khanikaev]. In photonic systems, spin-orbit coupling naturally appears from the energy splitting between the TE and TM optical modes and from structural anisotropies. Both effects can be described in terms of effective magnetic fields acting of the photon (pseudo)-spin [@Shelykh2010]. In planar cavity systems, the TE-TM effective field breaks the rotational symmetry, but preserves both time reversal and spatial inversion symmetries. It is characterized by a $k^2$ scaling and a double azimuthal dependence. This spin-orbit coupling is at the origin of the optical spin Hall effect (OSHE)[@Kavokin2005; @Leyder2007] and of the acceleration of effective magnetic monopoles [@Hivet; @Bramwell2012; @Solnyshkov2013]. As recently shown [@Tercas2014], the specific TE-TM symmetry can be locally transformed into a non-Abelian gauge field in a structure with a reduced spatial symmetry. In this work, we calculate the band structure of photonic graphene in the presence of the intrinsic spin-orbit coupling induced by the TE-TM splitting. We derive an effective Hamiltonian which allows to extract an effective magnetic field acting on the photon pseudo-spin only. We find that the low symmetry ($\mathrm{D_{3h}}$) induced by the honeycomb lattice close to the Dirac points transforms the TE-TM field in a emergent field with a Dresselhaus symmetry. Particles become massive but no gap opens. The dispersion topology shows larges similarities with the one of bilayer graphene [@McCann2006] and of monolayer graphene with Rashba spin-orbit coupling [@Rakyta2010], featuring trigonal warping [@Dresselhaus1974] and Lifshitz transition [@Lifshitz1960]. The symmetry of these states is revealed by the optical spin Hall effect (OSHE) which we describe by simulating resonant optical excitation of the $\Gamma$, K and K’ points. The OSHE at the $\Gamma$ point shows four spin domains associated with the TE-TM symmetry. The OSHE at the K and K’ shows two domains characteristic of the Dresshlauss symmetry. The spin domains at the K and K’ points are inverted, which is a signature of the restored $\mathrm{D_{6h}}$ symmetry when the two valleys are included. In what follows, in order to be specific, we consider a honeycomb lattice based on a patterned planar microcavity similar to the one recently fabricated and studied [@Jacqmin2014]. This does not reduce the generality of our description, which can apply to other physical realizations of honeycomb lattices, in optical and non-optical systems. In [@Jacqmin2014], quantum wells were embedded in the cavity which provided the strong coupling regime and the formation of cavity exciton-polaritons. Here, we will consider the linear regime, a parabolic in-plane dispersion, and no applied magnetic field. In such case, photons and exciton-polaritons behave in a similar way and our formalism applies to both types of particles. \[fig1\] ![(color online) A schematic sketch of the tight-binding model. (a) Photon tunneling between microcavity pillars is described as photon propagation through “waveguide”-like links. (b) Polarization dependence of tunneling probability due to TE-TM energy splitting: $L$ state which is polarized longitudinally to the “waveguide” link is closer in energy to the degenerate pillar-pinned states than the transversely-polarized state $T$, resulting in higher L-photon tunneling probability through the link. ](fig1.pdf "fig:") Tight-binding model First, we describe the spin-orbit coupling in photonic graphene structure (figure 1a) within the tight-binding approximation. We take a basis of $\sigma\pm$ polarized photon states localized on each pillar of the lattice as a zeroth approximation for the tight-binding model and introduce the hopping of photons from a pillar to one of its nearest neighbors as a perturbation $\hat{V}$ on this basis. To illustrate the polarization dependence of the hopping probability, let us consider two neighbouring pillars $A$ and $B$, shown in Figure (1b). The photon hopping between them may be described as propagation through a “waveguide”-like link. TE-TM energy splitting imposes a slight difference $\delta J$ in tunneling matrix elements for states linearly-polarized longitudinally ($L$) and transversely ($T$) to vector $\mathbf{d}_\varphi$ linking the pillars [@suppl], as it was recently shown for the eigenstates in a photonic benzene molecule [@Vera]. In that framework, the matrix elements read: $$\langle A, L \vert \hat{V} \vert B, L\rangle \equiv -J-\delta J/2, \quad \langle A, T \vert \hat{V} \vert B, T\rangle \equiv -J+\delta J/2. \notag$$ While a photon is in a link, TE-TM field does not rotate its eigenstate polarizations $L$ and $T$, implying no cross-polarization matrix elements: $$\langle A, L \vert \hat{V} \vert B, T\rangle = \langle A, T \vert \hat{V} \vert B, L\rangle = 0. \notag$$ In $\sigma\pm$ basis, the probability of spin flip during hopping is linear in $\delta J$ and its phase gain depends on the angle $\varphi$ between the link and the horizontal axis: $$\langle A, \pm \vert \hat{V} \vert B, \pm \rangle = -J, \quad \langle A, + \vert \hat{V} \vert B,- \rangle = - \delta J e^{-2\mathrm{i}\varphi}. \notag$$ This phase factor reflects the fact that when a link is rotated by 90 degrees, $L$ and $T$ polarization basis is inverted: if $L$ was horizontal, it becomes vertical and vice versa. A photon state may be described in the bispinor form $\Phi = \left( \Psi_A^+, \Psi_A^-, \Psi_B^+, \Psi_B^- \right)^{\mathrm{T}}$, with $\Psi_{A(B)}^\pm$ being the wave function on both sublattices in both spin components. The effective Hamiltonian acting on a plane wave bispinor $\Phi_\mathbf{k}$ then has a block matrix form: $$\label{Hamiltonian} \mathrm{H}_\mathbf{k} = \left( \begin{matrix} \mathrm{0} & \mathrm{F}_{\mathbf{k}} \\ \mathrm{F}_{\mathbf{k}}^\dagger & \mathrm{0} \end{matrix} \right), \quad \mathrm{F}_{\mathbf{k}} = - \left( \begin{matrix} f_{\mathbf{k}} J & f_{\mathbf{k}}^+ \delta J \\ f_{\mathbf{k}}^- \delta J & f_{\mathbf{k}} J \end{matrix} \right),$$ where complex coefficients $f_{\mathbf{k}}$,$f_{\mathbf{k}}^\pm$ are defined by: $$f_{\mathbf{k}}=\sum_{j=1}^3 \exp(-\mathrm{i}\mathbf{k d}_{\varphi_j}),\quad f_{\mathbf{k}}^\pm = \sum_{j=1}^3 \exp(-\mathrm{i}\left[\mathbf{k d}_{\varphi_j} \mp 2 \varphi_j \right]), \notag$$ and $\varphi_j = 2 \pi (j-1) / 3$ is the angle between the horizontal axis and the direction to the $j$th nearest neighbor of a type-A pillar. Its diagonalization results in a biquadratic equation on the photon dispersion, having two pairs of solutions $\pm E_\mathbf{k}^\pm$, given by: $$\begin{aligned} \label{disp_sol} &2 (E_\mathbf{k}^\pm)^2 = 2 \vert f_\mathbf{k} \vert^2 J^2 + \left( \vert f_\mathbf{k}^+ \vert^2 + \vert f_\mathbf{k}^-\vert^2 \right) \delta J^2 \pm \\ &\pm \sqrt{(\vert f_\mathbf{k}^+ \vert^2 - \vert f_\mathbf{k}^- \vert^2)^2 \delta J^4 + 4 \vert f_\mathbf{k} f_\mathbf{k}^{+}+f_\mathbf{k}^ f_\mathbf{k}^- \vert ^2 J^2 \delta J^2}. \notag\end{aligned}$$ The dispersion is plotted along the principal direction in Figure (2a), and the trigonal warping effect which is a characteristic of bilayer graphene [@McCann2006] and of monolayer graphene with Rashba spin-orbit coupling [@Rakyta2010] is shown on the Figure (2b) in the vicinity of the K point. When $\delta J = J /2$, trigonal warping disappears. The crossing points originating from different Dirac points meet and annihilate. The dispersion topology changes – a phenomenon associated with the so-called Lifshitz transition in Fermionic systems [@Lifshitz1960]. If $\delta J \ll J$, the distance $\delta K$ between a K point and the additional pockets is approximately given by $(\delta J / J)^2 a^{-1}$. \[fig2\] ![(color online) The eigenstates of the effective Hamiltonian due to the honeycomb potential and the TE-TM splitting. (a) Photon dispersion branches along principal directions. The inset demonstrates a zoomed K valley – the dispersion is gapless. (b) Isoenergetic lines around K point, illustrating the trigonal warping: Dirac cones split in four pockets. (c) and (d) Linear polarization map and the pseudospin texture of the lowest energy state.](fig2.pdf "fig:") The effective Hamiltonian may be expressed in terms of pseudospin operators $\boldsymbol{\sigma}$ and $\mathbf{s}$, having the same matrix form of Pauli matrices vector and corresponding to sublattice (A/B) and polarization (H/V) degrees of freedom. It may further be separated into polarization-independent part $H^{(0)}_\mathbf{k}$, coupling $\boldsymbol{\sigma}$ with momentum and giving a standard graphene dispersion with two Dirac valleys K and K$^\prime$, and a spin-orbit term $H^{\mathrm{SO}}_\mathbf{k}$, coupling $\mathbf{s}$ with $\boldsymbol{\sigma}$ and momentum: $$\begin{aligned} H_\mathbf{k}^{\mathrm{(0)}} =& - J \sigma_+ f_\mathbf{k} +h.c., \label{H_0}\\ H_\mathbf{k}^{\mathrm{SO}} =& - \delta J \sigma_+ \otimes \left( f_\mathbf{k}^+ s_+ + f_\mathbf{k}^- s_- \right) + h.c.. \label{H_SO}\end{aligned}$$ where $\sigma_\pm = (\sigma_x \pm \mathrm{i} \sigma_y)/2$, $s_\pm = (s_x \pm \mathrm{i} s_y)/2$, and the $\otimes$ symbol denotes Kronecker product. Expanding expressions (\[H\_0\],\[H\_SO\]) and keeping the main order in $\mathbf{q=k-K}$, we further isolate the momentum-independent part $H^\mathrm{SO}_{\mathbf{K}}$ coupling $\mathbf{s}$ with $\boldsymbol{\sigma}$ and rewrite both terms in the low-energy approximation: $$\begin{aligned} H_\mathbf{q}^{\mathrm{(0)}} =& \hbar v_F \left( \tau_z q_x \sigma_x + q_y \sigma_y \right), \label{H_01} \\ H^{\mathrm{SO}} =& \Delta \left( \tau_z \sigma_y s_y - \sigma_x s_x \right), \label{H_SOK}\end{aligned}$$ where $v_F = 3 J a / (2 \hbar)$, $\Delta = 3 \delta J / 2$ and $\tau_z$ equals $+1$ and $-1$ for K and K$^\prime$ valleys respectively. Here we use the same basis as the the one of Kane and Mele [@Kane2005] in order to allow for a direct comparison with their Hamiltonian. This basis is different from the original basis of Wallace [@Wallace] which is used in the eq.\[Hamiltonian\]. The passage from Wallace to Kane is obtained by writing $q_x\rightarrow q_y$,$q_y\rightarrow -q_x$). If one restricts state space by locally fixing the sublattice $\boldsymbol{\sigma}(\mathbf{k})$ pseudospin and valley $\tau_z$, the spin-orbit term may be treated as an interaction with an emergent field. As an example, if one considers eigenstates of the main term (\[H\_01\]) in one Dirac valley with a fixed energy sign $c=\pm 1$, spin-orbit term (\[H\_SOK\]) transforms to a symmetry-allowed Dresselhaus-like emergent field: $$\begin{aligned} H_{c}^{\mathrm{SO}} =& -\Delta c \left( q_x s_x + q_y s_y\right) /q. \label{H_SOKt}\end{aligned}$$ This term, having a well-defined physical origin, is similar in spirit with the Rashba term introduced by Kane and Mele [@Kane2005; @Kane2005b]. The effective field described by the spin-orbit term (\[H\_SOKt\]) splits the degenerate massless photon branches by $3 \delta J$, and their linear polarization only depends on the direction of $\mathbf{q}$ and not on its absolute value. However, if $q < \Delta / \hbar v_F = (\delta J / J) a^{-1}$, the spin orbit term cannot be considered as a perturbation of the main term (\[H\_01\]), the interplay between the two terms gives an effective photon mass $m^=(2 c \hbar^2 \delta J)/(3 a^2 J^2)$ in this region of reciprocal space. The pseudospin pattern (defining the linear polarization of light) of the lowest energy eigenstate reflects the effective field acting on the particles, because the pseudospin aligns with this field. The the whole reciprocal space is shown in Figure (2c). The figure (2d) shows a zoom on the K-point where the emergent Dressehlaus like field is clearly identified. Figure (2c) also clearly shows that the effective fields have an opposite sign close to the K and K’ points respectively. From this analytical calculation of the dispersion, we can conclude that the particular type of spin-orbit coupling we consider does not open a gap in the K point of the Brillouin zone, but leads to the appearance of massive particles. This, among other consequences, should induce a strong modification of the Klein tunneling effect. As shown in [@Liu2012], where Klein tunneling in the presence of a Rashba term was considered, the tunneling is suppressed for energies close to the K point, where the dispersion is not linear anymore, but is recovered for higher energies. The best evidence of the presence of a spin-orbit coupling inducing an effective magnetic field of a specific symmetry is the optical spin-Hall effect: rotation of the particle spin around the effective wavevector-dependent field during their propagation. The resonant excitation around the $\Gamma$ point with linearly polarised light should lead to a radial expansion of the wave-packet accompanied by a precession of the photon pseudo-spin. The double azimuthal dependence of the effective field orientation should lead, as in the planar case, to the formation of four spin domains [@Kavokin2005; @NOSHE]. Close to the K and K’ points, the Dressehlaus effective field orientation follows the azimuthal angle and two spin domains only should form [@Vishnevsky2013; @Tercas2014]. Numerical simulation* In the following, in order to check the validity of the tight-binding approximation, and the observability of the OSHE, in realistic structures and experiments, namely including the broadening induced by the finite life time, we study numerically the propagation of polarised light in the photonic graphene structure. We consider a structure etched out of a planar microcavity, where the graphene atoms are represented by overlapping pillars (fig 3a). The equation of motion for the photonic spinor wavefunction reads : $$\begin{aligned} & i\hbar \frac{{\partial \psi _ \pm }} {{\partial t}} = - \frac{{\hbar ^2 }} {{2m}}\Delta \psi _ \pm + U\psi _ \pm - \frac{{i\hbar }} {{2\tau }}\psi _ \pm + \\ & + \beta {\left( {\frac{\partial }{{\partial x}} \mp i\frac{\partial }{{\partial y}}} \right)^2}{\psi _ \mp } +P_0 e^{ { - \frac{{\left( {t - t_0 } \right)^2 }} {{\tau _0^2 }}}}e^{ { - \frac{{\left( {{\mathbf{r}} - {\mathbf{r}}_0 } \right)^2 }} {{\sigma ^2 }}}}e^{ {i\left( {{\mathbf{kr}} - \omega t} \right)} } \notag\end{aligned}$$ where $\psi(r)={\psi_+(r), \psi_-(r)}$ are the two circular components of the photon wave function, $m$ is the cavity photon mass, $\tau$ the lifetime. This equation is similar with the one describing the photon motion in a planar cavity in the presence of TE-TM splitting [@Shelykh2010], described by the parameter $\beta ={% \hbar ^{2}}\left( {m_{l}^{-1}-m_{t}^{-1}}\right) /4m$ where $% m_{l,t}$ are the effective masses of TM and TE polarized particles respectively and $m=2\left( {{m_{t}}-{m_{l}}}\right) /{m_{t}}{% m_{l}}$. We have taken $m_t=5\times10^{-5}m_0$, $m_l=0.95m_t$, where $m_0$ is the free electron mass. The only difference lies in the introduction of the honeycomb lattice potential $U(r)$ shown on the figure 3a (24x24 elementary cells). $P_{0}$ is the amplitude of the pulsed pumping (identical for both components, corresponding to horizontal polarization), the pulse duration is $\tau_0=1$ ps, the size of the spot $\sigma=15$ $\mu$m. Pumping is localized in real space and in reciprocal space close to the selected point ($\Gamma$, K or K’). The lifetime was taken $\tau=25$ ps. \[fig3\] ![(color online). Optical spin Hall effect in photonic graphene. Circular polarization degree as a function of coordinates: a) the potential used in the simulations; b) excitation at $\Gamma$ point (TE-TM field); c) excitation at K point (Dresselhaus effective field); d) excitation at K’ point (field inverted with respect to K’).](fig3.pdf "fig:") We have performed numerical simulation of optical spin Hall effect in photonic graphene using a high-resolution (512x512) representation of a potential, similar to the one already studied in experiments [@Jacqmin2014]. The nVidia CUDA graphical processor was used to carry out the integration of the 2D spinor Schroedinger equation. Figure (3-b,c,d) shows the snapshots taken at $t=30 ps$ of the circular polarization degree as a function of coordinates. Panel b) shows the polarization degree for the excitation in the $\Gamma$ point, where the field has the typical TE-TM texture, evidenced by the 4 polarization domains [@Kavokin2005; @Leyder2007; @NOSHE]. Panel c,d) demonstrate the optical spin Hall effect for the $K$ and $K'$ points respectively, where the field has the texture of the Dresselhaus spin-orbit coupling. This is evidenced by 2 polarization domains in real space [@Vishnevsky2013; @Tercas2014] being inverted between the $K$ and $K'$ points which reflects the fact that the fields around $K$ and $K'$ are respectively opposite. The texture of the optical spin-Hall effect is a clear demonstration of the different nature of the effective magnetic field due to the spin-orbit coupling in the two Dirac points (K and K’) of the Brillouin zone. From this numerical experiment, we clearly see the advantage of photonic systems, which allow to excite and analyze any point of the dispersion, much easier than in solid state systems. Other very interesting consequences of our work rely on the possibilities offered by the manipulation of the lattice geometry in photonic systems and by the mixed exciton-photon nature of exciton-polaritons. The system geometry is the tool which has been used to create a photonic topological insulator [@Rechtsman2013]. Combined with spin-orbit coupling, it opens very broad perspectives. The mixed nature of exciton-polaritons provides a magnetic response of the system at optical frequencies, which is of interest to realize a photonic topological insulator [@Haldane2008; @Soljacic09]. It also induces a very strong non-linear optical response. Non-linear spin Hall effect associated with the transmutation of topological defects and focusing of spin currents have been already described in planar structures [@NOSHE]. The behaviour of soliton states in photonic topological insulators was recently considered [@Segev]. More generally, the interactions allow an exciton-polariton gas to behave as a quantum fluid [@Carusotto2013] with spin-anisotropic interactions [@Shelykh2010]. Polaritonic graphene [@Jacqmin2014] therefore opens very large possibilities for the studies of interacting spinor quantum fluids, in the presence of different types of real and effective magnetic fields which suggest accessibility to different types of quantum phases. To conclude, we have studied the spin-orbit coupling induced by the TE-TM splitting in a microcavity etched in the shape of a graphene lattice. Within the tight-binding approximation we found the eigenstates of the system, derived an effective Hamiltonian and found the effective fields acting on the photon spin. The symmetry of the field is lowered close to the Dirac points where it takes the form of a Dressehlauss field. The experimental observability of the optical Spin Hall effect induced by this spin-orbit coupling is verified by numerical simulations. We acknowledge discussions with M. Glazov, A. Amo, I. Carusotto, and J. Bloch. This work has been supported by the ITN INDEX (289968), ANR Labex GANEX (Grant No. ANR-11-LABX-0014), ANR Quandyde (ANR-11-BS10-001) and IRSES POLAPHEN (246912).	ArXiv
--- abstract: 'Unbiased data collection is essential to guaranteeing fairness in artificial intelligence models. Implicit bias, a form of behavioral conditioning that leads us to attribute predetermined characteristics to members of certain groups and informs the data collection process. This paper quantifies implicit bias in viewer ratings of TEDTalks, a diverse social platform assessing social and professional performance, in order to present the correlations of different kinds of bias across sensitive attributes. Although the viewer ratings of these videos should purely reflect the speaker’s competence and skill, our analysis of the ratings demonstrates the presence of overwhelming and predominant implicit bias with respect to race and gender. In our paper, we present strategies to detect and mitigate bias that are critical to removing unfairness in AI.' author: - 'Rupam Acharyya\,Shouman Das\,Ankani Chattoraj, Oishani Sengupta,Md Iftekar Tanveer' bibliography: - 'reference.bib' title: Detection and Mitigation of Bias in Ted Talk Ratings --- Introduction ============ Machine-learning techniques are being used to evaluate human skills in areas of social performance, such as automatically grading essays [@alikaniotis2016automatic; @taghipour2016neural], outcomes of video based job interviews [@chen2017automated; @Naim2016], hirability [@Nguyen2016], presentation performance [@Tanveer2015; @Chen2017a; @Tanveer2018] etc. These algorithms automatically quantify the relative skills and performances by assessing large quantities of human annotated data. Companies and organizations world-wide are increasingly using commercial products that utilize machine learning techniques to assess these areas of social interaction. However, the presence of implicit bias in society reflected in the annotators and a combination of several other unknown factors (e.g. demographics of the subjects in the datasets, demographics of the annotators) creates systematic imbalances in human datasets. Machine learning algorithms (neural networks in most cases) trained on such biased datasets automatically replicate the imbalance [@o2016weapons] naturally present in the data and result in producing unfair predictions. Examining the impact of implicit bias in social behavior requires extensive, diverse human data that is spontaneously generated and reveals the perception of success in social performance. In this paper, we analyze ratings of TED Talk videos to quantify the amount of social bias in viewer opinions. TED Talks present a platform where speakers are given a short time to present inspiring and socially transformative ideas in an innovating and engaging way. In its mission statement, the TED organization describes itself as a “global community, welcoming people from every discipline and culture” and makes an explicit commitment to “change attitudes, lives, and ultimately the world” [@tedtalk]. Since TED Talks offer a platform to speakers from diverse backgrounds trying to convince people of their professional skills and achievements, the platform lends itself to a discussion of several critical issues regarding fairness and implicit bias: How can we determine the fairness of viewer ratings of TED Talk videos? Can we detect implicit bias in the ratings dataset? Are ratings influenced by the race and gender of the speaker? Ideally, these ratings should depend on the perception of the speaker’s success and communicative performance; not on the speaker’s gender or ethnicity. For instance, our findings show that while a larger proportion of viewers rate white speakers in a confidently positive manner, speaker of other gender identities and ethnic backgrounds receive a greater number of mixed ratings and elicit wider differences of opinion. In addition, men and women are rated as positive or negative with more consistency, while speakers identifying with other gender identities are rated less consistently in either direction. With this assumption, we conducted computational analysis to detect and mitigate bias present in our data. We utilize a state of the art metric “Disparate Impact” as in [@feldman2015certifying] for measuring fairness, and three popular methods of bias correction— 1. pre-processing [@calmon2017optimized; @kamiran2012data], 2. in-processing [@calders2010three; @kamishima2011fairness], and 3. post-processing [@hardt2016equality] . We compared such predictions of the ratings with the actual ratings provided by the viewers of the TED talks and found that our model prediction performs better w.r.t. a standard fairness metric. Our experiments show that if the traditional machine learning models are trained on a dataset without any consideration of the data bias, the model will make decision in an unwanted way which could be highly unfair to an unprivileged group of the society. In short, major contributions of the paper are as follows, 1. We show that public speaking ratings can be biased depending on the race and gender of a speaker. We utilize the state-of-the-art fairness measuring metric to identify the biases present in the TED talk public speaking rating. 2. We propose a systematic procedure to detect unfairess in the TEDTalk public speaking ratings. This method can be adopted by any machine learning practitioner or data scientist at industry who are concerned with fairness in their datasets or machine learning models. Related Works ============= With the increased availability of huge amount of data, data-driven decision making has emerged as a fundamental practice to all sorts of industries. In recent years, data scientists and the machine learning community put conscious effort to detect and mitigate bias from data sets and respective models. Over the years, researchers have used multiple notions of fairness as the tools to get rid of bias in data that are outlined below: - ‘individual fairness’, which means that similar individuals should be treated similarly [@dwork2012fairness] - ‘group fairness’, which means that underprivileged groups should be treated same as privileged groups [@pedreschi2009measuring; @pedreshi2008discrimination]. - ‘fairness through awareness’, which assumes that an algorithm is fair as long as its outcome or prediction is not dependent on the use of protected or sensitive attributes in decision making [@grgic2016case]. - ‘equality of opportunity’, mainly used in classification task which assumes that the probability of making a decision should be equal for groups with same attributes [@hardt2016equality]. - ‘counterfactual fairness, very close to equality of opportunity but the probability is calculated from the sample of counterfactuals [@russell2017worlds; @kusner2017counterfactual] which ensures that the predictor probability of a particular label should be same even if the protected attributes change to different values . The fairness measures mentioned above can be characterized as both, manipulation to data and implementation of a supervised classification algorithm. One can employ strategies of detecting unfairness in a machine learning algorithm, observe [@zliobaite2015survey] and removing them by, - Pre-processing: this strategy involves processing the data to detect any bias and mitigating unfairness before training any classifiers [@calmon2017optimized; @kamiran2012data]. - In-processing: this technique adds a regularizer term in the loss function of the classifier which gives a measurement of the unfairness of the classifier [@calders2010three; @kamishima2011fairness]. - Post-processing: this strategy manipulates predictor output which makes the classifier fair under the measurement of a specific metric [@hardt2016equality]. For our analysis, we follow this well established paradigm and use an open-source toolkit AIF360 [@aif360-oct-2018] to detect and mitigate bias present in the data set and classifiers at all three stages: the pre-processing, the in-processing and the post-processing step. Data Collection =============== We analysed the TedTalk data collected from the [ted.com](ted.com) website. We crawled the website and gathered information about TedTalk videos which have been published on the website for over a decade (2006-2017). These videos cover a wide range of topics, from contemporary political, social issues to modern technological advances. The speakers who delivered talks at the TedTalk platform are also from a diverse background; including but not limited to, scientists, education innovators, celebrities, environmentalists, philosophers, filmmakers etc. These videos are published on the [ted.com](ted.com) website and are watched by millions of people around the world who can give ratings to the speakers. The rating of each talk is a collection of fourteen labels such as beautiful, courageous, fascinating etc. In this study we try to find if there is any implicit bias in the rating of the talks with respect to the race and gender of the speaker. Some properties of the full dataset is given in table \[tab:datasize\]. Each viewer can assign three out of fourteen labels to a talk and we use the total count for each label of rating for our analysis. In figure \[fig:avg\_rating\], average number of ratings in each of the fourteen categories is shown as a bar plot. Our preliminary observation reveals some disparities among the rating labels e.g. the label ‘inspiring’ has significantly higher count than other labels. Property Quantity ---------------------------------- -------------- Total number of Talks 2,383 Average number of views per talk 1,765,071 Total length of all talks 564.63 Hours Average rating labels per talk 2,498.6 : TED talk Dataset Properties: Information about the TED talk videos that are used in our method of detecting unfairness[]{data-label="tab:datasize"} ![Average number of ratings per video in each of the fourteen categories in the TedTalk dataset. observe that the the positive ratings outnumbers the negative rating on average which inform us about the pattern of the viewer’s rating. Viewer’s are more inclined to give positive ratings. []{data-label="fig:avg_rating"}](figures/rating_barplot.pdf){width="1.\columnwidth"} Data Annotation and Normalization --------------------------------- Our raw data includes information about the topic of the talk, number of views, date of publications, the transcripts used by the speakers, count of rating labels given by the viewers etc. However, this raw data does not come with the protected features such as gender and race of the speakers. We identified gender and race as potential sensitive attributes from our preliminary analysis. Thereby, to annotate the data for these protected features, we used amazon mechanical turk. We assigned three turkers for each talk, and annotated the race and gender of the speaker. To estimate the reliability of the annotation, we performed the Krippendorff’s alpha reliability test [@krippendorff2018content] on the annotations. We found a value of 93% for the Krippendorff’s alpha on our annotations. If there is a disagreement among the annotations of different turkers, we take the majority vote whenever possible or, we manually investigate to do the annotations. For gender, we used three categories: male, female, others, and for race, we used four categories, White, African American or Black, Asian, others. In our analysis, we used the total number of views $(V)$, the transcripts of the talks $(T)$ and the rating $(Y)$ given by the viewers. For preprocessing we employ the following steps, - First we normalize the number of views using the min-max normalization technique. $$V = \frac{V-V_{\min}}{V_{\max}-V_{\min}}$$ - For handling the transcripts, we utilize a state-of-the-art text embedding method ‘doc2vec’ as described in [@le2014distributed] - We also scale the rating $(Y)$ of each video. Note that $Y$ consists of fourteen numbers $(y_1,\cdots, y_{14})$ which represents the number of fourteen rating categories. We scale it as, $$Y_{\textrm{scaled}} = \frac{Y}{\sum_{i=1}^{14}y_i}$$ Finally, to train our classification model, we binarize our rating lables with a threshold of the median across all talks. This means that for a given talk the rating label beautiful can be 0 or 1, courageous label can be 0 or 1 and so on for all fourteen possible lables per talk. Here 0 indicates the label does not hold and 1 indicates that the label holds true for the respective talk. Gender Count Race Count ------------ ----------- ---------- ----------- Female 768 White 1901 Male 1596 Asian 210 Other 19 Black 169 Other 103 : Total count of protected attributes (race and gender) in the TedTalk dataset. One preliminary obeservation is white male outnumbers all other groups by a huge margin, also third gender speakers are underrepresented in the data set.[]{data-label="tab:protected_attrinutes"} Observation in Data =================== Before we train our machine learning models for predicting TedTalk rating, we performed several exploratory analysis on the data set. First, we counted the total number of different gender and races of the speakers, see Table \[tab:protected\_attrinutes\]. We notice that there is a significant imbalance in the gender and race count. The count of ‘white male’ speakers outnumbers all the other groups combined. Also, the number of speakers from the third gender community is very small which shows an noteworthy imbalance in the dataset.\ ![image](figures/key_race.pdf){width="1.\linewidth"} ![image](figures/key_gender.pdf){width="1.\linewidth"} There is also a clear and visible difference in the distribution of the ratings for different groups corresponding to the protected attributes: race and gender. Figure \[fig:race\_smooth\_hist\] and figure \[fig:gender\_smooth\_hist\] represents smoothed histogram across different groups for several ratings where we observed substantial difference in the distribution. Interestingly we find one expected bias and one counter intuitive bias when considering ratings based on race. The expected bias is that talks of white speakers are rated to be beautiful and courageous with high confidence as compared to speakers of other races, the sharp blue distributions in top row of Figure \[fig:race\_smooth\_hist\] are indicative of that. The wider green and red distributions on the top row of Figure \[fig:race\_smooth\_hist\] confirm that the society is more confused and less confident while rating the talk of a non-white speaker as beautiful or courageous. On the other hand, counter intuitively, we find that viewers rate talks of speakers of other race as more fascinating than other races (sharpness of red distribution in \[fig:race\_smooth\_hist\] in bottom row, left panel). Though counter intuitive, such bias is also not acceptable and clearly not fair. This highlights the diversity and difficulty of the issues of fairness and bias, since bias may not always stand out to be against the “expected” unprivileged class. Our work hence highlights the need to be careful when accounting for fairness as in fair society all types of biases should be removed.\ Furthermore we also looked at the way viewers rate speakers based on gender. Even under this category we observed that male speakers are confidently rated to have given beautiful and courageous talks when compared to any other gender (sharpness, and less width of blue distributions in figure \[fig:gender\_smooth\_hist\] indicates that). Though the confidence and tendency of rating in favor of male speakers drops substantially under informative and inspiring category (comparable width of green blue and red distributions) but is still slightly more than that for the speakers that are female (bi-modality of the distribution is an indication of confused rating behaviour) and of other gender. Besides highlighting the prevalence of biases with respect to gender and race, these results point out the need consider both expected and unexpected biases in data.\ ![image](figures/correlation_heatmap.png){width="1.\linewidth"} To dive deep into the nature of prevalent bias with respect to gender of a speaker we further divided the rating labels into positive and negative, where positive includes say, “beautiful", “courageous" etc. and negative includes say, “confusing", “unconvincing" etc. Now if we carefully notice \[fig:correlation\], we will see that there is lot more structure in ratings for male and female speakers and a lot more variability in the matrix for other speakers when positive and negative rating labels are considered. Note that, in our dataset a viewer can choose to rate with any of three possible labels. The chaos in the third matrix (rightmost) of figure \[fig:correlation\] indicates that, for speakers of other genders, viewers tend to choose a mix of positive and negative labels. However for male and female speakers viewers are more sure and choose either all positive or all negative labels depending on their liking of the talk. This indicates that in general there is extreme confusion among viewers about their decision of whether or not they like a talk given by a speaker of other gender. This can also be identified as an indicator for the viewer’s double minded intention when they rate those talks. Methodology =========== In this section we will describe the methods we used to design a bias free rating prediction. We explored all three methods as suggested in [@d2017conscientious], 1) Pre-Processing, 2) In-Processing and 3) Post Processing. We utilized the aforementioned toolkit AIF360 to implement these methods. This is explained in Figure \[fig:pipeline\] which is adopted from [@aif360-oct-2018].For each of the three steps, we quantify the amount of fairness with respect to a well defined metric, in this case ‘Disparate Impact’ [@biddle2006adverse]. In our TedTalk dataset, the fourteen rating categories are transformed to binary labels as described in Data section, i.e the category ’beautiful’ can be 0 or 1 depending on if it was rated true or not. Using these binary labels we calculate the disparate impact in the rating of a video. Disparate impact can be understood with an example, let’s say we consider the rating category ‘beautiful’. For female speakers, we can estimate how likely are they to get a beautiful rating. Disaprate impact compares such probabilities for all possible groups like male with female, female with other gender, male with other gender and so on. Disparate impact equals to 1 identifies as a fair case since it means both groups that are compared are equally likely to be rated beautiful. ![image](figures/pipeline.pdf){width="1.\linewidth"} Pre Processing -------------- As a first step we modify the original data before training the rating classifier. We then feed the modified and bias free data to the classifier for training as in \[fig:pipeline\] top row. The main driving force for this step is the fact that raw data is inherently biased in almost all problems pertaining to social science. The pre-processing step attempts to make the data bias free, so that the classifier trained with this unbiased data makes fair predictions. When we look into our data in details, we see from Table \[tab:protected\_attrinutes\] that the there is a strong imbalance of bias in the data with respect to race and gender. We observe that our data has a stronger bias related to race than for gender. So with the goal to make fair predictions, we first preprocess the data and attempt to remove the dominant bias with respect to race.\ There are several ways to remove bias from the training data before feeding it to the classifier [@calmon2017optimized; @feldman2015certifying; @kamiran2012data; @zemel2013learning]. We have chosen the disparate impact remover method [@feldman2015certifying] as the metric we have used is disparate impact metric. We use the disparate impact remover from AIF360 for this purpose. ![image](figures/3comparison_race_fixed.png){width="0.7\linewidth"} The result of the preprocesing step is shown in Figure \[fig:di\_remover\]. This figure is showing the disparate impact across all rating labels of the dataset. We computed the disparate impact metric for the original label of the dataset and observed that it is far from 1 for most of the rating categories as shown by the blue bars. This exhibits the amount of unfairness present in the original dataset. The huge existing bias in original data highlights the need to design a fair predictor of the rating. Hence, we trained a logistic regression model with both the original dataset and the pre-processed dataset. We observed that prediction on preprocessed data has disparate impact closer to 1 (Figure \[fig:di\_remover\]) than when it is trained with the orginal biased data. Our result hence shows that it is important to identify the cause of dominant bias in the data and remove it by preprocessing to gain substantial improvement in fair predictions. In Processing ------------- In the last section we showed that it is possible to build a fair model by removing the bias present in the original data before training a classification model. However it is not always possible to do preprocessing, because there may not be access to the original data or it might be hard to identify the cause of bias in the data or it may be time consuming to re-annotate data and so on. Under such scenarios the solution is to design a fair classifier that uses a fair algorithm instead. So this allows us to still train the model with a biased dataset as input but the predictions employed by the fair algorithm are fair. The algorithm and the classifier in this case identifies the presence of bias in the dataset used for training and adjusts appropriately (based on the amount of bias) while predicting corresponding labels. Examples of such in processing techniques for bias mitigation can be found in [@art2018; @kamishima2012fairness; @zhang2018mitigating]. For example, in our case, we hope that even if there is some bias in our dataset in favor of male, our rating classifier will make sure it accounts for that and weighs females and speakers of other genders equally to get rid of such unwanted unfairness. This will then generate fair rating predictions.\ For the TED talk data we have used the “Prejudice Remover" to do in processing [@kamishima2012fairness]. The intuition is that we attempt to make the dependence of rating on sensitive attributes as strong as the nonsensitive attributes. For example, assume that female speaker is unprivileged and male speaker is privileged, that is ratings are heavily biased by male speakers in the real data. In that case the goal of the classifier and the learning algorithm is to make sure that rating predictions depend equally on male and female speaker and is not strongly influenced by male speakers only instead even if that is the case in the biased data.\ The result of in-processing is shown in Figure \[fig:inproc\]. This figure is showing the disparate impact across all rating labels of the dataset. Similar to the pre-processing part we computed the disparate impact metric (in this case we compute the metric for gender bias) for the original label of the dataset and observed that it is far from 1 for most of the rating categories as shown by the blue bars,. This exhibits the amount of unfairness present in the original dataset w.r.t gender. We then trained a logistic regression model and a Prejudice Remover model [@kamishima2012fairness] with the original dataset. We observed that prediction of the Prejudice Remover model has disparate impact closer to 1 (Figure \[fig:di\_remover\]) than the prediction of the logistic regression model for most of the categories. ![image](figures/male_female_gender_other_pr.pdf){width="0.7\linewidth"} Post Processing --------------- We tried many popular methods [@hardt2016equality; @pleiss2017fairness] available in literature to employ post processing in our data. However, none of the methods gained significant improvement in making fair predictions. This highlights two important issues: 1) Nature of bias strongly drives what technique works best in terms of making fair predictions, in our case gender and race bias cannot be removed by just post processing technique 2) Not all types of dataset can used to make fair prediction with a fixed processing technique, in our case, as shown, post processing is unsuitable for the TED talk data videos. Conclusion from analysis ------------------------ We have employed a principled technique for removing bias and making fair rating predictions for TED talk videos. We first identify the dominant cause of the bias (in our case it is racial bias) in the data and remove it by pre processing, we then remove the other non-dominant causes of bias (gender bias in our case) by in-processing and we show that post processing does not make any improvement in fair predictions for our data. This highlights that choice of processing to achieve fair predictions heavily depends of types of bias and the dataset being used. This establishes the need to explore all possible biases and all possible techniques to obtain best results in terms of fairness. Discussion ========== Ensuring fairness in a machine learning model is a complex, multi-faceted issue which depends not only on the challenges presented by the immediate task but also on the definition of fairness that is in use. Unfairness or bias may arise at any stage of a machine learning pipeline. Thus, while developing machine learning algorithms which can make fair predictions for a classification task is important, creating a data collection process that is free from bias is essential for the model’s ultimate success. For example, Holstein et al. [@holstein2019improving] show that commercial machine-learning product teams express the need for tools that can actively guide the data collection process to ensure fairness in the downstream machine learning task. In our study, we demonstrate a systematic method of detecting bias using the AIF360 toolkit and show that various bias mitigation techniques can be applied to our data set at various stages of a machine learning pipeline (i.e. pre-processing, in-processing, post-processing). This procedure can be followed by any machine learning practitioner or data scientist who has to deal with real world data to make machine learning models for decision making.\ We have applied a state of the art fairness measuring metric ‘Disparate impact’ to a new and diverse dataset ‘TedTalk’, revealing sharp differences in the perception of speakers based on their race and gender. Based on our findings, we demonstrate that in TED Talk ratings, viewers rate ‘white male’ speakers confidently, while all other speakers are rated in a weaker manner. On the other hand, viewers rating shows a great deal variability when rating any other groups combined.\ Our study shows that detecting and erasing bias in the data collection process is essential to resolving issues related to fairness in AI. While the existing literature about fairness in AI [@dwork2012fairness; @pedreschi2009measuring; @russell2017worlds] have focused almost exclusively on detecting bias in specific domains such as recidivism prediction, admission decision-making, face detection etc, the analysis of implicit bias and . In all these scenarios, researchers come up with a quantifiable measurement of fairness and train the model which ensures the model is fair. However, in a more diverse domain where the users and system interacts in a complex way (e.g. recommendation system, chatbots etc), there is a lot of work left to do to detect and erase unfairness. While we have evaluated the Ted Talk dataset with specific metrics that are appropriate to it, there is an urgent need to develop metrics that are adapted just as uniquely to the needs of each dataset and its possible implementations. Identifying different kinds of social bias that affects in applications directly engaging with social influence and opinion formation is absolutely necessary for creating fair and balanced artificial intelligence systems.	ArXiv
--- abstract: 'In this paper I describe the history of the surreal trajectories problem and argue that in fact it is not a problem for Bohm’s theory. More specifically, I argue that one can take the particle trajectories predicted by Bohm’s theory to be the actual trajectories that particles follow and that there is no reason to suppose that good particle detectors are somehow fooled in the context of the surreal trajectories experiments. Rather than showing that Bohm’s theory predicts the wrong particle trajectories or that it somehow prevents one from making reliable measurements, such experiments ultimately reveal the special role played by position and the fundamental incompatibility between Bohm’s theory and the relativity.[^1]' --- =10000 The Persistance of Memory: Surreal Trajectories in Bohm’s Theory Jeffrey A. Barrett Bohm’s theory[^2] has become increasingly popular as a nonrelativistic solution to the quantum measurement problem. It makes the same empirical predictions for the statistical distribution of particle configurations as the standard von Neumann-Dirac collapse formulation of quantum mechanics whenever the latter makes unambiguous predictions. Bohm’s theory also treats measuring devices exactly the same way it treats other physical systems. The quantum-mechanical state of a system always evolves in the usual linear, deterministic way, so one does not encounter the problems that arise in collapse formulations of quantum mechanics when one tries to stipulate the conditions under which a collapse occurs. And Bohm’s theory does not require one to postulate branching worlds or disembodied minds or any of the other extravagant assumptions that often accompany no-collapse formulations of quantum mechanics. While Bohm’s theory avoids many of the problems associated with other formulations of quantum mechanics, it does have its own problems. One problem, it has been argued, is that the particle trajectories it predicts are not the real particle trajectories. This is the surreal trajectories problem. If Bohm’s theory does in fact make false predictions concerning particle trajectories, then this is presumably a serious problem. I will argue, however, that there is no reason to suppose that Bohm’s theory makes false predictions concerning the trajectories of particles. Indeed, I will argue that a good position measuring device need never be mistaken concerning the actual position of a particle [at the moment that the particle’s position is in fact recorded]{}. While surreal trajectories are not a problem for Bohm’s theory, the way that it accounts for the results of the surreal trajectories experiments reveals the sense in which it is fundamentally incompatible with relativity, and this is a problem. On Bohm’s theory the quantum-mechanical state $\psi$ evolves in the usual linear, deterministic way, but one supposes that every particle always has a determinate position and follows a continuous, determinsitic trajectory. The motion of a particular particle typically depends on the evolution of $\psi$ and the positions of other (perhaps distant) particles. The particle motion is described by an auxiliary dynamics, a dynamics that supplements the usual linear quantum dynamics. In its simplest form, what one might call the [minimal version]{} (the version of the theory described by Bell 1987, 127), Bohm’s theory is characterized by the following basic principles: 1\. State Description: The complete physical state at a time is given by the wave function $\psi$ and the determinate particle configuration $Q$. 2\. Wave Dynamics: The time evolution of the wave function is given by the usual linear dynamics. In the simplest case, this is just Schrödinger’s equation $$i \hbar \frac{\partial \psi}{\partial t} = \hat{H} \psi$$ More generally, one uses the form of the linear dynamics appropriate to one’s application (as in the spin examples discussed below). 3\. Particle Dynamics: The particles move according to $$\frac{d Q_k}{dt} = \frac{1}{m_k} \frac{\mbox{Im}(\psi^* \nabla_k \psi)}{\psi^* \psi} \mbox{ evaluated at }Q$$ where $m_k$ is the mass of particle $k$ and $Q$ is the current particle configuration. 4\. Distribution Postulate: There is a time $t_0$ when the epistemic probability density for the configuration $Q$ is given by $\rho(Q, t_0)= \|\psi(Q, t_0)\|^2$. If there are $N$ particles, then $\psi$ is a function in $3N$-dimensional configuration space (three dimensions for the position of each particle), and the current particle configuration $Q$ is represented by a single point in configuration space (in configuration space a single point gives the position of every particle). Again, each particle moves in a way that depends on its position, the evolution of the wave function, and the positions of the other particles. Concerning how one should think of the role of the wave function in Bohm’s theory, John Bell once said that “[no one can understand this theory until he is willing to think of $\psi$ as a real objective field rather than just a ‘probability amplitude.’ Even though it propagates not in $3$-space but in $3N$-space]{}” (1987, 128). While the ontology suggested by Bell here is at best puzzling, the practical idea behind it is a good one: The best way to picture what the particle dynamics does is to picture the point representing the $N$-particle configuration being carried along by the probability currents generated by the linear evolution of the wave function $\psi$ in configuration space. Once one has this picture firmly in mind one will understand how Bohm’s theory accounts for quantum-mechanical correlations in the context of the surreal-trajectory experiments and the sense in which the theory is fundamentally incompatible with relativity. Since the total particle configuration can be thought of as being pushed around by the probability current in configuration space, the probability of the particle configuration being found in a particular region of configuration space changes as the integral of $\|\psi\|^2$ over that region changes. More specifically, the continuity equation $$\frac{\partial \rho}{\partial t} + \mbox{div}(\rho v^\psi) = 0$$ is satisfied by the probability density $\rho=\|\psi\|^2$. And this means that if the epistemic probability density for the particle configuration is ever $\|\psi\|^2$, then it will always be $\|\psi\|^2$, unless one makes an observation. That is, if one starts with an epistemic probability density of $\rho(t_0)=\|\psi(t_0)\|^2$, then, given the dynamics, one should update this probability density at time $t$ so that $\rho(t)=\|\psi(t)\|^2$. And if one makes an observation, then the epistemic probability density will be given by the system’s [effective]{} wave function, the component (in the configuration space representation) of the total wave function that is in fact responsible for the post-measurement time evolution of the system’s configuration. The upshot is that if the distribution postulate is ever satisfied, then the most that one can learn from a measurement is the wave packet that the current particle configuration is associated with and the epistemic probability distribution for the actual configuration over this packet.[^3] This is why Bohm’s theory makes the same statistical predictions for particle configurations as the standard collapse formulation of quantum mechanics. While it makes the same statistical predictions as the standard formulation of quantum mechanics, Bohm’s theory is deterministic. More specifically, given the energy properties of a simple closed system, the complete physical state at any time (the wave function and the particle configuration) fully determines the physical state at all other times.[^4]. It follows that, given a particular evolution of the wave function, possible trajectories for the configuration of a system can never cross at a time in configuration space. And this feature of Bohm’s theory will prove important later. Another feature of Bohm’s theory that will prove imporant later is the special role played by position in accounting for our determinate measurement results. In order for Bohm’s theory to explain why we get the determinate measurement records that we do (which is presumably a precondition for it counting as a solution to the measurement problem), one must suppose, as a basic interpretational principle, that, given the usual quantum mechanical state, making particle positions determinate provides determinate measurement records. Since particle positions are always determinate on Bohm’s theory, this would guarantee determinate measurement records. And, at least on the minimal version of Bohm’s theory, position is the only determinate, noncontextual property that could serve to provide determinate measurement records.[^5] The distinction between noncontextual and contextual properties deserves some explanation. Whether a system is found to have a particular contextual property or not typically depends on how one measures the property: one might get the result “Yes” if the contextual property is measured one way and “No” if it is measured another. Consequently, contextual properties are not intrinsic properties of the system to which they are typically ascribed. One might say that contextual properties serve to prop up our talk of those properties that we are used to talking about but which arguably should not count as properties at all in Bohm’s theory. While the language of contextual properties provides a convenient (but often misleading!) way of comparing the predictions of Bohm’s theory with the predictions of other physical theories, the predictions of Bohm’s theory are always ultmately just predictions about the evolution of the wavefunction and the positions of the particles relative to the wavefunction. The upshot of all this is just that position relative to the wave function, or more precisely configuration relative to the wave function, is ultimately the only property that one can appeal to in the minimal version of Bohm’s theory to explain how it is that we end up with the determinate measurement records we do. And this means that [for an interaction to count as a measurement, it must produce a record in terms of the position of something]{}—it must correlate the position of something with some aspect of the quantum-mechanical state of the system being measured. So in order to explain our determinate measurement records on Bohm’s theory one must suppose that all measurement records are ultimately position records, records represented in the relationship between the particle configuration and the wave function. And since Bohm’s theory predicts the right quantum statistics for particle positions relative to the wave function, it predicts the right quantum statistics for our measurement records.[^6] In their 1992 paper Englert, Scully, Süssman, and Walther (ESSW) argued that the trajectories predicted by Bohm’s theory are not the real trajectories followed by particles, but rather are “surreal”. The worry is that the observed trajectories of particles are not the trajectories that the particles actually follow in Bohm’s theory. And if our observations are reliable and if Bohm’s theory predicts the wrong particle trajectories, then this is presumably a problem for the theory. ESSW describe the surreal trajectories problem in the context of a two-path, delayed-choice interference experiment. John Bell (1980, reprinted in 1987) was perhaps the first to consider such an experiment in the context of Bohm’s theory. Consider an experiment where a spin-$1/2$ particle $P$ starts at region $S$ in a $z$-spin up eigenstate, has its wave packet split into an $x$-spin up component that travels from $A$ to $A'$ and an $x$-spin down component that travels from $B$ to $B'$.[^7] \[Figure 1: Crossing-Paths Experiment\] The wave function evolves as follows: Initial state: $$\|{\uparrow}_z\rangle_P\|S\rangle_P = 1/\sqrt{2} (\|{\uparrow}_x\rangle_P + \|{\downarrow}_x\rangle_P)\|S\rangle_P$$ After the initial wave packet splits: $$1/\sqrt{2} (\|{\uparrow}_x\rangle_P\|A\rangle_P + \|{\downarrow}_x\rangle_P\|B\rangle_P)$$ Final state: $$1/\sqrt{2} (\|{\uparrow}_x\rangle_P\|A'\rangle_P + \|{\downarrow}_x\rangle_P\|B'\rangle_P)$$ Bell explained that if one measures the properties of $P$ in region $I$, then one would observe interference phenomena (in this experiment, for example, one would observe $z$-spin up with probability one). The observation of interference phenomena is usually taken to entail that $P$ could not have followed path $A$ and could not have followed path $B$ since, in either case, the probably of observing $z$-spin up in $I$ would presumably be $1/2$ (as predicted by the standard collapse formulation of quantum mechanics). In such a situation, one would say, on the standard view, that $P$ followed a [superposition]{} of the two trajectories (which, on the standard interpretation of states is supposed to be neither one nor the other nor both trajectories). But according to Bohm’s theory, $P$ determinately follows one or the other of the two trajectories: that is, it either determinately follows $A$ or it determinately follows $B$. On Bohm’s theory, one might say that the interference effects that one observes at $I$ are the result of the wave function following both paths. If we do not observe the particle in region $I$, then $P$ will arrive at one of the two detectors to the right of the interference region: either the one at $A'$ or the one at $B'$. If it arrives at $A'$, then one might suppose that the particle traveled path $A$; and if it arrives at $B'$, then one might suppose that it traveled path $B$. But such inferences do not work in the standard collapse formulation of quantum mechanics, where (according to the standard eigenvalue-eigenstate link) under these circumstances $P$ would have traveled a superpostion of the two paths. And such inferences do not work in Bohm’s theory either, but for a very different reason. In Bohm’s theory the particle really does travel one or the other of the two paths, it is just that its trajectory is not what one might at first expect. In figuring out what trajectory Bohm’s theory predicts, the first thing to note is that, by symmetry, the probability current across the line $L$ is always zero.[^8] This means that if $P$ starts in the top half of the initial wave packet, then it must move from $S$ to $A$ to $I$ to $B'$; and if it starts in the bottom half of the initial wave packet, then it must move from $S$ to $B$ to $I$ to $A'$. That is, whichever path $P$ takes, Bohm’s theory predicts that it will [bounce]{} when it gets to region $I$—in order to follow either trajectory, the particle $P$ must accelerate in the field-free region $I$. Concerning this odd bouncing behavior Bell says that “it is vital here to put away the classical prejudice that a particle moves in a straight path in ‘field free’ space” (1987, 113). But certainly, one might object, this is more than a prejudice. After all, this particle bouncing nonsense is a direct violation of the conservation of momentum, and we have very good empirical reasons for supposing that momentum is conserved. Isn’t this alone reason enough to dismiss Bohm’s theory? Put another way, whenever we observe which path the particle in fact travels, if we find it at $A'$, then we also observed it traveling path $A$ and if we find it at $B'$, then we also observed it traveling path $B$. That is, whenever we make the appropriate observations, we never observe the crazy bouncing behavior (or any of the other violations of the conservation of momentum) predicted by Bohmian mechanics. This puzzling situation is the basis for ESSW’s surreal trajectories argument. The argument goes something like this: [Assumption 1]{} (explicit): Our experimental measurement records tell us that in a two-path interference experiment like that described above each particle either travels from $A$ to $A'$ or from $B$ to $B'$; that is, they never bounce. [Assumption 2]{} (implicit): Our measurement records reliably tell us where a particle is at the moment the record is made. [Assumption 3]{} (implicit): One can record which path a particular particle takes without breaking the symmetry in the probability currents that prevents the particle from crossing the line $L$. [Conclusion]{}: The trajectory predicted by Bohm’s theory, where the particle bounces, cannot be the particle’s actual trajectory; that is, Bohm trajectories are not real, they are “surreal.” And if the trajectories predicted by Bohm’s theory are not the actual particle trajectories, then Bohm’s theory is false, and this constitutes very good grounds for rejecting it. Dürr, Fusseder, Goldstein, and Zanghi (DFGZ) immediately responded to defend Bohm’s theory against the surreal trajectories argument: > In a recent paper \[ESSW (1992)\] it is argued that dispite its many virtues—its clarity and simplicity, both conceptual and physical, and the fact that it resolves the notorious conceptual difficulties which plague orthodox quantum theory—BM \[Bohmian mechanics\] itself suffers from a fatal flaw: the trajectories that it defines are “surrealistic”. It must be admitted that this is an intriguing claim, though an open minded advocate of quantum orthodoxy would presumably have preferred the clearer and stronger claim that BM is [incompatible]{} with the predictions of quantum theory, so that, despite its virtues, it would not in fact provide an explanation of quantum phenomena. The authors are, however, aware that such a claim would be false. (1993, 1261) And since Bohm’s theory makes the same predictions as the standard theory of quantum mechanics, DFGZ argue that ESSW cannot possibly provide, as ESSW describe it, “an experimentum crucis which, according to our quantum theoretic prediction, will clearly demonstrate that the reality attributed to Bohm trajectories is rather metaphysical than physical.” And with this DFGZ dismiss ESSW’s argument against Bohmian mechanics: > On the principle that the suggestions of scientists who propose pointless experiments cannot be relied upon with absolute confidence, with this proposal the \[ESSW\] paper self-destructs: The authors readily agree that the “quantum theoretical predictions” are also the predictions of BM. Thus they should recognize that the \[experimental\] outcome on the basis of which they hope to discredit BM is precisely the outcome predicted by BM. Under the circumstances it would appear prudent for the funding agencies to save their money! (1261) DFGZ conclude their defense of Bohm’s theory by making a point about the theory-ladenness of talk of particle trajectories and a point about the theory-ladenness of observation itself. But we will return to these two (important) points later, when we have the conceptual tools hand to make sense of them (Section 5). In their reply to DFGZ’s comment, ESSW want to make it perfectly clear that they did not anywhere conceed that Bohm’s theory had “many virtues” nor did they admit that the orthodox formulation of quantum mechanics was “plagued by notorious conceptual difficulties.” But, for their part, ESSW do seem to conceed, as DFGZ insisted, that Bohmiam mechanics makes the same empirical predictions as standard quantum mechanics: “Nowhere did we claim that BM makes predictions that differ from those of standard quantum mechanics” (1263).[^9] Rather than argue that Bohm’s theory made the wrong empirical predictions, ESSW claim that the purpose of their original paper was “to show clearly that the interpretation of the Bohm trajectory—as the real retrodicted history of the \[test particle that travels through the interferometer\]—is implausible, because this trajectory can be macroscopically at variance with the detected, actual way though the interferometer” (1263). This last clause identifies the detected path with the actual path traveled by the test particle. This is their (implicit) assumption that particle detectors would be reliable (in a perfectly straightforward way) on the delayed-choice interference experiments that they discuss. ESSW conclude, “Irrespective of what can be said in addition, we think that we have done a useful job in demonstrating just how artificial the Bohm trajectories can be” (1264). Again, ESSW’s claim is not that Bohm’s theory makes the wrong empirical predictions nor it is that the theory is somehow logically inconsistent; rather, they argue (on the implicit assumption that our particle detectors reliably tell us where particles are) that Bohm’s theory makes the wrong predictions for the actual motions of particles—that the predicted particle trajectories are “artificial,” “metaphysical,” and, at best, “implausible.” While I agree with DFGZ that surreal trajectories are not something that a proponent of Bohm’s theory should worry about, the full story is a bit more involved than the sketch given in their comment on ESSW’s paper. In order to get everything straight, let’s return to Bell’s original analysis of the delayed-choice interference experiment in the context of Bohm’s theory. Bell’s analysis of the delayed-choice interference experiment provides a good first step in explaining why conservation-of-momentum-violating “surreal” trajectories do not pose a problem for Bohm’s theory. While Bohm’s theory does indeed predict that momentum (in the usual sense) is not conserved in experiments like that described above, Bell explained why one would never detect violations of the conservation of momentum. The short story is this: while the [ actual]{} momentum (mass times particle velocity) is typically not conserved in Bohm’s theory, the [measured]{} momentum (as expressed by the results of what one would ordinarily take to be momentum measurements) is always conserved. In order to detect a momentum-violating bounce in an experiment like that described above, one would have to perform [two]{} measurements: one to show which path the particle travels, ($A$ or $B$) and another to show where the particle ends up ($A'$ or $B'$). One might then try to show that a particle that travels path $A$, say, ends up at $B'$, and thus violates the conservation of momentum. But one will never observe such a bounce in Bohm’s theory [because measuring which path the particle follows will destroy the symmetry in the probability currents that generate the bounce]{}. That is, particles only exhibit their crazy bouncing behavior in Bohm’s theory when no one is looking! Suppose (following Bell 1980) that one puts a detector on path $B$ designed to correlate the position of a flag with the position of the test particle $P$ (see figure 2). More specifically, consider a single flag particle $F$ whose position (as represented by the quantum-mechanical state) gets correlated with the position of $P$ as follows: (1) if $P$ is in an eigenstate of traveling path $A$, then $F$ remains in an eigenstate of pointing at “No” and (2) if $P$ is in an eigenstate of traveling path $B$, then $F$ ends up in an eigenstate of pointing at “Yes”. That is, the detector is designed so that the position of $F$ will record the path taken by $P$. \[Figure 2: Experiment where $F$’s position is correlated with the position of $P$\] While this experiment may look like the earlier one, introducing such a detector requires one to tell a very different story than the one told without the detector. Given the nature of the interaction between $P$ and $F$ and the linearity of the dynamics, if $P$ begins in the $z$-spin up state (a superposition of $x$-spin eigenstates), then the effective wave function of the composite system would evolve as follows: Initial state: $$\|{\uparrow}_z\rangle_P\|S\rangle_P \|\mbox{``No''}\rangle_F = \|S\rangle_P\|\mbox{``No''}\rangle_F 1/\sqrt{2} (\|{\uparrow}_x\rangle_P + \|{\downarrow}_x\rangle_P)$$ $P$’s wave packet splits: $$\|\mbox{``No''}\rangle_F 1/\sqrt{2} (\|{\uparrow}_x\rangle_P\|A\rangle_P + \|{\downarrow}_x\rangle_P\|B\rangle_P)$$ $M$’s position is correlated with the position of $P$: $$1/\sqrt{2} (\|{\uparrow}_x\rangle_P\|A\rangle_P \|\mbox{``No''}\rangle_F + \|{\downarrow}_x\rangle_P\|B\rangle_P\|\mbox{``Yes''}\rangle_F)$$ The two wave packets appear to pass though each other in region $I$ (but they [miss]{} each other in configuration space!): $$1/\sqrt{2} (\|{\uparrow}_x\rangle_P\|I\rangle_P \|\mbox{``No''}\rangle_F + \|{\downarrow}_x\rangle_P\|I\rangle_P\|\mbox{``Yes''}\rangle_F)$$ Final state: $$1/\sqrt{2} (\|{\uparrow}_x\rangle_P\|A'\rangle_P \|\mbox{``No''}\rangle_F + \|{\downarrow}_x\rangle_P\|B'\rangle_P\|\mbox{``Yes''}\rangle_F)$$ Note that the position of $F$ does in fact reliably record where $P$ was when the position record was made. Because the wave function associated with the two possible positions for $F$ do not overlap in configuration space, the position correlation between $P$ and $F$ destroys the symmetry that prevents $P$ from crossing $L$. While the two wave packets both appear to pass through region $I$ at the same time, they in fact miss each other in configuration space. In order to see how $P$ and $F$ move, consider the evolution of the wave function and the two-particle configuration in configuration space. \[Figure 3: The Last Experiment in Configuration Space\] If the two-particle configuration starts in the top half of the initial wave packet (as represented in Figure 3), then $P$ would move from $S$ to $A$ to $I$ to $A'$ and $F$ would stay at “No”. If the configuration starts in the bottom half of the initial wave packet, then $P$ would move from $S$ to $B$ then $F$ would move to “Yes” then $P$ would move from $B$ to $I$ to $B'$. That is, regardless of where $P$ starts, it will pass though the region $I$ without bouncing. Moreover, $F$ will record that $P$ was on path $A$ if and only if $P$ ends up at $A'$ and that $P$ was on path $B$ if and only if $P$ ends up at $B'$. That is, if one makes a determinate record of $P$’s position before $P$ gets to $I$, then $P$ will follow a perfectly natural trajectory, and the record will be reliable. Again, recording the position of $P$ destroys the symmetry that prevents $P$ from crossing $L$. This experiment illustrates why a measurement record is reliable in Bohm’s theory whenever there is a strong correlation between the position of the system being observed and the position of the recording system. And since all measurements are ultimately position measurements on the minimal Bohm’s theory, one might simply conclude that all determinate records produced by strong correlations are reliable in Bohm’s theory and dismiss the surreal-trajectories problem as a problem that was solved by Bell before it was even posed by ESSW. This is not such a bad conclusion, but the right thing to say about surreal trajectories is slightly more subtle. Note that in order to tell a story like the one above, one must record the path taken by the test particle in terms of the position of something. Here the record is in terms of the position of the flag particle $F$. It is this position correlation that breaks the symmetry in the probability currents, which then allows the test particle $P$ to follow a momentum-conserving trajectory. All it takes is a strong position correlation with even a single particle. And it is this that makes the final position record reliable.[^10] So what would happen if one tried to record the position of $P$ in terms of some physical property other than position? This is something that is important to our making sense of the history of the surreal trajectories problem, but it is something that Bell did not consider. In order to avoid Bell’s (preemptive) dissolution of the surreal trajectories problem ESSW must have had in mind a different sort of which-path detector than the one considered by Bell. Indeed, the experiments that ESSW describe in their 1992 paper employ detectors that record the path followed by the test particle in the creation of photons. A proponent of Bohm’s theory might point out that since the theory is explicitly nonrelativistic and since the very statement of its auxillary dynamics requires there to be a fixed number of particles, these experiments are simply outside the domain of the theory. But perhaps it is possible to capture at least the spirit of ESSW’s experiments with experiments that are well within the domain of the minimal Bohm’s theory.[^11] Consider what happens when one [tries]{} to record $P$’s position in something other than position (one might naturally, and quite correctly, object that there is no other quantity in Bohm’s theory that one [could]{} use to record $P$’s position, but with the aim of trying to revive the surreal trajectories problem, read on). Suppose, for example, that one tries to record $P$’s position in a particle $M$’s $x$-spin: that is, suppose that the interaction between $P$ and $M$ is such that if $P$’s initial effective wave function were an $x$-spin up eigenstate, then nothing would happen to $M$’s effective wave function; but if $P$’s initial effective wave function were an $x$-spin down eigenstate, then the spin index of $M$’s effective wave function would be flipped from $x$-spin up to $x$-spin down (since $x$-spin is a contextual property in the minimal Bohm’s theory, the value of the $x$-spin record depends, as we will see, on how it is read, and one might thus, quite correctly, argue that it is not a record of the position of $P$ at all, but read on). In the standard von Neumann-Dirac collapse formulation of quantum mechanics (once a collapse had eliminated one term or the other of the correlated superposition!) one might naturally think of this interaction as recording $P$’s position in $M$’s $x$-spin. On this view, $M$ might be thought of as a sort of which-path detector. Continuing with the experimental set up, suppose further that $M$’s $x$-spin might then be converted to a position record by a detector with a flag particle $F$ designed to point at “No” if $M$ is in the $x$-spin up state and to point at “Yes” if $M$ is in the $x$-spin down state. The conversion of the $x$-spin record (though it will turn out that there is no determinate $M$-record [until after this conversion is made]{} in the delay-choice interference experiments!) to a position record here consists in correlating the position of $F$ with the $x$-spin of $M$ (as represented by the quantum-mechanical state). The idea is that if $P$ is in an eigenstate of traveling path $B$, say, then $M$ will record this fact by the $x$-spin index on its wave function being flipped to $x$-spin down, which is something that might then be converted into a record in terms of the position of $F$ through the interaction between $M$ and $F$. In this case, $F$ would move to record the measurement result “Yes”. \[Figure 4: One tries to record the position of $P$ in the $x$-spin of $M$\] The effective wave function of the composite system then evolves as follows: Initial state: $$\|{\uparrow}_z\rangle_P\|S\rangle_P \|{\uparrow}_x\rangle_M \|\mbox{``No''}\rangle_F = \|S\rangle_P \|{\uparrow}_x\rangle_M\|\mbox{``No''}\rangle_F 1/\sqrt{2} (\|{\uparrow}_x\rangle_P + \|{\downarrow}_x\rangle_P)$$ $P$ wave packet is split: $$\|{\uparrow}_x\rangle_M\|\mbox{``No''}\rangle_F 1/\sqrt{2} (\|{\uparrow}_x\rangle_P\|A\rangle_P + \|{\downarrow}_x\rangle_P\|B\rangle_P)$$ The $x$-spin component of $M$’s wave packet is correlated to the position of $P$’s: $$\|\mbox{``No''}\rangle_F 1/\sqrt{2} (\|{\uparrow}_x\rangle_P\|A\rangle_P\|{\uparrow}_x\rangle_M + \|{\downarrow}_x\rangle_P\|B\rangle_P\|{\downarrow}_x\rangle_M)$$ The two wave packets pass through each other in configuration space: $$\|\mbox{``No''}\rangle_F 1/\sqrt{2} (\|{\uparrow}_x\rangle_P\|I\rangle_P\|{\uparrow}_x\rangle_M + \|{\downarrow}_x\rangle_P\|I\rangle_P\|{\downarrow}_x\rangle_M)$$ Then they separate: $$\|\mbox{``No''}\rangle_F 1/\sqrt{2} (\|{\uparrow}_x\rangle_P\|A'\rangle_P\|{\uparrow}_x\rangle_M + \|{\downarrow}_x\rangle_P\|B'\rangle_P\|{\downarrow}_x\rangle_M)$$ Then the position of the $F$ is correlated to the $x$-spin component of $M$’s wave packet: $$1/\sqrt{2} (\|{\uparrow}_x\rangle_P\|A'\rangle_P\|{\uparrow}_x\rangle_M \|\mbox{``No''}\rangle_F + \|{\downarrow}_x\rangle_P\|B'\rangle_P\|{\downarrow}_x\rangle_M \|\mbox{``Yes''}\rangle_F)$$ Note that here the symmetry in the probability current that prevents $P$ from crossing $L$ is preserved. That is, $P$ bounces just as it did in the first experiment we considered. \[Figure 5: Last experiment in configuration space\] If the three-particle configuration begins in the top half of the initial wave packet (as represented in Figure 5), then $P$ will move from $S$ to $A$ to $I$ to $B'$ then, when $M$ and $F$ interact, $F$ will move to “Yes”. If the three-particle configuration begins in the bottom half of the initial wave packet, then $P$ will move from $S$ to $B$ to $I$ to $A'$ and, when $M$ and $F$ interact, $F$ will stay at “No”. That is, the final position of the $F$ will be at “No” if $P$ traveled along the lower path and it will be at “Yes” if $P$ traveled along the upper path. In other words, $F$’s final position does not tell us which path $P$ followed in the way that it was intended. One might naturally conclude that the which-path detector is fooled by the late measurement, and defend Bohm’s theory against ESSW by denying their implicit assumption that the which-path detectors are reliable.[^12] There is, however, another way of looking at a delayed-choice interference experiment where one tries to record the path in some property other than position. One might claim both that Bohm’s theory is true [and]{} that one’s detectors are perfectly reliable. This, it seems to me, is an option suggested by DFGZ’s discussion of the theory-ladenness of talk of trajectories and of observation itself near the end of their response to ESSW’s original paper. Given their contention that the experiments described by ESSW could provide no empirical reason for rejecting Bohmian mechanics, DFGZ ask the question “So what on earth is going on here?” > The answer appears to be this: The authors \[ESSW\] distinguish between the Bohm trajectory for the atom and the [detected]{} path of the atom. In this regard it would be well to bear in mind that before one can speak coherently about the path of a particle, detected or otherwise, one must have in mind a theoretical framework in terms of which this notion has some meaning. BM provides one such framework, but it should be clear that within this framework the \[test particle\] can be detected passing only through the slit through which its trajectory in fact passes. More to the point, within a Bohmian framework it is the very existence of trajectories wich allows us to assign some meaning to this talk about detection of paths. (1261–2) It seems that there are two points here. The first point concerns the theory-ladenness of talk about trajectories. On the orthodox formulation of quantum mechanics, there is no matter of fact at all concerning which path the test particle traveled since it simply fails to have any determinate position whatsoever before it is detected. Indeed, insofar as ESSW’s description of the surreal trajectories experments presupposes that there are determinate particle trajectories, they are presupposing something that is [incompatible]{} with the very quantum orthodoxy they seek to defend! The point here is that any talk of determinate trajectories is talk within a theory. A precondition of such talk is that one have a theory where there are determinate trajectories, a theory like Bohmian mechanics. DFGZ’s second point, if I understand it correctly, concerns the theory-ladenness of observation, but this will first require some clarification. Their claim that in Bohmian mechanics a test particle “can be detected passing only through the slit through which its trajectory in fact passes” suggests that they were considering only experiments like those in the last section where the which-path detector indicates in a perfectly straighforward way the path that the test particle in fact followed. But DFGZ do in fact grant that there are situations where Bohm’s theory predicts that a late observation of a which-path detector would find that the detector registers that the test particle traveled one path when it in fact traveled the other. They also grant that this is somewhat surprising. But they explain that “if we have learned anything by now about quantum theory, we should have learned to expect surprises!” And DFGZ maintain that even in such experiments the measurement performed by the which-path detector “can indeed be regarded as a measurement of which path the \[test particle\] has taken, but one that conveys information which contradicts what naively would have been expected.” DFGZ then draw the moral that “BM, together with the authors \[ESSW\] of the paper on which we are commenting, does us the service of making it dramatically clear how very dependent upon theory is any talk of measurement or observation” (1262). While it is not entirely clear what DFGZ have in mind, one way to read this is that, contrary to what is later argued by Dewdney, Hardy, and Squires (1993), DFGZ take the which-path detector to be perfectly reliable even in experiments where it “records” that the test particle traveled one path when it in fact (according to Bohm’s theory) traveled the other [once one understands what the detector is detecting]{}. On this reading, then, the point here is that since observation is itself a theory-laden notion, what one is detecting can only be determined in the context of a theory that explains what it is that one is detecting. But if this is what DFGZ had in mind, then what exactly does Bohm’s theory tell us that a which-path detector is detecting in the context of a late-measurement experiment? (And is it really possible to tell a plausible story where the detectors are perfectly reliable here?) Perhaps the easiest answer would be to insist that when one tries to record the path taken by the test particle in a property other than position (in a delayed-choice interference experiment), one’s which-path detector simply works in exactly the opposite way that one would expect. The detector is perfectly reliable—it is just that when it records that the test particle traveled path $A$, the detector record (under such circumstances) really [means]{}, according to Bohm’s theory, that the test particle in fact traveled path $B$; and, similarly, on this view a $B$ record [means]{}, according to Bohm’s theory, that the test particle traveled path $A$. It seems, however, that this cannot be quite right. When one tries to record the path that the test particle traveled in a property other than position (in the delayed-choice interference experiment), there is no determinate record whatsoever (on the minimal Bohm’s theory) before the test particle passes through the interference region $I$ because the which-path detector has not yet correlated the position of anything with the position of the test particle. The right thing to say, it seems to me, is that while the which-path detector does not detect anything before one correlates the position of the flag $F$ with $M$’s $x$-spin (on the minimal Bohm’s theory), whenever one makes a determinate record in Bohm’s theory using a device that induces a strong correlation between the measured position of the object system and the position that records the outcome, then that record will be perfectly reliable [at the moment the determinate record is made]{}. On this view, there is still a sense in which one can think of the detectors in the delayed-choice interference experiments as being perfectly reliable, but this will take some explaining. As DFGZ suggest, we naturally rely on our best physical theories to tell us what it is that our measuring devices in fact measure, so what does Bohm’s theory tell us about the [late-measurement]{} of the which-path detector in the delayed-choice interference experiment? Note, again, that there is no determinate record whatsoever before the late measurement. Also note that while the final position of $F$ does not tell us where $P$ [was]{} when $P$ interacted with $M$, it does reliably tell us where $P$ [is]{} at the moment that the x-spin correlation is converted into a determinate measurement record (when the position of $F$ is correlated with the $x$-spin of $M$): if one gets the result “No” ($x$-spin up), then the theory tells us that $P$ is currently associated with the $x$-spin up wave packet [wherever that wave packet may be]{}, and if one gets the result “Yes” ($x$-spin down), then it tells us that $P$ is currently associated with the $x$-spin down wave packet [wherever that wave packet may be]{}. So this is how it works. Since the only determinate noncontextual records in Bohm’s theory are records in terms of the position of something, there is, stictly speaking, no determinate record of $P$’s position until we convert the correlation between the position of $P$ and the $x$-spin of $M$ into a correlation between the position of $P$ and the position of $F$. And whenever this position correlation is made, we reliably, and [nonlocally]{}, generate a record of $P$’s position [at that moment]{}. If we wait until after $P$ has passed through region $I$, then if $F$ stays at “No”, this means that $P$ is associated with the $x$-spin up component which means that it is at position $A'$, and if $F$ moves to “Yes”, this means that $P$ is associated with the $x$-spin down component which means that it is at position $B'$. The moral is that one cannot use a record in Bohm’s theory to figure out which path $P$ took unless one knows how and when the record was made. But note that in this Bohm’s theory is arguably better off than the standard von Neumann-Dirac collapse formulation of quantum mechanics. On the standard eigenvalue-eigenstate link (where a system determinately has a property if and only if it is in an eigehstate of having the property) one can say [nothing whatsoever]{} about which trajectory a particle followed since it would typically fail to have [any]{} determinate position until it was observed. If one does not worry about the unreliability of retrodiction in the context of the standard collapse theory (and ESSW do not seem to be worried about this!), then I can see no reason at all to worry about it in the context of Bohm’s theory. Further, there is no reason to suppose that Bohmian particle trajectories are not the actual particle trajectories. Nor is there any reason to conclude that our good particle detectors are somehow unreliable. Rather than saying that a detector is fooled by a late measurement, one should, I suggest, say that the late measurement [reliably]{} detects the position of the test particle [nonlocally]{}. On this view the surreal trajectories experiments simply serve to reveal the special role played by position and, ultimately, the nonlocal structure of Bohm’s theory. As Bell explained, “The fact that the guiding wave, in the general case, propagates not in ordinary three space, but in a multi-dimentional configuration space in the origin of the notorious ‘nonlocality’ of quantum mechanics. It is a merit of the de Broglie-Bohm version to bring this out so explicity that it cannot be ignored” (1987, 115). But one should note that it is not some subtle sort of nonlocality involved in the account of quantum-mechanical correlations here. The configuration space particle dynamics that accounts for the nonlocal correlations in the late-measurement experiments makes Bohm’s theory incompatible with relativity. But there is one more point that I would like to make before turning to a discussion of the relationship between how Bohm’s theory accounts for surreal trajectories and its incompatibility with relativity. As suggested above (on the minimal Bohm’s theory) whenever the position of one system is recorded in the position of another system via a strong correlation between the effective wave functions of the two systems (one that produces the appropriate separation of the wave function in the recording parameter in configuration space), then that record will reliably indicate where the measured particle is [at the moment the determinate record is made]{}. It is also the case (on the minimal Bohm theory) that all determinate records are ultimately position records. One can only take these facts to provide a solution to the surreal trajectories if one allows for Bohm’s theory to tell one something about what one is observing when one observes (or, in somewhat different language, what constitues a good measuring device). But it seems that this is precisely the sort of thing that one must be willing to do when entertaining a new theoretical option. One might dogmatically insist on holding to one’s pre-theoretic intuitions concerning what one’s detectors detect come what may, but this would certainly be a methodological mistake. Consider again the late-measurement experiment of the last section (see figures 4 and 5). If $P$ begins in the top half of the wave function at $S$, it will travel path $A$ to $I$ in the $x$-spin up wave packet. That is, before $P$ gets to $I$, the three-particle configuration will be associated with the $x$-spin up component of the wave function in configuration space. And this means that if one converts the spin record into a position record [before]{} the two wave packets interfer at $I$, one will get the result “No”. But if $P$ continues to $I$, bounces, and the two-particle configuration is picked up by the $x$-spin down wave packet, then, since the two-particle configuration is now associated with the $x$-spin down wave packet, if one now converts the $x$-spin record into a position record, one will get the result “Yes” . This means that one might instantaneously determine the value of the converted record at $B$ (the record one gets by converting the $M$ $x$-spin “record” into an $F$ position record) by choosing whether or not to interfer the two wave packets at $I$. If the two wave packets pass through each other, then $F$ will move to “Yes” when the spin record is converted; if not, then $F$ will stay at “No” when the spin record is converted. So, if someone at $I$ knew which path $P$ was on (something, as explained earlier, that is prohibited in Bohm’s theory if the distribution postulate is satisfied), then he or she could use this information to send a superluminal signal to a friend on path $B$ by deciding whether or not to interfere the wave packets at $I$. But regardless of whether one knows which path $P$ is on, the theory predicts (insofar as one is comfortable with the relavant counterfactuals in the context of a deterministic theory) that one can instantaneously affect the result of a measurement of $M$ from region $I$, and one might take the possibility of superluminal effects here to illustrate the incompatibility of Bohm’s theory and relativity. This incompatibility is more clearly illustrated by considering the role that the temporal order of events plays in Bohm’s theory. Consider the late-measurement experiment one more time. If one converts the spin record [before]{} the two wave packets interfer at $I$, then one will get the result “No”; and if one converts the spin record [after]{} the wave packets interfer, then one will get the result “Yes”. But if the conversion of the spin record and the interference of the wave packets are space-like separated events, then the conversion event occurs before the interference event in some inertial frames and after the interference event in others. So in order to get any empirical predictions whatsoever out of Bohm’s theory for this experiment whenever the conversion and interference events are space-like separated, one must choose a perfered inertial frame that imposes a perfered temporal order on the conversion and interference events. But having to choose a perfered intertial frame here is a direct violation of the basic principles of relativity. This is the sense in which the account that Bohm’s theory provides of the late-measurement experiment is fundamentally incompatible with relativity. If the distribution postulate is satisfied, then Bohm’s theory makes the same empirical predictions as the standard von Neumann-Dirac formulation of quantum mechanics (whenever the latter makes unambiguous predictions) and the standard quantum statistics do not allow one to send superluminal messages (given the usual quantum statistics, one can prove a no-signaling theorem). So while Bohm’s theory is not Lorentz-covariant, it explains why one would never notice this fact (just as it explains why one would never notice violations in the conservation of momentum). A proponent of Bohm’s theory might argue that nonlocally correlated motions like the correlated motions in the conversion and interference events describe above is too weak of a relationship to be causal, and that Bohm’s theory thus does not in fact allow for nonlocal causation. While such a conclusion would do nothing to make Bohm’s theory compatible with relativity even if it were granted, I do not think that it should be granted. It seems to me that if any correlated motions should count as causally connected in Bohm’s theory, then nonlocal correlated motions should as well. Nonlocal correlated motions, like local correlated motions (insofar as their are any truly [local]{} correlated motions!), are simply the result of the configuration space evolution of the physical state. The point here is that Bohm’s theory handles nonlocal correlated motions precisely the same way that it handles events that one would presumably want to count as causal—like the correlated motion produced between a football and the foot that kicks it. Of course, one might resist the conclusion that nonlocal correlated motions are causally related by denying that there are any causal relationships whatsoever in Bohm’s theory. But this would mean that even those explanations that one gives that look like causal explanations are not, and this seems to me to be putting things the wrong way around. Just as we look to our best theories to tell us how to build good detectors and to explain what it is that they detect, it seems that we should also look to our best theories to tell us something about the nature of causal relations. There is nothing inherently wrong with sitting down and deciding once and for all the necessary and sufficient conditions for events to be causally related. It is just that one risks adopting a notion of causation that is irrelavant to the sort of explanations provided by our best physical theories.[^13] But regardless of what one thinks about causality, the particle trajectories predicted by Bohm’s theory depend on one’s choice of inertial frame, which means that the theory is incompatible with the basic principles of relativity. And this is the real problem. It is the configuration space dynamics that makes Bohm’s theory incompatible with relativity. But it is also the instantaneous correlated motion predicted by the configuration space dynamics that explains the quantum-mechanical correlations in Bohm’s theory and makes the theory empirically adequate. And it is the configuration space dynamics that allows one to say that whenever the position of one system is recorded in the position of another system via a strong correlation between the effective wave functions of the two systems, then that record will reliably indicate where the measured system is [at the moment the determinate record is made]{}, which, it seems to me, is ultimately the best response to the supposed surreal trajectory problem. But this leaves a proponent of Bohm’s theory with a difficult choice. One might try to find some new way to account for quantum-mechanical correlations, one that does not require a preferred temporal order for space-like separated events where objects exhibit correlated properties. But it should be clear from the the configuration-space stories told above that such a theory would have to explain quantum-mechanical correlations in a way that is fundamentally different from the configuration-space way in which they are explained by Bohm’s theory. And, of course, actually finding such an alternative is much easier said than done.[^14] Or one might simply drop the requirement of Lorentz covariance as a feature of a satisfactory dynamics and settle for something weaker, perhaps something like [appearant]{} Lorentz covariance. But this would be an enormous theoretical sacrifice—presumably one that few physicists would seriously entertain. REFERENCES Aharonov, Y. and L. Vaidman: 1996, “About Position Measurements which do not show the Bohmian Particle Position,” in J. T. Cushing et al. (eds), (1996, 141–154). Albert, D. Z.: 1992, [Quantum Mechanics and Experience]{}, Harvard University Press, Cambridge. Arntzenius, F.: 1994, ‘Relativistic Hidden-Variable Theories?’ [Erkenntnis]{} 41, 207–231. Bacciagaluppi, G. and M. Dickson: 1996, ‘Modal Interpretations with Dynamics’ In Dieks and Vermaas eds. Barrett, J. A.: 1999 The Quantum Mechanics of Minds and Worlds, Oxford University Press. 1996, ‘Empirical Adequacy and the Availability of Reliable Records in Quantum Mechanics,’ [Philosophy of Science]{} 63, 49–64. 1995, ‘The Distribution Postulate in Bohm’s Theory,’ [Topoi]{} 14, 45–54. Bell, J. S.: 1987, [Speakable and Unspeakable in Quantum Theory]{}, Cambridge University Press, Cambridge. 1982, ‘On the Impossible Pilot Wave,’ [Foundations of Physics]{} 12:989-899. Reprinted in Bell (1987,159–168). 1981, ‘Quantum Mechanics for Cosmologists,’ in [Quantum Gravity]{} 2, C. Isham, R. Penrose, and D. Sciama (eds.), Oxford: Clarendon Press, 611–637. Reprinted in Bell (1987,117–138). 1980, ‘de Broglie-Bohm, Delayed-Choice Double-Slit Experiment, and Density Matrix,’ [International Journal of Quantum Chemistry]{}: Quantum Chemistry Symposium 14, 155–9. Reprinted in Bell (1987, 111–6). 1976b, ‘The Measurement Theory of Everett and de Broglie’s Pilot wave,’ in [Quantum Mechanics, Determinism, Causality, and Particles]{}, M. Flato et al. (eds.), D. Reidel, Dordrecht, Holland, 11–17. Reprinted in Bell (1987, 93–99). Berndl, K., M. Daumer, D. Dürr, S. Goldstein, and N.Zanghí: 1995, “A Survey of Bohmian Mechanics,” [Il Nuovo Cimento]{}, vol. 110B, n. 5–6, 737–750. Bohm, D.: 1952, ‘A Suggested Interpretation of Quantum Theory in Terms of “Hidden Variables”, ’ Parts I and II, [Physical Review]{} 85, 166–179, 180–193. Bohm, D. and B. J. Hiley: 1993, [The Undivided Universe: An Ontological Interpretation of Quantum Theory]{}. London: Routledge. Cushing, J. T.: 1996, ‘What Measurement Problem?’ in R. Clifton (ed.) (1996). 1994, [Quantum Mechanics: Historical Contingency and the Copenhagan Hegemony]{}. Chicago: University of Chicago Press. Cushing, J. T., A. Fine, and S. Goldstein: 1996, [Bohmian Mechanics and Quantum Theory: An Appraisal]{}, Boston Studies in the Philosophy of Science, vol. 184, Kluwer Academic Publishers, Dordrecht, The Netherlands. Dewdney, C., L. Hardy, and E. J. Squires: 1993, ‘How Late Measurements of Quantum Trajectories Can Fool a Detector,’ [Physics Letters A]{} 184, 6–11. Dickson, M.: 1996, “Is the Bohm Theory Local?”, in J. T. Cushing et al. (eds) [Bohmian Mechanics and Quantum Theory: An Appraisal]{}, (1996, 321–30). Dickson, M., R. Clifton: 1998, “Lorentz-Invariance in Modal Interpretations,” forthcoming in Dieks and Vermaas (eds) (1998). Dieks, D. G. B. J., and P. E. Vermaas (eds): 1998, [The Modal Interpretation of Quantum Mechanics]{}, Kluwer Academic Press, forthcoming. Dirac, P. A. M.: 1958, [The Principles of Quantum Mechanics]{}, fourth edition, Clarendon Press, Oxford. Dürr, W. Fussender, S. Goldstein, and N. Zanghì: 1993, “Comment on ‘Surrealistic Bohm Trajectories’, ” [Zeitschrift für Naturforschung]{} 48a, 1261–1262. Dürr, D., S. Goldstein, and N. Zanghí: 1993, “A Global Equilibrium as the Foundation of Quantum Randomness,” [Foundations of Physics]{} 23, no. 5, 721–738. 1992, ‘Quantum Mechanics, Randomness, and Deterministic Reality’, [Physics Letters A]{} 172, 6-12. 1992, “Quantum Equilibrium and the Origin of Absolute Uncertainty,” [Journal of Statistical Physics]{}, vol. 67, nos. 5–6, 843–907. Englert, B. G., M. O. Scully, G. Süssmann, and H. Walther: 1993, “Reply to Comment on ‘Surreal Bohm Trajectories’, ”[Zeitschrift für Naturforschung]{} 48a, 1263–1264. 1992, “Surrealistic Bohm Trajectories,” [Zeitschrift für Naturforschung]{} 47a, 1175–1186. Maudlin, T.: 1994, [Quantum Nonlocality and Relativity]{}, Oxford, Blackwell. Phillipidas, C. C. Dewdney, and B. H. Hiley: 1979, ‘Quantum Interference and the Quantum Potential’, [Il Nuovo Cimento]{} 52B, pp. 15–28. von Neumann, J.: 1955, [Mathematical Foundations of Quantum Mechanics]{}, Princeton University Press, Princeton; translated by R. Beyer from [Mathematische Grundlagen der Quantenmechanik]{}, Springer, Berlin, 1932. [^1]: This paper is an extension of the discussion of surreal trajectories in Barrett (1999, 127–40). That section of the book was based on talk I gave at the Quantum Mechanics Workshop at the University of Pittsburgh in the Spring of 1997. And that talk owed much to conversations with Peter Lewis, David Albert, and Rob Clifton. [^2]: This is the theory described by David Bohm in 1952. Bohm’s most complete description of his theory is found in Bohm and Hiley (1993). [^3]: See Dürr, D., S. Goldstein, and N. Zanghí (1993) for a discussion of the equivariance of the statistical distribution $\rho$ and the notion of the effective wave function in Bohm’s theory. [^4]: We will say that a closed system is simple if the Hamiltonian is bounded and if the particle configuration always has positive wave function support. [^5]: There is a sense in which one might say that some dynamical properties, like momentum, are also noncontextual in Bohm’s theory, but, as we will see, the noncontextual momentum is not the [measured]{} momentum. [^6]: The point here is that making the right statistical predictions concerning [particle configurations]{} is not necessarily a sufficient condition for making the right statistical predictions for our [measurement records]{}. One needs to make an extra assumption about the relationship between particle configurations and measurement records. [^7]: Since the standard line is that position is the only observable physical quantity in Bohm’s theory, this does not mean that $P$ has a determinate $z$-spin; rather, it is just a description of the spin index associated with $P$’s effective wave function. [^8]: See Phillipidas, Dewdney, and Hiley (1979) for an explicit calculation of the trajectories in for a similar experiment. The explicit calulations, of course, show that possible particle trajectories never cross $L$. Bell cites this paper at the end of his 1980 paper on delayed-choice experiments in Bohm’s theory. [^9]: But ESSW later make the following argument in favor of actually funding the surreal trajectories experiments that they describe: “Funding agencies were and are well advised to support experiments that have probed or would probe the “surprises” of quantum theory. Imagine the (farfetched) situation that the experimenter finds the photon always in the resonator through which the Bohm trajectory passes rather than the one predicted by quantum theory. Wouldn’t that please the advocates of BM?” (1263–4). This is, of course, very puzzling talk indeed once EWWS conceed that Bohmian mechanics makes the same empirical predictions as the standard theory—a proponent of Bohm’s theory would most certainly [not]{} be pleased if experiments showed that the standard quantum-mechanical predictions were false because this would mean that Bohm’s theory was itself false! When ESSW say things like this, it is easy to understand DFGZ’s frustration. [^10]: Bell explained that a good measurement record must make a macroscopic difference. He emphasized that a discharged detector is macroscopically different from an undischarged detector. This is also something emphasized by DFGZ (1993, 1262) in order to argue that one would not expect one of ESSW’s detectors to generate a sensible record of which path the test particle followed “until an interaction with a suitable macroscopic device occurs.” But note that all that really matters here is that the wave packets that correspond to different measurement outcomes (in terms of the position of $F$) be well-separated in configuration space in the $F$-position-record direction. This is not to say that Bell’s (and DFGZ’s) point concerning macroscopic differences irrelevant. If the flag is a macroscopic system that makes a macroscopic movement, then this will obviously help to provide the wave packet separation required for a reliable record. But while a macroscopic position correlation with a macroscopic system sufficient, it is not a necessary condition for generating a reliable record in Bohm’s theory. See Aharonov, Y. and L. Vaidman (1996) for a discussion of partial measurements in Bohm’s theory, measurements where the separation between the post-measurement wave packets in configuration space is incomplete. [^11]: The experiment below shows what would happen if one tried to record the path taken by the particle in the $x$-spin of another particle. Dewdney, Hardy, and Squires (1993) tried to capture the spirit of ESSW’s experiments by showing in graphic detail what would happen if one tried to record the path in terms of energy. [^12]: In their 1993 paper, Dewdney, Hardy, and Squires argue precisely this. [^13]: Michael Dickson (1996) has argued that it does not make any sense to ask whether a deterministic theory like Bohm’s theory is local because of the difficulty supporting counterfactual conditionals in such a theory. He also suggests that the notion of causality may not make make sense in such a theory either (1996, 329). I agree that some intuitions concerning what it would mean for a theory to be local or what it would mean for one event to cause another cannot be supported in a deterministic theory, but this does not mean that we can make no sense at all of what it would be for a deterministic theory to be local or for one event to cause another. Indeed, whether Bohm’s theory is Lorentz covariant is a perfectly good sensible question concerning its locality—and it isn’t. [^14]: For other discussions of the incompatibility of Bohm’s theory and relativity see Albert (1992) and Artnzenius (1994). For a recent discussion concerning the difficulty in getting a Bohm-like auxiliary quantum dynamics that is compatible with relativity see Dickson and Clifton (1998).	ArXiv
--- abstract: 'We consider a one dimensional elliptic distributed optimal control problem with pointwise constraints on the derivative of the state. By exploiting the variational inequality satisfied by the derivative of the optimal state, we obtain higher regularity for the optimal state under appropriate assumptions on the data. We also solve the optimal control problem as a fourth order variational inequality by a $C^1$ finite element method, and present the error analysis together with numerical results.' address: - 'Susanne C. Brenner, Department of Mathematics and Center for Computation and Technology, Louisiana State University, Baton Rouge, LA 70803, USA' - 'Li-yeng Sung, Department of Mathematics and Center for Computation and Technology, Louisiana State University, Baton Rouge, LA 70803, USA' - 'Winnifried Wollner, Department of Mathematics, Technische Universität Darmstadt, 64293 Darmstadt, Germany' author: - 'S.C. Brenner' - 'L.-Y. Sung' - 'W. Wollner' title: A One Dimensional Elliptic Distributed Optimal Control Problem with Pointwise Derivative Constraints --- Ø [^1] Introduction {#sec:Introduction} ============ Let $I$ be the interval $(-1,1)$ and the function $J:L_2(I)\times L_2(I)\longrightarrow \R$ be defined by $$\label{eq:JDef} J(y,u)=\frac12 \big(\\|y-y_d\\|_\LT^2+\beta\\|u\\|_\LT^2\big),$$ where $y_d\in L_2(I)$ and $\beta$ is a positive constant. The optimal control problem is to $$\label{eq:OCP} \text{find}\quad(\bar y,\bar u)=\argmin_{(y,u)\in \bbK}J(y,u),$$ where $(y,u)\in H^1_0(I)\times L_2(I)$ belongs to $\bbK$ if and only if $$\begin{aligned} {3} \int_I y'z'dx&=\int_I (u+f)z\,dx&\qquad&\forall\,z\in H^1_0(I), \label{eq:PDEConstraint}\\ y'&\leq \psi &\qquad&\text{a.e. on $I$}.\label{eq:DerivativeConstraint}\end{aligned}$$ We assume that $$\label{eq:DataRegularity} f\in H^1(I), \; \psi\in H^2(I)$$ and $$\label{eq:psiConstraint} \int_I \psi\,dx>0.$$ \[rem:Literature\] The optimal control problem defined by – is a one dimensional analog of the optimal control problems considered in [@CB:1988:Gradient; @CF:1993:Gradient; @DGH:2009:Gradient; @OW:2011:Gradient; @Wollner:2012:Gradient]. It was solved by a $C^1$ finite element method in [@BSW:2020:OneD] under the assumptions that $$\label{eq:OldData} f\in H^{\frac12-\epsilon}(I)\quad\text{and}\quad \psi\in H^{\frac32-\epsilon}(I).$$ Since the constraint implies $y\in H^2(I)$ by elliptic regularity, we can reformulate the optimization problem – as follows: $$\label{eq:ROCP} \text{Find}\quad \bar y=\argmin_{y\in K}\frac12\big(\\|y-y_d\\|_\LT^2 +\beta\\|y''+f\\|_\LT^2\big),$$ where $$\label{eq:KDef} K=\{y\in H^2(I)\cap H^1_0(I):\,y'\leq \psi \;\text{on}\;I\}.$$ According to the standard theory [@ET:1999:Convex; @KS:1980:VarInequalities], the minimization problem defined by - has a unique solution characterized by the fourth order variational inequality $$\beta\int_I (\bar y''+f)(y''-\bar y'')dx+ \int_I (\bar y-y_d) (y-\bar y)dx\geq 0 \qquad\forall\,y\in K,$$ which can also be written as $$\label{eq:VI} a(\bar y,y-\bar y)\geq \int_I y_d(y-\bar y)dx-\beta\int_I f(y''-\bar y'')dx \qquad\forall\,y\in K,$$ where $$\label{eq:aDef} a(y,z)=\beta\int_I y'z'dx+\int_I yz\,dx.$$ \[rem:VI\] The reformulation of state constraint optimal control problems as fourth order variational inequalities was discussed in [@PS:1996:OC], and a nonconforming finite element based on this idea was introduced in [@LGY:2009:Control]. Other finite element methods can be found in [@GY:2011:State; @BSZ:2013:OptimalControl; @BDS:2014:PUMOC; @BOPPSS:2016:OC3D; @BGPS:2018:Morley; @BGS:2018:Nonconvex; @BSZ:2019:Neumann]. \[rem:Nontrivial\]Note that implies $$\int_I\psi\,dx\geq \int_I y'\,dx=0 \qquad \forall\,y\in K$$ and hence $\int_I \psi\,dx\geq0$ is a necessary condition for $K$ to be nonempty. It is also a sufficient condition because the function $y$ defined by $$y(x)=\int_{-1}^x (\psi(t)-\bar\psi)dt$$ belongs to $K$, where $\bar\psi$ is the mean of $\psi$ over $I$. Furthermore, $$0=\int_I \psi\,dx=\int_I(\psi-y')dx$$ together with implies $\psi=y'$ identically on $I$ and hence $K=\{\psi\}$ is a singleton. Therefore we impose the condition to ensure that the optimization problem defined by – is nontrivial. Our goal is to show that $\bar y\in H^3(I)$ under the assumptions in and consequently / can be solved by a $C^1$ finite element method with $O(h)$ convergence in the energy norm. Note that previously $\bar y\in H^{\frac52-\epsilon}$ was the best regularity result in the literature for Dirichlet elliptic distributed optimal control problems on smooth/convex domains with pointwise constraints on the gradient of the state. The rest of the paper is organized as follows. The $H^3$ regularity of $\bar y$ is obtained in Section \[sec:DerivativeVI\] through a variational inequality for $\bar y'$ that can be interpreted as a Neumann obstacle problem for the Laplace operator. The $C^1$ finite element method for / is analyzed in Section \[sec:Discrete\], followed by numerical results in Section \[sec:Numerics\]. We end with some remarks on the extension to higher dimensions in Section \[sec:Conclusions\]. A Variational Inequality for $\bar y'$ {#sec:DerivativeVI} ====================================== Observe that the set $\{y':\,y\in K\}$ is the subset $\cK$ of $H^1(I)$ given by $$\begin{aligned} \label{eq:cKDef} \cK&=\{v\in H^1(I):\,\int_I v\,dx=0\quad\text{and}\quad v\leq\psi \;\text{on}\;I\}, %=\{v\in H^1(I)/\R:\,v\leq\psi \;\text{on}\;I\},\end{aligned}$$ and the variational inequality is equivalent to $$\begin{aligned} \label{eq:NeumannVI} &\int_I (\Phi-f')(q-p)dx+\int_I p'(q'-p')dx\\ &\hspace{40pt}+[f(1)(q(1)-p(1))-f(-1)(q(-1)-p(-1))]\geq0 \qquad \forall\,q\in \cK,\notag\end{aligned}$$ where $p=\bar y'$, $q=y'$, and $\Phi\in H^1(I)$ is determined by $$\begin{aligned} \beta\Phi'&=y_d-\bar y \label{eq:PhiDef1} \intertext{and} \int_I\Phi\,dx&=0.\label{eq:PhiDef2}\end{aligned}$$ Moreover is the variational inequality that characterizes the solution of the following minimization problem: $$\label{eq:NeumannObstacle} \text{Find}\quad p=\argmin_{q\in \cK}\Big[\frac12 \int_I (q')^2dx +\int_I(\Phi-f')q\,dx +f(1) q(1)- f(-1) q(-1)\Big].$$ A Neumann Obstacle Problem {#subsec:Neumann} -------------------------- The minimization problem , which is a Neumann obstacle problem, can be written more conveniently as $$\label{eq:ObstacleProblem} p=\argmin_{q\in \cK}\Big[\frac12 b(q,q)+(\phi,q)+ \tau q(1) -\sigma q(-1)\Big],$$ where $\sigma=f(-1)$, $\tau=f(1)$, $$\label{eq:Shorthands} b(q,r)=\int_I q'r'\,dx,\quad (\phi,q)=\int_I \phi\, q\,dx \quad\text{and}\quad \phi=\Phi-f'.$$ Note that we have a compatibility condition $$\label{eq:Compatibility} \int_I \phi\,dx+\tau-\sigma=0$$ that follows from , and the Fundamental Theorem of Calculus for absolutely continuous functions. Since $b(\cdot,\cdot)$ is coercive on $H^1(I)/\R$, the obstacle problem defined by and has a unique solution $p$ characterized by the variational inequality $$\label{eq:VI1} b(p,q-p)+(\phi,q-p)+\tau(q(1)-p(1))-\sigma(q(-1)-p(-1))\geq 0 \qquad\forall\,q\in \cK.$$ \[thm:pRegularity\] The solution $p=\bar y'\in \cK$ of / belongs to $H^2(I)$. We begin by observing that $$\label{eq:VI2} b(p,q-p)+(\phi,q-p)+\tau(q(1)-p(1))-\sigma(q(-1)-p(-1))\geq 0 \qquad\forall\,q\in\tilde K,$$ where $$\label{eq:tKDef} \tilde K=\{q\in H^1(I):\,q\leq\psi\;\;\text{in}\;I\;\; \text{and}\;\int_I q\,dx\geq0\}.$$ Indeed, $q\in \tilde K$ implies $q-\bar q\in K$, where $\bar q$ is the mean of $q$ over $I$, and hence, in view of , the definition of $b(\cdot,\cdot)$ in and , $$\begin{aligned} &b(p,q-p)+(\phi,q-p)+\tau(q(1)-p(1))-\sigma((q(-1)-p(-1))\\ &\hspace{10pt}=b(p,q-\bar q-p)+(\phi,q-\bar q-p)+ \tau(q(1)-\bar q-p(1))-\sigma(q(-1)-\bar q-p(-1))\\ &\hspace{10pt}\geq 0\end{aligned}$$ for all $q\in\tilde{K}$. Let $\fK\subset H^1(I)$ be defined by $$\label{eq:bKDef} \fK=\{q\in H^1(I):\;q\leq \;\psi\;\;\text{in}\;I\},$$ and $G:H^1(I)\longrightarrow [0,\infty)$ be defined by $$\label{eq:gDef} G(q)=\int_I q\,dx.$$ Then the function $\psi$ belongs to $\fK$ and $$\label{eq:Slater} G(\psi)>0$$ by . It follows from the Slater condition and the theory of Lagrange multipliers [@IK:2008:Lagrange Chapter 1, Theorem 1.6] that there exists a nonnegative number $\lambda$ such that $$\label{eq:VI3} b(p,q-p)+(\phi,q-p)+\tau((q(1)-p(1))-\sigma(q(-1)-p(-1))-\lambda\int_I (q-p)dx\geq 0$$ for all $q\in \fK$. Finally we observe that $$\label{eq:VI4} \tilde b(p,q-p)+(F,q-p)+\tau(q(1)-p(1))-\sigma(q(-1)-p(-1)) \geq 0\qquad\forall\,q\in\fK,$$ where $$\label{eq:taDef} \tilde b(q,r)=\int_I q'r'dx+\int_I qr\,dx$$ and $$\label{eq:FDef} F=\phi-\lambda-p.$$ The variational inequality defined by , and characterizes the solution of a coercive Neumann obstacle problem on $H^1(I)$. Since $F\in \LT$ and $\psi\in H^2(I)$, we can apply the result in [@Rodrigues:1987:Obstacle Chapter 5, Theorem 3.4] to conclude that $p\in H^2(I)$. We can deduce the regularity of $(\bar y,\bar u)$ from the relations $p=\bar y'$ and $\bar u=-(\bar y''+f)$. \[cor:Regularity\] Under the assumption on the data, the solution $(\bar y,\bar u)$ of the optimal control problem – belongs to $H^3(I)\times H^1(I)$. The result in [@Rodrigues:1987:Obstacle], which is for dimensions $\geq 2$, requires a compatibility condition between $\p\psi/\p n$ and the Neumann boundary condition so that the boundary trace of the normal derivative of the solution of the obstacle problem belongs to the correct Sobolev space. This is not needed in one dimension since the boundary values of the normal derivative are just numbers. The Karush-Kuhn-Tucker Conditions {#subsec:KKT} --------------------------------- It follows from , Theorem \[thm:pRegularity\] and integration by parts that $$\label{eq:ObstacleLM} b(p,q)+(\phi,q)+\tau q(1)-\sigma q(-1)-\lambda\int_I q\,dx +\int_I q\,d\nu=0 \qquad\forall\,q\in H^1(I),$$ where the regular Borel measure $\nu$ is given by $$\label{eq:ObstaclenuDef} d\nu=(p''-\phi+\lambda)dx+[p'(-1)+\sigma]d\delta_{-1}-[p'(1) +\tau]d\delta_{1},$$ and $\delta_{-1}$ (resp., $\delta_1$) is the Dirac point measure at $-1$ (resp., $1$). Let $\fA$ be the active set of the derivative constraint , i.e., $$\label{eq:ActiveSet} \fA=\{x\in [-1,1]:\,\bar y'(x)=\psi(x)\}=\{x\in[-1,1]:\,p(x)=\psi(x)\}.$$ By a standard argument, $p$ satisfies if and only if $$\label{eq:ObstaclenuProperties} \text{$\nu$ is nonnegative and supported on $\fA$.}$$ We can translate – into Karush-Kuhn-Tucker (KKT) conditions for the solution $\bar y'=p\in \cK$ of /, which is summarized in the following theorem. \[thm:KKT\] There exists a nonnegative number $\lambda$ such that $$\begin{aligned} &\int_I p'q'dx+\int_I(\Phi-f')q\,dx+f(1)q(1)-f(-1)q(-1)+ \int_I q\,d\nu\label{eq:KKT1}\\ &\hspace{100pt}=\lambda\int_I q\,dx&\qquad\forall\,q\in H^1(I),\notag\\ &\int_{[-1,1]}(p-\psi)d\nu=0,\label{eq:KKT2}\\[4pt] &d\nu=\rho\,dx+\gamma d\delta_{-1}+\zeta d\delta_1,\label{eq:KKT3}\end{aligned}$$ where $$\begin{aligned} &\text{$\rho=p''+f'-\Phi+\lambda\in L_2(I)$ is nonnegative a.e.},\label{eq:KKT4}\\ &\text{$\gamma=p'(-1)+f(-1)$ and $\zeta=-[p'(1)+f(1)]$ are nonnegative numbers},\label{eq:KKT5}\end{aligned}$$ and $\Phi\in H^1(I)$ satisfies –. \[rem:KKT\] The KKT conditions – are also sufficient conditions for . Indeed, they imply, for any $q\in\cK$, $$\begin{aligned} &\int_I p'(q'-p')dx+\int_I(\Phi-f')q\,dx+f(1)\big(q(1)-p(1)\big) -f(-1)\big(q(-1)-p(-1)\big)\\ &\hspace{50pt}= \lambda\int_I(q-p)dx-\int_I (q-p)d\nu\\ &\hspace{50pt}=-\int_I (q-\psi)d\nu\geq 0,\end{aligned}$$ which then also implies $\bar y(x)=\int_{-1}^x p(t)dt$ is the solution of . Finally we observe that Theorem \[thm:KKT\] implies $$\label{eq:KKT} \beta\int_I (\bar y''+f)z''dx+\int_I(\bar y-y_d)z\,dx+\int_{[-1,1]}z'd\mu=0 \qquad\forall\, z\in H^2(I)\cap H^1_0(I),$$ where $$\label{eq:muDef} \text{$\mu=\beta\nu$ is a nonnegative Borel measure},$$ and $$\label{eq:Complementarity} \int_{[-1,1]} (\bar y'-\psi)d\mu=0.$$ The Discrete Problem {#sec:Discrete} ==================== Let $V_h\subset H^2(I)\cap H^1_0(I)$ be the cubic Hermite finite element space (cf. [@Ciarlet:1978:FEM; @BScott:2008:FEM]) associated with a triangulation/partition $\cT_h$ of $I$ with mesh size $h$. The discrete problem is to find $$\label{eq:DiscreteProblem} \bar y_h=\argmin_{y_h\in K_h}\frac12\big(\\|y_h-y_d\\|_\LT^2 +\beta\\|y_h''+f\\|_\LT^2\big),$$ where $$\label{eq:KhDef} K_h=\{y_h\in V_h:\, P_hy_h'\leq P_h\psi\},$$ and $P_h$ is the interpolation operator associated with the $P_1$ finite element space associated with $\cT_h$, i.e., the constraint is only enforced at the grid points. The nodal interpolation operator from $C^1([-1,1])$ onto $V_h$ is denoted by $\Pi_h$. We will use the following standard estimates for $P_h$ and $\Pi_h$ (cf. [@Ciarlet:1978:FEM; @BScott:2008:FEM]) in the error analysis: $$\begin{aligned} {3} \\|\zeta- P_h\zeta\\|_\LT&\leq Ch\|\zeta\|_{H^1(I)} &\qquad&\forall\,\zeta\in H^1(I),\label{eq:PhError1}\\ \\|\zeta-P_h\zeta\\|_\LT&\leq Ch^2\|\zeta\|_{H^2(I)} &\qquad&\forall\,\zeta\in H^2(I),\label{eq:PhError2}\\ \|\zeta-\Pi_h\zeta\|_{H^1(I)}+h\|\zeta-\Pi_h\zeta\|_{H^2(I)}&\leq Ch^2\|\zeta\|_{H^3(I)} &\qquad&\forall\,\zeta\in H^3(I).\label{eq:PihEst}\end{aligned}$$ Here and below we use $C$ to denote a generic positive constant that is independent of the mesh size $h$. The unique solution $\bar y_h\in K_h$ of the minimization problem defined by and is characterized by the discrete variational inequality $$\beta\int_I (\bar y_h''+f)(y_h''-\bar y_h'')dx +\int_I (\bar y_h-y_d)(y_h-\bar y_h)dx\geq0 \qquad\forall\,y_h\in K_h,$$ which can also be written as $$\label{eq:DVI} a(\bar y_h,y_h-\bar y_h)\geq \int_I y_d(y_h-\bar y_h)dx -\beta\int_I f(y_h''-\bar y_h'')dx\qquad\forall\,y_h\in K_h,$$ where the bilinear form $a(\cdot,\cdot)$ is defined in . The error analysis of the finite element method is based on the approach in [@BSung:2017:State] for state constrained optimal control problems that was extended to one dimensional problems with constraints on the derivative of the state in [@BSW:2020:OneD]. We will use the energy norm $\\|\cdot\\|_a$ defined by $$\label{eq:EnergyNorm} \\|v\\|_a^2=a(v,v)=\\|v\\|_\LT^2+\beta\|v\|_{H^2(I)}^2.$$ Note that $$\label{eq:NormEquivalence} \\|v\\|_a\approx \\|v\\|_{H^2(I)} \qquad\forall\,v\in H^2(I)\cap H^1_0(I)$$ by a Poincaré-Friedrichs inequality [@Necas:2012:Direct]. An Abstract Error Estimate {#subsec:Abstract} -------------------------- In view of , and the Cauchy-Schwarz inequality, we have $$\begin{aligned} \label{eq:Preliminary} \\|\bar y-\bar y_h\\|_a^2&=a(\bar y-\bar y_h,\bar y-y_h)+ a(\bar y-\bar y_h,y_h-\bar y_h)\notag\\ &\leq \frac12\\|\bar y-\bar y_h\\|_a^2+\frac12\\|\bar y-y_h\\|_a^2 +a(\bar y,y_h-\bar y_h)\\ &\hspace{60pt} -\int_I y_d(y_h-\bar y_h)dx+ \beta\int_I f(y_h''-\bar y_h'')dx\qquad\forall\,y_h\in K_h.\notag\end{aligned}$$ It follows from , and that $$\begin{aligned} \label{eq:KKTRelation} &a(\bar y,y_h-\bar y_h)-\int_I y_d(y_h-\bar y_h)dx+ \beta\int_I f(y_h''-\bar y_h'')dx\notag\\ &\hspace{40pt}=\int_{[-1,1]} (\bar y_h'-y_h')d\mu\notag\\ &\hspace{40pt}=\int_{[-1,1]} (\bar y_h'-P_h\bar y_h')d\mu+ \int_{[-1,1]} (P_h\bar y_h'-P_h\psi)d\mu+\int_{[-1,1]} (P_h\psi-\psi)d\mu\\ &\hspace{90pt}+\int_{[-1,1]} (\psi-\bar y')d\mu+ \int_{[-1,1]} (\bar y'-y_h')d\mu\notag\\ &\hspace{40pt}\leq \int_{[-1,1]} (\bar y_h'-P_h\bar y_h')d\mu +\int_{[-1,1]} (P_h\psi-\psi)d\mu +\int_{[-1,1]} (\bar y'-y_h')d\mu \notag\end{aligned}$$ for all $y_h\in K_h$. Putting and together, we arrive at the abstract error estimate $$\begin{aligned} \label{eq:Abstract} \\|\bar y-\bar y_h\\|_a^2&\leq 2\Big(\int_{[-1,1]} (\bar y_h'-P_h\bar y_h')d\mu +\int_{[-1,1]}(P_h\psi-\psi)d\mu\Big)\\ &\hspace{50pt} +\inf_{y_h\in K_h}\Big(\\|\bar y-y_h\\|_a^2+ 2\int_{[-1,1]} (\bar y'-y_h')d\mu\Big).\notag\end{aligned}$$ Concrete Error Estimates {#subsec:Concrete} ------------------------ The three terms on the right-hand side of can be estimated as follows. First of all, we have $$\begin{aligned} \label{eq:Est1} \int_{[-1,1]}(\bar y_h'-P_h\bar y_h')d\mu& =\int_{[-1,1]} \big[(\bar y_h'-\bar y')-P_h(\bar y_h'-\bar y')\big]d\mu+ \int_{[-1,1]} (\bar y'-P_h\bar y')d\mu\notag\\ &=\beta \Big(\int_I \big[(\bar y_h'-\bar y')-P_h(\bar y_h'-\bar y')\big]\rho\,dx+ \int_I (\bar y'-P_h\bar y')\rho\,dx\Big)\\ &\leq C\big(h\\|\bar y-\bar y_h\\|_a+h^2\|y\|_{H^3(I)}\big),\notag\end{aligned}$$ by Corollary \[cor:Regularity\], , , , , and the fact that $\zeta-P_h\zeta$ vanishes at the points $\pm1$ for any $\zeta\in H^1(I)$. Similarly we can derive $$\label{eq:Est2} \int_{[-1,1]}(P_h\psi-\psi)d\mu=\beta\int_I (P_h\psi-\psi)\rho\,dx \leq Ch^2\|\psi\|_{H^2(I)}$$ by and . Finally we have $$\begin{aligned} \label{eq:Est3} &\inf_{y_h\in K_h}\Big(\\|\bar y-y_h\\|_a^2+ 2\int_{[-1,1]} (\bar y'-y_h')d\mu\Big)\notag\\ &\hspace{50pt}\leq \\|\bar y-\Pi_h \bar y\\|_a^2+2\int_{[-1,1]} \big[\bar y'-(\Pi_h\bar y)'\big]d\mu\\ &\hspace{50pt}= \\|\bar y-\Pi_h \bar y\\|_a^2+2\beta\int_I \big[\bar y'-(\Pi_h\bar y)'\big]\rho\,dx \leq Ch^2\big[\|\bar y\|_{H^3(I)}^2+\|\bar y\|_{H^3(I)}\big],\notag\end{aligned}$$ by Corollary \[cor:Regularity\], , , , and the fact that $\bar y'-(\Pi_h\bar y)'$ vanishes at $\pm1$. It follows from – and Young’s inequality that $$\label{eq:EnergyError} \\|\bar y-\bar y_h\\|_a\leq Ch,$$ which immediately implies the following result, where $\bar u_h=-(\bar y_h''+f)$ is the approximation for $\bar u=-(\bar y+f)$. \[thm:ErrorEstimates\] Under the assumptions on the data in , we have $$\|\bar y-\bar y_h\|_\HO+\\|\bar u-\bar u_h\\|_\LT\leq Ch.$$ \[rem:Sharp\] Numerical results in Section \[sec:Numerics\] indicate that the estimate for $\\|\bar u-\bar u_h\\|_\LT$ in Theorem \[thm:ErrorEstimates\] is sharp. \[rem:Comparison\] For comparison, the error estimate $$\|\bar y-\bar y_h\|_\HO+\\|\bar u-\bar u_h\\|_\LT\leq C_\epsilon h^{\frac12-\epsilon}$$ was obtained in [@BSW:2020:OneD] under the assumptions in . A Numerical Experiment {#sec:Numerics} ====================== We begin by constructing an example for the problem / with a known exact solution. An Example {#subsec:Example} ---------- Let $\beta=1$, $$\label{eq:psiExample} \psi(x)=\begin{cases} 1-\frac92 x^2&\qquad -1\leq x\leq 0\\[2pt] 1&\qquad \phantom{-}0\leq x\leq 1 \end{cases},$$ and $$\label{eq:ExactSolution} \bar y(x)=\int_{-1}^x p(t)dt,$$ where $$\label{eq:pDef} p(x)=\begin{cases} 1-\frac{81}{32}(x-\frac13)^2&\qquad -1\leq x\leq \frac13 \\[2pt] 1&\qquad\hspace{10pt} \frac13\leq x\leq 1 \end{cases}.$$ We have $\psi\in H^2(I)$, $$\label{eq:psiExampleConstraint} \int_I\psi\, dx =\frac12,$$ $p\in H^2(I)$, $$\label{eq:pSD} p''(x)=\begin{cases} -81/16& \qquad -1<x<\frac13\\[2pt] 0&\qquad\hspace{10pt} \frac13<x<1 \end{cases},$$ $p'(1)=0$, $p'(-1)=27/4$, $$\label{eq:pProperties} \int_I p\,dx=0, \quad p\leq\psi \quad\text{and}\quad \fA=\{-1\}\cup[1/3,1].$$ Let $f\in H^1(I)$ be defined by $$\label{eq:fDef} f(x)=\begin{cases}\d \frac{2}{9\pi}\sin (\pi(3x-1))&\qquad -1< x\leq \frac13 \\[8pt] -(x-\frac13)^2 &\qquad \hspace{10pt} \frac13\leq x<1 \end{cases}.$$ We have $f(-1)=0$, $f(1/3)=0$, $f_-'(1/3)=2/3$, $f'_+(1/3)=0$ and $f(1)=-4/9$. Therefore the function $$\label{eq:PhiExample} \Phi(x)=\begin{cases} f'(x)&\qquad -1<x< \frac13\\[2pt] f'(x)+\frac23&\qquad\hspace{10pt} \frac13< x<1 \end{cases}$$ belongs to $H^1(I)$ and $$\label{eq:PhiIntegralExample} \int_I\Phi\,dx=\int_I f'(x)+\int_\frac13^1\frac23\,dx =f(1)-f(-1)+\frac49=0.$$ Finally we take $\lambda=81/16$ and $y_d=\bar y+\Phi'$. Then the KKT conditions – are satisfied with $$\begin{aligned} d\nu&=[p''+f'-\Phi+\lambda]dx+[p'(-1)+f(-1)]d\delta_{-1} -[p'(1)+f(1)]d\delta_1\\ &=(211/48)\chi_{[1/3,1]} dx+(27/4)d\delta_{-1}+(4/9)d\delta_1,\end{aligned}$$ where $\chi_{[1/3,1]}$ is the characteristic function of the interval $[1/3,1]$. Numerical Results {#subsec:Results} ----------------- We solved the problem in Section \[subsec:Example\] by the finite element method in Section \[sec:Discrete\] on uniform meshes. The results are displayed in Table \[table:Results\]. [\|l\|c\|c\|c\|c\|]{} ------------------------------------------------------------------------ $2/h$ &$\\|\bar y-\bar y_h\\|_{L_2(I)}$&$\\|\bar y-\bar y_h\\|_{L_\infty(I)}$ &$\|\bar y-\bar y_h\|_{H^1(I)}$ &$\|\bar y-\bar y_h\|_{H^2(I)}$\ &&&&\ $1+2^0$ & 1.430334e-01& 1.625937e-01& 2.581989e-01& 8.660252e-01\ $1+2^1$ & 1.216070e-01& 1.385037e-01& 2.199480e-01& 7.486796e-01\ $1+2^2$ & 4.306657e-02 & 4.679253e-02 & 8.061916e-02 & 4.485156e-01\ $1+2^3$ & 1.613494e-02 & 1.850729e-02 & 2.919318e-02 & 2.573315e-01\ $1+2^4$ & 3.439341e-03 & 3.849954e-03 & 6.315816e-03 & 1.266029e-01\ $1+2^5$ & 9.590453e-04 & 1.087740e-03 & 1.741244e-03 & 6.470514e-02\ $1+2^6$ & 2.256478e-04& 2.542346e-04 &4.125212e-04 & 3.223430e-02\ $1+2^7$ & 5.874304e-05 & 6.639870e-05& 1.067193e-04&1.618687e-02\ $1+2^8$ & 1.425640e-05 &1.608790e-05 &2.549283e-05 & 8.086258e-03\ $1+2^9$ & 3.657433e-06 & 4.124680e-06 & 6.430499e-06 & 4.047165e-03\ We observe $O(h)$ convergence in the $H^2$ norm which agrees with Theorem \[thm:ErrorEstimates\]. On the other hand the convergence in the $H^1$ norm is $O(h^2)$, better than the $O(h)$ convergence predicted by Theorem \[thm:ErrorEstimates\]. The convergence in $L_2$ and $L_\infty$ is also $O(h^2)$. Concluding Remarks {#sec:Conclusions} ================== We have shown that higher regularity for the solutions of one dimensional Dirichlet elliptic distributed optimal control problems with pointwise constraints on the derivative of the state can be obtained through a variational inequality satisfied by the derivative of the optimal state. A similar result for one dimensional optimal control problems with mixed boundary conditions was obtained earlier in [@BSW:2020:OneD]. A natural question is: Can these results be extended to higher dimensions? For analogs of – on a smooth/convex domain $\O\in \R^d$ ($d=2,3$), where $f\in H^1(\O)$ and $\bm{\Psi}\in [H^2(\O)]^d$, one can also derive a variational inequality for the gradient of the optimal state. Observe that the space $\bG$ of the gradients of the states is characterized by (cf. [@GR:1986:NS Chapter I, Section 2.3]) $$\begin{aligned} \bG&=\{\nabla y:\,y\in H^2(\O)\cap H^1_0(\O)\}\notag\\ &=\{\bq\in[ H^1(\O)]^d:\,\Curl\bq=0 \;\text{on}\;\O\;\text{and} \;\bn\times\bq=0 \; \text{on}\;\p\O\},\end{aligned}$$ where $\bn$ is the unit outward normal along $\p\O$. Let $\bK$ be the subset of $\bG$ defined by $$\bK=\{\bq\in\bG:\,\bq\leq\bm{\Psi}\;\text{a.e. in $\O$}\}.$$ We assume that $\bK$ is nonempty, which is the case for example if $\bm{\Psi}\geq{\bf 0}$. The analog of is given by $$\label{eq:DirichletHigher} \int_\O (\bPhi-\nabla f)\cdot(\bq-\bp)dx +\int_\O \Div\bp\,\Div(\bq-\bp)\,dx +\int_{\p\O} f(\bq-\bp)\cdot\bn\,dS\geq 0$$ for all $\bq\in \bK$, where $\bp=\nabla\bar y\in\bK$, and $\bPhi\in \bG$ is defined by $ \beta\Div\bPhi=y_d-\bar y$, which is an analog of . The variational inequality is uniquely solvable because (cf. [@GR:1986:NS Chapter I, Sections 3.2 and 3.4]) $$\int_\O (\Div \bq)^2dx\geq C_\O\|\bq\|_{H^1(\O)}^2\qquad \forall\,\bq\in \bG.$$ We can also write as $$\begin{aligned} \label{eq:DirichletVector} &\int_\O (\bPhi-\nabla f)\cdot(\bq-\bp)dx+ \int_\O \big[\Div\bp\,\Div(\bq-\bp) +\Curl\bp\cdot\Curl(\bq-\bp)\big]\,dx\\ &\hspace{100pt}+\int_{\p\O} f(\bq-\bp)\cdot\bn\,dS \geq 0\qquad\forall\,\bq\in \bK,\notag\end{aligned}$$ which can be interpreted as an obstacle problem for the vector Laplacian operator with natural boundary conditions. In order to obtain higher regularity for the optimal state $\bar y$, one will need regularity results for /, which unfortunately are not available. Therefore the problem of extending the results in this paper to higher dimensions remains open. [10]{} S.C. Brenner, C.B. Davis, and L.-Y. Sung. . , 276:612–626, 2014. S.C. Brenner, J. Gedicke, and L.-Y. 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[^1]: The work of the first and second authors was supported in part by the National Science Foundation under Grant No. DMS-19-13035.	ArXiv
--- bibliography: - 'Planck\_bib.bib' - 'Zodi.bib' - 'SSO4ken/SSO\_extra.bib' title: ' 2013 results. XIV. Zodiacal emission' --- ,[–]{} Introduction {#sec:introduction} ============ This paper, one of a set associated with the 2013 release of data from the [^1] mission [@planck2011-1.1], describes the measurement of Zodiacal emission using data. Zodiacal light, or the reflection of sunlight from small dust particles in our Solar System, can be seen by eye at dawn or dusk in dark locations, and contributes significantly to the diffuse sky brightness at optical and near-infrared wavelengths. The study of Zodiacal emission, or the thermal re-emission of absorbed energy from these interplanetary dust (IPD) particles has been enabled by the advent of infrared astronomical techniques, and it is now known to dominate the diffuse sky brightness over most of the sky between 10 and 50$\micron$ [see, for example, @Leinert1997]. Full-sky, infrared satellite surveys, in particular, have allowed us to begin to determine the structure of the density of the IPD [@Hauser84; @kelsall1998; @fixsendwek2002; @Pyo2010]. One of the full-sky models of Zodiacal emission most easily adapted for and most commonly used at longer wavelengths is based on work from the Cosmic Background Explorer Diffuse Infrared Brightness Experiment (COBE/DIRBE) team [@kelsall1998 hereafter K98]. Others are presented in [@Good1986], [@Rowan-Robinson1990; @Rowan-Robinson1991], [@Jones1993], [@Vrtilek1995], [@Wright1998], and [@Rowan-Robinson2012]. The K98 model comprises the well-known diffuse cloud, three sets of dust bands first discovered by IRAS [@Low1984], and a circumsolar ring and Earth-trailing feature, hinted at in IRAS and confirmed in DIRBE [called a ‘blob’ in K98. See @Reach1995 and references therein]. @fixsendwek2002 have used data from the Far Infrared Absolute Spectrophotometer (FIRAS) to extend measurements of the diffuse cloud to longer wavelengths, but given its modest angular resolution and large uncertainties in the submillimetre region, could not say more about the smaller angular-scale Zodiacal features. ’s sensitivity allows it to detect and measure the emissivity of the diffuse Zodiacal cloud at long wavelengths, and its angular resolution also allows it to characterize the smaller-scale components of the Zodiacal. This paper continues as follows: in Sect. \[sec:planck:mission\] we describe the relevant aspects of the mission for this analysis, along with it’s observation strategy and data processing; in Sect. \[sec:detection\] we describe how detects Zodiacal emission; and in Sect. \[sec:model\] we briefly describe the COBE Zodiacal emission model. The fit to the /HFI data is described in Sect. \[sec:fit\], and the results of the fit are discussed in Sect. \[sec:discussion\]. We conclude in Sect. \[sec:conclusion\]. The Mission {#sec:planck:mission} ============ , comprised of the High Frequency Instrument, or HFI, and the Low Frequency Instrument, or LFI, was launched in May of 2009. The mission as a whole is described in [@planck2013-p01]. This work uses only data at frequencies of 100GHz and higher. At these frequencies, observed the entire sky in six broad frequency bands from 100 to 857GHz, with corresponding angular resolutions from roughly 97 to 46 [@planck2013-p03c]. Orbit, Scanning Strategy and Dates of Observation ------------------------------------------------- While ’s orbit and scanning strategy are described in depth in [@planck2011-1.1] and [@planck2013-p01], we give a synopsis of the elements relevant to our analysis here. orbits around the second Sun-Earth Lagrange point, and is thus always close to the Ecliptic plane and about 1.01 AU from the Sun and 0.01 AU from the Earth. Its focal plane scans the sky once per minute, with each detector always observing on a circle approximately $85^\circ$ from its spin axis. A simplified video showing this scanning strategy can be found at the ESA website[^2]. In addition, and not visible in the video noted above, the spin axis traces the Sun-Earth vector, but with an additional “cycloid” component, so that in the Sun-Earth frame the spin axis is always $7.5^\circ$ degrees from the Sun-Earth vector and circles around it twice per year. This cycloid component results in differing total amounts of IPD in ’s line of sight for different observations of the same point on the distant celestial sphere. This is shown schematically in Fig. \[fig:Scan\]. ![ Schematic representation of the geometry of ’s measurements, which shows that it can view different amounts of Zodiacal emission while looking at the same point on the distant sky. The plane of the ecliptic is in the plane of the diagram. The Sun is in the center of the circles. The solid black line represents the orbit of the Earth and . The dashed line at the outer edge of the shaded ring represents the orbit of Jupiter, beyond which we assume there is no contribution to the Zodiacal emission from IPD. Panel (a) shows a case where the phase of the scan cycloid and the location of the observed point on the sky yield two measurements for which the lines of sight through the IPD is roughly equal, and the same Zodiacal signal is seen. Panel (b) shows a case where the phase of the scan cycloid and the location of the observed point on the sky yield different total columns of IPD along the lines of sight, and thus a different Zodiacal signal is seen in each of the two measurements. Note that this figure is highly stylized and not to scale. []{data-label="fig:Scan"}](Scan.pdf){width="88mm"} As nearly the entire sky is seen twice each year, the Planck team divides the observations into “surveys” of approximately six-month duration. The exact definition of the beginning and end of each survey was agreed upon within the team. The basic characteristics are that each survey lasts about six months and covers a maximum of sky, with a minimum of overlap between the beginnings and ends of the survey. During any single one of these surveys, some pixels near the ecliptic poles are observed multiple times, as are the pixels near the ecliptic plane which are seen both at the beginning and at the end of the survey. The bulk of the sky, however, is observed only during well-defined periods, usually less than a single week. In Fig. \[fig:jd\] we show the Julian dates of observations of those pixels on the sky for which the observation times during survey 1 spanned one week or less. The analogous plot for survey 2 is similar in nature, and the corresponding maps for surveys 3 and 4 are quite similar to those of surveys 1 and 2, as the scanning strategy for surveys 3 and 4 were almost identical to those of surveys 1 and 2, respectively. ![The Julian date of observation of pixels on the sky during survey 1, for a single detector, in Galactic coordinates. There are only very small differences between maps for different detectors. The grid lines show ecliptic coordinates, with the darker lines representing the ecliptic plane and the line of zero ecliptic longitude. Undefined pixels, which were either not observed at all, or which were observed multiple times over a period that spanned more than one week and are thus not used in this analysis, are shown as the uniform gray band.[]{data-label="fig:jd"}](jd_h857-1_s1_d07_v53.jpg){width="88mm"} Data Processing --------------- The overall HFI data processing is described in [@planck2011-1.7] and [@planck2013-p03]. Given the time-dependent nature of the Zodiacal signal seen with the scanning strategy, this analysis is done using the individual survey 1-4 maps. This allows us to exclude from the analysis regions of the sky and periods of time where the column of IPD viewed by is not constant. The HFI instrument has a number of horns at each measurement frequency [@planck2013-p03c Fig. 9]. Working with individual horn maps, rather than the co-added frequency maps, allows us to adjust the response of each detector so that they are uniform for a source with a Zodiacal spectrum, rather than that used for a CMB spectrum, as is done in the standard processing [@planck2013-p03d]. At 100, 143, 217, and 353GHz, some of these horns contain two polarization sensitive bolometers [PSBs; @Jones2003]. As we are not addressing polarisation here, for those horns that have PSBs, we combine the maps from each of the two detectors within the given horn with a simple average. As the evaluation of the model to be presented in Sect. \[sec:model\] involves calculating emission from a number of points along each line of sight and summing them, the computations are time consuming. To mitigate this to some extent, we use $13\parcm7\times13\parcm7$ pixels, rather than the original $1\parcm7\times1\parcm7$ HFI pixels, reducing the number of map pixels from 50 million to a bit less than 800 thousand [that is, we use HEALPix pixels with $N_\mathrm{side}$ of 256 rather than 2048; @gorski2005]. While this does reduce our sensitivity to finer scale structures, it is not a serious hindrance, since we will be making comparisons with DIRBE, which had a still larger beam. Smaller pixels will be used in future work as more detail is teased out of the data. Pre-launch estimates of ’s ability to detect Zodiacal emission and an estimate of the possible level of contamination at the highest frequencies were presented in [@maris2006]. More recent predictions have addressed the possibility of Zodiacal contamination at lower frequencies [@diego2010] and speculated that emission from dust in the outer Solar System might contribute to large-scale anomalies which have been reported in data from the Wilkinson Microwave Anisotropy Probe (WMAP) at large angular scales [@maris2011; @Hansen2012]. Detection {#sec:detection} ========= The existence of the Zodiacal emission in the maps is straightforward to demonstrate by exploiting the fact, noted above, that different surveys often sample different columns of IPD while observing the same location on the distant celestial sphere. Fig. \[fig:surveymaps\] shows the first (top) and second (middle) survey maps for the 857-1 detector, along with the difference of these two maps (bottom). Similar differences for all HFI frequencies, using data from all horns at each frequency, are shown in the left-hand panel of Fig. \[fig:beforeAndAfterJackknives\]. There are at least three features that stand out in the map difference: (1) The scale has been reduced immensely, but the Galactic plane is still visible in this difference map; (2) the “arcs” at the top and the bottom of the difference map are the images of the Galactic centre as seen through the instrument’s far sidelobes (FSLs); and (3) the Zodiacal emission can be seen as the variations following the ecliptic plane. We begin with the Zodiacal emission. ![Single-survey maps in Galactic coordinates for the 857-1 detector. Top: Survey 1 map. Middle: Survey 2 map. Bottom: Survey 2 minus survey 1 difference map. This bottom image shows the Zodiacal emission and the residual Galactic emission effects discussed in the text. The units are MJy/sr, assuming a spectrum inversely proportional to frequency. Undefined pixels are shown in gray. These occur in pixels which either have not been observed during the survey, were observed during the passage of a planet, or a small number of other events. In the top two plots, pixels which were observed over periods longer than a week were not masked, and thus the masked regions are smaller in the top two images than that in the difference (bottom) panel.[]{data-label="fig:surveymaps"}](857-1_W_TauDeconv_survey1_I.jpg "fig:")\ ![Single-survey maps in Galactic coordinates for the 857-1 detector. Top: Survey 1 map. Middle: Survey 2 map. Bottom: Survey 2 minus survey 1 difference map. This bottom image shows the Zodiacal emission and the residual Galactic emission effects discussed in the text. The units are MJy/sr, assuming a spectrum inversely proportional to frequency. Undefined pixels are shown in gray. These occur in pixels which either have not been observed during the survey, were observed during the passage of a planet, or a small number of other events. In the top two plots, pixels which were observed over periods longer than a week were not masked, and thus the masked regions are smaller in the top two images than that in the difference (bottom) panel.[]{data-label="fig:surveymaps"}](857-1_W_TauDeconv_survey2_I.jpg "fig:")\ ![Single-survey maps in Galactic coordinates for the 857-1 detector. Top: Survey 1 map. Middle: Survey 2 map. Bottom: Survey 2 minus survey 1 difference map. This bottom image shows the Zodiacal emission and the residual Galactic emission effects discussed in the text. The units are MJy/sr, assuming a spectrum inversely proportional to frequency. Undefined pixels are shown in gray. These occur in pixels which either have not been observed during the survey, were observed during the passage of a planet, or a small number of other events. In the top two plots, pixels which were observed over periods longer than a week were not masked, and thus the masked regions are smaller in the top two images than that in the difference (bottom) panel.[]{data-label="fig:surveymaps"}](survey_jack_857-1_year1.jpg "fig:") Since these are difference maps, the Zodiacal emission is seen in the ecliptic plane as both positive and as negative, depending upon the relative geometry of the IPD and the satellite when a given location was observed in the two surveys. While the amplitude of the Zodiacal emission signal is reduced in this differential process [this will be described in Sect. \[sec:model\], and in particular Fig. \[fig:zodiMaps\]; see also @maris2006], it has the advantage that the bulk of the Galactic and extra-galactic signal, the main contaminants in the analysis at high frequencies, is efficiently removed with little effort. What remains of the Galactic signal arises from effects such as beam asymmetries and imperfections in the transfer function removal [@planck2013-p03c]. To mitigate these residual Galactic effects, we generally do not use data within 10 degrees of the Galactic plane in this analysis. We have confirmed that changing this cut to 5 or to 20 degrees does not change the conclusions of this analysis, and emphasize that contrary to other analyses, the fact that we are analysing difference maps makes this work less sensitive to Galactic contamination. It should be noted that the Zodiacal emission is much dimmer than many other background components in the data. Whereas the Zodiacal emission dominates the sky in some IRAS and COBE bands, this is never the case for , with the cosmic infrared background and Galactic dust dominating at high frequencies, and the CMB itself dominating at lower frequencies. This makes our differencing scheme appealing, but also restricts the analysis in some ways. It is, for example, difficult to look at individual scans, or slices of the sky, as has been done successfully with IRAS. We are obligated to use almost the entire sky, and use a model of the Zodiacal emission to interpret variations, rather than being able to directly interpret the total Zodiacal emission on the sky. Model {#sec:model} ===== The goal of this section is to describe the Zodiacal and far sidelobe templates we will create to fit to the map at the bottom of Fig. \[fig:surveymaps\]. Zodiacal Components ------------------- The COBE/DIRBE Zodiacal emission model is described in depth in K98, but we review the salient parts here. ### Diffuse cloud {#sec:cloudmath} The density of the diffuse IPD cloud, having both radial and vertical dependence, is taken to be of the form $$n_c\left(\mathbf{R}\right) = n_0 R_c^{-\alpha} \left\{ \begin{array}{cl} e^{-\beta\left(\zeta^2/2\mu\right)^\gamma} & \mathrm{if}\ \zeta < \mu \\ e^{-\beta\left(\zeta -\mu/2\right)^\gamma} & \mathrm{if}\ \zeta\geq\mu \\ \end{array} \right.,$$ where $$R_c = \sqrt{\left(x-x_0\right)^2 + \left(y-y_0\right)^2 + \left(z-z_0\right)^2},$$ $$\zeta = \left\|Z_c\right\|/R_c,$$ $$\begin{aligned} Z_c & = & \left(x-x_0\right) \cdot\sin\left(\Omega_R\right) \cdot\sin\left(i_R\right) \nonumber \\ & - & \left(y-y_0\right) \cdot\cos\left(\Omega_R\right) \cdot\sin\left(i_R\right) + \left(z-z_0\right)\cdot\cos(i_R), \end{aligned}$$ and $\alpha$, $\beta$, $\gamma$, $\mu$, $x_0$, $y_0$, $z_0$, $\Omega_R$, and $i_R$ are parameters describing the location and shape of the cloud. This form (and others used elsewhere which are similar) is based on an approximation of a model of particles orbiting the Sun, accounting for drag from the Poynting-Robertson effect, and with a modified fan distribution used to describe the changes in density above and below the plane of the ecliptic. See K98 for more details and references. The numerical values for the parameters can be found in K98, or from the LAMBDA website[^3]. This is shown, for both survey 1 and survey 2, as well as their difference, assuming an emissivity of one, in the bottom row of Fig. \[fig:zodiMaps\]. $\begin{array}{cc\|c} \includegraphics[width=60mm]{Z857JackSim_survey1_band3.jpg} & \includegraphics[width=60mm]{Z857JackSim_survey2_band3.jpg} & \includegraphics[width=60mm]{Z857JackSim_surveyD_band3.jpg} \\ \includegraphics[width=60mm]{Z857JackSim_survey1_band2.jpg} & \includegraphics[width=60mm]{Z857JackSim_survey2_band2.jpg} & \includegraphics[width=60mm]{Z857JackSim_surveyD_band2.jpg} \\ \includegraphics[width=60mm]{Z857JackSim_survey1_band1.jpg} & \includegraphics[width=60mm]{Z857JackSim_survey2_band1.jpg} & \includegraphics[width=60mm]{Z857JackSim_surveyD_band1.jpg} \\ \includegraphics[width=60mm]{Z857JackSim_survey1_cring.jpg} & \includegraphics[width=60mm]{Z857JackSim_survey2_cring.jpg} & \includegraphics[width=60mm]{Z857JackSim_surveyD_cring.jpg} \\ \includegraphics[width=60mm]{Z857JackSim_survey1_tblob.jpg} & \includegraphics[width=60mm]{Z857JackSim_survey2_tblob.jpg} & \includegraphics[width=60mm]{Z857JackSim_surveyD_tblob.jpg} \\ \hline \includegraphics[width=60mm]{Z857JackSim_survey1_cloud.jpg} & \includegraphics[width=60mm]{Z857JackSim_survey2_cloud.jpg} & \includegraphics[width=60mm]{Z857JackSim_surveyD_cloud.jpg} \end{array}$ ### Dust bands {#sec:bandmath} The Zodiacal dust bands were first seen by IRAS [@Low1984], and appear as pairs of bright, parallel bands equally spaced above and below the ecliptic plane. They were quickly associated with asteroid families, and then understood to be the relics of asteroid collisions or collapses [@Dermott1984; @sykes86]. [@Reach1997] study them in detail. The K98 model contains three bands[^4] called, reasonably enough, bands 1, 2 and 3. They appear at ecliptic latitudes around $\pm 1\pdeg4$, $\pm 10\deg$, and $\pm 15\deg$. IRAS, having higher angular resolution than DIRBE, was able find more bands, dubbed $\alpha$, $\beta$, $\gamma$, E/F, G/H, J/K, and M/N [@Sykes1990], though some are more firmly detected than others. The K98 band 1, or IRAS band $\gamma$, was originally associated with the Eos family of asteroids [@Dermott1984], but [@Grogan2001] called this into question and @Nesvorny2003 [hereafter N03] found better correspondence with the Veritas family of asteroids. K98 note that their band 2 is a blend of IRAS bands $\alpha$ and $\beta$ [@Sykes1990]. [@sykes86] tentatively associated the $\alpha$ band with the Themis family of asteroids, and @Nesvorny2008 [hereafter N08] has narrowed this association to a cluster within this family associated with the Beagle asteroid. The $\beta$ band was associated by [@sykes86] with the Koronis family of asteroids, and N03 narrowed this to the Karin cluster within the Koronis family. Anticipating the discussion in Sect. \[subsec:Bands\], we note also that the $\beta$ band appears brighter than the $\alpha$ band [@Sykes1988; @Reach1997; @Nesvorny2008]. K98 states that their band 3, that furthest from the ecliptic plane, has been associated with both the Io and Maria families of asteroids [@Sykes1988; @Reach1997], corresponding to IRAS bands J/K and M/N. N03 have more recently noted, however, that 4652 Iannini and/or 845 Naema may be better asteroid associations for the J/K band-pair and that 1521 Seinajoki may work better for the M/N pair. We summarize these associations in table \[tab:bandInfo\]. K98 $\delta_\zeta$ IRAS $i_p$ $i_p$ Type Age a ----- ---------------- ---------- ------- ------ ----------- ------- ------------- -------------- ------- 1 8.78 $\gamma$ 9.35 490 Veritas 9.26 C / Ch/... 8.3 3.169 $\alpha$ 1.34 656 Beagle 1.34 .../.../ C $\lesssim$10 3.157 $\beta$ 2.11 832 Karin 2.11 .../.../ S 5.8 2.866 845 Naema 11.96 .../ C /... $\gtrsim$20 ... 4562 Iannini 12.17 .../.../ S $\lesssim$5 2.644 M/N 15.0 1521 Seinajoki 15.02 .../.../... ... 2.852 For each of the three dust bands in the COBE model, the density is given by $$n_{B}\left(\mathbf{R}\right) = \frac{3N_0}{R} e^{-\left(\zeta/\delta_\zeta\right)^6} \left(1 + \frac{(\zeta/\delta_\zeta)^{p}}{v_B}\right) (1-e^{-(R/\delta_R)^{20}}), \label{eq:bandDensity}$$ where we have used a simplified notation based on that of K98, where one can also find the numerical values for the parameters. Note that Eqn. \[eq:bandDensity\] matches the code used for the Zodiacal model (which can be found on the LAMBDA website), but that there is a factor of $1/v_B$ difference between Eqn. \[eq:bandDensity\] and Eqn. 8 of K98 ($v_B$ is a shape parameter, not a frequency). Also, K98 assumed that the emissivities of the three sets of bands were all equal. We relax this assumption below and allow the emissivities of each of the sets of bands to be different. The bands are shown, assuming unit emissivity, as the first, second, and third rows in Fig. \[fig:zodiMaps\]. ### Circumsolar ring and Earth-trailing feature {#sec:ringmath} The functional form for the density of the circumsolar ring is taken to be $$n_{B}\left(\mathbf{R}\right) = n_{SR}\cdot e^{-\left(R-R_{SR}\right)^2/\sigma_{rSR}^2-\left\|Z_R\right\|/\sigma_{zSR}}.$$ Similar to the treatment of the bands, K98 assumed that the emissivity of the circumsolar ring was the same as that of the Earth-trailing feature, below. We also relax this assumption, and allow them to be different. The shape of the expected signal from the circumsolar ring is shown in the fourth row of Fig. \[fig:zodiMaps\]. The density of the Earth-trailing feature is given by $$n_{B}\left(\mathbf{R}\right) = n_{TB}\cdot e^{-\left(R-R_{TB}\right)^2/\sigma_{rTB}^2-\left\|Z_R\right\|/\sigma_{zTB} -\left(\theta-\theta_{TB}\right)^2/\sigma_{\theta TB}^2}.$$ We note that for both the circumsolar ring and the Earth-trailing feature, there is a typo in the text of K98 – a factor of 2 in the denominator of the first and third terms in the exponential has been added in the text, compared to what is in the code. We follow the code. The expected signal from the Earth-trailing feature is shown in the fifth row of Fig. \[fig:zodiMaps\]. ### Integrated Emission The total Zodiacal emission is calculated as $$I_x\left(\nu\right) = \epsilon_x\int d\mathbf{R} \cdot n_x\left(\mathbf{R}\right) \cdot B\left(\nu, T\left(\mathbf{R}\right)\right),$$ where $x$ is the Zodiacal component, $\nu$ is the frequency, $\mathbf{R}$ is a location in the Solar System, and $B\left(\nu, T\left(\mathbf{R}\right)\right)$ is the Planck function for the given frequency and temperature at the given location, given by $T_0/R^\delta$, with $T_0$ and $\delta$ being parameters. $\epsilon$ is the emissivity for the given component, which we will be finding with our fit. $n_x$ is the density for the given component, described above. The integral is done along the line of sight, from the location of the satellite to 5.2AU. Galactic Emission Seen Through Sidelobes {#subsec:galFSL} ---------------------------------------- Like all real telescopes, has some small sensitivity to radiation arriving at the telescope from off-axis, often called spillover or ‘far sidelobes’. Note that here we are referring to light coming from more than five degrees from the instrument’s nominal pointing direction, which should not be confused with ‘near sidelobes’, discussed in the HFI transfer function and beams papers [@planck2013-p03c see also [@tauber2010b]]. Figure \[fig:sidelobeDiagram\] shows the main stray light routes. ![Origin of far sidelobes. The ‘SR Spillover’ (for ‘Secondary Reflector Spillover’; the lowest set of rays on the left of the figure) arrives at the focal plane from outside the secondary mirror, directly from the sky. The ‘PR Spillover’ (for ‘Primary Reflector Spillover’), arrives at the focal plane from above the primary mirror and reflects off of the secondary to arrive at the focal plane. The set of rays between these two contributions represents the main beam. The ‘baffle’ contribution, light reaching the focal plane after reflecting from the inner sides of the baffles, is not shown here. It is often included as part of the SR Spillover. Adopted from [@tauber2010b]. []{data-label="fig:sidelobeDiagram"}](Rays-sketch-2.pdf){width="88mm"} The secondary reflector (SR) spillover arises from radiation that reaches the focal plane without reflecting off of the primary reflector. As such, a major component is radiation from the general direction of the telescope boresight, though well outside the main beam. The ‘baffle’ contribution to the SR spillover results from radiation which reflects off of the baffles to arrive at the focal plane. The primary reflector (PR) spillover arises from radiation that comes to the satellite from just above the primary mirror, reflects off the secondary mirror and arrives at the detectors. At the highest frequencies, the Galactic centre is bright enough to be seen through these far sidelobes, even if faintly. Since the orientations of these sidelobes change as the instrument scans, the survey differences done to detect the Zodiacal emission are also sensitive tests of the FSLs. Though the Galactic emission mechanism and amplitude is different, analogous effects are discussed for the LFI in [@planck2013-p02d]. To this point, resource constraints have limited this study to a single far sidelobe calculation for all detectors. We use a GRASP calculation of the far sidelobes for the 353-1 horn [see Fig. 9 of @planck2013-p03c] and do not attempt to correct for differences in frequency or location for other horns. While this is not optimal, and will be improved in later releases, we note that the primary, large-scale features of the far sidelobes are defined by the telescope, rather that the horns or their placement, so first-order effects should be captured. Some of the limitations imposed by this are discussed in Sect. \[sec:farSidelobeResults\]. To make templates of what we might see from the Galactic centre through the far sidelobes, we use the simulation software described in [@reinecke2006], but use GRASP calculations of the the various far sidelobe components instead of main beams. As inputs to the simulations, we use the actual maps at the appropriate frequency. This allows us to account for differences in the brightness of the Galaxy as a function of frequency with minimal effort – the Galactic templates should already be calibrated in the correct units at the given frequency. The far sidelobe templates are made at the timeline level and run through the relevant parts of the pipeline software. In particular, the offset removal, or ‘destriping’ must be done on the templates before fitting in order to get reasonable fit results. The resulting templates made using these FSL calculations with a 857GHz sky as input are shown in the bottom three rows of Fig. \[fig:FSLs\]. $\begin{array}{cc\|c} \includegraphics[width=60mm]{Z857JackSim_survey1_Dipolar_direct.jpg} & \includegraphics[width=60mm]{Z857JackSim_survey2_Dipolar_direct.jpg} & \includegraphics[width=60mm]{Z857JackSim_surveyD_Dipolar_direct.jpg} \\ \includegraphics[width=60mm]{Z857JackSim_survey1_Dipolar_secondary.jpg} & \includegraphics[width=60mm]{Z857JackSim_survey2_Dipolar_secondary.jpg} & \includegraphics[width=60mm]{Z857JackSim_surveyD_Dipolar_secondary.jpg} \\ \includegraphics[width=60mm]{Z857JackSim_survey1_Dipolar_baffle.jpg} & \includegraphics[width=60mm]{Z857JackSim_survey2_Dipolar_baffle.jpg} & \includegraphics[width=60mm]{Z857JackSim_surveyD_Dipolar_baffle.jpg} \\ \hline \includegraphics[width=60mm]{Z857JackSim_survey1_Galactic_direct.jpg} & \includegraphics[width=60mm]{Z857JackSim_survey2_Galactic_direct.jpg} & \includegraphics[width=60mm]{Z857JackSim_surveyD_Galactic_direct.jpg} \\ \includegraphics[width=60mm]{Z857JackSim_survey1_Galactic_secondary.jpg} & \includegraphics[width=60mm]{Z857JackSim_survey2_Galactic_secondary.jpg} & \includegraphics[width=60mm]{Z857JackSim_surveyD_Galactic_secondary.jpg} \\ \includegraphics[width=60mm]{Z857JackSim_survey1_Galactic_baffle.jpg} & \includegraphics[width=60mm]{Z857JackSim_survey2_Galactic_baffle.jpg} & \includegraphics[width=60mm]{Z857JackSim_surveyD_Galactic_baffle.jpg} \end{array}$ One factor for which we do not account with these templates is the difference in spillover between the different frequencies [@Lamarre2010; @tauber2010b]. Since we illuminate more of the telescope at lower frequencies than at higher frequencies, if our templates simply scale with the spillover, we will expect different fit values for our templates at different frequencies. To illustrate the relative contributions of these various templates, in Fig. \[fig:evolution\] we show a series of maps. Maps in each row are similar to those in the previous row, except that one more template or group of templates has been added to form the new row. For all rows, the first column corresponds to data from the first survey, the second column corresponds to data from the second survey, and the third column is the difference of the second column minus the first. The first row shows the sum of all the far sidelobes; the second row shows the result when we add dust band 1 to the far sidelobes – note that the scales change from the first to the second row; the third row shows the sum of the far sidelobes and the first two bands; the fourth row shows the sum of the far sidelobes and all the dust bands; the fifth row shows what’s in the fourth row, plus the circumsolar ring and Earth-trailing feature; finally, the bottom row shows the sum of the far sidelobes and all Zodiacal templates (and the scales have again changed). As this is simply illustrative, we have assumed unit emissivities for the Zodiacal components, and multiplied the far sidelobe components by a factor of 15 (which we will see in Sect. \[sec:fit\] is representative of the most extreme, multi-moded case). With these caveats in mind, the survey difference of the sums of all the components, the lower right image, can be compared to the bottom image in Fig. \[fig:surveymaps\]. $\begin{array}{cc\|c} \includegraphics[width=60mm]{Z857JackSim_survey1_5.jpg} & \includegraphics[width=60mm]{Z857JackSim_survey2_5.jpg} & \includegraphics[width=60mm]{Z857JackSim_surveyD_5.jpg} \\ \hline \includegraphics[width=60mm]{Z857JackSim_survey1_6.jpg} & \includegraphics[width=60mm]{Z857JackSim_survey2_6.jpg} & \includegraphics[width=60mm]{Z857JackSim_surveyD_6.jpg} \\ \includegraphics[width=60mm]{Z857JackSim_survey1_7.jpg} & \includegraphics[width=60mm]{Z857JackSim_survey2_7.jpg} & \includegraphics[width=60mm]{Z857JackSim_surveyD_7.jpg} \\ \includegraphics[width=60mm]{Z857JackSim_survey1_8.jpg} & \includegraphics[width=60mm]{Z857JackSim_survey2_8.jpg} & \includegraphics[width=60mm]{Z857JackSim_surveyD_8.jpg} \\ \includegraphics[width=60mm]{Z857JackSim_survey1_10.jpg} & \includegraphics[width=60mm]{Z857JackSim_survey2_10.jpg} & \includegraphics[width=60mm]{Z857JackSim_surveyD_10.jpg} \\ \hline \includegraphics[width=60mm]{Z857JackSim_survey1_11.jpg} & \includegraphics[width=60mm]{Z857JackSim_survey2_11.jpg} & \includegraphics[width=60mm]{Z857JackSim_surveyD_11.jpg} \end{array}$ Spectrum {#sec:fit} ======== We fit the data shown in the bottom panel of Fig. \[fig:surveymaps\], as well as the analogous data at other frequencies and for the second year of observations, to a constant plus combinations of the templates shown in Sect. \[sec:model\], Figs. \[fig:zodiMaps\] and \[fig:FSLs\]. For each fit, we assume that the survey difference map at sky pixel $p$, called $D_{p}$, can be modelled as $$D_{p} = \sum_t \left( \epsilon_{2,t}\cdot T_{2,t,p}-\epsilon_{1,t}\cdot T_{1,t,p} \right) + \mathrm{constant},$$ where $\epsilon_{\left[1\|2\right],t}$ is the emissivity fit for template $t$ during survey 1 or 2 at the given frequency and $T_{\left[1\|2\right],t,p}$ is the value of the $t^\mathrm{th}$ template at pixel $p$ for survey 1 or 2, calculated as described in Sect. \[sec:model\]. For example, for our ‘basic’ fit, we will have 19 templates – one each for the diffuse cloud, circumsolar ring, Earth-trailing feature and each of the three dust bands, as well as one each for the Galactic far sidelobes. All of these are repeated twice; once for each survey in a yearly difference map. Finally we also fit to an overall constant, to which is not sensitive. We then minimize $$\chi^2 = \sum_p \left(\Delta_{p} - D_{p}\right)^2.$$ Separating the templates into surveys has the disadvantage of increasing the number of parameters in our fits. While we do not expect either the emissivity of the Zodiacal emission or the far sidelobe calculations to change from survey-to-survey, we separate them in this way for two reasons. The first is simply as a basic reality check – if we see significant differences between fits to two different surveys, we should be sceptical. Beam asymmetries or transfer function effects, for example, might cause differences from survey-to-survey, as might imperfections in the model itself. The second reason is to calculate the error bars. As just noted, we are often as concerned by systemic effects as much as by “random” noise. By separating the data by survey, we may calculate error bars using the standard error of the successive measurements as a proxy for the uncertainties, rather than propagating white noise estimates. This should allow us a more conservative estimate of our uncertainties, which accounts for model deficiencies or low levels of systematics which change by survey (the aforementioned beam asymmetries and transfer functions, for example). A note about weighting: the bulk of the Zodiacal emission is, of course, in the ecliptic plane. , on the other hand, has more statistical weight, on a pixel-by-pixel basis, at the ecliptic poles. We therefore use uniform weights over pixels, rather than statistical weights, since this would down-weight specifically the regions with our signal. As mentioned above, we fit each of the two years’ survey difference maps to a cloud, circumsolar ring, Earth-trailing feature, three bands and three far sidelobe templates, plus a constant. The results for the four 857 GHz horns are shown in Fig. \[fig:fit857\]. Averaging over horns and surveys at all six HFI frequencies yields Fig. \[fig:noDipoleSidelobesFit\]. Numerical values are given in tables \[tab:fullFit\], \[tab:fullFitBands\] and \[tab:galacticSidelobeFit\]. ![image](1111101111000__857__10.0){width="180mm"} ![image](Emis_1111101111000_bcut10.0_T0286.0_delta0.4669_alpha1.337_beta4.142_gamma0.942_mu0.189) $\nu$ (GHz) ------------- -------------------- -------------------- -------------------- 857 0.301 $\pm$ 0.008 0.578 $\pm$ 0.359 0.423 $\pm$ 0.114 545 0.223 $\pm$ 0.007 0.591 $\pm$ 0.203 -0.182 $\pm$ 0.061 353 0.168 $\pm$ 0.005 -0.211 $\pm$ 0.085 0.676 $\pm$ 0.149 217 0.031 $\pm$ 0.004 -0.185 $\pm$ 0.143 0.243 $\pm$ 0.139 143 -0.014 $\pm$ 0.010 -0.252 $\pm$ 0.314 -0.002 $\pm$ 0.180 100 0.003 $\pm$ 0.022 0.163 $\pm$ 0.784 0.252 $\pm$ 0.455 : Emissivities of the diffuse cloud, circumsolar ring, and Earth-trailing feature from fit result averages.[]{data-label="tab:fullFit"} $\nu$ (GHz) Band 1 Band 2 Band 3 ------------- ------------------- ------------------- ------------------- 857 1.777 $\pm$ 0.066 0.716 $\pm$ 0.049 2.870 $\pm$ 0.137 545 2.235 $\pm$ 0.059 0.718 $\pm$ 0.041 3.193 $\pm$ 0.097 353 2.035 $\pm$ 0.053 0.436 $\pm$ 0.041 2.400 $\pm$ 0.100 217 2.024 $\pm$ 0.072 0.338 $\pm$ 0.047 2.507 $\pm$ 0.109 143 1.463 $\pm$ 0.103 0.530 $\pm$ 0.073 1.794 $\pm$ 0.184 100 1.129 $\pm$ 0.154 0.674 $\pm$ 0.197 1.106 $\pm$ 0.413 : Emissivities of the three dust bands from fit result averages.[]{data-label="tab:fullFitBands"} ------- ----------------- ----- ----------------- ---------------- ------- $\nu$ (GHz) Direct Baffle 100 -25.8 $\pm$ 5.7 7 26.3 $\pm$ 15.7 56.7 $\pm$ 5.7 10 143 -9.1 $\pm$ 4.1 6 13.0 $\pm$ 4.8 23.8 $\pm$ 5.4 10 217 0.6 $\pm$ 1.1 5 6.3 $\pm$ 2.1 6.5 $\pm$ 1.3 6 353 -1.2 $\pm$ 0.5 1 -4.3 $\pm$ 2.1 3.6 $\pm$ 0.7 1 545 7.7 $\pm$ 1.7 15 8.8 $\pm$ 3.1 7.9 $\pm$ 1.0 1 857 17.1 $\pm$ 3.4 1.5 23.9 $\pm$ 4.2 16.7 $\pm$ 3.1 0.005 ------- ----------------- ----- ----------------- ---------------- ------- : Fit coefficients for the Galaxy seen through the far sidelobes.[]{data-label="tab:galacticSidelobeFit"} $\begin{array}{cc\|c} \includegraphics[width=60mm]{Z857SurveyJackknifeWiZodi.jpg} & \includegraphics[width=60mm]{Z857SurveyJackknifeNoZodi.jpg} & \includegraphics[width=60mm]{Z857Correction.jpg} \\ \includegraphics[width=60mm]{Z545SurveyJackknifeWiZodi.jpg} & \includegraphics[width=60mm]{Z545SurveyJackknifeNoZodi.jpg} & \includegraphics[width=60mm]{Z545Correction.jpg} \\ \hline \includegraphics[width=60mm]{Z353SurveyJackknifeWiZodi.jpg} & \includegraphics[width=60mm]{Z353SurveyJackknifeNoZodi.jpg} & \includegraphics[width=60mm]{Z353Correction.jpg} \\ \includegraphics[width=60mm]{Z217SurveyJackknifeWiZodi.jpg} & \includegraphics[width=60mm]{Z217SurveyJackknifeNoZodi.jpg} & \includegraphics[width=60mm]{Z217Correction.jpg} \\ \includegraphics[width=60mm]{Z143SurveyJackknifeWiZodi.jpg} & \includegraphics[width=60mm]{Z143SurveyJackknifeNoZodi.jpg} & \includegraphics[width=60mm]{Z143Correction.jpg} \\ \includegraphics[width=60mm]{Z100SurveyJackknifeWiZodi.jpg} & \includegraphics[width=60mm]{Z100SurveyJackknifeNoZodi.jpg} & \includegraphics[width=60mm]{Z100Correction.jpg} \end{array}$ Discussion {#sec:discussion} ========== As discussed in the HFI processing paper [@planck2013-p03] and in the explanatory supplement [@planck2013-p28], these fits are used to create the implied Zodiacal and sidelobe emission in each HFI observation, which can then be removed and the maps are recreated. The survey 2 minus survey 1 difference maps for the 857-1 horn both before and after Zodiacal and far sidelobe removal, are shown in Fig. \[fig:beforeAndAfterJackknives\]. In this section we discuss the fit implications further. Far Sidelobes {#sec:farSidelobeResults} ------------- In addition to fit values obtained for the Galaxy seen in each component of the far sidelobes, the Cols. labelled ‘Pred.’ in table \[tab:galacticSidelobeFit\] show the expected values of the spillover, normalized to that of the 353GHz channel, from [@tauber2010b]. These are the ratios of the expected spillover in each frequency, compared to that at 353GHz, the frequency for which the sidelobe calculations were done. Since the fit values account for the changes in Galactic emission with frequency, if our predictions and data were perfect, the fit values would match those of the predictions. The FSL signature is clearly visible at 857GHz in the bottom panel of Fig. \[fig:surveymaps\], and this is reinforced in table \[tab:galacticSidelobeFit\]. As the 857 and 545GHz channels are multi-moded, the differences are not that surprising; it is difficult to do the calculations necessary for the prediction. In addition, the specifications for the horn fabrication were quite demanding, and small variations, though still within the mechanical tolerances, could give large variations in the amount of spillover. For the lower frequency, single-moded channels, however, the situation is different. There is no clear detection of PR spillover. While the significant negative values may indicate some low-level, large-scale systematic, there seems to be nothing with the distinctive signature of primary spillover at frequencies between 100 and 353GHz. For the direct contribution of the secondary SR spillover, the situation is similar at 353GHz, but at 217 and 143GHz we are finding a 3$\sigma$ detection at about the level expected, while at 100GHz the value is about 2.5 times higher than expected, though the signal-to-noise on the detection is less than 2$\sigma$. The baffle contribution to the SR spillover seems to be in accord with expectations at 353 and 217GHz, and higher than what is predicted at 100GHz. The values for the PR spillover, which is the most distinctive of the far sidelobe patterns, and presumably the easiest to disentangle from other effects, would indicate that the PR spillover values in table 2 of [@tauber2010b] may be slightly overestimated. The values for the direct contribution of the SR spillover roughly confirm the far sidelobe calculations. The baffle contribution to the SR spillover seems a bit high. We take the ensemble of these numbers as rough confirmation that our beam calculations are not drastically incorrect, but do not use the specific numbers in either [@planck2013-p03c] or [@planck2013-p03b]. Similar conclusions are drawn for the LFI in [@planck2013-p02d]. We have also included a template of the Dipole as seen through the far sidelobes in the fits in some fits to check to see if they are detected. As expected, they are not. The results quoted above are from fits which do not include these Dipole templates. Diffuse cloud {#diffuse-cloud} ------------- Figure \[fig:noDipoleSidelobesFit\] shows the emissivity of the diffuse cloud falling off with increasing wavelength, as would be expected for particles with characteristic sizes of order 30. The dashed line in this figure shows a flat emissivity to 150, with values proportional to the frequency-squared at longer wavelengths, for comparison with Fig. 2 of [@fixsendwek2002], who used FIRAS data to investigate the far-infrared/sub-millimetre behaviour of the Zodiacal cloud. These results are consistent with their conclusions[^5]. As the diffuse cloud is so much brighter in the mid-infrared than in the CMB, its relatively low level at wavelengths has been exploited in [@planck2013-p03d] to set in-flight limits on any possible out-of-band leaks in the instrument’s spectral transmission. Circumsolar ring and Earth-trailing feature {#circumsolar-ring-and-earth-trailing-feature} ------------------------------------------- We draw no conclusions about the circumsolar ring or the Earth-trailing feature. The fit values obtained for their emissivities are inconsistent from frequency-to-frequency, their values often being negative. This remains true for data when the fits are done requiring that the two components have the same emissivities, as was done in K98. Inspection of the panels labelled ‘Ring’ and ’Feat.’ (i.e., the middle- and lower-left panels) of Fig. \[fig:fit857\] show systematic differences between results from even- and odd-numbered surveys (that is, the circles and squares seem to be systematically different, regardless of whether they are blue or red). ’s observing pattern was different for odd and even numbered surveys, but very similar for either even numbered surveys alone, or for odd numbered surveys alone. Random noise is not an issue, since the measurements with similar observations are repeatable, so this is either an indication that the circumsolar ring and Earth-trailing model templates themselves need improvement at wavelengths, or that systematic errors are affecting these specific components. We note that the conclusions presented elsewhere in this work remain essentially the same whether or not we include the circumsolar ring and Earth-trailing feature. Bands {#subsec:Bands} ----- An interesting feature of Fig. \[fig:noDipoleSidelobesFit\], and the primary result of this work, is the difference in the emissivities of the bands compared to that of the diffuse cloud, pointing towards differences in the particles in these different components. While there may be hints of this in the longest wavelength DIRBE data, the effect becomes clear at wavelengths. This is not necessarily unexpected. The composition of the diffuse cloud is still disputed, but is often claimed to be both asteroidal and cometary [see, for example, @Kortenkamp1998; @Nesvorny2010; @Tsumura2010]. Since the bands, on the other hand, are understood to be asteroidal debris only [@sykes86], the difference may simply be a reflection of these different origins. The fact that the fitted emissivities of bands 1 and 3 rise above unity is perplexing. At first glance, one might imagine some new, cold component in the cloud which would cause an excess that might be interpreted as an excess in emissivity in some other component. However, to peak around 545GHz, this component would have to have a temperature of the order of 10K, and therefore be much more distant than most of the dust usually associated with the Zodiacal cloud. While enticing, it is difficult to understand how such a component could survive the differencing process used in this analysis, which reduces signals from distant sources more than those nearby, or how such a component could mimic an excess in two dust bands above from the ecliptic plane, but not do the same in the other Zodiacal cloud components. One might worry that covariance between between the various components might be causing problems in the fitting procedure. To check this, we have repeated the fit including and omitting various combinations of the the circumsolar ring and Earth-trailing feature, or both, and assuming their emissivities were independent or equal. In no case did the difference between the diffuse cloud and the dust bands disappear. The excess may ultimately be explained by degeneracies in the model for the density of the bands. As presented in Sect. \[sec:bandmath\], the normalisation of the density of particles is completely degenerate with the emissivity for each band. In addition, the emission is also roughly proportional to the temperature normalisation, because we are observing in the Rayleigh-Jeans tail of the Zodiacal emission. While any overall change in the temperature of the IPD particles would scale all components of the Zodiacal emission, because temperature is nearly inversely proportional to distance from the Sun, the location of the bands are important. While the excess over 1 is too large to be explained by errors in distance and thus temperatures alone, one might appeal to a change in a combination of distance, particle density normalisation and emissivity of these bands in particular to arrive at mutually consistent results for both and DIRBE. As this will involve simultaneous work with both and DIRBE data, it is beyond the scope of this paper. Bands 1 and 3 also seem to show different behaviour than band 2. Since bands 1 and 3 are both at high ecliptic latitude, while band 2 is not, one might again worry that one of the other templates to which we are fitting might have significant overlap with a subset of the bands, which in turn causes an apparent difference in emissivities. To check this, we have repeated the fits with and without various combinations of the cloud, circumsolar ring and Earth-trailing feature, as well as the far sidelobes. In all cases, bands 1 and 3 are always significantly different than dust band 2. When the diffuse cloud itself is omitted from the fit, the emissivity of dust band 2 goes up, but is still distinctively different from that of dust bands 1 and 3. As dust band 2 is a combination of the IRAS $\alpha$ and $\beta$ bands, one may also worry that one of these two is more important for the shorter IRAS and COBE wavelengths, but that the other might be more important for the longer wavelengths. We note that its emission is dominated by contributions from the Karin/Koronis family [see @Nesvorny2008 Fig. 1], but have therefore confirmed specifically that varying the $\delta_\zeta$ parameter of the second band between values appropriate for either $\alpha$ or $\beta$ does not remove this difference (see table \[tab:bandInfo\]). If the age of band 2 were significantly different from those of bands 1 and 3, we might argue that Poynting-Robertson drag had depleted some of the bands of more smaller particles than the others [@Wyatt2011]. N03 and N08, however, have estimated the ages of most of the asteroid families which might be associated with the bands (reproduced in table \[tab:bandInfo\]), and the age of any of the associations with band 2 is between those of any of the possible associations with bands 1 or 3. These same figures tend to rule out modifications of the material properties due to photo-processing or solar wind exposure for differing periods. Band 2 also seems to be roughly the same distance from the Sun as the other two bands, so it is difficult to appeal to differences in environment as the cause. We speculate on the following to explain any differences: Veritas, the asteroid family proposed to be associated with dust band 1, is classified as carbonaceous [@Bus2002]. As noted above, the IRAS $\beta$ band, associated with the Karin family of asteroids, seems to dominate the emission from dust band 2. Karin and its larger sibling, Koronis, are classified as siliceous, or stony, objects [@Bus2002; @Carvano2010]. While the dust band 3 has a number of asteroid families which may be contributing to it (see table \[tab:bandInfo\]), we propose that the emission is dominated by carbonaceous-based asteroid families (three quarters of the asteroids in the Solar System are carbonaceous), and that the difference in emissivities between dust band 2 and dust bands 1 and 3 arises from this difference in composition. The differing emissivities may be either due to this intrinsic composition difference, or the size-frequency distribution of particles that results from different kinds of asteroids colliding [@Grogan2001 for example]. This explanation would not be valid if it were to turn out that dust band 3 was dominated by dust associated with the Iannini asteroid, for example, as it is siliceous. Implications for the CMB ------------------------ The right-hand column of Fig. \[fig:beforeAndAfterJackknives\] shows the Zodiacal emission implied by the fits, created by subtracting the maps made applying the Zodiacal emission correction from those that were made without it applied. One can see here the difference in the relative amplitudes of the bands versus the diffuse cloud, the bands being relatively more important at low frequencies than at high. Figure \[fig:Cls\] shows the power spectra of the Zodiacal correction maps, all in units of $\left(\mu\mathrm{K}_\mathrm{CMB}\right)^2$. Here, the cloud is seen at multipoles of less than about 10, while the bands and other structures are see in higher multipoles. ![Power spectra of the Zodiacal correction maps shown in the right-hand column of Fig. \[fig:beforeAndAfterJackknives\]. Black plus signs, grey crosses, cyan triangles, orange diamonds, blue circles, and red squares represent, respectively, 857, 545, 353, 217, 143, and 100GHz. For the CMB channels, even multipoles are shown with filled symbols, while odd multipoles are shown with empty symbols. This “even-odd” pattern is a reflection of the symmetry around the ecliptic plane – odd multipoles are almost absent, as they would indicate structure in the maps that were anti-symmetric about the ecliptic plane. A standard $\Lambda$CDM CMB temperature spectrum is shown as the grey line roughly half-way down the plot, orders of magnitude above the Zodiacal spectrum in the CMB channels.[]{data-label="fig:Cls"}](ZCorrectionCls.pdf){width="88mm"} At 143GHz, The signal reaches a few $\mu\mathrm{K}_\mathrm{CMB}$ in the map, while the power spectrum has values of the order of one$\left(\mu\mathrm{K}_\mathrm{CMB}\right)^2$. The absence of power in the odd multipoles is a reflection of the north-south symmetry of the signal. These can be compared with the CMB temperature spectrum, which is shown as the black line about halfway between the top and the bottom of the plot[^6]. The Zodiacal emission correction spectra are orders of magnitude smaller than the CMB spectrum. The Zodiacal emission does not compromise ’s cosmological results. Conclusion {#sec:conclusion} ========== Zodiacal emission has long been an important foreground for searches for the extra-galactic background at infrared wavelengths. With the ever-increasing sensitivity of CMB experiments, it will soon become important to account for in sub-millimetre and CMB analyses as well. The K98 model does fairly well in modelling the diffuse Zodiacal cloud emission at wavelengths, as long as appropriate emissivities are assumed. It does less well, however, in modelling the other features. Because they appear to be more emissive at these frequencies than the cloud, the bands contribute more to the Zodiacal emission relative to the diffuse cloud at CMB frequencies (i.e., near 143GHz, or 2.1mm). The 2013 release includes both maps which have had Zodiacal emission removed, and maps which have not had Zodiacal emission removed. We note that [@planck2013-p06] found better results using maps that had not had the Zodiacal emission removed to make estimations of the CMB and other astrophysical components in the HFI maps. The component separation methods used there naturally correct for a large amount of Zodiacal emission, as it is spectrally similar to Galactic dust emission in the CMB channels. In addition, one can see in Fig. \[fig:noDipoleSidelobesFit\] and table \[tab:fullFit\] that the diffuse cloud is not actually detected at 143GHz, the frequency which carries the most weight in CMB analyses. The Zodiacal emission removal, of course, depends delicately on how this non-detection is treated. This will be addressed in future releases. Improvements in modelling of the circumsolar ring, Earth-trailing feature, and the dust bands, as well as inclusion of fainter and partial bands [e.g., @Espy2009] should be done to make truly “clean” CMB maps. As these bands are believed to be the products of asteroid collisions, further study of these bands at these wavelengths may also inform us on the nature of the intermediate-sized particles created during the destruction of the associated asteroids. We may hope to learn not only more about the size distribution, but also the differences between, for example, the results of a collisions involving siliceous and carbonaceous asteroids. Just as material from asteroid collisions contributes to Zodiacal emission, material shed from comets must also contribute. As part of the HFI data reduction process, we mask out solar system objects which would cause “noise” in the final sky maps. We searched for comets as part of this process, but found only one (Christensen). For completeness, appendix \[sec:Ast\] presents this information, as it is not possible to extract from the data in the 2013 map release. We have not yet detected extended tails of comets. One of the primary goals of the next stage of analyses, once this “nearby” IPD has been completely removed, will be to search for dust associated with the Kuiper Belt, or set limits upon it. The full-mission data release will include polarisation information. While it is not expected, limits will be put on possible contamination of the polarisation of the CMB by polarised Zodiacal emission, as has been done here for temperature. Work is now underway to address all these points for the next data release. While the signal is quite small – at CMB wavelengths the signal we are discussing is orders of magnitude smaller than the primary CMB anisotropies, it is detectable and should be subtracted from the data. There will be improvements in dust modelling, improvements in satellite modelling, and additions to address polarisation. The ultimate goal will be simultaneous analyses with IRAS, COBE, and Akari and other to understand the large-scale Zodiacal emission from the near-infrared to the microwave. This paper benefited from exchanges with Dale Fixsen, Tom Kelsall and Janet Weiland. We acknowledge the IN2P3 Computer Center (<http://cc.in2p3.fr>) for providing a significant amount of the computing resources and services needed for this work. We acknowledge the use of the Legacy Archive for Microwave Background Data Analysis (LAMBDA). Support for LAMBDA is provided by the NASA Office of Space Science. The development of has been supported by: ESA; CNES and CNRS/INSU-IN2P3-INP (France); ASI, CNR, and INAF (Italy); NASA and DoE (USA); STFC and UKSA (UK); CSIC, MICINN, JA and RES (Spain); Tekes, AoF and CSC (Finland); DLR and MPG (Germany); CSA (Canada); DTU Space (Denmark); SER/SSO (Switzerland); RCN (Norway); SFI (Ireland); FCT/MCTES (Portugal); and PRACE (EU). A description of the Collaboration and a list of its members, including the technical or scientific activities in which they have been involved, can be found at <http://www.sciops.esa.int/index.php?project=planck&page=Planck_Collaboration>. Basic Behaviour --------------- Figure \[fig:Asts\] demonstrates some of the basics of the asteroid detections. For each measurement, we denote the measurement as $f$, the distance between and the object as $d$, and the distance between the Sun and the object as $s$. These quantities for the first measurement of each asteroid will be denoted $f_0$, $d_0$, and $s_0$. In the top panel, we show asteroids that were detected in multiple surveys. Assuming the temperature of the object at any time is inversely proportional to $\sqrt{s}$, we would expect $$f\sqrt{s} \propto \frac{1}{d^2},$$ which is roughly seen in the data. ![image](Asts.pdf) In the bottom panel, we show the 545-to-857GHz spectral indices, as well as the 545-to-353GHz spectral indices for Ceres and Vesta. [^1]: (http://www.esa.int/Planck) is a project of the European Space Agency (ESA) with instruments provided by two scientific consortia funded by ESA member states (in particular the lead countries France and Italy), with contributions from NASA (USA) and telescope reflectors provided by a collaboration between ESA and a scientific consortium led and funded by Denmark. [^2]: <http://sci.esa.int/science-e/www/object/index.cfm?fobjectid=47345>. [^3]: Legacy Archive for Microwave Background Data Analysis – NASA (<http://lambda.gsfc.nasa.gov/>). [^4]: While three bands are described in the text of K98, there are actually four in the code. There is, however, no ambiguity, as the density of the fourth band is set to zero in the code at the LAMBDA site. [^5]: Care must be taken with direct comparisons between the emissivities quoted in [@fixsendwek2002], who quote emissivities relative to a 245K cloud, and those in this work, which follow K98 and assume a cloud of temperature 286K at 1AU from the Sun. [^6]: Made with CAMB – <http://camb.info>	ArXiv
--- abstract: 'We experimentally study a vacuum-induced Autler-Townes doublet in a superconducting three-level artificial atom strongly coupled to a coplanar waveguide resonator and simultaneously to a transmission line. The Autler-Townes splitting is observed in the reflection spectrum from the three-level atom in a transition between the ground state and the second excited state when the transition between the two excited states is resonant with a resonator. By applying a driving field to the resonator, we observe a change in the regime of the Autler-Townes splitting from quantum (vacuum-induced) to classical (with many resonator photons). Furthermore, we show that the reflection of propagating microwaves in a transmission line could be controlled by different frequency single photons in a resonator.' author: - 'Z.H. Peng' - 'J.H. Ding' - 'Y. Zhou' - 'L.L. Ying' - 'Z. Wang' - 'L. Zhou' - 'L.M. Kuang' - 'Yu-xi Liu' - 'O. Astafiev' - 'J.S. Tsai' title: 'Vacuum-induced Autler-Townes splitting in a superconducting artificial atom' --- Electromagnetic waves propagating through a medium of identical atoms are resonantly absorbed. The absorption can be eliminated when a strong driving field couples other atomic transitions in, for example, three-level atoms, creating a transparency window for the waves. This results in either Autler-Townes splitting (ATS) [@Autler1955] or electromagnetically induced transparency (EIT) [@Harris1990; @Harris1997] and is extensively studied in quantum optics [@Fleischhauer2005; @Scully-book]. The same phenomena can be observed even if the medium is replaced by a single atom or a molecule which is coupled to either a cavity [@Muller2007; @Mucke2010] or an open space [@Tey2008; @Hwang2009]. However, the strong coupling between the driving field and a single natural atom is difficult to achieve. Recently, the strong coupling has been experimentally achieved between a superconducting artificial atom and non-quantized microwave fields confined in a one-dimensional transmission line [@Astafiev2010]. This enables ATS and EIT to be observed using a single superconducting artificial atom [@Murali2004; @Ian2010; @Sun2014; @Gu2016]. Several experiments have already shown ATS [@Baur2009; @Sillanpaa2009; @Abdumalikov2010; @Li2012; @Hoi2013PRL; @Hoi2013; @Novikov2013; @Liu2017] and state manipulation [@Kelly2010; @Xu2016] in a single three-level artificial atom. ATS [@Suri2013] and EIT [@Sovikov2015] have been demonstrated in a Jaynes-Cummings ladder system of a single two-level artificial atom dispersively coupled to a cavity. ![([a]{}) Micrograph of the device, comprising a superconducting artificial atom capacitively coupled to a transmission line resonator (meandering structure). Additionally, the artificial atom is coupled to a transmission line (on the left side) used to directly measure the reflection spectrum from the atom. ([b]{}) Magnified micrograph of the artificial atom of the tunable-gap superconducting flux qubit geometry. ([c]{}) Schematic of the three-level artificial atom coupled to a resonator. The transition from the ground state to the second excited state is probed at frequency $\omega_p$. ([d]{}) Left panel: The dressed state picture of the vacuum induced (quantized) ATS due to coupling to a resonator in $\|e\rangle \leftrightarrow \|f\rangle$. Right panel: The dressed state picture of the classical ATS with many resonator photons. []{data-label="picture"}](Fig1.pdf) The observation of both ATS and EIT in a medium (an ensemble of atoms) usually requires strong classical driving fields. However, the conditions are different: the driving field required for the observation of EIT is weaker than that for ATS, and there is a crossover from ATS to EIT [@Sanders; @Yang]. In particular, in ATS the splitting is determined by the Rabi frequency of the driving field, whereas in EIT the dip between the two peaks is a result of quantum interference. These effects take place even if the medium is scaled down to a single atom and the driving field is quantized. The condition for the observation of the effects can then be reduced to a small number of photons or even without photons owing to coupling to a vacuum mode. Theoretical investigations show that the transparency for the classical probing field can still occur [@Field1993]. This has potentially important applications, for example, in single-photon switches and transistors, all-optical quantum logic, quantum communication, and metrology [@Chang2014]. However, vacuum-induced transparency has only recently been observed in an ensemble of three-level atoms [@Tanji-Suzuki2011]. Here, we demonstrate ATS with a single three-level artificial atom, which is controlled by a quantized single-mode field in a transmission line resonator. Our device, shown in Fig. \[picture\](a), consists of a three-level artificial atom capacitively and strongly coupled to two macroscopic objects simultaneously: a coplanar waveguide resonator (CPWR) and a transmission line. Although the device is reminiscent of the one studied in Ref. [@Peng2016], where the atom was coupled to two transmission lines, it is essentially different. We emphasize that in spite of the strong coupling of the atom to the transmission line and the atom to the resonator, the line and resonator are decoupled from each other (details are given in Supplementary Material [@Suppl]). This is a peculiarity of the device owing to the unique property of superconducting quantum systems: they are micrometer-scale electronic circuits. Namely, the two macroscopic objects (i.e. open lines, resonators) can be effectively decoupled from each other but, at the same time, strongly coupled to the quantum circuit. The artificial atom has the geometry of a tunable-gap flux qubit: a superconducting loop with two Josephson junctions and a dc-SQUID ($\alpha$-loop) [@Fedorov2010; @Peng2016], shown in Fig. \[picture\](b), is fabricated near the voltage antinode (open end) of the CPWR using electron-beam lithography and Al/AlO$_x$/Al shadow evaporation (see the SEM image in [@Suppl]). A weak probe signal with frequency $\omega_p$ is applied from the transmission line (left-hand side). Both the half-wavelength CPWR and the transmission line are etched from a $50\,\rm nm$ thick Nb thin film sputtered on an oxidized undoped silicon wafer. The lines have a center conductor of $10\,\mu\rm m$ width separated from the ground planes by $6\,\mu\rm m$ gaps, resulting in a $50\,\Omega$ wave impedance. Our experiment is carried out in a dilution refrigerator with a base temperature of about 30 mK. The fundamental frequency of the CPWR is $\omega_r=2\pi\times8.794\,$GHz with the decay rate $\kappa=2\pi\times3.6\,$MHz. The three lowest energy levels of the artificial atom, denoted by $\|g\rangle$ for the ground state, $\|e\rangle$ for the first excited state, and $\|f\rangle$ for the second excited state, are controlled by the magnetic flux $\Phi$ in the loop. The minimum transition energies from the ground to both excited states occur at half-integer flux quanta $\Phi_{N}=(N+\frac{1}{2})\Phi_{0}$, where $N$ is an integer and $\Phi_{0}$ is the flux quantum. The minimal energy gap $\Delta$ for the transition from $\|g\rangle$ to $\|f\rangle$ can be tuned by controlling the magnetic flux $\Phi_{\alpha}$ penetrating through the $\alpha$-loop. When the biased flux $\Phi$ is close to $\Phi_{N}$, the two lowest-energy eigenstates ($\|g\rangle$ and $\|e\rangle$) are in superposition of the two oppositely circulating persistent current states of the loop. The energy between the two lowest levels $\hbar\omega_{ge}$ in the vicinity of $\Phi_{N}$ is approximated by the expression $\sqrt{(2I_{p}\delta\Phi)^2+\Delta(\Phi_{N})^2}$, where $\delta\Phi=\Phi-\Phi_{N}$ and $I_{p}$ is the persistent current in the qubit loop. The weak dependence of $\Delta$ on $\Phi$ can be safely neglected when $\delta\Phi\ll\Phi_0$, which means that $\Delta(\Phi)\approx\Delta(\Phi_{N})$. As shown in Fig. \[picture\](c), by choosing $\Phi_{N}$ and $\delta\Phi$, we can tune both transition frequencies $\omega_{ef}$ and $\omega_{gf}$. The ATS can be observed by measuring directly the reflection spectrum through the left transmission line. As shown in Fig. \[picture\](c), the transition energy between the $\|e\rangle$ and $\|f\rangle$ states is aligned to the resonator resonance. The probe field is applied at the $\|g\rangle$ to $\|f\rangle$ transition. Thus, the effective Hamiltonian of the whole system driven by classical waves with amplitude $\Omega$ at frequency $\omega_{p}$ close to the $\|g\rangle \leftrightarrow \|f\rangle$ transition is given by $$\begin{aligned} \label{effectHami} H &=&\hbar \omega_{r} a^{\dag}a+\hbar\omega_{gf} \sigma_{ff}+\hbar \omega_{ge}\sigma _{ee} \\ \nonumber &&+\hbar g_{0}\left( a^{\dag}\sigma _{ef}+a\sigma _{fe}\right) + 2\hbar \Omega(\sigma_{fg}+\sigma_{gf}) \cos\omega_p t.\end{aligned}$$ Here, the atomic operator $\sigma_{jk}$ is defined as $\sigma_{jk}=\|j\rangle\langle k\|$ with $\{\|j\rangle,\|k\rangle\}=\{\|g\rangle,\|e\rangle, \|f\rangle\}$ and $a^\dag$ and $a$ are the photon creation and annihilation operators in the single-mode resonator, respectively. The first four terms of Eq. (1) represent the Jaynes-Cummings Hamiltonian for the two-level system with the following specific features: it presents the interaction of two excited states ($\|e\rangle$ and $\|f\rangle$) with the resonator rather than the ground and excited states. Nevertheless, the Jaynes-Cummings physics can be applied here [@JC1963; @Haroche2006; @Rempe1987]. Namely, in the vicinity of the $\|e\rangle \leftrightarrow \|f\rangle$ resonance with the resonator, $\|f\rangle$ level splits due to the interaction with the resonator vacuum mode, which can be described in terms of zero-photon dressed states as $\|v_{0}\rangle=\cos\theta\|f0\rangle-\sin\theta\|e1\rangle$ and $\|u_{0}\rangle=\sin\theta\|f0\rangle+\cos\theta\|e1\rangle$, where $\tan{2\theta} = 2g_{0}/(\omega_{r}-\omega_{f}+\omega_{e})$ and the state $\|jn\rangle=\|j\rangle\otimes\|n\rangle$ with the two quantum numbers $j$ and $n$, denoting the three-level atomic states $\|j\rangle$ and the Fock states of resonator $\|n\rangle=\lbrace\|0\rangle,\|1\rangle,\|2\rangle,...\rbrace$, respectively. The dressed states are schematically shown in the left panel of Fig. \[picture\](d), while the right panel shows the general case of a system with $n$ photons in the resonator. The last term represents the probing field which couples the ground and the second excited states ($\|g\rangle \leftrightarrow \|f\rangle$, where the vacuum mode $\|0\rangle$ is omitted for simplification). ![(a) Spectrum of the artificial atom vs biased flux plotted as a reflection coefficient $\|r\|$. The transition $\|e\rangle \leftrightarrow \|f\rangle$ is visible due to the thermal population of the $\|e\rangle$ level. (b) The spectrum with theoretically calculated energy levels (dashed lines) from the full system Hamiltonian. At $\delta\Phi\sim2.15\times10^{-3}\Phi_{0}$ (indicated by arrows), the quantum regime of ATS (vacuum splitting) is observed due to resonant coupling between the $\|e\rangle \leftrightarrow \|f\rangle$ transition and the resonator. (c) The anticrossing in the $\|g\rangle \leftrightarrow \|f\rangle$ spectroscopy line. (d) The anticrossing measured in the transition $\|e\rangle \leftrightarrow \|f\rangle$. The splitting sizes are essentially the same because both are determined by the atom-resonator coherent interaction strength $g_{0}$. The insets of (c) and (d) show the splitting at the positions indicated by arrows. []{data-label="ATS"}](Fig2.pdf) In our experiment, we first preset the flux bias to $\Phi_{N} = 0.5\Phi_{0}$ ($N$ = 0) and characterize the energy bands of our system by measuring the reflection with a vector network analyzer. A weak probe signal is applied to the transmission line from the left-hand side in Figs. \[picture\](a-b) and scanned in the frequency range from $6$ to $12.5$ GHz. By measuring the reflection amplitude $r$ versus the probe frequency $\omega_p$ and a narrow range of flux bias $\delta\Phi$ ($\delta\Phi \ll \Phi_0$), the transitions $\|g\rangle \leftrightarrow \|e\rangle$ and $\|g\rangle \leftrightarrow \|f\rangle$ are revealed as sharp dips. This is presented in Fig. \[ATS\](a) as reflection coefficient $\|r\|$ – the normalized reflection amplitude. The transition frequencies of the atom at $\delta\Phi = 0$ reach minima $\omega_{ge}=2\pi\times1.3\,\rm GHz$ and $\omega_{gf}=2\pi\times11.2\,\rm GHz$, respectively (the data below $2\pi\times6\,$GHz are not shown). In addition to the transitions from the ground state, we are able to directly measure the transitions between excited states $\|e\rangle$ and $\|f\rangle$. This is possible because of the finite population of the first excited state $\|e\rangle$ due to the non-equilibrium background radiation from the control lines. The $\|e\rangle \leftrightarrow \|f\rangle$ line is ‘brightest’ at the degeneracy point ($\delta\Phi = 0$), where $\hbar\omega_{eg}$ is minimal. It is difficult to avoid the unwanted population of the $\|e\rangle$ level with an energy gap of about 1 GHz, but in our experiment it is useful as it allows to reveal the full spectroscopy picture. The calibrated system effective temperature is around 75 mK (details in Supplementary Material [@Suppl]), which is similar to the results of other groups [@Geerlings2013; @Jin2015]. The transitions $\|g\rangle \leftrightarrow \|e\rangle$ and $\|e\rangle \leftrightarrow \|f\rangle$ are bright due to large transition matrix elements. However, the $\|g\rangle \leftrightarrow \|f\rangle$ and $\|g0\rangle \leftrightarrow \|e1\rangle$ lines vanish at $\delta\Phi = 0$ as they are prohibited due to selection rules [@Abdumalikov2010; @Inomata2012]. The dashed lines in Fig. \[ATS\](b) represent the calculations from the full-system Hamiltonian with Josephson energy $E_{J}=h \times242\,\rm GHz$ of the junctions, charging energy $E_{c}= h \times4.2\,\rm GHz$, and the geometrical factor $\alpha=0.8$ of the SQUID Josephson junctions [@Inomata2012]. Our artificial atom is coupled to the resonator through capacitance $C_{c} \approx 2.0\, \rm fF$. Note that the $\|e\rangle \leftrightarrow \|f\rangle$ transition between the two excited states is resonant with the resonator at $\delta\Phi\sim2.15\times10^{-3}\Phi_{0}$. Next, we measure the reflection from the system in the vicinity of the resonance $\omega_{ef} \approx \omega_r$. The result ($\|r\|$ in the frequency range from 6 to 12.5 GHz vs $\delta\Phi$) is shown in Figs. \[ATS\](c) and (d). The anticrossings are clearly observed in the two spectroscopic lines corresponding to the transitions $\|g\rangle \leftrightarrow \|f\rangle$ in Fig. \[ATS\](c) and $\|e\rangle \leftrightarrow \|f\rangle$ in Fig. \[ATS\](d), respectively. This confirms the strong coupling between the excited state transition ($\|e\rangle \leftrightarrow \|f\rangle$) of the atom and the resonator vacuum mode ($\|0\rangle \leftrightarrow \|1\rangle$). The line shapes at the exact resonance, indicated by blue arrows in Figs. \[ATS\](c) and (d), are presented in the insets. The splitting sizes are $2g_{0}=2\pi\times132\,$MHz because the observed effect is the quantized ATS due to the coupling between transition $\|e\rangle \leftrightarrow \|f\rangle$ and the vacuum mode of the resonator. We estimate the probing amplitude to be $\Omega/2\pi \approx$ 7 MHz. After confirming the vacuum-induced ATS, we study the dependence of the ATS on the resonant driving to the resonator field at $\omega_{r} \approx 2\pi\times$ 8.794 GHz using a probe signal at $\omega_{p} \approx 2\pi\times$ 12.2 GHz, which corresponds to the transition between the ground state and the second excited state ($\|g\rangle \leftrightarrow \|f\rangle$). The dynamics of this driven atom-resonator system can be described by the Markovian master equation [@Ding2017] $$\begin{aligned} \label{MasterEQ} \dot{\rho} &=&-\frac{i}{\hbar}[H_{F},\rho]+\kappa(2a\rho a^{+}-a^{+}a\rho-\rho a^{+}a)\\ \nonumber & &+\sum_{j,k=g,e,f}^{j\geq k}\frac{\gamma_{jk}}{2}[2\sigma_{kj}\rho\sigma_{jk}-\sigma_{jk}\sigma_{kj}\rho-\rho\sigma_{jk}\sigma_{kj}],\end{aligned}$$ for the density matrix $\rho$ of the coupled system, where $\gamma_{jk}$ is the relaxation/dephasing rate of the atom. Here, $H_{F}=H+\hbar \Omega_{d}(a^{+}e^{-i\omega_{r}t}+ae^{i\omega_{r}t})$ is the Hamiltonian when the driving with amplitude $\Omega_d$ to the cavity field is considered and the Hamiltonian $H$ is given by Eq.(\[effectHami\]). The reflection coefficient versus frequency is shown in Fig. \[ATSVsPower\](a), where the $x$-axis is the driving power in the resonator with the probing amplitude $\Omega/2\pi \approx 7$ MHz for the $\|g\rangle \leftrightarrow \|f\rangle$ transition close to the weak driving limit [@Suppl]. In this regime, the splitting is determined by the coupling to the vacuum mode of the resonator. The vacuum-induced ATS is a quantum mechanical phenomenon, which cannot be observed in the classically driven ATS. At larger power, the splitting starts to increase according to $2 g_{0}\sqrt{\langle n\rangle + 1}$, where $\langle n\rangle$ is the mean photon number inside the resonator. We call quantum regime the one, where the vacuum splitting gives the main contribution to the ATS. Note that there are other quantum effects that can be observed beyond the subject of the present work (e.g., resolving photon states due to quantized fields in the resonator). The asymmetry of the branches is explained by the ac Zeeman shift (or ac Stark shift) [@Abdumalikov2008; @Schuster2005] from the $\|g\rangle \leftrightarrow \|f\rangle$ transition dispersively coupled to the resonator, and the frequency shift $\zeta \langle n\rangle$ is found to be $\zeta$ = 2.5 MHz per photon. The mechanism is different from that of the asymmetric ATS observed in superconducting qubits with weak anharmonicity [@Baur2009; @Sillanpaa2009; @Li2012]. The extracted dip positions versus calibrated photon number $\langle n\rangle$ inside the resonator are presented in the inset. The solid lines show the calculated dip positions $\omega_{n}$ from the relations $\omega_{n}=\omega_{fg}\pm g_0 \sqrt{\langle n\rangle+1} + \zeta \langle n\rangle$ including the correction of the ac Zeeman shift. ![(a) Dependence of ATS on driving power of the resonator. The orange guidelines show the splitting. The inset shows dip positions versus calibrated photon numbers. The solid lines are fits accounted the ac Zeeman shift. (b) Simulated ATS in the three-level atom coupled to a resonator. (c) Observed splittings at $\langle n\rangle = 0$ (quantum regime), $\langle n\rangle = 1$ and $\langle n\rangle = 5$ (upper three panels). The solid lines show the results of calculations. The bottom panel shows the difference between the traces with $\langle n\rangle= 1$ and $\langle n\rangle = 0$. (d) The splitting at $P = -113$ dBm, corresponding to $\langle n\rangle \approx 35$. []{data-label="ATSVsPower"}](Fig3.pdf) The simulated of the reflection coefficient is shown in Fig. \[ATSVsPower\](b) with a simplified model by numerically solving Eq. (\[MasterEQ\]) in a steady state with the artificial atom parameters: $\omega_{eg}=2\pi\times3.379\,$GHz, $\omega_{fg}=2\pi\times12.173\,$GHz, $\gamma_{eg}=2\pi\times2\,$MHz, $\gamma_{fg}=2\pi\times4\,$MHz, $\gamma_{fe}=2\pi\times25\,$MHz, $\gamma_{ff}=2\pi\times1\,$MHz, and $\gamma_{ee}=2\pi\times20\,$MHz. The dephasing effect comes from $\|f\rangle \leftrightarrow \|g\rangle$ fluctuations, which induced by the photon shot noise inside the cavity, is also added to $\gamma_{ff}$ [@Schuster2005]. The other second-order effects in the $\|f\rangle \leftrightarrow \|g\rangle$ interaction with the resonator are neglected. We simplify our model for the numerical calculations because of limited computer resources insufficient to operate with the full Hamiltonian in the dissipative regime with many photons because of the too large Hilbert space. Note that although there is a weak signature of zero-one state photon splitting, it cannot be resolved in the experiment because of the measurement noise. In our experiment, it is not yet reach the reliable resolution of the two states, which can be expressed as $\Delta\omega_{10} > 2\lambda_{fg}$ – the half-spacing between two dips is larger than $\lambda_{fg}$ which is the decay rate of the atomic off-diagonal terms $\rho_{fg}$, where $\Delta\omega_{10}=g_0 (\sqrt{2}-1) = 2\pi\times$ 27 MHz and $\lambda_{fg}=2\pi\times$ 16 MHz(details are given in [@Suppl]). Figure \[ATSVsPower\](c) shows the dependence of reflection power with $\langle n\rangle = 0$, $\langle n\rangle = 1$ and $\langle n\rangle = 5$. The bottom panel shows the difference between the traces with $\langle n\rangle = 0$ and $\langle n\rangle = 1$. It demonstrates the reflection of propagating microwaves in the transmission line could be controlled by microwave fields in the resonator at the single-photon level. It may find potential applications in single-photon switches, single-photon transistors, and so on [@Chen2013; @Shomroni2014; @Baur2014; @Zhou2013]. The intrinsic linewidth broadening with increasing driving power when $\langle n\rangle \gg 1$ is explained by the dephasing of $\|f\rangle \leftrightarrow \|g\rangle$ from the fluctuations of the photon inside the cavity [@Schuster2005]. This is contrast to the linewidth narrowing with increasing driving power in ATS driven with classical control fields [@Abdumalikov2010]. The upper branch of the spectrum is broadened in the vicinity of 12.4 GHz, probably due to an unwanted low quality mode in the environment, which is difficult to remove at high (above 10 GHz) frequencies. Figure 3(d) shows the reflection power when $\langle n\rangle \approx 35$. The red line is fitted by a double-Gaussian curve of the form $A\exp{(-2\big(\delta\omega/\Delta\omega}\big)^2)$, where the fitting parameter $\Delta\omega/2\pi$ is found to be $121$ MHz at 11.876 GHz and $186$ MHz at 12.655 GHz. The linewidths are about twice the coupling strength $g_0$, which can be mainly determined by the Poisson photon distribution of coherent states formed inside the resonator [@Schuster2005]. Finally, we change the coupling strength $g_{0}$ between the transition $\|e\rangle \leftrightarrow \|f\rangle$ and the resonator by tuning the magnetic field inside the device $\alpha$-loop [@Fedorov2010]. We confirm that the splitting in the reflection spectrum of the coupled system around the transition frequency $\omega_{fe}$ is also twice the coupling strength $g_{0}$. In conclusion, we engineer an interaction between a superconducting artificial atom and a resonator and observe the vacuum-induced (quantized) Autler-Townes doublet by direct measurement of the reflection spectrum from the atom. Furthermore, we study the transition of the quantum-to-classical Autler-Townes doublet with changing the number of photons inside the resonator. Our work is an important step towards the control of propagating microwaves in a transmission line by a single photon confined in a different frequency resonator. With the further improvements, the effect may have potential applications in single-photon switches, single-photon transistors, and so on. This work was funded by ImPACT Progam of Council for Science, Technology and Innovation (Cabinet Office, Government of Japan) and NEDO IoT program. O. V. Astafiev is supported by Russian Science Foundation, grant No. 16-12-00070. Z. W. is supported by the Strategic Priority Research Program (B) of the Chinese Academy of Sciences (XDB04010100 and XDB040104000). L. M. Kuang is supported by the National Natural Science Foundation of China under Grants Nos.11775075 and 11434011, the National Fundamental Research Program of China (the 973 Program) under Grant No. 2013CB921804. L. Zhou is supported by the National Natural Science Foundation of China under Grants Nos.1142250 and 11374095. Y. X. 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